Fast Method Radial Basis Functions

FastRBF Toolbox M ATLAB Interface Version 1.4 4th August 2004

FarField Technology

Copyright c 2001 FarField Technology Ltd. All rights reserved.

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Preface Welcome to the FastRBF interpolation toolbox. This manual describes the functions offered by the FastRBF Toolbox and introduces some of the theory behind Radial Basis Function approximation. It illustrates how FastRBF can be used to interpolate and visualise scattered data. Special attention is paid to applications involving the reconstruction of surfaces from range data (from laser scanners) and incomplete mesh data (meshes containing holes).

What does the FastRBF toolbox do?

The FastRBF toolbox allows scattered 2D and 3D data sets to be described by a single mathematical function, a Radial Basis Function (RBF). The resulting function and its gradient can be evaluated anywhere, for example, on a grid or on a surface. RBFs are a natural way to interpolate scattered data particularly when the data samples do not lie on a regular grid and when the sampling density varies. The ability to fit an RBF to large data sets has previously been considered impractical for data sets consisting of more than a few thousand points. FastRBF overcomes these computational limitations and allows millions of measurements to be modelled by a single RBF on a desktop PC. The toolbox supports a variety of RBFs including the polyharmonic (thin-plate) splines. The latter are known as smoothest interpolants in the sense that they minimise certain energy functionals based upon the 2nd, 3rd, or higher derivatives.

In addition to exact interpolation of data, the FastRBF toolbox offers several techniques for fitting RBFs to noisy data including error-bar fitting, spline smoothing and linear filtering. FastRBF’s optimised surface-following isosurfacer and implicit surface modelling tools mean that FastRBF is particularly useful for reconstructing surfaces from incomplete meshes and from point cloud range data.

Who should be using FastRBF? FastRBF is ideal for engineers and scientists faced with interpolating large scattered data sets that may be non-uniformly sampled or contain large regions of missing data. It is particularly appropriate when a smooth differentiable underlying function is desirable. As a teaching tool, FastRBF offers students and researchers a means of processing irregularly sampled signals and interpolating scattered data. For those already working with Radial Basis Functions

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(RBFs), FastRBF allows them to solve extremely large systems very rapidly — a previously impossible task.

How to use this manual The following manual is organised into three parts. We suggest that even the reader familiar with RBFs read these sections carefully and work through the tutorial examples prior to using FastRBF in earnest.

Part 1, “Theory and Concepts”, introduces basic RBF concepts and describes how surfaces and incomplete surface data can be modelled with RBFs. It also describes the ideas behind the FastRBF iso-surfacer which has been developed especially for deriving a polygonal surface representation from an RBF surface model.

Part 2 of the manual, “The User’s Guide and Tutorial”, begins with an installation guide followed by a detailed description of the key toolbox functions. These descriptions are illustrated with tutorial examples which include examples of interpolating geophysical data, fitting surfaces to scattered range data and interpolating noisy density data.

Part 3 of the manual contains a quick-reference summary of the toolbox commands, a trouble-shooting guide, and a glossary of terms.

New features in this version FastRBF is a continually evolving suite of functions for scattered data interpolation and approximation. This current release features new methods for processing noisy data and improved general functionality, as well as more comprehensive documentation and examples.

• Smoothest restricted range fitter, or ‘error-bar’ fitter, offers an intuitive way to approximate noisy data with a smooth RBF – Finds the smoothest function which passes within the tolerances (error bars) specified at the data points • Implicit low pass filtering avoids evaluating an RBF on a grid and performing computationally intensive convolutions – Allows users to vary smoothing after fitting – Acts as an anti-aliasing filter during evaluation

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• Iterative refinement of an RBF model can speed up the processing of very large data sets and avoid unnecessary computation – Progressive refinement of fitting accuracy — start with a coarse fit and iteratively refine the accuracy of the the fitted RBF – The fitter uses the larger of the Global and per-point accuracies to achieve the progressive fit when per-point values are present – Progressively add new data to an RBF — improve an RBF model by adding new while avoiding computing the new solution from scratch • Addition of generalised multiquadric basic function – Users may now fit and evaluate with the multiquadric and the inverse multiquadric basic functions – The fitter also permits users to alter the degree of the polynomial component of an RBF from the minimum required up to a quadratic.

• New default behaviour – FastRBF default fitting behaviour has been changed to perform a ‘direct’ fit – Interface for multiple attributes has been changed, functions now use ‘attr’ option to specify a particular attribute rather than ‘alt’ – Improved import and export functions for formatted text data – New utility functions

Acknowledgement The FastRBF toolbox is based on research conducted in the Department of Mathematics and Statistics at the University of Canterbury by Prof. Rick Beatson. Significant contributions have also been made by Rick Fright, Jon Cherrie, Tim Mitchel, Tim McLennan, Tim Evans, Bruce McCallum and Jonathan Carr at Applied Research Associates. A bibliography of relevant research papers is provided for the interested reader at the end of the Theory section of this manual (Part 1).

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Contents I

Theory and Concepts

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1 Radial Basis Functions 1.1 RBF Description and Terminology 1.2 Basic Functions . . . . . . . . . . .

1.3 Centre Reduction . . . . . . . . . .

1.4 Anti-aliasing . . . . . . . . . . . . .

1.5 Data Smoothing . . . . . . . . . . .

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5 6 9 12 14 15

2 Fitting RBFs To Surface Data in 3D 2.1 Creating density data from Surface Normals . . . . . . . . . . .

2.2 Ensuring consistent distance to surface data . . . . . . . . . . . .

21 22 23

3 Isosurfacing 3.1 Isosurface Seed Points . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Mesh Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Boundary Capping . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 32 33

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User’s Guide and Tutorial

4 Using the M ATLAB Interface 4.1 Installation . . . . . . . .

4.2 Licensing . . . . . . . . .

4.3 Utilities . . . . . . . . . .

4.4 Online Help . . . . . . .

4.5 Commands . . . . . . . .

4.6 Messages . . . . . . . . .

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CONTENTS

4.7 4.8

Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Getting started: Importing and Exporting Data 5.1 How is data represented in FastRBF? . . . . .

5.2 Importing and exporting formated text data .

5.3 Importing and exporting 3D data . . . . . . .

5.4 Importing and Exporting Surface Data . . . .

5.5 Summary . . . . . . . . . . . . . . . . . . . . .

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6 Fitting and Evaluating RBFs 6.1 Fitting and Evaluating 2D Data 6.2 Fitting and Evaluating 3D data 6.3 Anti-aliasing . . . . . . . . . . .

6.4 Noise reduction . . . . . . . . .

6.5 Summary . . . . . . . . . . . . .

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7 Fitting 3D Surface Data 7.1 Fitting Surfaces to Mesh Data . . .

7.2 Fitting surfaces to point cloud data 7.3 Colour surface fitting . . . . . . . .

7.4 Surfacing Noisy 3D Scanner Data .

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8 3D Isosurfacing and Mesh Simplification 8.1 Isosurfacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2 Multi attribute data . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3 Mesh Simplifying . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 115 117

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Reference Manual

9 Data Types 9.1 Attribute Type . . .

9.2 PointList Type . . .

9.3 Grid Type . . . . .

9.4 Mesh Type . . . . .

9.5 Scan Type . . . . .

9.6 RBF Solution Type

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125 125 126 127 127 127 128

CONTENTS

10 Functions 10.1 Options . . . . . . . . . . . . .

10.2 Data Processing . . . . . . . .

10.3 Fitting and Evaluating . . . .

10.4 3D Surface Fitting . . . . . . .

10.5 3D Isosurfacing . . . . . . . .

10.6 Data Import and Export . . .

10.7 Load and Save FastRBF Files

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11 File Formats 159 11.1 A RANZ File Format . . . . . . . . . . . . . . . . . . . . . . . . . . 159 11.2 FastRBF File Formats . . . . . . . . . . . . . . . . . . . . . . . . . 162 11.3 FastRBF File Viewer . . . . . . . . . . . . . . . . . . . . . . . . . . 165 12 Troubleshooting 167 12.1 FastRBF License file not found . . . . . . . . . . . . . . . . . . . 167 12.2 RBF Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 12.3 Isosurfacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 A Direct methods for fitting and evaluating RBFs 175 A.1 Demonstration code for fitting and evaluating an RBF . . . . . . 175 B Glossary

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CONTENTS

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Part I

Theory and Concepts

1

In the following three chapters we will introduce the terminology and the concepts behind the functions found in the FastRBF library. Chapter 1 introduces interpolation and approximation with Radial Basis Functions (RBFs) and the principles behind the fast methods used in the FastRBF library. Chapter 2 describes how surface data can be modelled by RBFs and the FastRBF functions which facilitate this. Chapter 3 describes the FastRBF iso-surfacing routines. In particular, it describes FastRBF’s surfacing-following method which is optimised for evaluating surfaces and objects modelled by RBFs.

3

Theory and Concepts

4

Chapter 1

Radial Basis Functions Radial Basis Functions (RBFs) are popular for interpolating scattered data since the associated system of linear equations is guaranteed to be invertible under very mild conditions on the location of the data points. For example, the thin-plate spline used in this library only requires that the points are not co-linear. In particular, Radial Basis Functions do not require that the data lie on any sort of regular grid.

In a large scale comparison of methods for interpolating scattered data in two dimensions, Franke1 identifies radial basis functions as “often giving the most accurate results of all tested methods” and producing surfaces that are “usually pleasing and very smooth.”

Hardy2 lists a large number of areas where Radial Basis Functions have been employed, usually with better results than competing techniques. These applications include geodesy, geophysics, signal processing, and hydrology.

Radial Basis Functions have also been successfully employed for medical imaging and morphing of surfaces in three dimensions.

Widespread adoption of RBFs for data interpolation has been delayed because of their apparent extreme computational cost. Conventional methods for fitting an RBF by interpolation require O (N 3 ) arithmetic operations, where N is the number of data points. This makes computation when N exceeds 10 000 impractical even on mainframe systems. The FastRBF library, however, uses 1 R. Franke, ”Scattered Data Interpolation: Tests of Some Methods”, Mathematics of Computation, Vol. 38 No. 157, pp181-200, January, 1982.

2 R. L. Hardy, ”Theory and applications of the multiquadric biharmonic method”, Comput. Math.

Applic., Vol. 19, pp163–208, 1990.

5

Theory and Concepts

recent algorithmic advances involving hierarchical and fast multipole methods to reduce the computational cost to O (N log N). This makes RBFs a practical method for N as large as 1 000 000 or more, on a desk-top PC. To quantify the FastRBF difference, we give M ATLAB code for the conventional fitting and evaluation of the 3D thin-plate spline RBF in Appendix A.

1.1 RBF Description and Terminology 1.1.1

What is a Radial Basis Function?

A Radial Basis Function (RBF) is a function of the form N

s ( x ) = p ( x ) + ∑ λi Φ ( x − xi ) ,

(1.1)

i =1

where: • s is the radial basis function (RBF for short), • p is a low degree polynomial, typically linear or quadratic, • the λi ’s are the RBF coefficients, • Φ is a real valued function called the basic function, • the xi ’s are the RBF centres.

The RBF consists of a weighted sum of a radially symmetric basic function φ located at the centres xi and a low degree polynomial p. Given a set of N points xi and values f i , the process of finding an interpolating RBF, s, such that, s ( xi ) = f i ,

i = 1, 2, . . . , N

(1.2)

is called fitting. The fitted RBF is defined by the λi , the coefficients of the basic function in the summation, together with the coefficients of the polynomial term p( x).

If we let { p1 , . . . , p` } be a monomial basis for polynomials of the degree of p, and c = (c1 , . . . , c` ) be the coefficients that give p( x) in terms of this basis, then the interpolation conditions (1.2), that s( xi ) = f i , can be rewritten in matrix form as a linear system,

A P λ f = , (1.3) PT 0 c 0

6

Radial Basis Functions

where Ai, j = φ(| xi − x j |),

i, j = 1, . . . , N,

Pi, j = p j ( xi ),

i = 1, . . . , N,

j = 1, . . . , `.

Solving the linear system (Equation (1.3)) determines λ and c, and hence s( x).

1.1.2

What does FastRBF do?

Solving Equation (1.3) by ordinary or direct methods is computationally expensive and rapidly becomes impossible as N, the number of points to be interpolated, becomes larger than a few thousand. This is particularly true for non-compactly supported basic functions such as the polyharmonic splines which have attractive energy minimisation properties and are consequently known as smoothest interpolators. Not only are the storage requirements for the system in Equation (1.3) proportional to N 2 and the work to solve the system proportional to N 3 , but the work associated with evaluating s( x) (Equation (1.1)) is also proportional to N. The latter becomes significant when N is large and s( x) is evaluated on a fine grid. In appendix A we give typical M ATLAB code for the solution of Equation (1.3) via direct methods.

The FastRBF library makes the solution and evaluation of very large systems possible on ordinary hardware. The advantage of using FastRBF becomes apparent after N exceeds 2-3000 points on normal hardware. Above 10,000 points only methods used in FastRBF are capable of solving such systems on normal desktop hardware. Above 1,000,000 points or even 100,000 points, fast methods are essential on any current platform.

In addition to this core ability, FastRBF also offers various ways of filtering (smoothing) noisy data and special functions for fitting surfaces to point cloud data and incomplete mesh data.

1.1.3

How does FastRBF work?

The FastRBF library uses fast approximation methods to fit RBFs and to evaluate RBFs. This means that true equality at the interpolation nodes is never achieved. The maximum difference between the fitted RBF value and the given values, max |s( xi ) − f i |, i =1,...,N

is called the fitting accuracy.

7

Theory and Concepts

Given a point x, calculating the value s( x) is called evaluating the RBF. Once again, since we use fast approximation methods, the true RBF value is not usually achieved. If ai are the approximate values of the RBF s at the points xi , then the evaluation accuracy is the value max |s( xi ) − ai |.

i =1,...,N

Generally, the evaluation accuracy should be much higher (that is, a smaller number) than the fitting accuracy. Figure 1.1 illustrates an exaggerated case of ‘interpolating’ 1D data with FastRBF. The data are shown with fitting accuracy bars. The RBF approximation, the associated evaluation accuracy envelope and the points evaluated within this tolerance are also shown.

input data points fitted RBF output eval points evaluation accuracy output surface

Figure 1.1: Fitter accuracy and evaluation accuracy.

8

Radial Basis Functions

1.1.4

Confidence

The FastRBF fitter offers an additional fitting parameter called the confidence.

This is a real number between 0 and 1 specifying the fraction of points at which the specified RBF fitting accuracy must be achieved. For example, suppose we have 100 000 points and a specified accuracy of . If a confidence of 0.99 is specified then the error will be less than for at least 0.99 Ă— 100 000 = 99 000 of the points. The error at the remaining 1 000 points is unconstrained. The confidence parameter is a means of speeding up computation and provides robustness to statistical outliers in the data, although the points at which the error exceeds are not strictly guaranteed to correspond to outliers. The default value is 1.0, however a value as high as 0.99 can speed up the fitting process significantly.

1.2 Basic Functions The basic function3 Ό is of the form Ό(¡) = φ(k ¡ k2 ), where φ : R+ → R. This library supports the basic functions in the following table: Basic function Biharmonic spline Triharmonic spline Generalised multiquadric

1.2.1

2D φ(r) = r2 ln r φ(r) = r4 ln r

3D φ (r ) = r φ (r ) = r 3

φ (r ) = (r 2 + c 2 ) k /2

φ (r ) = (r 2 + c 2 ) k /2

The polynomial component

The polynomial component of an RBF depends on the basic function used. Table 1.1 gives the minimum degree required of the polynomial term for the basic functions used in FastRBF. The default is a linear polynomial for the biharmonic spline, a quadratic for the triharmonic and a linear for the generalised multiquadric. The degree can be varied as long as the minimum requirements tabulated in Table 1.1 are met. FastRBF currently limits the degree of the polynomial component to quadratics i.e., a maximum of d = 2.

The polynomial is stored with respect to the power basis. The order of the polynomial coefficients in n dimensions is same as the row-wise natural ordering of elements in an n dimensional array A, whose entry A(i1 , ..., in ) 3 Basic functions are often referred to as basis functions. However they do not form a basis for the RBF in a mathematical sense so we prefer to call them basic functions instead.

9

Theory and Concepts

Basic function φ (r ) Biharmonic Triharmonic

Default degree 1 2

Multiquadric

1

Minimum required degree 2D 3D 1 0 2 1 k < 0 : No poly required, i.e., d = −1 k ≥ 0 : d = k−2 1

Table 1.1: The minimum degree of the polynomial component required by an RBF. i

is the coefficient of x11 × ... × xinn , where elements with too great a degree are omitted. For example, if the dimension is 2, and the polynomial degree is 2 (quadratic), then the order of the coefficients is 1, x, x2 , y, yx, y2 , and in 3 dimensions it is 1, x, x2 , y, yx, y2 , z, zx, zy, z2 .

1.2.2

Fast methods

Table 1.2 summarises the computational advantages of FastRBF when fitting an RBF. Ordinarily, fitting an RBF requires O( N 2 /2) storage and O( N 3 ) computation, where N is the number of data points to be interpolated. With the current release of FastRBF these requirements are substantially reduced, particularly in 3D for all three basic functions, and in 2D for the biharmonic (thinplate) spline. Actual computation time also depends on the accuracy required by a fit.

Basic function φ (r ) Biharmonic Triharmonic Multiquadric

Storage

O( N ) O( N ) O( N )

Computation 2D 3D O( N log N ) O( N log N ) O( N 2 ) O( N log N ) O( N 2 ) O( N log N )

Table 1.2: Computation and storage required for fitting an RBF with FastRBF as a function of N, where N is the number of data points to be interpolated. Standard methods require O( N 2 /2) storage and O( N 3 ) computation, irrespective of the basic function and dimension of the data.

10

Radial Basis Functions

Table 1.3 summarises the computational advantages of FastRBF when evaluating an RBF. Ordinarily the computation required is proportional to N, where N is the number of RBF centres. Again FastRBF makes a significant difference in 3D4 and in 2D with the biharmonic spline.

Basic function Biharmonic Triharmonic Multiquadric

2D O(1) O( N ) O( N )

3D O(log N ) O(log N ) O(log N )

Table 1.3: Computation required for evaluating an RBF with FastRBF as a function of N, where N is the number of RBF centres. Standard methods require O( N ) computation per evaluation.

1.2.3

“Smoothest Interpolants”

The biharmonic and triharmonic splines are known as “smoothest” interpolators in the sense that they minimise certain energy functionals and interpolate the data. For example, given a set of nodes { xi }iN=1 ⊂ R3 and a set of function values { f i }iN=1 ⊂ R, the function s? ( x) that satisfies the interpolation conditions s? ( xi ) = f i and minimises the integral of the second derivative squared, s? = arg min ksk,

(1.4)

where

ksk2 =

Z R3

2 2 2 ∂2 s( x) ∂ s( x) + ∂y2 ∂z2 2 2 2 2 2 2 ∂ s( x) ∂ s( x) ∂ s( x) +2 +2 +2 dx. ∂x∂y ∂x∂z ∂y∂z

∂2 s( x) ∂x2

2

+

(1.5)

is the biharmonic or thin-plate spline. ksk2 is a measure of the energy in the second derivative of s( x). The triharmonic spline minimises the corresponding energy in the 3rd derivative of s( x).

4 O (1) computation efficiency is achievable in 3D through a full multipole implementation of the RBF evaluator, but the high setup costs mean that the advantage is gained only for very large RBFs with more than 100,000 centres. The current release of FastRBF only implements the full multipole method in 2D.

11

Theory and Concepts

These RBFs will smoothly change in value between the interpolation points. However, the RBF may not appear smooth if the data are noisy since it is constrained to interpolate the noisy values. Close inspection will reveal the smooth properties of the interpolant nevertheless. For example, consider the data shown in Figure 1.2(a), which is the M ATLAB peaks function with noise of magnitude 0.5 added. Fitting an RBF to this data and evaluating on a grid gave the surface in Figure 1.2(b). The surface interpolates the noisy data faithfully. However, at the scale of the sampling interval, the surface smoothly varies between the sample points, as Figure 1.2(c) shows.

An RBF fitted to noisy data can be smoothed by applying a low pass filter or by fitting an approximating, rather than interpolating, RBF to the data. The latter can be achieved in FastRBF by using spline smoothing or by employing a smoothest restricted range fit. Figure 1.2(d) illustrates the result of fitting an approximating RBF to our noisy data example. These methods and low pass filtering are discussed in Section 1.5 and Section 1.4, respectively.

1.3 Centre Reduction Traditionally, to interpolate a set of values f i at points xi , all the xi â&#x20AC;&#x2122;s are used as RBF centres and the resulting system of equations is solved for the RBF coefficients. In FastRBF this standard type of fit is called a direct fit. We use fast approximations to make the computation of the RBF possible. This library, however, also offers a refined greedy algorithm for fitting RBFs which reduces the number of centres used to interpolate the data which in turn results in even faster fitting and evaluation times. This method is made possible by the limited accuracy required of the interpolating RBF. Centre reduction is most suited to data which has significant regions where the data is smooth compared to the sampling density, i.e. the data is over-sampled relative to the level of detail present and the accuracy required. This is not least squares fitting as other packages use on large data sets. Rather, it is fitting so that the infinity norm error at the input points is less than a specified accuracy. For example, in Figure 1.3 we have fitted an RBF to a 600mm tall garden statue to an accuracy of 0.2mm. The signed-distance data, to which an RBF is fitted, contains 660 000 data points whereas the RBF contains only 59 000 centres. Centre reduction can greatly reduce the RBF fitting time in many cases; however, for sparsely sampled data centre reduction is ineffective and computing a direct fit, using all the data points as RBF centres, is faster. Reducing the number of RBF centres where possible can significantly reduce the time it takes to evalu-

12

Radial Basis Functions

(a) Noisy data points.

(b) Fitted RBF surface.

(c) Close up of front corner.

(d) RBF fitted with data smoothing (approximates at data points).

Figure 1.2: Illustration of smooth interpolation and data smoothing. (a) Data with uniform random noise of magnitude 0.5 added, (b) Standard RBF fit to the data. (c) Close-up view reveals that the RBF smoothly interpolates the data. (d) RBF fitted with spline smoothing approximates at the data points rather than interpolating.

13

Theory and Concepts

(a) Original Points.

(b) RBF Centres.

(c) Surfaced RBF.

Figure 1.3: Illustration of centre reduction when fitting an RBF to surface data. (a) An RBF is fitted to 660 000 distance-to-surface data points computed from a laser scan data of a 600mm tall statue. (b) The subset of 59 000 RBF centres required to interpolate the data in (a) to within an accuracy of 0.2mm.

The scanner accuracy is 1mm. (c) The corresponding isosurface of the RBF smoothly interpolates the original surface data.

ate the RBF. It is worthwhile comparing fitting options on various types of data to see which method works best in each situation, keeping in mind evaluation considerations.

1.4 Anti-aliasing The Nyquist sampling theorem says that we must sample at no less than twice the highest frequency present in a signalâ&#x20AC;&#x2122;s spectrum to avoid aliasing artifacts.

Consequently, good practice says that any signal should be low pass filtered prior to sampling to ensure that the Nyquist condition is met. The same is true when evaluating an RBF, which can also be considered a signal. The FastRBF

14

Radial Basis Functions

library offers a low-pass filter option in all the routines that evaluate RBFs. The filter kernel used in the RBF evaluator is similar to convolving the signal with a Gaussian, i.e. computing the Gaussian-weighted average over a local neighbourhood at each evaluation point. This convolution is realised by analytic means in the FastRBF library. The computational cost of otherwise achieving this convolution, by spatial or Fourier means, is avoided and the data need not be resampled on a regular grid.

The single parameter associated with the low-pass smoothing option determines the frequencies that are attenuated in the RBF. In the spatial domain this parameter corresponds to the Full Width at Half Maximum (FWHM) of the filter — a measure of the filter support or spatial extent. Detail or structure finer than this width will be lost by smoothing. In the frequency domain this approximately corresponds to the 3dB cut-off frequency for the filter. Frequencies higher than this cut-off are reduced in power by at least half.

Figure 1.4 illustrates both the spatial and frequency characteristics of the filter kernel used in the evaluator. The FWHM and 3dB points are indicated.

The smoothing parameter c is approximately the FWHM distance. The 3dB half-power frequency corresponds to approximately 1c .

Note The computational cost of a smooth evaluation in 3D is approximately twice that of an ordinary evaluation and is independent of the size of the smoothing kernel. In the current release of FastRBF, the 2D computational cost is not so efficient and is O( N ) rather than O(2 log N ), where N is the number of centres.

1.5 Data Smoothing In this section we consider the task of smoothing a signal to remove detail. For convenience we will refer to unwanted detail generally as ‘noise’, whether it be false detail due to artefacts or genuine structure in the data. Data smoothing can be achieved by smoothing an RBF fitted to ‘noisy’ data during evaluation, or alternatively, by approximating the data when fitting the RBF, rather than forcing the RBF to interpolate the data exactly. FastRBF offers two methods for fitting an approximating RBF. These are referred to as spline smoothing (also known as ρ smoothing) and smoothest restricted range approximation or error-bar fitting for short. The two methods optimise different smoothness criteria.

15

Theory and Concepts

0

0.7

c=1.0

−2

0.6

20 log(| h(u) |) dB

−4 0.5

0.4

0.3

0.2

−6

−8

−10

−12

−14

−16 0.1

FWHM 0 −3

−2

−1

0

−18

1

2

3

−20

(a) Spatial domain

0

1

u 3dB

2

3

4

5

6

u

(b) Frequency domain

Figure 1.4: Illustration of filter response. (a) Spatial domain characteristics, (b) Frequency domain characteristics.

1.5.1

Low Pass Filtering

In the previous section we discussed anti-aliasing. There the purpose of filtering an RBF was to faithfully represent the signal when sampled at frequencies below the Nyquist limit. Low pass filtering beyond the Nyquist limit can be used in the evaluator to smooth an RBF fitted to noisy data. High frequencies, irrespective of their origin and magnitude, will be globally attenuated. The convenience of low pass filtering is that an RBF is fitted once to the data, to the maximum desired precision, and smoothing is varied a posteriori during evaluation of the RBF. The detail in the original RBF, and hence in the original data, is retained in the RBF.

1.5.2

Smoothest Restricted Range Smoothing

Restricted range smoothing assumes that there exists a range of values at each data point between which the true function value lies. This value range is analogous to the error bars associated with experimental data. The restricted range fitter finds the smoothest function which passes within the error bars of all the data points. The key parameter is an estimate of the noise magnitude at each data point (the permitted range of values). Mathematically, we can

16

Radial Basis Functions

express this by modifying the interpolation conditions in Equation (1.2) to seek an RBF such that,

`i â&#x2030;¤ s ( xi ) â&#x2030;¤ ui ,

i = 1, . . . , N.

(1.6)

The bounds `i and ui analogous to error bars for the value f i . If we set `i = f i â&#x2C6;&#x2019; and ui = f i + , then we are seeking the smoothest spline within of the data.

Note The smoothest restricted range RBF may lie anywhere between the range of values specified at each data point. It is not weighted towards the centre of the value range or the specified data value, rather the fitted RBF is governed solely by the criterion that it is the smoothest function which passes between the error bars.

It can be shown that only the binding constraints (the non-redundant errorbars which most constrain the function) contribute centres to the resulting RBF under these conditions. As a result, significant reductions in the number of RBF centres are possible when using an error-bar fit compared to a direct fit followed by low pass filtering or compared to a spline smoothing fit as described in the following section.

1.5.3 Spline Smoothing Data smoothing can also be achieved in the RBF fitter by modifying the smoothness criterion (Equation (1.4)) such that we seek minimize

1 Ď ksk + N 2

!

N

â&#x2C6;&#x2018;

s ( xi ) â&#x2C6;&#x2019; f i

2

,

(1.7)

i =1

where Ď â&#x2030;Ľ 0 and ksk2 denotes the smoothness penalty defined by Equation (1.5). Ď may be thought of in terms of an inverse spring constant. When zero, the springs are inextensible and the RBF is forced to pass through the data points. When non-zero the springs can stretch and the RBF is pulled close to the data points, but not forced to pass through them. The RBF which satisfies Equation (1.7) is also found by solving a linear system,

A + Ď I P Îť f = .

(1.8) PT 0 c 0

17

Theory and Concepts

The meaning of the accuracy argument in this context requires some further explanation. Usually the accuracy argument means,

|s( xi ) â&#x2C6;&#x2019; f i | < ,

i = 1, 2, . . . , N.

(1.9)

However, this is not the case when data smoothing with Ď . If an accuracy of is specified to the RBF fitter when Ď smoothing, then is the maximum error allowed in Equation (1.8), which means that

A + Ď I P Îť f

< . â&#x2C6;&#x2019; (1.10) T

P 0 c 0 â&#x2C6;&#x17E; Rearranging this gives the actual error bounds for the RBF values at each data point,

| s ( xi ) â&#x2C6;&#x2019; f i | < + Ď | Îť i | ,

for all i.

(1.11)

When Ď = 0 this is same as Equation (1.9), however when Ď 6= 0 the error in the RBF is not bounded by . As Ď increases we are essentially trading off fidelity to the data in favour of smoothness.

1.5.4

Smoothing Summary

Linear low-pass filtering offers a convenient, and computationally efficient way to smooth an RBF during evaluation. The controlling parameter is a cutoff frequency corresponding to the sampling interval or the frequency of noise present in the data or the desired level of detail. Smoothing can be varied a posteriori.

The smoothest restricted range (error-bar) fitter fits a smooth RBF to the data. The detail lost in the raw data can not be recovered from the RBF during evaluation. However, a more compact RBF, with fewer centres may result.

This method requires an estimate of the magnitude of noise present at each data point. High frequencies are not necessarily uniformly attenuated.

Spline (Ď ) smoothing is a traditional RBF smoothing technique. It also approximates the raw data and is applied during the fitting process. Though it can produce very good results, choosing the parameter Ď can be difficult since it does not have a straightforward physical analogy that the tolerance parameter of the error-bar method or the frequency parameter of the low pass filter approach have. Unlike the smoothest restricted range technique, compression is not inherent in the resulting smoothed RBF and the number of RBF centres will correspond to the number of data points (constraints). However, as with

18

Radial Basis Functions

the direct fit in Section 1.3, we can employ a Greedy algorithm to reduce the number of centres. This is achieved by specifying the centre reduction option when performing a spline-smoothing fit.

19

Theory and Concepts

20

Chapter 2

Fitting RBFs To Surface Data in 3D This chapter describes the process of fitting a 3D RBF, s, to a set of N points { xi }iN=1 that lie on a surface. s is chosen such that the set of points where s( x) = 0 forms a surface that smoothly interpolates the points xi . The interpolating surface is an isosurface of the function s. Surfaces defined as an isosurface are often called implicit surfaces1 , because of the implicit nature of their definition. They are also known as f-rep’s (short for functional representation). The process of forming an explicit representation of the surface, as a set of points or as a mesh of polygons, is called isosurfacing. This is discussed in the following chapter. This chapter focuses on how we construct the function s.

We choose s to approximate a signed distance-to-surface function, where s( x) is the signed distance from x to the surface. Points on one side of the surface are assigned positive distances while points on the other side are assigned negative distances. The surface therefore corresponds to the set of points where the distance to the surface is 0, i.e., { x ∈ R3 : s( x) = 0}. All the given data points xi lie on the surface. We therefore seek s such that, s( xi ) = 0,

i = 1, . . . , N.

Solving this system of equations will yield the trivial solution s( x) = 0 everywhere. Additional constraints, corresponding to points where s is non-zero, are required to achieve a non-trivial solution. By definition, points with a nonzero distance to the surface are off-surface points. Hence we choose a set of M 1 For a good introduction to implicit surfaces see J. Bloomenthal (Ed.), “Introduction to Implicit Surfaces”, Morgan Kaufmann Publishers, Inc., San Francisco, CA., 1997

21

Theory and Concepts

off-surface points { yi }iM=1 , setting s( yi ) = f i 6= 0,

i = 1, . . . , M,

where the f i ’s are the signed distances from yi to the nearest point on the surface. This results in a system of equations with N + M points.

To generate these off surface points, surface normals (or estimates thereof) are required.2 For now we will assume that surface normals exist for most of the points.

(a) Surface points with normals.

(b) Off-surface points added on both sides.

(c) Off-surface points added on one side only.

Figure 2.1: How surface normals are used to generate off-surface points.

2.1 Creating density data from Surface Normals When a surface normal n exists for a surface point p, off-surface points po (outward) and pi (inward) are generated by po = p + αn and pi = p − βn, where α and β are the projection distances of the off-surface points. Figure 2.1(a) shows a 1D example of a surface with normals, the resulting augmented data set is shown in Figure 2.1(b). Alternatively we can create off-surface points on only one side as in Figure 2.1(c), however this can sometimes cause ripples in the isosurface s( x) = 0 corresponding to the interpolated surface.

When normals on both sides are used, every normal used generates two off-surface points. This can increase the size of the density data set to 3 times the original number of points.

Since tripling the number of interpolation points is rarely desirable, a maximum increase in the number of points may be set. The default is 1.0, which 2

Other methods may be used but these generally require knowing the topology of the object.

22

Fitting RBFs To Surface Data in 3D

means that given N surface points, no more than N off-surface points will be added. Figure 2.2 demonstrates the effect of this parameter.

(b) Max Increase = 2.0. 30 density points

(d) Max Increase = 1.0. 20 density points

(c) Max Increase = 1.0. 20 density points

(e) Max Increase = 0.5. 15 density points

(a) 10 surface points with normals

Figure 2.2: (a) 10 surface points with normals. (b) Off-surface points added on both sides with max increase = 2. (c) Off-surface points added on both sides with max increase = 1. (d) Off-surface points added on one side with max increase = 1. (e) Off-surface points added on one side with max increase = 0.5

2.2 Ensuring consistent distance to surface data Using normals of a fixed length can cause problems when there are thin sections or sharp corners, creating inconsistent distance-to-surface data. The old adage â&#x20AC;&#x153;garbage in, garbage outâ&#x20AC;? is true of RBF fitting, so we try to ensure the consistency of the data either by validating the projection distance along normals, or by validating the density value at the projected point.

Figure 2.3 shows the effect of validating the projection distance and density values where two parts of the surface are close together. Figure 2.3(a) shows the original data. Figure 2.3(b) shows the undesirable results that can occur if

23

Theory and Concepts

(a) Surface points with normals

(b) Projecting a fixed distance with fixed values

(c) Projecting a fixed distance with validated values

(d) Validated projection distance

Figure 2.3: The effect of normal length validation. Points are coloured by the distance to the surface (blue points are on the surface).

we do not validate the off-surface points. Figure 2.3(c) shows the validation of density values at the off-surface points. This strategy finds the closest point, q to the off-surface point and sets the density value to the signed distance to that point. The sign is determined by the surface normal at q. Figure 2.3(d) shows the projection distance validation method. When we generate an off-surface point poff from a surface point p, we find it’s closest surface point q. If q is not p then the projection distance is reduced until p is the closest point to poff .

Figure 2.4(a) shows an example of density data created from a hand where a fixed normal length of 15mm was used without any validation. The points are coloured by their distance to the surface; the original surface points are green, the “outside” points are red and the “inside” points are blue. This is not

24

Fitting RBFs To Surface Data in 3D

(a) No validation

(b) Value validation

(c) Distance validation

(d) Resulting surface

(e) Resulting surface

(f) Resulting surface

Figure 2.4: The need to validate the generated distance-to-surface data is demonstrated using a projection distance of 15mm along surface normals to generate off-surface points for a hand.

25

Theory and Concepts

good “distance-to-surface” data, as is shown in the resulting RBF isosurface in Figure 2.4(d).

In Figure 2.4(b) we validate the density values. Because of the unfortunate choice of projection distance, many of the points almost coincide with the original surface points and the result is not as good as expected (Figure 2.4(e)).

In Figure 2.4(c) normal projection distance validation has been used. The normals around the fingers have been shortened and the (blue) inside points remain inside each finger. Similarly the outside normals have been shortened between the fingers. This gives much better distance-to-surface data as Figure 2.4(f) shows.

2.2.1

Projection distance validation and noise

When validating the projection distance of noisy data, undesirably short normals may occur. This can cause dimpling in the surface and reduce the efficacy of data smoothing when the fitting accuracy is less than the noise level. If you have an accurate bound on the noise, and you fit with the errorbar option with a fitting accuracy at the level of noise, changing the minimum projection distance has no effect. However, if the fitting accuracy is too tight then having a minimum projection distance around twice the expected level of noise can improve surface smoothness. Points generated at a distance less than the minimum distance are discarded.

In Figure 2.5 we have added uniform noise to a segment of a sphere. The sphere has radius 1, the noise had magnitude 0.03, and we have fitted with the errorbar option and an accuracy of 0.02 — less than the level of noise.

Figure 2.5(a)-(c) shows the effect of using longer minimum projection distances while Figure 2.5(d) shows an RBF fitted with the correct fitting accuracy of 0.03, and a minimum normal length of 0. The surfaces are coloured by vertex distance to the true sphere.

26

Fitting RBFs To Surface Data in 3D

(a) Minimum normal length of 0.0

(b) Minimum normal length of 0.03 – the level of noise

(c) Minimum normal length of 0.06 – twice the level of noise

(d) Fitting to an accuracy of 0.03 – the noise level, with a minimum normal length of 0.0

Figure 2.5: (a)-(c) Using a non-zero minimum normal length can mitigate the effects of fitting too tightly when noise level is not accurately known. (d) Specifying a minimum projection distance is unnecessary when a fitting accuracy equal to or larger than the magnitude of the noise is used.

27

Theory and Concepts

28

Chapter 3

Isosurfacing Displaying iso-valued surfaces is a useful way of visualising a 3D function.

Given a density function s, an isosurface at value a is defined as the set of all points x where s( x) = a. In the context of surface reconstruction, where s represents a signed-distance function, the reconstructed surface corresponds to the isosurface where s = 0. The process of extracting an isosurface of a function, either as a set of points or as a mesh of polygons, is called isosurfacing.

FastRBF offers two isosurfacing functions. A conventional Marching Cubes implementation and an optimised surface-following implementation. Both sample the function s at regular intervals to construct a manifold mesh of polygons representing the desired isosurface at a specified resolution. Facet vertices are ordered such that the cross product of adjacent edges (the facet normal) is consistent with gradient of the density function s. The marching cubes algorithm is a well-known general purpose algorithm and described elsewhere1 . In this chapter we focus on FastRBF’s optimised surface-following algorithm which is particularly suited to isosurfacing signed-distance data.

The conventional means of extracting a polygonal representation (a mesh) of an isosurface is with a Marching Cubes or Marching Tetrahedra algorithm.

Common implementations require the data to be discretely sampled on an n × n × n regular grid, a process also known as voxelisation. When the data is represented by a function, this requires evaluating the function s at every voxel resulting in O (n3 ) computation (Figure 3.1(a)). The surface following technique used in FastRBF minimises the number of evaluations of s by only evaluating s near the surface. This results in only O (n2 ) computation and is 1

See Foley and van Dam’s “Computer Graphics – Principles and practice”, 1996, Addison Wesley.

29

Theory and Concepts

suited to situations where the computational cost of evaluating s is high (Figure 3.1(b)). The use of a tetrahedral sampling lattice ensures a more even spatial sampling.

(a) Marching cubes full-grid evaluation.

(b) Evaluation points for surface-following tetrahedra.

Figure 3.1: Comparison of RBF evaluations for (a) full grid Marching Cubes(O (n3 ) evaluations) and (b) surface-following tetrahedra (O (n2 ) evaluations).

3.1 Isosurface Seed Points Seed points near the isosurface are required to initialise the surface-following process because we do not evaluate s over the whole volume. A gradient search from each of the seed points finds the closest surface point. Thus, the seed points do not have to lie exactly on the isosurface. A mesh wavefront spreads out from each seed point until it either intersects itself or meets a wavefront associated with another seed.

At least one seed is required to extract each separate surface in the volume. However, specifying a large number of seeds causes many more function

30

Isosurfacing

evaluations at setup and increases memory usage which can cause a dramatic decrease in performance.

In the surface modelling context, described in Chapter 2, the raw data consist of points lying near the surface of interest. The locations of RBF centres correspond to these raw data points or are derived from them. Seed points for isosurfacing are therefore selected from the RBF centres. For general density data, however, the isosurfaces of interest may lie far from the RBF centres and even the original data points. Consequently, there is no guarantee that using all the centres or all the data points will result in detecting all the isosurface components. For this reason, as well as the impractical overhead associated with using all data points as seeds, the user is advised that full grid evaluation followed by FastRBFâ&#x20AC;&#x2122;s Marching Cubes algorithm may be more appropriate when visualising arbitrary density data.

1 2 1 2

1.0

z y

x Figure 3.2: Tetrahedral sampling lattice.

3.1.1

Sampling Lattice

Figure 3.2 illustrates the tetrahedral sampling lattice used by the isosurfacing algorithm. Sampling on a tetrahedral lattice results in a more uniform sampling of space compared to a conventional cubic lattice. Neighbouringâ&#x2C6;&#x161;sample points in the tetrahedral lattice are separated by either 1.0 or 0.866 ( 3/2) units. Throughout this documentation the resolution parameter specifies the

31

Theory and Concepts

sample spacing between xy planes in the lattice (i.e., the parameter describes the sampling resolution along the z-axis of the lattice).

(a) without optimisation.

(b) with optimisation.

Figure 3.3: 2D illustration of face optimisation. The small dots are the output mesh vertices, (a) without optimisation, (b) with optimisation.

3.2 Mesh Optimisation The FastRBF isosurfacer incorporates mesh optimisation during surfacing.

Mesh optimisation improves the aspect ratio of triangular facets and minimises the number of faces produced. Optimisation is distinct from the postprocessing step known as mesh reduction or simplification, since every sample point still contributes at least one face to the mesh. Surface topology (as defined by the function values at the lattice points) is always preserved when mesh optimisation is used.

Figure 3.3 illustrates in 2D the effects of mesh optimisation. Without optimisation, as in Figure 3.3(a), contour vertices lie on the edges of the triangular 2D lattice resulting in some very short contour segments. When optimisation is used, as in Figure 3.3(b), the contour vertices do not generally lie on edges of the lattice, resulting in fewer segments.

Figure 3.4 illustrates optimisation in 3D. Figure 3.4(a) and (b) compare no optimisation with full optimisation. Figure 3.4(c) illustrates constrained optimisation where facet vertices have been constrained, where possible, to lie on parallel planes. These planes coincide with the x â&#x2C6;&#x2019; y planes of the lattice coordinate system in Figure 3.2. An arbitrary plane axis can be achieved by rotating the lattice coordinate system relative to the data.

32

Isosurfacing

(a) No optimisation.

(b) Full optimisation.

(c) Optimisation constrained to parallel planes at 30â&#x2014;Ś .

Figure 3.4: The results of different mesh optimisation methods.

3.3 Boundary Capping The surface following strategy employed by the isosurfacer does not require a 3D array of sample points to be stored. This means that a surface of infinite extent, irrespective of sampling resolution, can in theory be extracted. In practice we rarely want an infinite surface so the specification of a bounding box is required. Furthermore, unlike the FastRBF C library, the M ATLAB interface requires storing the mesh prior to exporting. A surface clipped by a bounding box can be left open (Figure 3.5(a)) or capped to create a water-tight surface.

When a surface is to be capped, the value of the RBF outside the bounding box can be considered to be either positive or negative with respect to the isosurface threshold, i.e. either above or below the threshold. Figure 3.5(b) illustrates capping where points outside the bounding box are considered to have density values above the threshold (positive) and are therefore not part of the object. Figure 3.5(c) illustrates capping where the density value outside the box is considered negative, i.e. the value is below the threshold and the point therefore is inside the object.

33

Theory and Concepts

(a) Open surface with no boundary capping.

(b) Surface closed with closeplus. Points outside the boundary are above the threshold.

(c) Surface closed with negative capping. Points outside the boundary are below the threshold.

Figure 3.5: The results of different surface capping methods.

34

Bibliography [1] R. K. Beatson, J. B. Cherrie, and C. T. Mouat. Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration. Advances in Computational Mathematics, 11:253–270, 1999.

[2] R. K. Beatson, J. B. Cherrie, and D. L. Ragozin. Fast evaluation of radial basis functions: Methods for four-dimensional polyharmonic splines.

SIAM J. Math. Anal., 32(6):1272–1310, 2001.

[3] R. K. Beatson and N. Dyn. Multiquadric b-splines. J. Approximation Theory, 87:1–24, 1996.

[4] R. K. Beatson and L. Greengard. A short course on fast multipole methods. In M. Ainsworth, J. Levesley, W.A. Light, and M. Marletta, editors, Wavelets, Multilevel Methods and Elliptic PDEs, pages 1–37. Oxford University Press, 1997.

[5] R. K. Beatson and W. A. Light. Fast evaluation of radial basis functions: Methods for two-dimensional polyharmonic splines. IMA Journal of Numerical Analysis, 17:343–372, 1997.

[6] R. K. Beatson, W. A. Light, and S. Billings. Fast solution of the radial basis function interpolation equations: Domain decomposition methods.

SIAM J. Sci. Comput., 22(5):1717–1740, 2000.

[7] R. K. Beatson, W. A. Light, and S. Billings. Fast solution of the radial basis function interpolation equations: Domain decomposition methods.

SIAM J. Sci. Comput., 22:1717–1740, 2000.

[8] R. K. Beatson and G. N. Newsam. Fast evaluation of radial basis functions: I. Computers Math. Applic., 24(12):7–19, 1992.

35

BIBLIOGRAPHY

[9] R. K. Beatson, A. M. Tan, and M. J. D. Powell. Fast evaluation of radial basis functions: Methods for 3-dimensional polyharmonic splines.

In preparation.

[10] J. Bloomenthal, editor. Introduction to Implicit Surfaces. Morgan Kaufmann, San Francisco, California, 1997.

[11] J. C. Carr and R. K. Beatson and B. C. McCallum and W. R. Fright and T. J. McLennan and T. J. Mitchell. Smooth surface reconstruction from noisy range data. In proceedings of GRAPHITE 2003 - To appear, ACM Press/ACM SIGGRAPH, 11-14 February 2003.

[12] J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. McCallum, and T. R. Evans. Reconstruction and representation of 3D objects with radial basis functions. In Computer Graphics SIGGRAPH 2001 proceedings, ACM Press/ACM SIGGRAPH, pages 67–76, 12-17 August 2001.

[13] J. C. Carr, W. R. Fright, and R. K. Beatson. Surface interpolation with radial basis functions for medical imaging. IEEE Trans. Medical Imaging, 16(1):96–107, February 1997.

[14] E. W. Cheney and W. A. Light. A Course in Approximation Theory. Brooks Cole, Pacific Grove, 1999.

[15] J. B. Cherrie, R. K. Beatson, and G. N. Newsam. Fast evaluation of radial basis functions: Methods for generalised multiquadrics in R. Technical Report UCDMS2000/1, Dept. of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand, January 2000.

[16] J. B. Cherrie, R. K. Beatson, and G. N. Newsam. Fast evaluation of radial basis functions: Methods for generalized multiquadrics in Rn . SIAM J. Sci. Comput. - To appear, 2001.

[17] J. Duchon. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In W. Schempp and K. Zeller, editors, Constructive Theory of Functions of Several Variables, number 571 in Lecture Notes in Mathematics, pages 85–100, Berlin, 1977. Springer-Verlag.

[18] N. Dyn. Interpolation and approximation with radial and related functions. In C. K. Chui, L. L. Schumaker, and J. D. Ward, editors, Approximation Theory VI, volume 1, pages 211–234. Academic Press, 1989.

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[19] N. Dyn, D. Levin, and S. Rippa. Numerical procedures for surface fitting of scattered data by radial functions. SIAM J. Sci. Stat. Comput., 7(2):639– 659, 1986.

[20] G. E. Fasshauer. On smoothing for multilevel approximation with radial basis functions. In C. K. Chui and L. L. Schumaker, editors, Approximation Theory IX, volume 2, pages 55–62. Vanderbilt University Press, 1999.

[21] W. E. Lorensen and H. E. Cline. Marching cubes: A high resolution 3D surface construction algorithm. Computer Graphics, 21(4):163–169, July 1987.

[22] K. Ritter. Generalized spline interpolation and non-linear programming.

In I. J. Schoenberg, editor, Approximation with a special emphasis on spline functions, pages 75–117, New York and London, 1969. Academic Press.

[23] M. Villalobos and G. Wahba. Inequality constrained multivariate smoothing splines and application to the estimation of posterior probabilities. J. American Statistical Association, 387:239–248, 1987.

[24] G. Wahba. Spline Models for Observational Data. Number 59 in CBMS-NSF Regional Conference Series in Applied Math. SIAM, 1990.

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38

Part II

Userâ&#x20AC;&#x2122;s Guide and Tutorial

39

The following chapters will help new users to get started with the FastRBF Toolbox. The installation guide is followed by a series of tutorials which introduce the basic functions of the toolbox using practical examples. These begin with a chapter on importing and exporting data followed by a chapter on interpolating and approximating scalar data and a special chapter on modelling surfaces. The functions are illustrated with worked problems which have been restricted to a few thousand points for speed and compatibility with the demo version of the toolbox. Sometimes the benefits of FastRBF are not always apparent when working with these small sized problems as compared to very large problems. Part III of this manual, the reference guide, contains a full summary of all functions and their associated parameters.

Sample datasets, .m files and additional code can be found under the Tutorial directory and its sub-folders.

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Userâ&#x20AC;&#x2122;s Guide and Tutorial

42

Chapter 4

Using the M ATLAB Interface 4.1 Installation 4.1.1

Windows

Installation of the FastRBF M ATLAB interface is performed via an installation program. There is a separate installation package for each release of M ATLAB. Currently versions R12.1 and R13 are supported.

The installation includes the M ATLAB Toolbox, a PDF copy of this document, and the scripts and data sets for the tutorials. Note that the tutorials are stored in sub-folders of the Tutorial directory. The RunTutorialScripts.m file in the root Tutorial directory will run through all the examples in the tutorial sub-folders. The installer will also add a folder to the start menu containing links to the documentation and utilities. The default installation directory and start menu folder are: • C:\Program Files\FarField Technology\FastRBF v1.4\Matlab • Start/Programs/FastRBF v1.4/Matlab For M ATLAB to be able to find the FastRBF Toolbox, you need to add the installation directory to the M ATLAB search path. You can do this by using the addpath command. To change the path permanently, put this command in your startup.m file. For example: ... addpath ’C:\Program Files\FarField Technology\FastRBF v1.4\...

Matlab’

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4.1.2

Linux

The toolbox, which will appear as a file with a name similar to FastRBF_Matlab_Linux_v1.4.1.tar.gz, should be placed in the desired FastRBF installation directory. A typical location might be /usr/local/fastrbf/.

Note, you may require root priviledges depending where you install FastRBF. Next unpack the the .tar.gz file, eg., gunzip FastRBF_Matlab_Linux_v1.4.1.tar.gz tar -xvf FastRBF_Matlab_Linux_v1.4.1.tar If the toolbox were unpacked in to the directory /usr/local/fastrbf, then the resulting directory, /usr/local/fastrbf/FastRBF_Matlab_Linux_v1.4.1/toolbox, should be added to the Matlab path. This can be done in Matlab using the path command.

4.2 Licensing 4.2.1

Windows

Once the Toolbox is installed you will need to install the license file: 1. Start the license file manager by clicking on the “License File” link in the start menu folder.

2. Click on “Install License”.

3. Select the new license file, FastRBF.lic, in the file dialog, and press “Open”.

4. If the new license file installed correctly, its details will be displayed in the license file manager.

To run the FastRBF Toolbox, a license file FastRBF.lic is required. An evaluation license allows the Toolbox to operate for a limited period of time, on a limited number of points. An error is reported if the evaluation period has elapsed, or the point limit is exceeded. Once a registered license is purchased, the Toolbox operates indefinitely, up to the point limit chosen at the time of purchase.

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Using the M ATLAB Interface

4.2.2

Linux

The license file (FastRBF.lic) should be placed in the same directory as the FastRBF_MEXr13.mexglx executable. This is stored in the toolbox subdirectory /private. In the case of the above example this would be, /usr/local/fastrbf/FastRBF_Matlab_Linux_v1.4.1/toolbox/private.

FastRBF will, however first search for the license file specified by the enviornment variable FASTRBFLICENSE provided that both the variable, and the file it refers to, are present (you can define this variable in your .bashrc/.cshrc/.zshrc etc). FastRBF will lastly search for FastRBF.lic in the current directory where you are running Matlab.

4.3 Utilities FastRBF file viewer FastRBF format files are created when you use the FastRBF_Save command and are loaded with the FastRBF_Load command. This file format is useful for sharing your data with other FastRBF users. Installed with the Toolbox is a program called FastRBFFile which displays the items in the index of a FastRBF file. The values of small items are also displayed. We suggest adding a shortcut to FastRBFFile.exe to your SendTo folder. You can then view a file from Windows Explorer by right-clicking the file and choosing Send to|FastRBFFile from the pop-up menu.

License File Manager You start the license file manager by clicking on the “License Manager” link in the start menu folder. If there is a valid license file called FastRBF.lic in the executable directory it is displayed. To copy a new license file into the executable directory click the “Install License” button.

4.4 Online Help A list of the commands provided in the FastRBF Toolbox is given by typing: help fastrbf Help on a particular command is obtained in the usual M ATLAB manner; e.g. for help on the fastrbf_fit command, type:

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help fastrbf_fit Help on FastRBF data types is given by typing: help fastrbf_types

4.5

Commands

The commands provided in the FastRBF Toolbox are listed in Table 4.1. See Chapter 10, or the online help, for a detailed description of each command.

The syntax of FastRBF commands follows the usual Matlab conventions.

Note that many commands may take a variable number of arguments.

4.6

Messages

The level of messages produced by each FastRBF command can be varied. The available message levels are: Level 1 2 3 4 5

Messages errors warnings success messages progress bar (default) informational messages

The message level is set using the following options, which are accepted by all FastRBF functions: Option ’quiet’ ’verbose’ ’messages’, N

4.7

Effect set message level to 2 set message level to 5 set message level to N

Data Structures

The FastRBF Toolbox commands operate on various data structures, which are stored in Matlab as 1x1 struct arrays. These data structures are described in Chapter 9.

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Using the M ATLAB Interface

Group Core

Data processing

Isosurface

Surface Fit

Visualisation FastRBF Files Other File Formats

Command fastrbf_fit fastrbf_pointeval fastrbf_grideval fastrbf_unique fastrbf_crop fastrbf_trim fastrbf_makerbf fastrbf_transformrbf fastrbf_gridcoords fastrbf_gridvec fastrbf_makepointlist fastrbf_isosurf fastrbf_mcubes fastrbf_simplify fastrbf_simplifyrgb fastrbf_normalsfrommesh fastrbf_normalsfromscan fastrbf_normalsfromsigns fastrbf_densityfromnormals fastrbf_mesh2mesh fastrbf_checkmesh fastrbf_view fastrbf_save fastrbf_load fastrbf_export fastrbf_import

Table 4.1: FastRBF Toolbox commands.

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Userâ&#x20AC;&#x2122;s Guide and Tutorial

4.8 Files Any of the FastRBF data structures may of course be saved to or loaded from disk with the Matlab save and load commands. For compatibility with the command line version of FastRBF, which is file-based, the Matlab FastRBF has commands (fastrbf_save and fastrbf_load) to save and load FastRBF format files. The format of FastRBF files is documented in Chapter 11. The FastRBF Toolbox also includes functions (fastrbf_export and fastrbf_import) for saving to or loading from various industry-standard file formats.

48

Chapter 5

Getting started: Importing and Exporting Data In this chapter we illustrate how data can be imported into FastRBF and then saved or exported. FastRBF’s native format is described in Chapter 11. FastRBF also supports four specific graphics file formats for interpolating surface data. For general 2D and 3D data, FastRBF provides the facility to import and export formatted text files directly. This provides maximum flexibility and avoids writing files in the FastRBF data format. It also avoids the intermediate step of reading ASCII data into the Matlab environment. Besides using the FastRBF import and export functions, users can also load and save data as .mat files within Matlab and pass vectors to FastRBF functions as they can with any ordinary Matlab function.

We begin the chapter by considering importing and exporting general scattered data from formatted text files. Next we consider surface data and the specific graphics and CAD formats supported by FastRBF.

5.1 How is data represented in FastRBF? FastRBF is primarily an interpolation toolbox. The basic data type is a pointlist.

Each point in a point list consists of a location and an array of attributes to be interpolated or approximated by an RBF. Each attribute has a name associated with it and may consist of any, none or all of the following: a scalar value, an accuracy, an upper error-bar on the value, a lower error-bar on the value and a ‘gradient’ vector field. The accuracy and error-bar values are positive scalar

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quantities. The error-bar values are relevant to the error-bar fitting strategy described in Section 6.4.2. They specify the bounds on the data value relative to the given value, i.e., they are the absolute difference between the given value and the corresponding limit on that value.

Currently, FastRBF supports 2D and 3D data. The dimension of the data is determined from the number of components in the location vector. The dimension of the gradient vector field must match that of the location data. So for 3D data the gradient attribute must have three components. Originally the gradient field was only used in an output context to return the gradient of an attribute when evaluating its RBF at the associated point. Consequently the field is referred to as the gradient. However, in an input context, the gradient field can be used to represent any type of vector data. For example, when importing surface data to be interpolated by an RBF in Chapter 7, Section 7.4, the ‘gradient’ field is used to specify scanner position. The gradient field is often used to specify point normals or camera/scanner directions to assist fitting surfaces to point cloud data.

In the case of pure point cloud data, the input data may consist only of the locations of surface points and have no associated attribute data. No value need be specified when FastRBF’s implicit surface modelling functions are used in this context since it is understood that the RBF is zero at all surface points. When an accuracy value is present with such point cloud data, it is interpreted as an uncertainty on the location of the surface point.

5.2 Importing and exporting formated text data Consider the 2D data file ‘testdata2d.txt’ which can be found in the FastRBF Tutorial folder: ..\fastrbf_v1.4\matlab\tutorial\ImportData The first few lines of ‘testdata2d.txt’ are as follows % Example 2D data Loc 0.41 0.31 Tval 6.72e-001 Pval -5.49e-001 Loc 0.49 0.59 Tval 1.03e-001 Pval -8.95e-002 Loc 0.16 0.04 Tval 6.03e-001 Pval 7.61e-001 . . .

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Getting started: Importing and Exporting Data

The file consists of lines of 2D coordinates followed by two attributes which we wish to interpolate. This data can be read into a FastRBF data structure using fastrbf_import: data2d=fastrbf_import(’testdata2d.txt’,’format’,’%(*s) %x ... %y %(*s) %<Tvalue>v %(*s) %<Pvalue>v\n’,’comment’,’%’) data2d = Location: [2x10000 double] Tvalue: [1x1 struct] Pvalue: [1x1 struct] The resulting data2d structure consists of a 2x10000 matrix of point locations, data2d.Location, a 1x10000 vector of ‘Tval’ values, data2d.Tvalue.Value, and a 1x10000 vector of ‘Pval’ values, data2d.Pvalue.Value.

The full syntax for fastrbf_import is described in the reference guide in Section 10.6. In this example the comment option tells the parser to skip any line beginning with the associated string, which in this case is the character ‘%’.

Consequently, the first line of the file is ignored as a comment by the parser.

The string argument of the format option tells FastRBF how to the parse a noncomment line in the file. The syntax for the format string, is similar to fscanf in C and described fully in Section 10.6. %(*s) tells the parser to skip any string found in the corresponding position within the line. Thus, the strings ‘Loc’, ‘Tval’ and ‘Pval’ are ignored in this example. The %x and %y fields denote location coordinates. There is no %z string in this example and so FastRBF interprets the data as being 2D. The %<Tvalue>v and %<Pvalue>v strings each denote an attribute with a scalar value at the location coordinates. The string within the <> brackets denotes the arbitrary name assigned to the attribute and the v indicates that the attribute type is a scalar value. In FastRBF all values are read as double precision numbers. Both attributes must be present on each line of the file for the line to be correctly parsed. If a line is not correctly parsed then it is skipped and none of the data within that line is stored. We can use this feature as a more flexible alternative to using the comment option.

Rather than specifying the ‘%’ string to designate a comment line in the above example, we can force FastRBF to read only those lines starting with ’Loc’ as follows, ... data2d=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y... %(*s) %<Tvalue>v %(*s) %<Pvalue>v\n’)

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User’s Guide and Tutorial

Since all data lines in this example are consistent with their attribute labels we could write, data2d=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %<Tvalue>v Pval %<Pvalue>v\n’) If a line does not match the format string, i.e. it can not be parsed by the parser, then that line is discarded — no data is read and the parser advances to the next line in the file.

We will now consider the case where only one of the attributes in the file is of interest, say ’Tval’. We can simplify the previous command even further: data2d=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %<Tvalue>v’) data2d = Location: [2x10000 double] Tvalue: [1x1 struct] The line is parsed successfully since the format string still matches the first part of each line read. No constraints are placed by the format string on the remainder of the line after the ‘Tval’ value is read and so any trailing data is ignored. Thus, the ‘Pval’ attribute is not parsed.

If only the ‘Pval’ attribute was of interest then the ‘Tval’ values could be skipped using the ‘don’t care’ code %(*s) in the format string, data2d=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %(*s) Pval %<Pvalue>v\n’) data2d = Location: [2x10000 double] Pvalue: [1x1 struct]

5.2.1

Default attribute

There is a default attribute associated with each point in a FastRBF point list.

If no specific attribute name is specified when reading or accessing data then the default attribute is used. For example, ‘Tval’ is read as the default value stored at each location in the following command since no attribute name is specified, i.e. we have used %v instead of %<attribute name>v.

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Getting started: Importing and Exporting Data

data2d=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %v Pval %<Pvalue>v\n’) data2d = Location: [2x10000 double] Value: [1x10000 double] Pvalue: [1x1 struct] In this case data2d.Value is the column vector of ‘Tval’ values and data2d.Pvalue.Value is the column vector of ‘Pval’ values.

5.2.2

Putting it all together

The following Matlab code is a straightforward example of loading a 2D data set, fitting an RBF and then evaluating on a regular grid. The fitting and grid evaluation functions are fully explained in the following chapter. The raw data for both attributes, and the corresponding result of fitting an RBF to each, are shown in Figure 5.1. %Read formated text data data=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %v Pval %<Pvalue>v\n’) data=fastrbf_unique(data) %Save data in FastRBF native format fastrbf_save(data,’tut2ddata’) %Fit RBF to Tval & evaluate on grid rbf_Tval=fastrbf_fit(data,0.0001) Tgrid=fastrbf_grideval(rbf_Tval,’spacing’,0.01) %Display interpolated data [x y] = fastrbf_gridcoords(Tgrid); surf(x, y, Tgrid.Value, ’facecolor’, ’interp’); %Fit RBF to Pval & evaluate on grid rbf_Pval=fastrbf_fit(data,0.0001,’attr’,’Pvalue’) Pgrid=fastrbf_grideval(rbf_Pval,’spacing’,0.01)

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User’s Guide and Tutorial

%Display interpolated data [x y] = fastrbf_gridcoords(Pgrid); surf(x, y, Pgrid.Value, ’facecolor’, ’interp’); Note that in the above code ‘Tval’ was read as the default value at each point while ‘Pval’ was stored as a named attribute. The attr option is used with fastrbf_fit to specify a named attribute. Consequently, when the second RBF is fitted to the ‘Pval’ data, the attr option is used with the argument "Pvalue". Attention is also drawn to the use of the fastrbf_unique command. This command removes duplicate points from the data. A nonparametric function can not be fitted to an attribute which is multivalued at a point in space. This occurrence is not as rare as it might first appear, particularly when reading data stored in a text format. The limited precision of many data formats, and sometimes of intermediate processing steps prior to writing the file, means that otherwise distinct points can be assigned the same location in an output file. This can result in an apparently multi-valued distribution.

Various CAD formats for graphics data are prone to producing apparent multiple vertices in mesh data.

Hint It is good practice to apply fastrbf_unique before fitting an RBF to any data set.

5.2.3

Exporting data as formatted text

Data can be exported from FastRBF as formatted text using the fastrbf_export function. The syntax is similar to fastrbf_import. A format string is specified which determines which data attributes are written to the file and how the line is formatted. The syntax for the format string is similar to printf and described in detail in Section 10.6.2. fastrbf_export currently permits only point list data to be written out as formatted text. In the previous example, the fastrbf_grideval function returned a grid data structure. A point list could be constructed from this using the fastrbf_gridvec function as follows, [x y] = fastrbf_gridvec(Tgrid); [xx, yy]=ndgrid(x,y); Tgrid_plist.Location=Pgrid_plist.Location=[xx(:), yy(:)]’; Tgrid_plist.Value=Tgrid.Value(:)’; Pgrid_plist.Value=Pgrid.Value(:)’;

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Getting started: Importing and Exporting Data

%Export point list data as formatted text fastrbf_export(Tgrid_plist,’Tgrid.txt’,’txt’,’format’,’%x ... %y %v\n’) fastrbf_export(Pgrid_plist,’Pgrid.txt’,’txt’,’format’,’%x ... %y %v\n’) Here we have stored the result of grid evaluations as the default value in two separate pointlists and exported them as two simple ASCII files. More efficient use of memory can be achieved by storing the data as a single point list. An advantage of doing this is that fastrbf_export can write out both attributes in a single file. [x y] = fastrbf_gridvec(Tgrid); [xx, yy]=ndgrid(x,y); TPgrid_plist.Location=[xx(:), yy(:)]’; TPgrid_plist.Tvalue.Value=Tgrid.Value(:)’; TPgrid_plist.Pvalue.Value=Pgrid.Value(:)’; %Export point list data as formatted text ... fastrbf_export(TPgrid_plist,’TPgrid.txt’,’txt’,’format’,’%x... %y %<Tvalue>v %<Pvalue>v\n’) Alternatively, the fastrbf_grideval function can produce a point list in the first place by using the pointlist option. %Generate grid data as a point list ...

Tgrid_plist=fastrbf_grideval(rbf_Tval,’pointlist’,’spacing’... ,0.01) ...

Pgrid_plist=fastrbf_grideval(rbf_Pval,’pointlist’,’spacing’... ,0.01) %Export point list data as formatted text fastrbf_export(Tgrid_plist,’Tgrid.txt’,’txt’,’format’,’%x ... %y %v\n’) fastrbf_export(Pgrid_plist,’Pgrid.txt’,’txt’,’format’,’%x ... %y %v\n’) A further use for fastrbf_export is simply to reformat ASCII file data.

For example, we can reformat our 2D test data as follows,

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User’s Guide and Tutorial

data=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %<Tvalue>v Pval %<Pvalue>v\n’) ... fastrbf_export(data,’newdata.txt’,’txt’,’format’,’%x %y... %(.4f)<Tvalue>v %(.4f)<Pvalue>v\n’) The result is a new file of the following form, 0.41 0.31 0.6720 -0.5490 0.49 0.59 0.1030 -0.0895 0.16 0.04 0.6030 0.7610 . .

In this example we have specified a format in which values are written as doubles with 4 decimal places of precision. In the previous example the default format, (.10g), was used. Note that for many Windows applications \r\n should be used for correct line termination.

5.3

Importing and exporting 3D data

We will now consider importing the example 3D data file ‘testdata3d.txt’ to illustrate importing vector and accuracy values. The first few lines of the file are as follows, % Example 3D data 11.5 2.5 3.5 val 0.3 acc 0.001 grad 0.5 3.5 8.7 col 1 5 7 11.5 23.5 13.5 val 4.3 acc 0.001 grad 1.0 30.0 20.0 col 30 5 8 3.3 22.5 5.0 val 9.3 acc 0.003 grad 0.5 0.5 0.707 col 2 4 9 . . .

At each point two vector fields are specified along with a value and an accuracy. This data could be read using the following command, ... data3d=fastrbf_import(’testdata3d.txt’,’format’,’%x %y %z... %(*s) %v %(*s) %a %(*s) %dx %dy %dz %(*s) %<col>dx ... %<col>dy %<col>dz\n’,’comment’,’%’) In this example the scalar value, accuracy and the first vector are read as default attribute values and the second vector is read as the col attribute. The

56

Getting started: Importing and Exporting Data

(a) ‘Tval’ attribute data

(b) ‘Pval’ attribute data

(c) ‘Tval’ RBF

(d) ‘Pval’ RBF

Figure 5.1: The example 2D data set read as formatted text and the corresponding grid evaluations of an RBF fitted to each data attribute.

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vector data are stored in the gradient field. The gradient components are denoted by %dx, %dy and %dz. The code %a denotes an accuracy value. In this example the second vector field represents an RGB colour tuple at a point in 3space, rather than a gradient. These colour components could also have been read and stored as individual attributes as follows, ... data3d=fastrbf_import(’testdata3d.txt’,’format’,’%x %y %z... ... %(*s) %v %(*s) %a %(*s) %dx %dy %dz %(*s) %<R>v %<G>v... %<B>v\n’,’comment’,’%’) Note that the scalar value, accuracy and components of the vector data are all stored as double precision floating point numbers by FastRBF. Error bars The FastRBF error-bar fitter described in the following Chapter finds the smoothest function within a range of values specified at each data point. This is useful for fitting a function to noisy data. The range is specified by a lower and upper bound on an attribute value at each point. These bounds can be specified in formatted text files and read using the %<attribute_name>l and %<attribute_name>u codes in the format string, or %l and %u if the bounds apply to the default value.

5.3.1

Exporting 3D data

3D pointlist data including vector and accuracy attributes can also be exported from FastRBF as formatted text files, just as in the previous 2D example. For example, the following code will load and convert the 3D data to a new format. data=fastrbf_import(’testdata3d.txt’,’format’,’%x %y %z ... ... val %v acc %a grad %dx %dy %dz col %<col>dx %<col>dy %<... col>dz\n’) fastrbf_export(data,’newdata3d.txt’,’txt’,’format’,’%x %y ... ... %z %(.4f)v %(.4f)a %(.4f)dx %(.4f)dy %(.4f)dz %(.4f)<... col>dx %(.4f)<col>dy %(.4f)<col>dz\n’) The result is a new file of the following form, 11.5 2.5 3.5 0.3 11.5 23.5 13.5 4.3 3.3 22.5 5.0 9.3

0.001 0.5 3.5 8.7 0.001 1.0 30.0 20.0 0.003 0.5 0.5 0.707

58

1 5 7 30 5 8 2 4 9

Getting started: Importing and Exporting Data

5.4 Importing and Exporting Surface Data FastRBF supports four specific types of surface data for RBF applications involving surface reconstruction, mesh repair and simplification. The currently supported formats are: Argument

Description

.obj

Alias|Wavefront point cloud and 3D mesh data with colour per vertex.

.ptx

Cyra PTX 3D LIDAR scan.

.pto

Maptek PTO 3D scan.

.stl

Stereo-lithography binary 3D mesh.

It is important to note that the current version of FastRBF only supports mesh data consisting of vertices and facets which are either triangles or quadrilaterals. N-agons are not yet supported. These data formats are discussed further in the reference guide (Section 10.6) and are illustrated in “Fitting 3D surface data” (Chapter 7).

5.4.1

Example

Importing an Alias|Wavefront obj file is as simple as, mesh=fastrbf_import(’Hand.obj’); mesh is a FastRBF data structure which represents the mesh as an array of vertex coordinates and facet indices (see Section 11.2.2). In this case the format of the imported file has been successfully determined from the suffix of the filename. We can explicitly instruct FastRBF to import a particular type using the appropriate option listed in the Reference guide (Section 10.6.1). For example, mesh=fastrbf_import(’Hand.obj’,’obj’); forces FastRBF to interpret the file as an obj file. Similarly, exporting the mesh as an stl file is as simple as, fastrbf_export(mesh,’hand.stl’); Again, the output format was determined from the filename suffix. The stl option could have been used to explicitly enforce an stl output. fastrbf_export(mesh,’hand.stl’,’stl’);

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5.5 Summary 5.5.1

Importing text files

• Data are stored as Pointlists • A Pointlist consists of a location vector and attribute list • Attributes may consist of a scalar value, an accuracy, upper and lower bounds (specified relative to the value) and a vector • Any type of vector data can be stored in the gradient field of an attribute • All values, precisions, locations and vector components are stored as double precision numbers by FastRBF • A default attribute is associated with each location in a point list and is used when no attribute name is specified • No data is recorded from a line unless the whole line is successfully parsed

5.5.2

Importing surface data

• FastRBF only supports mesh data consisting of vertices and quadrilateral or triangular facets

5.5.3

Exporting text files

• The default format for writing doubles is (.10g) • \r\n should be used for line termination on Windows machines

5.5.4

General tips

• Beware of the finite precision of various data formats when reading and writing data to a file • Use fastrbf_unique to detect and remove duplicate points arising from finite precision • Use the attr option with subsequent FastRBF functions to process any attribute other than the default

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Chapter 6

Fitting and Evaluating RBFs In this chapter we illustrate interpolation and approximation of scattered data with Radial Basis Functions. The process of fitting a function to scattered data and resampling on a regular grid is also known as gridding. The chapter is organised in to two tutorial examples which explore the various options associated with the FastRBF fitter and evaluator. The first considers a 2D example and the second considers 3D data. There is no conceptual difference between modelling 2D and 3D data. However, the are some differences in speed and computation overhead.

6.1 Fitting and Evaluating 2D Data We begin by considering the straightforward case of interpolating a 2D data set and evaluating the resulting RBF on a regular grid.

First we import the 2D point data ‘testdata2d.txt’ using the FastRBF fastrbf_import function described in the previous chapter. In this tutorial we will consider just the ‘Tval’ attribute of the data set. This attribute is read as the default attribute using the following command, data=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %v Pval %(*s) \n’,’comment’,’%’) data = Location: [2x10000 double] Value: [1x10000 double]

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Next we use fastrbf_unique to remove duplicate points data=fastrbf_unique(data) data = Location: [2x6367 double] Value: [1x6367 double] This reduces the number of points from 10000 to 6367 points. fastrbf_unique removes points that are within a certain distance of another point. The default distance is 1e-10. It is advisable to use fastrbf_unique whenever an RBF is fitted to ensure there are no duplicate points in the data.

Note fastrbf_unique is order sensitive and removes ‘duplicate’ points as it encounters them. If the input pointlist is re-ordered then the points removed and the number of points removed may change. For example, if a point is close to two other points, but those points are sufficiently far apart then both will be preserved if the common point is encountered first and removed. Alternatively, two of the three points will be removed if the common point is not encountered first.

Figure 6.1 illustrates the raw data. Figure 6.1(a) shows the locations of the data. Figure 6.1(b) is a 3D view depicting the value at each data point as a height above the xy plane. These plots were generated using the following M ATLAB commands: plot(data.Location(1,:), data.Location(2,:), ’.’); plot3(data.Location(1,:), data.Location(2,:), data.Value, ... ’.’); Alternatively, the fastrbf_view command could be used, fastrbf_view(data) The data locations lie on the unit square. The attribute values are in the range -1,1. We can fit an RBF to 0.0001 accuracy using the fastrbf_fit function as follows. rbf=fastrbf_fit(data,0.0001)

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6.1.1

Accuracy

A fitting accuracy must always be specified when performing a FastRBF fit since FastRBF uses approximations to find the coefficients of the RBF. In this example the test data does not contain a vector of accuracies per point and so a global fitting accuracy has been specified. 0.0001 was deemed sufficient for the range of values present in the data. If a local accuracy per point is present, then FastRBF will use the per point value only where it exceeds the global value. This behaviour is exploited in Section 6.1.7 where an RBF is progressively fitted to the data by refining the global value from coarse to fine (i.e., progressively tightening the fit by reducing the accuracy value). To ensure that the per point accuracy is always used, the fitter can be called with a zero value for the global accuracy.

(a) Data samples

(b) Data values

Figure 6.1: Raw 2D data â&#x20AC;&#x2DC;Tvalâ&#x20AC;&#x2122; attribute.

Note The accuracy specified with the fitter is a maximum limit. The actual values achieved at most points will be less than this.

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6.1.2

‘Verbose’ option

The verbose option can reveal useful fitting information including the fitting time and the actual achieved accuracy (maximum deviation from the data).

Also indicated are the minimum and maximum memory used along with the desired maximum memory. This information can be used to diagnose performance problems and determine whether the hardware is sufficient for the size of the problem.

For example, the following command revealed that the greatest deviation of the RBF from any data value in the previos example was 1.8959e-005. rbf=fastrbf_fit(data,0.0001,’verbose’)

6.1.3

Grid evaluations

The fitted RBF can be evaluated anywhere in space using the FastRBF pointeval command. FastRBF also provides functions for evaluating on a grid (2D and 3D where appropriate) as well as an iso-surfacing command for 3D RBFs. Here we evaluate the RBF fitted in the previous section on a grid with a step size of 0.01 over the default domain of the raw data, which in this case is the unit square. g=fastrbf_grideval(rbf,’spacing’,0.01) The default evaluation accuracy is automatically chosen to be 100th the fitting accuracy of the RBF. This information is stored in the RBF. Alternatively, the user can explicitly specify an evaluation accuracy, g=fastrbf_grideval(rbf,’spacing’,0.01,’accuracy’,0.005) Specifying a looser accuracy than the default will generally result in faster evaluation times. Figure 6.2 depicts the resulting grid evaluation as (a) a 2D image, where colour corresponds to pixel value, and (b) as a surface above the xy plane, where height and colour depict the interpolated value. Figure 6.2(a) and Figure 6.2(b) were generated by the following M ATLAB commands followed by appropriate scaling of the axes. fastrbf_view(g); surf(g.Value’); Note that in the surf command, the transpose of the evaluated matrix is passed as the argument to make M ATLAB’s row-column representation consistent with the xy axes of the original data.

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To export the grid it is necessary to use the pointlist option because the export command does not support the grid type. The pointlist option causes grideval to return the grid as a point list type which can then be exported. For example, to export the grid in Microsoft Excel Character Separated Values format, we could use the following commands, p=fastrbf_grideval(rbf,’spacing’,0.01, ’pointlist’) fastrbf_export(p, ’grid.csv’,’txt’,’format’, ’%x,%y,%v\n’);

(a) Image view

(b) 3D view

Figure 6.2: 2D grid evaluation of an RBF fitted to scattered data.

6.1.4

Bounding boxes

In Figure 6.3 we demonstrate the extrapolation capabilities of the RBF by evaluating the fitted RBF over a larger domain which includes regions far from the original data points. This achieved simply by specifying a larger bounding box for grideval using the min and max options. The default bounding box in the previous example corresponded to that which just enclosed the raw data points, which in this case was approximately the unit square. gl=fastrbf_grideval(rbf,’spacing’,0.01,’min’,[-0.5 -0.5], ... ’max’,[1.5 1.5]) fastrbf_view(gl)

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surf(gl.Value’) Figure 6.3 illustrates the RBF evaluated over the larger domain. The smooth extrapolation properties of the RBF are apparent.

6.1.5

Anisotropic sampling

So far we have generated grids with isotropic sampling (we have evaluated at equal intervals along each grid axis). We can specify different sampling frequencies along the x and y axes by supplying a vector argument to the spacing option, g=fastrbf_grideval(rbf,’spacing’, [0.01 0.02])

6.1.6

Multiple attributes

The attr option allows the user to specify the particular attribute which a FastRBF function operates on when multiple attributes exist at each data point. In the above examples the default attribute was modelled by an RBF and evaluated on a grid. As mentioned in the preceding chapter, the alternative ‘Pval’ attribute could have been loaded and modelled in the following manner, data=fastrbf_import(’testdata2d.txt’,’format’,’Loc %x %y ...

Tval %v Pval %<Pval>v\n’,’comment’,’%’); data=fastrbf_unique(data); rbf=fastrbf_fit(data,0.0001,’attr’,’Pval’); g=fastrbf_grideval(rbf,’spacing’,0.01,’accuracy’,0.005);

6.1.7

Iterative fitting of an RBF

The initial option for the FastRBF fitter enables a user to supply an initial guestimate of the RBF solution when fitting an RBF to a data set. This can speed up fitting times significantly depending on the quality of the initial estimate. There are two common contexts in which this situation arises. The first is when a user progressively refines the accuracy of an RBF fit. The second is when new data is added to a data set already described by an RBF. The latter situation may occur with large problems where initially an RBF is fitted to subset of the data points.

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(a) Image view

(b) 3D view

Figure 6.3: RBF extrapolation beyond the original scattered data.

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Progressively refining the accuracy of an RBF fit A coarse fit to a data set is usually faster than a fine fit. The initial option can be used to iteratively refine the accuracy of a fit as follows, rbf=fastrbf_fit(data,0.005); rbf=fastrbf_fit(data,0.000001,’initial’,rbf); Warning In certain fitting modes, such as when the errorbar or reduce options are used, performing an incremental fit will result in a slower overall fitting time if the accuracy is relaxed rather than tightened. Incremental fitting is usually an advantage only when refining the accuracy from coarse to fine.

Adding new data to an RBF interpolant In some large problems it is desirable to fit to a subset of the data points available and then progressively add more data. In other problems new data might become available. In either case, a new fit can often be achieved faster by starting with the solution for the smaller set of data. In the following example we add new data to our 2D test data set and compute the new RBF using the initial option with the current RBF solution, % initial fit rbf=fastrbf_fit(data,0.0001); % load new data ... newdata=fastrbf_import(’newtestdata2d.txt’,’format’,’Loc %x... %y Tval %v Pval %<Pval>v\n’,’comment’,’%’); data.Location=[data.Location newdata.Location] data.Value=[data.Value newdata.Value] data.Pval.Value=[data.Pval.Value newdata.Pval.Value] % perform new fit - only on Tval attribute rbf=fastrbf_fit(data,0.0001,’initial’,rbf); Care must be taken when appending new data to an existing pointlist that the ordering of points in the pointlist remains consistent with the initial RBF solution. The length of all arrays past to the fitter must also be the same. Thus,

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if there are N elements in a point list then each attribute in the pointlist past to the fitter must have N corresponding values. If the new data which becomes available covers only a subset of the attributes in the previous pointlist, then the user must ensure that the new longer pointlist only contains those attributes which are complete for all the points. Care must also be taken to ensure that the new data added is not spatially coincident with existing points, resulting in a multi-valued distribution. This can be ensured by applying FastRBFâ&#x20AC;&#x2122;s unique function. In general, the user must ensure that the new data is spatially consistent (aligned) with the existing data. If not, then the result may appear very noisy and it is likely to take longer than a fit from scratch.

Warning The overhead for setting up the RBF system is a function of the total number of data points and is the same for an incremental fit using the initial option as for a new fit from scratch. For this reason it is not efficient to incrementally add small numbers of points to an RBF problem. Generally, an addition of 5% or more points is beneficial for large problems.

The example given here is purely illustrative. Only 3 new points are added to the system. Given the fixed setup costs, little benefit is observed from an incremental fit in this case.

Note The fitting parameters can be changed between one iteration of fitting and the next. This includes the fitting strategy, whether it is direct, reduce, errorbar, rho or rho reduce. However, the benefit of using an initial RBF solution is maximised when the final RBF is most similar to the initial estimate.

6.2 Fitting and Evaluating 3D data In this tutorial we explore further FastRBF fitting and evaluation parameters with a 3D data set. In particular we will consider the issues of under-sampling (anti-aliasing) and the presence of noise on the input data.

We begin by loading the 3D point data. In this case the data is already in the FastRBF format. The FastRBF import function can be used to load other formats.

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data=fastrbf_load(’data.p3d’) Figure 6.4 illustrates the raw data. Figure 6.4 shows the locations of the data in the unit cube and the corresponding value is indicated by colour. Values range from 0 to 1. This plot was generated using the fastrbf_view function: colormap(jet(256)) fastrbf_view(data) To be safe we process the data with unique function, data=fastrbf_unique(data) Since the data value range is 0,1, we fit an RBF to 0.001 accuracy using the fit function. rbf=fastrbf_fit(data,0.001)

Figure 6.4: Raw 3D data ’data.p3d’.

6.2.1

RBF centre reduction

Recall our basic definition of an RBF (Equation (1.1)) N

s ( x ) = p ( x ) + ∑ λi Φ ( x − xi ) , i =1

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The default operation of FastRBF’s fitter is to solve the corresponding linear system (Equation (1.3)) to find the coefficients λi which define s( x) such that s( xi ) = f i , where the f i are our known (i.e., measured or given) data values.

Equation (1.1) and Equation (1.3) imply that for each data point there will be a corresponding centre (term) in the fitted RBF. In FastRBF the straightforward solution of Equation (1.3) indeed results in a centre for each data point. This is called the direct method and is the default behaviour of the fitter. It can be specified explicitly using the direct option. Often the desired accuracy can be achieved by a fit with fewer centres. The FastRBF reduce option attempts to find an RBF which satisfies the same interpolation conditions as the direct method, but with fewer centres. Only a subset of the original data points are utilised as RBF centres. Consider the following fit, rbf=fastrbf_fit(data,0.001,’reduce’,’verbose’) Centre reduction results in 843 centres compared to 5000 centres for the direct method and it is achieved in one third the fitting time. The same fitting accuracy as for the direct method is achieved.

Centre reduction is very important when modelling large data sets with RBFs for two reasons, • Reducing the number of centres results in a more compact RBF which can be evaluated in less time.

Subsequent grid evaluations, or iso-surfacing operations (in 3D) can be greatly speeded up depending on the amount of reduction achieved.

• Centre reduction can result in significantly faster fitting times even with moderate amounts of centre reduction (i.e. a factor as low as 2-3).

In addition, when modelling surface data, a compact RBF can represent significant compression of the corresponding mesh or point cloud data. For these reasons, wherever possible the reduce option is recommended over a direct fit for low noise data. Due to the small size of this example problem, the benefits of the reduce fit are not as apparent as for larger 100,000 point plus problems.

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Note The reduce option is recommended for data with low noise (relative to the desired fitting accuracy). In such cases RBF fitting will generally be faster and subsequent evaluations of the resulting RBF will also be faster than if a direct fit had been used. If the data is very noisy or the underlying signal contains predominantly high frequencies (varies randomly) then the overall fitting time may be slower than a direct fit and an approximation technique such as the errorbar fit should be considered. If the reducer detects that sufficient reduction is not achievable (less than a factor of 2) then it reverts to a direct fit. For noisy data, the error-bar fitter can achieve a significant reduction in centres while smoothing the data (Section 6.4.2).

In our experience the reduce fit has been very useful for low noise surface data particularly data acquired from small scale laser scanners rather than large-scale LIDAR scanners. The errorbar fit is more suited to the latter. However, the noise in the data depends on the surface being scanned as well as the acquisition system. We have found that many types of noisy geophysical and empirical data are better suited to a straightforward direct fit.

The reduce option can also be used with the spline smoothing approximation method discussed in Section 6.4.3 and indicated by the rho option.

6.2.2

3D grid evaluation

Here we consider evaluating our 3D RBF on a grid at a step size of 0.02 over the default domain of the raw data. g=fastrbf_grideval(rbf,’spacing’,0.02) As with the 2D fitting and evaluation examples in the previous tutorial, the evaluation accuracy is automatically chosen as 100th the fitting accuracy of the RBF. Figure 6.5(a) depicts the resulting 3D grid evaluation as cut-away slices through the volume. Note that the M ATLAB slice command uses meshgrid data ordering whereas we use ndgrid data ordering, hence the need to permute the Value array and arguments. [gX, gY, gZ] = fastrbf_gridcoords(g); gV = permute(g.Value, [2 1 3]); slice(gY, gX, gZ, gV, .7, .7, .3); colorbar; axis tight;

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To export the grid it is necessary to use the pointlist option because the export command does not support the grid type. For example, to export the grid in Microsoft Excel Character Separated Values format we do the following. p=fastrbf_grideval(rbf,’spacing’,0.02, ’pointlist’) ... fastrbf_export(p, ’grid.csv’, ’txt’, ’format’, ’%x,%y,%z,%v... \n’); Figure 6.5(b) depicts the RBF evaluated over a larger domain using the min and max options, ... gl=fastrbf_grideval(rbf,’spacing’,0.02,’min’... ,[-0.5 -0.5 -0.5],’max’,[1.5 1.5 1.5])

(a)

(b)

Figure 6.5: 3D grid evaluation of an RBF fitted to scattered data. (a) Evaluation over domain of input data, (b) evaluation over an extended domain.

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6.3 Anti-aliasing In this section we consider evaluating an RBF at coarse resolutions relative to the frequencies present in the RBF. First we conduct a direct fit to a new data file, ‘noise .2 data.p3d’. This data was formed from the previous data set by adding uniform random noise of magnitude ±0.1 (±10%). noise_data=fastrbf_load(’noise_.2_data.p3d’) noise_rbf=fastrbf_fit(noise_data,0.001)

(a) Noisy data

(b) Grid evaluation

Figure 6.6: 3D data with 10% noise added. (a) Raw data, (b) grid evaluation of an RBF fitted to the data.

Figure 6.6(a) depicts the raw noisy data and Figure 6.6(b) the result of a grid evaluation of the fitted RBF on a 0.02 grid. g=fastrbf_grideval(noise_rbf,’spacing’,0.02) It is apparent from Figure 6.6(b) that the noise has contributed a high frequency component to the data. We will now consider evaluating the RBF on a coarse grid relative to the frequency of the noise.

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(a) Without anti-aliasing

(b) With anti-aliasing

(c) Noise-free grid evaluation

Figure 6.7: Coarse grid evaluation, (a) without anti-aliasing, (b) with antialiasing, (c) original noise-free data evaluated on a 0.1 grid.

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Consider evaluating the RBF fitted above on a grid with 0.1 step size. Figure 6.7(a) illustrates an ordinary grid evaluation without low pass filtering (smoothing), g=fastrbf_grideval(noise_rbf,’spacing’,0.1) Relative to the noise component of the signal, the RBF is being under-sampled.

Figure 6.7(b) illustrates the result of specifying a smoothing filter (Section 1.4) in the evaluation with a FWHM (Full Width Half Maximum) equal to the sampling interval 0.1. g=fastrbf_grideval(noise_rbf,’spacing’,0.1,’smooth’,0.1) For comparison Figure 6.7(c) illustrates a 0.1 grid evaluation of the original noiseless data.

The effects of noise and aliasing are difficult to see in a grid evaluation.

These are more apparent when an isosurface is extracted from the data. Figure 6.8 uses isosurfaces to illustrate the effects of smoothing in the following section.

In theory, every RBF evaluation should be low-pass filtered according to the sampling interval to ensure that aliasing is avoided, otherwise smooth data can appear spiky or rippled when under-sampled. However, it is left to the discretion of the user to employ smoothing during RBF evaluation since the Gaussian-like filter kernel is not an ideal low pass filter and sometimes further smoothing is desirable beyond the Nyquist limit.

Note Evaluation with smoothing is slower than an ordinary RBF evaluation by a factor of two in 3D and significantly slower in 2D where it is O( N ) rather than O(log N ), where N is the number of centres. so only use smoothing when required.

6.4 Noise reduction In the previous section a low pass (smoothing) filter was used to prevent aliasing when under-sampling an RBF during evaluation. The purpose was to obtain a more faithful representation of the signal on the under-sampled grid.

Now we will consider the task of smoothing the signal to remove unwanted detail or ‘noise’. FastRBF offers two approaches to processing noisy data. One method is to interpolate the data exactly with an RBF and then apply low pass

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filtering (smoothing) during RBF evaluation, while the other is to fit an RBF which approximates the input data rather than interpolating it. FastRBF offers two options in the latter case, one is called smoothest restricted range approximation, or error-bar fitting for short, and the other is spline-smoothing.

Figure 6.8 illustrates a series of isosurfaces evaluated on the example data.

The isosurface threshold applied is 0.5 and the mesh resolution (sampling frequency) is 0.03. Figure 6.8(a) is the isosurface corresponding to the noise-free data and is generated by the following commands, Mesh = fastrbf_isosurf(rbf, 0.03, 0.5); fastrbf_view(Mesh,’fe’); Figure 6.8(b) is the corresponding isosurface for the noisy data, Mesh = fastrbf_isosurf(noise_rbf, 0.03, 0.5);

6.4.1 Low pass filter By assuming that the noise in the data can be discriminated by its contribution to the higher frequencies, we can reduce it by applying a low pass filter (LPF) with a suitable cut-off frequency, during RBF evaluations. In this case we choose a cut-off corresponding to 0.1 (considerably more smoothing than required to achieved the Nyquist limit for the 0.03 sampling interval). Figure 6.8(c) illustrates the result.

Mesh = fastrbf_isosurf(noise_rbf, 0.03,0.5,’smooth’,0.1); Another interpretation of this result is that it is the consequence of applying a local average weighted by a Gaussian kernel with a FWHM of approximately 0.1. Features occurring at a scale finer than 0.1 are attenuated while structure occurring at coarser scales is relatively unaltered.

Note Low pass filtering (LPF) can be applied in conjunction with an errorbar or spline-smoothing fit to further smooth an RBF during evaluation.

A benefit of the the LPF approach is that it can be varied a posteriori and hence, ‘undone’. However, a similar, albeit irreversible, smoothing effect can often be achieved with an error-bar or spline smoothing fit, but with far fewer centres.

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(a) No noise

(b) Noise

(c) Smooth evaluation

(d) Restricted range fit

Figure 6.8: Comparison of data smoothing effects due to smoothing during evaluation (implicit smoothing) and restricted range RBF approximation.

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Hint For a given amount of smoothing, centre reduction can be achieved with LPF by evaluating the RBF at the original data points with LPF smoothing and then fitting a new RBF to this smoothed data using the reduce option.

6.4.2

Error-bar fitter

The error-bar fitter finds the smoothest RBF passing between an upper and lower bound on the value at each data point. Figure 6.8(d) illustrates the result of an errorbar fit. In this case we know a priori that the magnitude of the noise added to the data is 0.1 (10%) so a fit to within 0.1 of the data values at each data point was specified. rrfit_rbf = fastrbf_fit(noise_data, 0.1,’errorbar’) Mesh = fastrbf_isosurf(rrfit_rbf, 0.03,0.5,’seedlimit’,0); Note that the ‘accuracy’ parameter is interpreted in this context as a global bound on the value of the RBF at each data point, rather than the precision to which Equation (1.1) is fitted. A point list with individual per point error-bars (fitting tolerances) can also be approximated in this way. Asymmetric errorbars can be achieved by specifying both upper and lower values in the data, or symmetric per point error-bars can be specified with just the upper or lower bound being given.

Note When both a global accuracy and a per point error bound are specified, the greater (i.e. the loosest constraint) of the two at each point is applied.

Figure 6.8(d) and Figure 6.8(c) are not the same. In Figure 6.8(c) we attenuated high frequencies in the signal while in Figure 6.8(d) the controlling parameter was an estimate of the magnitude of the noise at each data value. In the latter, high frequencies are not necessarily attenuated uniformly throughout the data, as with the low pass filter (LPF).

The error-bar RBF has only 250 centres compared to the ‘exact’ fit RBF which has 5000 centres. In general, centre reduction is expected with an approximating RBF such as the error bar fitter since information is reduced. The direct fit RBF captures all the detail present in the raw data. The fine detail

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smoothed by an error bar fit is not represented in the RBF and can not be recovered. However, just as LPF is used to attenuate high frequencies during subsequent evaluations of a direct fit RBF, LPF can be used to further smooth an error-bar fit or, as in the following example, reduce aliasing artifacts.

Technically, with the error-bar fit we should also employ anti-aliasing when evaluating the isosurface. Filtering should be appropriate to the 0.03 sampling interval, Mesh = fastrbf_isosurf(rrfit_rbf, 0.03,0.5,’seedlimit’, 0, ... ’smooth’, 0.03); In this case the effect is not noticeable since the error-bar fit contains very few frequencies higher than the 0.03 sampling interval.

Note With the error-bar fitter only a subset of the input data points contribute centres to the RBF. A consequence of finding the smoothest approximator to the data is that at those data points which contribute centres to the RBF, the maximum permitted deviation will occur, i.e., the RBF will pass through one of the two errorbars associated with each centre.

Asymmetric error bars Asymmetric error bars can be specified using the Upper and Lower fields in the pointlist data structure. The file ’ebardata.p3d’ contains per-point upper and lower bounds on each value. The lower bound is 0.01 and the upper is 0.03. These bounds could vary at each data point, but in this example they are the same at all data points. data=fastrbf_load(’ebardata.p3d’) First we fit an RBF to the data with a global accuracy of 0.05. We evaluate the RBF at the original data points and look at the deviation of the fitted surface from the raw data. rbf=fastrbf_fit(data,0.05,’errorbar’,’verbose’) evpnts=fastrbf_pointEval(rbf,data.Location) min(evpnts.Value-data.Value) ans = -0.0490 max(evpnts.Value-data.Value) ans = 0.0494

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The fit is constrained by the looser 0.05 global bound which is greater than any of the per-point bounds. Thus, the greatest deviations are just within 0.05 of the given values. Now we refine the fit by reducing the global accuracy to 0.02. ... rbf2=fastrbf_fit(data,0.02,’errorbar’,’initial’,rbf,’... verbose’) evpnts=fastrbf_pointEval(rbf2,data.Location) min(evpnts.Value-data.Value) ans = -0.0197 max(evpnts.Value-data.Value) ans = 0.0298 The looser upper bound (0.03) takes effect since it is now greater than the global accuracy, but the tighter lower bound is ignored due to the looser global bound of 0.02. Finally, we force the fitter to use all the per-point error bounds by specifying a global accuracy of zero. ... rbf3=fastrbf_fit(data,0.0,’errorbar’,’initial’,rbf2,’... verbose’) evpnts=fastrbf_pointEval(rbf3,data.Location) min(evpnts.Value-data.Value) ans = -0.099 max(evpnts.Value-data.Value) ans = 0.0298 In this example the lower bound of 0.01 is achieved along with the upper bound of 0.03.

Note If a data set contains only per-point Upper or Lower bounds, then these are bounds are treated as being symmetric on the per-point values.

6.4.3

Spline smoothing

The rho option can also be used to fit an RBF to noisy data. This technique is known as spline smoothing and is described in Section 1.5.3. The value of rho determines the balance between fidelity to the data and smoothness. In the current release of FastRBF, only a single global value for rho can be specified. similar smoothing is achieved with our test data in the following call,

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rhofit_rbf = fastrbf_fit(noise_data, 0.1, ’rho’,0.1) Mesh = fastrbf_isosurf(rhofit_rbf, 0.03,0.5,’seedlimit’,0); Figure 6.9 illustrates two choices for ρ, 0.1 and 1.0. Choosing a suitable value for rho can be difficult as it is related to the energy in the RBF and therefore is a function of the data. The deviation of an RBF from a data value is a product of rho and the RBF weight for the corresponding centre (Equation (1.11)).

(a) ρ = 0.1

(b) ρ = 1.0

Figure 6.9: Comparison of choices for ρ with spline smoothing

6.4.4

Outliers & the confidence parameter

Even with good estimates of noise characteristics, a fitted RBF can be subject to statistical outliers or impulse noise, sometimes referred to as shot noise. The confidence parameter in the FastRBF fitter can help in such cases. The confidence parameter enforces a percentage of the constraints. Those which are hardest to meet are ignored with the assumption that these correspond to outliers. The default confidence value is 100%. However, it is usually safe to fit with a high confidence such as 99% without detrimental effect, even when outliers are not present. The confidence parameter can be used with all types of fits. For example,

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... rrfit_rbf = fastrbf_fit(noise_data, 0.1,’confidence’,0.99,’... errorbar’) Hint Specifying a high confidence, such as 99%, can result in faster fitting times with little or no detectable difference in the fitted RBF.

6.5 Summary Choosing appropriate fitting options and smoothing methods depends on the characteristics of the data to be modelled. Techniques such as centre reduction, error-bar fitting and incremental fitting, which have been found to offer speed improvements with many kinds of data (especially laser surface data), may not be appropriate for all data sets. In some cases, a straightforward direct fit may be the best option. Some experimentation is generally required with new types od data. All fitting methods should work regardless of the data type, however, some fits may be unnecessarily slower or result in larger RBFs.

6.5.1

Fitting

• An accuracy or fitting tolerance is always required by FastRBF. • It is advisable to use FastRBF unique to ensure data are not multivalued.

• Centre reduction can improve RBF evaluation and fitting times.

Fitting strategies • direct — solve the basic interpolation problem (Equation (1.3)), RBF centres coincide with the data points.

• reduce — solve the basic interpolation problem (Equation (1.3)), RBF centres form a subset of original data points.

• errorbar — find the smoothest RBF which passes within the data error bounds (Equation (1.6)), RBF centres form a subset of the error bounds.

• rho — solve modified interpolation problem (Equation (1.8)), ρ controls balance between smoothness and faithfulness to the data, RBF centres coincide with the data points.

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• rho & reduce — same as for rho, but RBF centres form a subset of the original data points.

Valid fitting options direct reduce errorbar rho rho, reduce

Interpolation or approximation Interpolation Interpolation Approximation Approximation Approximation

Compression None Yes Yes None Yes

• A reduce fit is recommended in preference to a direct fit when the data contains redundancy (is over-sampled) and has low noise relative to the fitting accuracy.

Incremental fit • New data can be added to an existing RBF without fitting from scratch.

• Maximum benefit from adding data to an existing RBF occurs when the new data is consistent with the old.

• Beware: setup overhead is proportional to total number of points in the RBF. It is inefficient to add only a small number of points at a time.

• Fitting accuracy can be progressively refined from coarse to fine

6.5.2

Evaluation

• An RBF, together with its gradient, can be evaluated at a point, on a plane or on a 3D grid.

• Low pass filtering (LPF) can be used to avoid aliasing during evaluation or to smooth an RBF. • LPF increases evaluation time approximately by two in 3D but is significanlty slower in 2D.

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6.5.3

Modelling noisy data

• FastRBF offers low pass filtering (LPF), error-bar fitting and spline smoothing.

• Spline smoothing requires an apprpriate choice for ρ which depends on the data.

• LPF assumes that noise can be discriminated by high frequency content alone and requires an estimate of the frequency above which the noise component of the data dominates.

• Error-bar requires an accurate estimate of noise magnitude rather than frequency.

• With error-bar fitting, the maximum deviation permitted occurs at the points which contribute centres to the RBF. • LPF is reversible since it is applied during RBF evaluation, whereas spline smoothing and error-bar fitting achieve smoothing during fitting and therefore can not be varied.

• Error-bar fits can achieve significant centre reduction.

• Spline smoothing uses a greedy fit to achieve centre reduction.

• LPF does not reduce the number of centres in an RBF. • Centre reduction can be achieved with LPF only by fitting a new RBF using the direct and reduce options to the existing RBF evaluated with LPF smoothing at its centres.

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Fitting 3D Surface Data Fitting an RBF to surface data with the FastRBF toolbox involves three steps.

The first converts the surface data to a signed distance-to-surface distribution in 3D. The second step fits an RBF to this 3D distribution. The final step isosurfaces the RBF to extract the interpolating surface. We illustrate this process by fitting a surface to 3D mesh data and point cloud data from two different types of laser scanner.

7.1 Fitting Surfaces to Mesh Data Mesh data is produced by many types of 3D scanners and often contains irregular holes, both small and large. Figure 7.1(a) shows a triangle mesh produced by the Polhemus FastSCAN hand-held 3D laser scanner 1 , doctored with holes for this tutorial, and Figure 7.1(b) the corresponding smooth shaded surface.

The first step is to import the file holey_face.obj.

Mesh = fastrbf_import(â&#x20AC;&#x2122;holey_face.objâ&#x20AC;&#x2122;) fastrbf_view(Mesh); Other FastRBF commands can now be applied to the Mesh struct.

As described in Chapter 2, our approach to fitting 3D surface data involves first determining some surface normals. Normal vectors are estimated from the mesh by: 1

(http://www.polhemus.com)

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(a)

(b)

Figure 7.1: Original triangle mesh, displayed as (a) triangles, and (b) smooth shaded surface.

MeshWithNormals = fastrbf_normalsfrommesh(Mesh) fastrbf_view(MeshWithNormals, â&#x20AC;&#x2122;fvâ&#x20AC;&#x2122;); The new normals are stored in the Gradient field of the MeshWithNormals structure along with the original mesh. Figure 7.2(a) shows the points and their corresponding normals.

The normals and the mesh points are now used to generate distance-tosurface density data. We do not define the distance to the surface everywhere only within a band surrounding the surface. The width of this band is set by the maximum normal length. RBF values at points further than the maximum normal length from the surface will not necessarily be the distance to the surface. A value of several times larger than the average mesh resolution is usually sufficient and must certainly be greater than the accuracy specified in the RBF fitter. In this example a maximum normal length of 5mm will be used. As described in Chapter 2, normal length validation may be used to re-

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duce the lengths of some normals, so as to produce a more consistent density distribution. Using normals whose length is less than the level of noise in the data may have a detrimental effect on the smoothness of the RBF solution. In this example, all normals shorter than 0.5mm are discarded. ...

Density = fastrbf_densityfromnormals(MeshWithNormals... , 0.5, 5.0) fastrbf_view(Density); The output contains the locations of the on-surface and off-surface points, and their corresponding values (distance-to-surface), as shown in Figure 7.2(b).

The mesh information is no longer required and is not stored in the output.

(a) Mesh points and normals.

(b) Density data created from normals.

Figure 7.2: Converting a mesh into density data. Surface normals are estimated from the mesh faces (a), then off-surface points are generated inside (blue) and outside (red) the surface (b).

The RBF fitter is unable to handle input data in which any points are either identical or very nearly identical (relative to the scale of the data), so it is

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recommended that the unique command is used before fitting: Density = fastrbf_unique(Density) The FastSCAN accuracy specification is 1mm, so we shall fit to half that accuracy here. To illustrate the benefit of using centre reduction (see Section 1.3), we shall first fit without reduction: rbf = fastrbf_fit(Density, 0.5, ’direct’) fastrbf_view(rbf); Doing so produces one RBF centre for each data point, as shown in Figure 7.3(a). The essence of centre reduction is to find a solution in which most of the RBF coefficients are zero, and can therefore be discarded, while still maintaining the specified accuracy. Fitting with reduction is achieved by: rbf = fastrbf_fit(Density, 0.5,’reduce’) fastrbf_view(rbf); The resulting centres are shown in Figure 7.3(b). In this case, fitting with reduction required 1157 centres, compared to 3984 without reduction.

As described in Chapter 2, the RBF describes the distance to the surface, and can be evaluated anywhere in space. For example, we can evaluate it on a 3D grid: g = fastrbf_grideval(rbf, 0.1, ’spacing’, 3); slice(g.Value, 40, 27, 15); The grid values on three orthogonal planes through the origin are shown in Figure 7.4.

The desired surface is simply the isosurface on which the RBF is zero. By following the isosurface through space, a triangular mesh representation can be generated. This is done (at a resolution of 2mm) by: NewMesh = fastrbf_isosurf(rbf, 2) fastrbf_view(NewMesh); The resulting mesh is shown in Figure 7.5. Note that all the holes have been filled in automatically.

Finally, we export the mesh to a Wavefront .obj format file, to allow it to be loaded into other 3D packages: fastrbf_export(NewMesh, ’NewMesh.obj’)

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(a)

(b)

Figure 7.3: RBF centres obtained by fitting (a) directly and (b) with reduction.

The centres are coloured by their coefficient value.

The RBF fitted to the distance-to-surface density field derived from the mesh can be viewed as a solid model of the object. The RBF smoothly extrapolates the distance-to-surface distribution according to the energy-minimisation characteristics of the thin-plate spline. The original mesh surface is thereby implicitly interpolated by the RBF.

7.2 Fitting surfaces to point cloud data If the surface data are only available as a point cloud without any additional information, i.e.,purely vertex data, then fastrbf_normalsfrompoints can be used to estimate the normals. In the previous tutorial we could have ignored the mesh connectivity data and estimated normals from the mesh vertices alone using the following commands, MeshWithNormals = fastrbf_normalsfrompoints(Mesh) The rest of the fitting process proceeds as in Section 7.1. In this example the point-cloud estimator works well. However, in general, without any addi-

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Figure 7.4: planes.

Density function grid values shown on x − y, y − z, and x − z

tional information, estimating normals is ambiguous and likely to be less robust than using normalsfrommesh, normalsfromsigns or normalsfromscan.

7.3 Colour surface fitting The FastRBF toolbox can repair meshes with colour-per-vertex information.

To obtain the colours at the new mesh vertices RBFs are fitted to the vertex colours, and these colour RBFs are then evaluated at the new mesh vertices. To illustrate this process we have taken a close up photo of a New Zealand Weta2 (Figure 7.6(a)) and projected it onto a hemispherical mesh (Figure 7.6(b)).

First we import the coloured mesh.

2 Photo courtesy of Jonathan Carr, j.carr@aranz.com. See www.kcc.org.nz/animals/weta.htm for more about Wetas.

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(a)

(b)

Figure 7.5: Final triangle mesh, displayed as (a) triangles, and (b) smooth shaded surface.

weta_dome = fastrbf_import(’weta_dome.obj’) Notice that the imported mesh has attributes named Red, Green and Blue. In matlab the attributes are just sub-structs of the mesh structure.

Hint To view colour meshes with the FastRBF_View command it is first necessary to extract the RGB values into an 3xN array and scaled to be on [0 1]. For example, to view the weta_dome mesh: rgb = [weta_dome.Red.Value; weta_dome.Green.Value; ... weta_dome.Blue.Value]/255; fastrbf_view(weta_dome, ’f’, rgb) For the purposes of this tutorial we will create a higher resolution mesh.

For further explanation of reconstructing surfaces see Section 7.1.

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(a) Close up photo of a New Zealand Weta.

(b) Weta photo projected onto a hemispherical mesh.

Figure 7.6: Colour surface input data

weta_dome = fastrbf_normalsfrommesh(weta_dome); density=fastrbf_densityfromnormals(weta_dome, 0.001, 0.05); weta.rbf = fastrbf_fit(density, 1e-3,’reduce’); weta.mesh = fastrbf_isosurf(weta.rbf, 0.01) We now fit RBFs to the RGB values on [0 255] at each point. Because the colour values do not generally change smoothly, we use the direct fitting method as centre reduction is unlikely to be effective. Note that different fitting methods can be applied to different attributes. The RGB data is integer valued, so an accuracy of 0.3 is sufficient to unambiguously interpolate the colour values. =fastrbf_fit(weta_dome,0.3,’direct’,’attr’, ... ’Red’); weta.Green=fastrbf_fit(weta_dome,0.3,’direct’,’attr’, ... ’Green’); weta.Blue=fastrbf_fit(weta_dome,0.3,’direct’,’attr’, ... ’Blue’); weta.Red

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Now we evaluate each of the colour RBFs at the mesh vertices. Note that 0.3 the default evaluation accuracy would be 100 = 0.003 which is unecessary given that the underlying values were integers. Setting the accuracy to 0.1 will improve the speed of evaluations. weta.mesh = ’attr’, weta.mesh = ’attr’, weta.mesh = ’attr’,

fastrbf_pointeval(weta.Red, weta.mesh, 0.1, ... ’Red’); fastrbf_pointeval(weta.Green, weta.mesh, 0.1, ... ’Green’); fastrbf_pointeval(weta.Blue, weta.mesh, 0.1, ... ’Blue’);

(a) Reconstructed colour mesh.

(b) Level surface at 0.03. The colour bleeds as you move away from the data points.

Figure 7.7: Colour surface output data

To view the reconstructed mesh (shown in Figure 7.7(a)) we first construct and 3xN RGB array to pass to the FastRBF_View. rgb = [weta.mesh.Red.Value; weta.mesh.Green.Value; ... weta.mesh.Blue.Value]/255; fastrbf_view(weta.mesh, ’f’, rgb);

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view([90 0]); If the reconstructed surface lies away from the original data points the reconstructed colours will bleed. To see this effect we will use the same colour RBFs to colour a mesh at iso-value 0.03. weta.iso = fastrbf_isosurf(weta.rbf, 0.01); weta.iso = fastrbf_pointeval(weta.Red, weta.iso, 0.1, ’attr’, ’Red’); weta.iso = fastrbf_pointeval(weta.Green, weta.iso, 0.1, ’attr’, ’Green’); weta.iso = fastrbf_pointeval(weta.Blue, weta.iso, 0.1, ’attr’, ’Blue’);

... ... ...

rgb = [weta.iso.Red.Value; weta.iso.Green.Value; ... weta.iso.Blue.Value]/255; fastrbf_view(weta.iso, ’f’, rgb); view([90 0]); The resulting mesh is shown in Figure 7.7(b).

7.4 Surfacing Noisy 3D Scanner Data This tutorial illustrates the steps involved in fitting a smooth surface to noisy scan data and builds upon the material covered in the previous tutorial, Section 7.1.

Figure 7.8 shows raw scan data captured by a Cyra Technologies3 Cyrax 2500 LIDAR scanner (data courtesy of Allen Instruments & Supplies4 ). A Cyra PTX file contains one or more scans of an object, and information about the scanner location during each scan. The file data is stored in metre units. The first step is to load the file Fountain.ptx into M ATLAB: Scan = fastrbf_import(’Fountain.ptx’) fastrbf_view( Scan ); To illustrate the effects of smoothing, we will operate on only a small subset of the data. This is done with the crop command. Note that the third component of the min and max arguments are omitted to not crop in the z direction.

Figure 7.9 shows the cropped region.

3 4

http://www.cyra.com http://www.alleninst.com

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Figure 7.8: LIDAR scan, displayed such that the colour of each point corresponds to its z value.

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Figure 7.9: Cropped LIDAR scan.

Scan = fastrbf_crop(Scan,[0.100 -0.900 NaN],[0.800 -0.200 ...

NaN]); fastrbf_view( Scan ); The Cyrax 2500 is specified as having a maximum error of 6mm, furthermore the scanner was not focused properly for this scan. Noise of this magnitude is going to be clearly visible, hence we need to use data smoothing.

As before, we now need to create distance-to-surface density data for the RBF fitter. First we need some surface normals. We estimate surface normals by fitting a plane to a subset of points in the neighbourhood of each point. The ambiguity in the sign of the normal is resolved by using the position of the scanner when that point was measured. We use the normalsfromscan function with default parameters to do this.

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Scan = fastrbf_normalsfromscan(Scan); The minimum normal length should be about the level of noise (or greater), so we use 0.006m (6mm). The maximum normal length is set to 0.030m (30mm).

The density data is derived from the points and normals by: Density = fastrbf_densityfromnormals(Scan, 0.006,0.030); As before, we ensure that no identical points exist before fitting by: Density = fastrbf_unique(Density); Usually we would proceed with a fit using the errorbar option (restricted range smoothing), but first let us see what happens if we don’t use data smoothing. We will pretend the data has no noise and interpolate it. We do this by doing a fit to 2mm accuracy, followed by isosurfacing to a 5mm resolution: Rbf = fastrbf_fit(Density, 0.002); Mesh = fastrbf_isosurf(Rbf, 0.005); fastrbf_view(Mesh); view(2); which gives the result shown in Figure 7.10(a).

As described in Section 1.5, data smoothing can be achieved by performing another fit using the errorbar option or ρ smoothing, or alternatively, by applying a low-pass filter to the ‘exact’ fit RBF we have just computed. We consider the latter case first.

Filtering during RBF evaluation comes at an extra cost (approximately twice as long to compute). However, this is usually faster than fitting another RBF and certainly much faster than performing a convolution by spatial or Fourier techniques which requires evaluating on a regular grid. Figure 7.10(b) was created using a filter of width 0.02m (20mm) with the Isosurf command: Mesh1 = fastrbf_isosurf(Rbf1, 0.005, ’smooth’, 0.02); FastRBF_view(Mesh1); view(2); The application of this filter can be viewed as attenuating (removing) details finer than 0.02m (20mm) from the surface.

We now consider performing a restricted range fit to the original data. Although the specified maximum error is 6mm, we will assume that most variations are within 4mm and that these are normally distributed about zero.

Furthermore, we assume that these assumptions apply throughout the data, and therefore globally specify an error bar range of ±4mm at each data point.

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(a) Surface obtained without smoothing.

(b) Noisy RBF with low-pass filter smoothing.

Figure 7.10: Comparison of ordinary evaluation of a surface fit and smooth evaluation with low-pass filtering.

Rbf2 = fastrbf_fit(Density, 0.004, ’errorbar’); Mesh2 = fastrbf_isosurf(Rbf2, 0.005); fastrbf_view(Mesh2); view(2); The resulting surface is shown in Figure 7.11(a). ρ smoothing is useful because it is similar to data regularisation methods used by engineers. However, it requires a good estimate (or guess) of the value of ρ, and fitting an RBF for each guess. We have guessed a value of 0.1 (by trial and error). The resulting surface is shown in Figure 7.11(b).

Rbf3 = fastrbf_fit(Density, 0.002, ’rho’, 0.1); Mesh3 = fastrbf_isosurf(Rbf3, 0.005); fastrbf_view(Mesh3); view(2);

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(b) Smoothing using ρ = 0.1.

(a) Smoothing using restricted range fitting.

Figure 7.11: Comparison of smoothest restricted range (error-bar) fit and spline smoothing.

Varying smoothing parameters An advantage of low pass filtering is that smoothing can be easily varied after fitting an RBF, during surface evaluation. With a error bar fit and ρ smoothing the process of varying the amount of smoothing or iteratively estimating the best fitting tolerance or ρ value, is computationally intensive since it requires fitting a new RBF with each iteration. The ‘initial’ option in the fitter can be used to reduce this computation by reusing the previously calculated RBF as a starting point for the next iteration of refining the parameter choice. For example, if we wanted to tighten the restricted range tolerance from 4mm to 3mm in the previous example, we could use the following code, ...

Rbf3 = fastrbf_fit(Density, 0.003, ’errorbar’,’initial’,...

Rbf2); Mesh3 = fastrbf_isosurf(Rbf3, 0.005); fastrbf_view(Mesh3); view(3);

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Often it is most advantageous to go from a coarse fit to a finer (less smooth) fit since the coarse fit is initially faster and generally tightening the tolerance adds new centres to the current RBF while relaxing the tolerance removes centres.

In the case of ρ smoothing the benefit may be less that with the error bar fit, depending on how different the choices of ρ are.

7.4.1

Adding new surface data

The ‘initial’ option in the fitter can also be used to add new surface data to an existing RBF model. Care must be taken so that • new data are appended to the old data • old data must maintain an ordering consistent with the associated RBF • the existence of any attributes, i.e., ’attr’ fields such as colour, must be consistent with those specified in the old data In addition to these technical requirements, there is the practical requirement that the new data is spatially consistent (aligned) with the existing data. If the new data ‘contradict’ the existing data, as missaligned data could be viewed as doing, then a fit from scratch might be faster than using the initial option.

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3D Isosurfacing and Mesh Simplification 8.1 Isosurfacing The FastRBF library includes two isosurfacing routines. A conventional Marching Cubes isosurfacer which evaluates an RBF everywhere on a regular voxel grid and a surface-following isosurfacer especially tailored to evaluating 3D surfaces modelled by RBFs. The surface-following implementation minimises the number of RBF evaluations while optimising the number and size of mesh facets. This tutorial focusses on the use of this latter algorithm and uses laser scan data of a hand to illustrate some of the options available to the user. A surface-following algorithm can be very beneficial in an application such as modelling and reconstructing 3D objects. However, it should be used with some understanding of the inherent limitations in order to achieve the best performance. Surface-following is not always the most suitable approach in other applications, such as visualising arbitrary isosurfaces from general density data. This tutorial describes the isosurf command and illustrates these situations. The mcubes command (Marching Cubes) is covered in Section 8.1.7.

8.1.1

Basic operation

We begin by loading laser scan data from a hand and fitting an RBF to the incomplete mesh. Note that the holes in the scan occur mostly between the

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(a) 1mm mesh

(b) 2mm mesh

Figure 8.1: Isosurface resolution options.

fingers and the units are millimetres. Off-surface points are generated by projecting between 0.5mm and 5mm off the hand surface according to the approach described in Chapter 2 and illustrated in the tutorial in Section 7.1. An RBF is fitted to the resulting density data to within 0.25mm.

Mesh = fastrbf_import(’Hand.obj’); Density = fastrbf_densityfromnormals(Mesh, 0.5, 5.0); Density = fastrbf_unique(Density); rbf = fastrbf_fit(Density, 0.25,’reduce’); Given an RBF description of the hand data, we now want to extract a polygonal mesh. A 1mm mesh, as illustrated in Figure 8.1(a), can be produced simply using the FastRBF isosurfacer.

NewMesh = fastrbf_isosurf(rbf, 1);

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figure; fastrbf_view(NewMesh); view([20 60]); The default isosurfacing parameters result in an open surface clipped by the default bounding box. The default box is determined by the maximum extent of the density data to which the RBF was fitted. This density data consists of our raw surface data augmented with projected off-surface data. In this example the isosurfacing threshold is zero, the default value. An alternative threshold, for example corresponding to 2mm, could be extracted as follows, NewMesh = fastrbf_isosurf(rbf, 1, 2); The accuracy, with which an RBF evaluation is computed within the isosurfacer, is automatically set as one hundredth the fitting accuracy. This number is recorded in the RBF data structure. The user can nevertheless specify a looser accuracy, which may result in faster evaluation times, or a finer accuracy. For example, we can specify an evaluation accuracy of 0.1mm as follows (the default would have been 0.25/100 = 0.0025mm), NewMesh = fastrbf_isosurf(rbf, 1, 2, 0.1); We can also vary the resolution of the mesh as desired. Figure 8.1(b) illustrates the mesh produced by sampling on a 2mm lattice.

NewMesh = fastrbf_isosurf(rbf, 2); Note that the number of facets generally increases with the square of the change in the sampling resolution, unless the surface has been under-sampled.

8.1.2

Mesh properties

The FastRBF isosurfacer generates manifold (non self-intersecting) surfaces.

Every edge has at most two incident faces and back-to-back faces should not occur.

Warning On rare occasions, the properties of a FastRBF mesh might be jeopardised by the precision with which the mesh is exported or by the internal precision of third-party software used to import the mesh. Finite precision in either case could cause distinct mesh vertices to become indistinct. Some formats are more prone to this problem than others, especially ASCII text formats.

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(a) Open mesh

(b) Closeplus mesh

(c) Closeminus mesh

Figure 8.2: Isosurface capping options.

8.1.3

Capping options

Figure 8.2 illustrates the capping options at the bounding box. Figure 8.2(a) is an open surface, the mesh is not closed at the wrist. This is the default operation.

NewMesh = fastrbf_isosurf(rbf, 1,’open’) Figure 8.2(b) illustrates a closed surface. The RBF is treated as being greater than the threshold outside the bounding box, i.e., in this case it is has a positive value outside the box. The result is a mesh closed along the bounding box faces.

NewMesh = fastrbf_isosurf(rbf, 1,’closeplus’) Figure 8.2(c) illustrates a closed surface where the RBF is treated as being less than the threshold outside the bounding box, i.e., in this case it has a negative value outside the box. The result is also a mesh closed along the bounding box faces, but now the surface is a mold in the shape of the hand.

NewMesh = fastrbf_isosurf(rbf, 1,’closeminus’)

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8.1.4

Mesh optimisation

By default, FastRBF meshes are optimised so as to minimise facets and to produce facets with even aspect ratios. Specifying the grid option results in no optimisation and a surface that is typical of a Marching Cubes style of algorithm. Specifying no optimisation typically results in 3-4 times the number of facets, many of which are thin and irregular. This option may give slightly better surfaces when evaluating at coarse resolutions in the vicinity of fine structures (relative to the sampling resolution). However, such situations imply that the surface is under-sampled.

Optimisation always preserves surface topology. Therefore, small holes and fine details are preserved. Small facets will necessarily be found in these regions. Similarly, where a surface is clipped by a bounding surface, optimisation will be compromised to preserve an accurate sharp clipping boundary.

In Figure 8.3(a) we illustrate constrained optimisation where facet vertices have been constrained to lie on parallel xy planes. The result is that the facets form closed contours in the xy planes of the sampling lattice described in Section 3.1.1. This is more clearly depicted in the close-up view shown in Figure 8.3(d).

NewMesh = fastrbf_isosurf(rbf, 2, ’plane’) By rotating the tetrahedral sampling lattice relative to the data, we can specify the planes along which vertices are forced to lie. Re-orienting the sampling lattice is achieved by specifying the corresponding 4x4 orientation matrix. Figure 8.3(b) and Figure 8.3(e) illustrate planar optimisation after rotating the sampling lattice by 90 degrees about the x-axis. rx = [ 1 0 0 0 NewMesh =

0 0 0; 0 -1 0; 1 0 0; 0 0 1 ] fastrbf_isosurf(rbf, 2, ’plane’, ’orient’, rx)

The xy plane of the sampling lattice now corresponds to the x − z plane of the data. The hand has been surfaced as if it were sliced along the length of the arm at 2mm intervals. Note that the hand itself has not been reoriented, its position in space is consistent with the raw surface data. The transform matrix operates only on the orientation of the sampling lattice.

A simpler way of specifying the lattice orientation, is by specifying the plane normal with the ’up’ option. However, with this option you have no

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control over the orientation of the sampling lattice (and hence bounding box angle) in the transformed lattice plane. But this is only a problem if you are also clipping the isosurface. The following is equivalent to the previous command.

NewMesh = fastrbf_isosurf(rbf, 2, ’plane’, ’up’, [0 1 0])

Effective bounding box Figure 8.3(c) illustrates an important point when re-orienting the sampling lattice. In this case we specify a rotation of 30 degrees (π /6 radians) about the x-axis. rx = [ 1 0 0 0; 0 cos(pi/6) -sin(pi/6) 0; 0 sin(pi/6) cos(pi/6) 0; 0 0 0 1 ] NewMesh = fastrbf_isosurf(rbf, 2,’plane’,’orient’,rx) Note that the scales are different in Figure 8.3(c). The bounding box is effectively larger than that specified (in this case the default box defined in the RBF data structure). The hemispherical behaviour of the RBF interpolated surface across the wrist has been revealed by the larger bounding box. This always existed, but was clipped in the previous figures. The larger box occurs because the actual bounding box is always in the coordinate system of the sampling lattice while the specified bounding box (in this case the default box) is specified in the original data (RBF) coordinate system. The bounding box coordinates are transformed into the new sampling lattice coordinate system and the smallest box which encloses the specified one is used.

Note that any homogeneous transformation matrix could be specified as the argument of the lattice orientation option. This allows the user to specify a non-uniform sampling along the different axes of the sampling lattice.

Similarly, any affine transform could be specified and the user should be wary of its consequent effect. The specified resolution and transformation matrix combined determine the actual sampling of the RBF.

8.1.5

Isosurface seeding

We will now return to the straightforward case of isosurfacing our RBF at 2mm (Figure 8.1(b)) and explore the seeding issue associated with surface-following.

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(a)

(b)

(d)

(c)

(e)

Figure 8.3: Varying the orientation of the sampling lattice while isosurfacing with the â&#x20AC;&#x2122;planeâ&#x20AC;&#x2122; optimisation option. (a,d) no lattice rotation, (b,e) 90 degrees x-axis rotation, (c) 30 degrees x-axis rotation.

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(a) 1 seed

(b) 10 seeds

(c) All seeds

Figure 8.4: Isosurface seeding variations.

This time we will truncate the default bounding box along the y-axis by specifying the lower left corner of the box using the min option. min = [NaN 35 NaN]; NewMesh = fastrbf_isosurf(rbf, 2,’min’,min) The result shown in Figure 8.4(a) is not what might be expected. A single finger has been isosurfaced. This is because the specified cropping of the native bounding box has resulted in five separate (unconnected) surface elements corresponding to the fingers of the hand. The default operation of the isosurfacer is to commence isosurfacing from just one seed point. The first valid seed in the available list of seeds is used. When an array of seeds is not provided by the user, the RBF centres are used. Because only one seed is used, only one connected surface element can be extracted. The user can rectify this by specifying their own seed limit. In Figure 8.4(b) we have specified a limit of 10 seeds. This means that surfacing propagates outwards from the first 10 valid seed points on the seed list (in this case the list of RBF centres).

NewMesh = fastrbf_isosurf(rbf, 2,’min’,min,’seedlimit’,10) Even so, in this example only four of the finger tips have been surfaced. If we specify the use of all the seeds (RBF centres) then we are likely to recover all surface elements. We can specify the use of all the seeds by either setting

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the seed limit to be equal to, or greater than, the number of seeds available, or simply by specifying a seedlimit of 0, NewMesh = fastrbf_isosurf(rbf, 2,’min’,min,’seedlimit’,0) Figure 8.4(c) illustrates the result of using all the seeds. Indeed all the fingers have been surfaced. However, this is not a recommended general strategy since the memory overhead associated with initialising all potential seeds and maintaining a large number of advancing wavefronts can be very high for a large number of seeds. Furthermore, it is usually unnecessary. The set-up time will also be very long, and the net running time may be slower in some cases. We advise the use of a small seed limit. This approach may seem somewhat haphazard, but it is rarely a problem in the intended surface modelling application. Real world objects, imaged by scanners, tend to have connectivity (if nothing else, due to gravity) and therefore consist of at most a few discrete components, and usually just one. The seeds, whether they be raw data (on-surface points) or RBF centres (near or on the surface), will all lie near the desired surface. Any RBF isosurface which does not lie near one of these points is probably a spurious surface.

Specifying seeds with the seeds option allows the user to specify and sort their seeds so that they are distributed over the region of interest in a spatially uniform way.

Hint When isosurfacing density data, use the raw data points as seeds but first prune this list by removing those points which are significantly above or below the threshold of interest.

Valid seeds A valid seed is a seed which lies near the isosurface, or results in a successful gradient search to find a point near the isosurface. A valid seed must also lie within the specified bounding box. Seeds in the seed list are processed in consecutive order with the bounding box test applied first. A seed can be rejected if a gradient search fails or if the result of a successful gradient search lies outside or within a sampling interval of the bounding box. It is also rejected if it is a duplicate or within two sampling intervals of another seed.

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(a) Marching cubes surface.

(b) Close-up view.

Figure 8.5: 3D grid evaluation at 2mm followed by Marching Cubes isosurface.

8.1.6

When is surface-following inappropriate?

When an RBF represents some kind of density function other than a signeddistance-to-surface distribution, there is no guarantee that using all the seeds, no matter how they are chosen, will extract all the isosurfaces of interest.

Therefore, in the case of isosurfacing density data with arbitrary thresholds a full grid evaluation followed by traditional Marching Cubes isosurfacing is recommended.

8.1.7

Marching Cubes

Marching Cubes is typically much slower in FastRBF than surface-following since it requires first evaluating an RBF on a fine 3D grid. It guarantees, however, that within the volume of interest, all isosurface elements larger than the

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sampling resolution will be found.

Figure 8.5 illustrates the marching cubes approach.

G=fastrbf_grideval(rbf,â&#x20AC;&#x2122;spacingâ&#x20AC;&#x2122;,2) mesh=fastrbf_mcubes(G); figure; fastrbf_view(mesh); Figure 8.5(b) is a close-up view of the resulting mesh.

Summary The isosurf command is ideally suited to extracting the surface of an object represented by an RBF model. With the judicious choice of a seed limit and an appropriate seed list, great efficiency can be achieved compared to a full 3D grid evaluation followed by Marching Cubes approach.

8.1.8

Smoothing

Smoothing can be applied during iso-surfacing using the smooth option. This option envokes low pass filtering during RBF evaluations and was demonstrated in Section 7.4 in the context of smoothing noisy surface data. Smoothing can also be envoked when iso-surfacing to ensure aliasing is avoided when the mesh resolution is below or close to the Nyquist sampling limit (twice the maximum frequency present in the data).

Note The RBF represents a distance field and so it is the 3D distance field which is smoothed when a low pass filter is applied. This is subtly different from smoothing the iso-surface directly, say with a morphological operator.

Recall the Fish data set from Section 7.4. Figure 8.6 shows the results of applying smoothing during iso-surfacing to the Fish data set for kernel widths of 0mm (no smoothing), 10mm, 20mm and 30mm. Iso-surfacing without smoothing is approximately twice as fast as with smoothing. Smoothing during evaluation, however, is generally much faster than fitting an approximating RBF to the data in the first place and has the advantage of being able to be varied.

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(a) no smoothing

(b) 10mm smoothing

(c) 20mm smoothing

(d) 30mm smoothing

Figure 8.6: Varying isosurface smoothing.

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8.2 Multi attribute data In this tutorial we will reconstruct the plastic watering can shown in Figure 8.7.

Firstly we fit an RBF which describes the geometry of the can and extract an isosurface which interpolates the can, as in Section 7.1. can = fastrbf_load(’can.msh’); can = fastrbf_normalsfrommesh(can) density=fastrbf_densityfromnormals(can,0.2,5,’increase’, ...

0.5) rbf = fastrbf_fit(density, 0.2) can_rbfmesh = fastrbf_isosurf(rbf, 8)

(a) With edges

(b) Without edges

Figure 8.7: The original triangle mesh. 15229 vertices and 30299 faces.

The can mesh file can.msh also contains colour information in the form of an RGB tripple associated with each vertex in the mesh. This information is stored in the Red, Green and Blue attributes. We can model the colour information with three additional RBFs: red = fastrbf_fit(can, 1e-3, ’attr’, ’Red’); green = fastrbf_fit(can, 1e-3, ’attr’, ’Green’); blue = fastrbf_fit(can, 1e-3, ’attr’, ’Blue’);

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(a) With edges

(b) Without edges

Figure 8.8: RBF isosurface at 8mm resolution. 4280 vertices and 8416 faces.

Note that the colour values in this case lie on the range 0 to 1. Furthermore, 1e-3 is excessive accuracy. Generally for colour camera data the fitting accuracy can be as low as 1e-1 with little perceptible difference.

The colour per vertex information is not stored in the output mesh, to add it in again we simply evaluate the colour RBFs at the mesh vertices. ... can_rbfmesh=fastrbf_pointeval(red,can_rbfmesh,’attr’,’Red’)... ; ... can_rbfmesh=fastrbf_pointeval(green,can_rbfmesh,’attr’,’...

Green’); ... can_rbfmesh=fastrbf_pointeval(blue,can_rbfmesh,’attr’,’Blue... ’); The only export formats that support colour are the non-standard Wavefront OBJ colour-per-vertex and VRML. So to view the colour mesh we will export it to VRML. The third argument, 1, to fastrbf_export is the range of the colour data. rgb = [can_rbfmesh.Red.Value; can_rbfmesh.Green.Value; ... can_rbfmesh.Blue.Value]; fastrbf_export(can_rbfmesh, rgb, 1, ’can_rbfmesh.wrl’); We can also use matlab FastRBF_View command to view the mesh.

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rgb = [can_rbfmesh.Red.Value; can_rbfmesh.Green.Value; ... can_rbfmesh.Blue.Value]; fastrbf_view(can_rbfmesh, ’f’, rgb); Note If the colour values did not lie on the range 0 to 1 when using fastrbf_view then they would have to be scaled so that they do. For example, RGB values on the range 0-255 would have to be divided by 255. Otherwise values above 1 are clipped to 1 and similarly values below 0 are clipped to 0. An RGB value whose components are all greater than 1 would appear as white.

The result is shown in Figure 8.8.

8.3

Mesh Simplifying

Our current procedure for reconstructing a surface involves isosurfacing an RBF at a specified resolution chosen to be consistent with the highest level of detail expected in the object. While this generates a beautifully regular mesh, and is faithful to the data in detailed areas, it can also generate a very large number of faces which may be redundant in relatively flat areas. Generally speaking, the fewer the number of faces the better, for reasons of storage, transmission and processing speed. The mesh simplifier is used to reduce the number of faces whilst preserving the geometry and colour-per-vertex information (if present) of the input mesh.

The FastRBF offers a specialised mesh simplifier which uses an RBF fitted to the input mesh data to constrain the mesh’s geometry as it is simplified.

The simplifier can be controlled in several ways. The simplifier can proceed until a desired number of vertices or faces is achieved, or, more usefully, the simplifier can also be constrained to remove only faces with a chordal error less than a given bound.

We will try this latter method first. We will use an error bound of 1mm: can_simp_e1 = fastrbf_simplify(can_rbfmesh, rbf, 1); figure; fastrbf_view(can_simp_e1, ’fe’); This produces the mesh shown in Figure 8.9. Alternatively we can specify a desired number of faces or vertices in the final mesh. The simplified mesh we have just created has 2760 faces, so lets try for 2000 faces:

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... can_simp_f2k=fastrbf_simplify(can_rbfmesh,rbf,-1,’faces’... ,2000); figure; fastrbf_view(can_simp_f2k, ’fe’); Giving the meshes shown in Figure 8.10.

(a) With edges

Figure 8.9: faces.

(b) Without edges

Simplified mesh with 1mm error limit. 1409 vertices and 2759

We could similarly set a target number of points in the mesh using the points option. However the faces and points options work best when combined with an error limit.

When multiple constraints are passed to the simplifier, mesh simplification stops when one of the constraints is met. So, if we issued the following command, can_simp_e1_f2k = fastrbf_simplify(can_rbfmesh, rbf, 1, ... ’faces’, 2000) figure; fastrbf_view(can_simp_e1_f2k, ’fe’); we again get a mesh with 2760 faces because to remove any more would violate the error condition.

A parameter not yet mentioned is the minimum enclosing angle limit. The minimum enclosing angle is the smallest angle between adjacent edges of a face.

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(a) With edges

(b) Without edges

Figure 8.10: Simplified mesh with face limit of 2000. 1027 vertices and 1999 faces.

This controls how long and narrow any new faces are allowed to be. The default value is 10 degrees. If the input mesh has faces not satisfying this criteria they will only be removed if, in doing so, no new face violates this criteria.

Hence, requiring a good aspect ratio will decrease the effectiveness of the simplification. If we were to require a very good aspect ratio with an angle limit of 40 degrees with: ... can_simp_fa=fastrbf_simplify(can_rbf, rbf, 1e100, ’faces’... , 2000, ’angle’, 40) figure; fastrbf_view(can_simp_fa, ’fe’); we would get a mesh with 7743 faces, close to the 8416 faces we started with.

This is illustrated further in Table 8.1 which shows the mesh sizes for different angles with an error bound of 1mm. Note however that the simplifier has the tendency to produce mostly faces with good aspect ratio and only a few long thin ones.

8.3.1

Simplifying a coloured mesh

The meshes in the previous section both faithfully reproduce the shape of the object but play havoc with the edges of our colour stripes. To overcome this

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Angle 5 10 20 30 40

Vertex Count 1343 1409 1664 2431 4020

Face Count 2646 2759 3222 4731 7907

Table 8.1: The effect of the minimum enclosed angle on the efficacy of mesh simplification. The input mesh had 4280 vertices and 8416 faces. problem, we can add a colour constraint to the simplifier. This sets a limit on the colour variation between vertices on a face. If the colour variation is above the limit then the face is not simplified.

We can pass the RBFs which we fitted to the colour data in can.msh to the simplifier p as well and specify a maximum deviation of 0.03 in colour space (that is, r2 + g2 + b2 < 0.03): can_simp=fastrbf_simplifyrgb(can_rbf, rbf, 1, red, green, ... blue, 0.03) figure; fastrbf_view(can_simp, ’fe’); The output mesh shown in Figure 8.11 has not been simplified along the colour boundaries preserving the colour information much better than Figure 8.9. The colour limit can be used with any other geometry limit. We will now combine it with the points option to specify a target of 1800 vertices, giving Figure 8.12. can_simp_p18_c003=fastrbf_simplifyrgb(can_rbf, rbf, 1e100, ... red, green, blue, 0.03, ’points’, 1800) figure; fastrbf_view(can_simp_p18_c003, ’fe’); The colour per vertex information is not stored in the output mesh, to add it in again we simply evaluate the colour RBFs at the mesh vertices as in the previous section.

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(a) With edges

(b) Without edges

Figure 8.11: Simplified mesh with 1mm error limit and 0.03 colour limit. 2119 vertices and 4165 faces.

(a) With edges

(b) Without edges

Figure 8.12: Simplified mesh with a point limit of 1800 and a colour limit of 0.03. 1800 vertices and 3549 faces.

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Chapter 9

Data Types

This chapter describes the various data types used by the FastRBF library. All data types are M ATLAB structures. Field names enclosed in square brackets, such as [Value], indicate an optional field that need not be present.

Point or gradient arrays are stored in dim-by-N fashion with each column being a 2 or 3 dimensional point. Similarly one-dimensional arrays are always 1-by-N (that is, row vectors).

9.1 Attribute Type Fields [Value] [Accuracy] [Upper] [Lower] [Gradient] (at least one must be present) Attributes are used as sub-structs of the point list type to store per-point data. The Gradient field must be the same length as the Location field in the parent point list struct.

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9.2 PointList Type Fields Location [Value] [Accuracy] [Upper] [Lower] [Gradient] (any number of Attribute structs) Point lists are used for passing arrays of 2-D and 3-D points to the library along with data values associated with those points. Points are stored in the Location field which is always present. The Value, Accuracy, Upper, Lower and Gradient fields are the default attribute fields for the point. They are optional. The Gradient field, if present, must have the same length (dimension) as the Location field and similarly, so must the Gradient field of any other attribute associated with the point. The attribute fields are used for supplying or returning information about the points in the Location field. For example, the PointEval function will put the values of an RBF at each point into the Value field of the named attribute struct, and the Fit function will use data in the named attribute’s Accuracy field as the fitting accuracy at each point. Individual Attribute sub-structs are specified using the ’attr’ option in FastRBF commands. If the ’attr’ option is not used when calling a command, then the default attribute field in the pointlist type is used. For example, fastrbf_fit(X,0.1,’attr’,’Temperature’) fits an RBF to the values stored in X.Temperature.Value using the greater of the global fitting accuracy 0.1 and the local per point accuracy X.Temperature.Accuracy, if present. However, fastrbf_fit(X,0.1) fits an RBF to the default attribute field X.Value using the greater of the global accuracy and the per point accuracy X.Accuracy, if present.

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9.3 Grid Type Fields Min Max Spacing Value [Gradient] Grid data types are returned by the grid evaluation functions. The Min and Max fields specify the spatial extents of the grid. The Spacing field specifies the gap between the grid points in each direction. The size of the grid can be obtained from the size of the Value field. The Value and Gradient fields store the RBF values and gradients at the grid points in a 2 or 3 dimensional array.

The indexing of these arrays complies with the ndgrid convention rather than the meshgrid convention. For example, the gradient value at the (i j k)-th position in a grid g is obtained by g.Gradient(:,i,j,k). Gradient values obtained in this way are column vectors.

9.4 Mesh Type Fields (all point list fields) [Tri] [Quad] The mesh type describes a 3D mesh of triangles and/or quadrilaterals. See Section 9.2 for more information on point lists. Indices in the Tri and Quad arrays start at 1. One of Tri or Quad must be present.

9.5 Scan Type Fields (all point list fields) Size Origin

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The scan type is used represent data which has been scanned from multiple locations such as data from a LIDAR scanner. The Origin array specifies the location of the scanner for each scan. The Size array specifies the number of points in each scan. The point list fields contain the points from all the scanner positions. The points belonging to the scan from Origin(:,k) are the Size(k) points starting at index 1+sum(Size(1:k-1)).

9.6

RBF Solution Type

Fields AchievedAcc [DefaultEvalAcc] Centres Coeffs DataMin DataMax Version FitType BasicFunc [BasicFuncParam] [BasicFuncParam2] Rho Confidence [Indices] The RBF solution type stores data that describes an RBF along with information about the RBF. The fields are described in the following table.

Field

Description

AchievedAcc

The achieved infinity-norm accuracy. Note that this value may be larger than the desired accuracy passed to the fitter if fixed smoothing was performed or the accuracy confidence was less than 1.0.

DefaultEvalAcc

The default evaluation accuracy. This value is passed to the evaluation routines when an accuracy is not specified.

Centres

The RBF centres.

Coeffs

The RBF coefficients with polynomial part last.

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DataMin

The minimum corner of the bounding box enclosing the RBF centres.

DataMax

The maximum corner of the bounding box enclosing the RBF centres.

Version

A string identifying the version of the library that created the RBF.

FitType

An Integer specifying the fit type.

0 Direct fitting algorithm 1 Centre reduction algorithm 2 Restricted range fitting algorithm

BasicFunc

Specifies the RBF basic function.

0 Biharmonic spline 1 Triharmonic spline 2 Generalised multiquadric

BasicFuncParam

Specifies the first basic function parameter. This is the constant value in the case of the multiquadric. The polyharmonic basic functions do not take a parameter.

BasicFuncParam2 Specifies the second basic function parameter. This is the multiquadric type (the exponent) in the case of the multiquadric. The polyharmonic basic functions do not take a parameter.

Rho

The Ď smoothing value used.

Confidence

The accuracy confidence used. See Section 1.1.4 for a description of accuracy confidence.

Indices

If FitType is 1 or 2, this gives the indices of the data points used as RBF centres.

Given the BasicFunc and, in the case of the multiquadric, the associated BasicFuncParam fields, an RBF is completely described by the Centres and Coeffs fields.

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Chapter 10

Functions Throughout this chapter, the symbol | is used to separate mutually exclusive arguments.

10.1 Options The following options may be passed to any of the FastRBF routines.

Common Options Description ’messages’, N

Set the terminal message level to N, where 0 ≤ N ≤ 5. The default is 4.

’quiet’

Equivalent to ’messages’,2.

’verbose’

Equivalent to ’messages’,5.

10.2 Data Processing 10.2.1 Unique [Y, U] = FastRBF_Unique(X) FastRBF_Unique(X, DIST, ’relative’|’absolute’) Remove duplicate points from a point list, mesh or scan X. Points are considered to be duplicate if they are closer than the specified minimum inf-norm

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distance. When two points are considered duplicate the point with the highest index is removed. When a point is removed, corresponding elements of all of the optional fields, including attr-array fields, are removed as well. If U is present it is set to 1 if all the points were unique (no points removed) and 0 if the points were not unique (some points were removed).

Note fastrbf_unique is order sensitive and removes ‘duplicate’ points as it encounters them. If the input pointlist is re-ordered then the points removed and the number of points removed may change. For example, if a point is close to two other points, but those points are sufficiently far apart then both will be preserved if the common point is encountered first and removed. Alternatively, two of the three points will be removed if the common point is not encountered first.

Argument

Description

X

A point list, mesh or scan structure.

DIST

The minimum allowed relative or absolute distance between two points. If omitted the default value of 1e-10 is used.

’relative’

Specifies that DIST is relative to the extent of the data. In this case DIST is scaled by the longest side of the point list bounding box.

’absolute’

Specifies that DIST is an absolute distance.

10.2.2

Crop

Y = FastRBF_Crop(X, MIN, MAX) Y = FastRBF_Crop(X, MIN, MAX, ’range’) Remove parts of a point list, scan or mesh X that are outside a specified cropping box. If X is a mesh, faces that reference removed vertices are also removed.

If X is a scan, the Size field is updated appropriately.

Individual elements in the MIN and MAX arguments can be omitted by putting NaN in the vector instead of a number. For example, passing [NaN NaN 1] for the MIN argument sets the minimum z value to 1, but does not crop in the x and

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y directions. Passing a scalar s for MIN or MAX is equivalent to passing [s s] in 2D and [s s s] in 3D. The ’range’ option may be used with scan types to crop on the distance of the data point from the scanner, in which case MIN and MAX are scalar.

Argument

Description

X

The point list, scan or mesh to crop.

MIN

The minimum corner of the crop box.

MAX

The maximum corner of the crop box.

10.2.3 Trim Z = FastRBF_Trim(X, Y, DIST) Z = FastRBF_Trim(X, Y, DIST, ’edges’) Remove parts of the point list, scan or mesh X that are further than the specified distance from the points in Y. The DIST argument specifies the trimming distance. If X is a mesh, faces that reference removed vertices are also removed. If X is a scan, the Size field is updated appropriately.

Argument

Description

X

The point list, scan or mesh to be trimmed.

Y

The point list, scan or mesh to trim to.

DIST

The trimming distance.

’edges’

If X is a mesh, then only the mesh edges are trimmed.

10.2.4 Grid Coordinates [X,Y,Z] = FastRBF_GridCoords(GRID) [X,Y] = FastRBF_GridCoords(GRID) Get the X, Y and Z coordinates of the elements in the grid structure GRID such that GRID[i] is the value at the point [X[i], Y[i], Z[i]]. If the grid is 2D then the Z values will be all zero. [X,Y] = FastRBF_GridCoords(GRID) gets only the X and Y coordinates of a 2D grid structure.

Argument

Description

GRID

The 2D or 3D grid structure.

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10.2.5

Grid Vectors

[X,Y,Z] = FastRBF_GridVec(GRID) [X,Y] = FastRBF_GridVec(GRID) Return the vectors X, Y and Z such that the value at GRID[k,j,i] has the coordinates X[i], Y[j], Z[k]. In 2D return the vectors X and Y such that the value at GRID[j,i] has the coordinates X[i], Y[j].

Argument

Description

GRID

The 2D or 3D grid structure.

10.2.6

Create RBF Structure

RBF = FastRBF_MakeRBF(CENTRES, COEFFS) FastRBF_MakeRBF(..., ’biharm’ | ’triharm’ | ’multi’) FastRBF_MakeRBF(..., ’multi’, C, K) FastRBF_MakeRBF(..., BASICFUNC, C, K, POLYBASE, DEGREE) Creates a FastRBF solution structure from the given data.

Argument

Description

CENTRES

A vector of RBF centres DIM x N where DIM is 2 or 3.

COEFFS

A vector of RBF basic function coefficients of length N followed by DIM+1 polynomial coefficients in the order 1, x, y, z for biharmonic spline basic function and (DIM+1)*(DIM+2)/2 coefficients in the order 1, x, x2 , y, yx, y2 , z, zx, zy, z2 for the tri-harmonic spline.

C

The multiquadric constant c in φ( x) = (| x|2 + c2 )k/2 . The default values is C = 0.

K

The multiquadric exponent k in φ( x) = (| x|2 + c2 )k/2 . The default values is K = 1.

BASICFUNC

The basic function (0=’biharmonic’, 1=’triharmonic’, 2=’multiquadric’). The default is 0, ’biharmonic’.

POLYBASE

The centre of the polynomial. The default polynomial centre is the origin.

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The degree of the polynomial component of the RBF. The degree is determined from the number of polynomial coeffs in COEFFS. See FastRBF_fit (Section 10.3.1) and Table 10.1 for minimum degree requirements for different basic functions.

DEGREE

10.2.7

Create PointList

PL = FastRBF_MakePointList(LOC) PL = FastRBF_MakePointList(LOC, VAL, ACC,

LOW,

UPP, GRAD)

Creates a pointlist struct PL from the specified data. If a field empty or not present then it is not created.

Argument

Description

LOC

The location of the points. LOC is DI M × N where DI M is 2 or 3 and N is the number of points.

VAL

The value at the points.

ACC

The per-point accuracy.

LOW

The per-point lower error bar.

UPP

The per-point upper error bar.

GRAD

The per-point gradient field. GRAD is DI M × N where DI M is the same dimension as the points in LOC.

10.2.8 Converting a Grid to a PointList Converting a FastRBF Grid to a PointList can be achieved using FastRBF_GridCoords.

This following example considers a 2D grid.

[X, Y]=fastrbf_gridcoords(grid); pl=fastrbf_makepointlist([X(:) Y(:)], grid.Value(:));

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10.2.9

Transform RBF

T = FASTRBF_TRANSFORMRBF(S, M) Applies the affine transformation matrix M to the RBF S and returns the transformed RBF T. Argument

Description

T

The transformed RBF.

S

The input RBF. S is transformed so that if S is an interpolant to data ( xi , f i ), then the new RBF T is an interpolant to the transformed data ([ Mxi ], f i ) where xi is a homogeneous column vector.

M

An affine (rigid body) transform matrix. It should be a square matrix of size 1 larger than the dimension of the RBF. Any smaller matrix will be expanded to this size by adding in the appropriate rows and columns of the identity matrix.

10.2.10 View FastRBF Data Structures H H H H

= = = =

FastRBF_View(DATA) FastRBF_View(DATA,’type’) FastRBF_View(DATA,’type’,COL) FastRBF_View(...,’--’,’param’,’value’,...)

Displays data structures using the display type ’type’ and colour COL. FastRBF_View(...,’--’,’param’,’value’,...) specifies patch parameter/value pairs for the main patch object. The ’--’ argument marks the start of these pairs.

Argument

Description

DATA

The data to be viewed. It can be a FastRBF pointlist, mesh, scan, RBF solution or grid.

H

The returned vector of patch handles. If the type contains ’p’, ’f’, or ’e’ then the first handle is the point/face/edge patch. If the type contains ’v’ the remaining handles will be the output of QUIVER or QUIVER3.

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A string that can contain the following characters:

type

’p’ make point locations visible. ’f’ make faces visible, if faces are present in DATA. ’e’ make edge visible, if faces are present in DATA. ’v’ make the vectors DATA.Gradient visible as arrows. ’n’ uses DATA.Gradient as vertex normals, if ’f’ is present.

If DATA is a mesh or a 2D grid, the default is ’f’, otherwise it is ’p’.

Can be one of several different matrices where Np is the number of points:

COL

1-by-Np indexed colour value per vertex.

3-by-Np RGB colour per vertex 4-by-Np RGBA colour per vertex (M ATLAB 6 only) 3-by-1 RGB colour for patch 4-by-1 RGBA colour for patch (M ATLAB 6 only) 1-by-M where M < N and M is whole specifies groups of points, each of which is given a separate indexed value The default value for COL is DATA.Size if DATA is a scan, if DATA.Value exists then COL is DATA.Value otherwise COL is DATA.Location(end,:) (the height of the point).

10.3 10.3.1

Fitting and Evaluating Fit

RBF = FastRBF_Fit(P, ACCURACY) FastRBF_Fit(..., ’direct’|’reduce’|’errorbar’) FastRBF_Fit(..., ’biharm’|’triharm’|’multi’) FastRBF_Fit(..., ’rho’, S) FastRBF_Fit(..., ’c’, CONST) FastRBF_Fit(..., ’k’, EXP)

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Basic function Ď&#x2020; (r ) Biharmonic Triharmonic

Default degree 1 2

Multiquadric

1

Minimum required degree 2D 3D 1 0 2 1 k < 0 : No poly required, i.e., d = â&#x2C6;&#x2019;1 k â&#x2030;Ľ 0 : d = kâ&#x2C6;&#x2019;2 1

Table 10.1: The minimum degree required of the polynomial component of an RBF. FastRBF_Fit(..., FastRBF_Fit(..., FastRBF_Fit(..., FastRBF_Fit(...,

â&#x20AC;&#x2122;degreeâ&#x20AC;&#x2122;, DEG) â&#x20AC;&#x2122;confidenceâ&#x20AC;&#x2122;, CONF) â&#x20AC;&#x2122;memlimitâ&#x20AC;&#x2122;, M) â&#x20AC;&#x2122;attrâ&#x20AC;&#x2122;, NAME);

Fit an RBF to a list of 2D or 3D point locations and values.

FastRBF supports 3 basic functions in 2D and 3D. These are the biharmonic spline, triharmonic spline and generalised multiquadric. FastRBF uses fast algorithms to fit RBFs with all of these basic functions. The triharmonic spline in 2D, however, is significantly slower to solve for. The fitted RBF includes a polynomial term. The minimum degree of the polynomial depends on the basic function used. Table 10.1 lists the minimum degree, along with the default degree used in each case. Any degree higher than the minimum is theoretically possible, however, FastRBF currently limits users to quadratics i.e.,a maximum of d = 2.

There are four fitting strategies in FastRBF. These are summarised in Table 10.3.1. direct uses fast algorithms to solve the traditional RBF fitting problem where each interpolation point is used as an RBF centre. reduce uses a greedy algorithm that selects a subset of interpolation points as RBF centres until the desired accuracy is met at all the points. In both these algorithms the returned RBF is â&#x20AC;?within of the smoothest interpolant of the dataâ&#x20AC;?, where is the fitting accuracy. errorbar uses a restricted range interpolation approach and the returned RBF approximates the data values. It is â&#x20AC;?the smoothest interpolant within of the data valuesâ&#x20AC;?. Hence when is a discernible distance then the errorbar algorithm will give a smoother RBF. rho also fits an approximating RBF determined by a parameter Ď which balances faithfulness to the data against smoothness. rho can be used with reduce to produce an approx-

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imating RBF with fewer centres.

Valid fitting options direct reduce errorbar rho rho, reduce

Interpolation or approximation Interpolation Interpolation Approximation Approximation Approximation

Compression None Yes Yes None Yes

The fitting accuracy can be a per-point value taken from the Accuracy field of P (P.Accuracy), if present, or a global value determined by the ACCURACY argument. Similarly, per-point values for upper and lower error bounds can be taken from the pointlist fields, P.Upper and P.Lower — if present. These fields are used with the error-bar fitting strategy to specify asymmetric error bounds. If only one of the Upper or Lower fields is specified, then the bound is applied symmetrically. If the per-point accuracy field (P.Accuracy) exists, but neither an Upper or Lower field is available, then the errorbar fitter will apply the per-point accuracy as a symmetric per-point error bound. If the global ACCURACY argument is specified with an errorbar fit, then it will be interpreted as a symmetric bound applied to all data values.

The ACCURACY argument and the Upper, Lower and Accuracy pointlist fields are all positive quantities specifying the absolute difference between a bound and the given data value. For example, if a value were 13.6 and had an upper value of 14.3 and lower value 13.2 then the upper bound would be +0.7 and the lower bound would be +0.4. When a global fitting tolerance (ACCURACY argument) is specified along with a local per point bound, the greater of the two (i.e., the loosest constraint) is applied at each data point. This facilitates incremental fitting where the closeness of a fit to the data is progressively refined from coarse to fine using the initial option. To force the per point fields to be used at all data points, specify an ACCURACY value of zero.

By default, an RBF is fitted to the values in P.Value using the per-point bounds P.Accuracy, P.Lower and P.Upper if present and appropriate to the fitting strategy. Use the ’attr’ option with the NAME argument to fit to another attribute. The corresponding P.NAME.Value, P.NAME.Accuracy, P.NAME.Lower and P.NAME.Upper fields will then be used.

Incremental fitting, either to improve the accuracy of a fit or to add new points, is possible using the initial option. Ensure that the ordering of the point list is consistent with that used to compute the initial RBF. Ensure that

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the presence of any attributes is consistent between newly added points and previous points.

The set-up overhead is proportional to the number of interpolation nodes.

Therefore it can be inefficient to use initial for the addition of small numbers of points at a time.

Argument

Description

P

The point list containing the data to fit to.

ACCURACY

Specifies the fitting accuracy or tolerance. The RBF value will be within this amount of the original value at each input point.

’direct’

Specifies the direct fitting algorithm. This is the default.

All input points are used as centres. Best for true density data.

’reduce’

Specifies the centre reduction fitting algorithm. Best for clean data with little or no discernible noise.

’rho’

Set the smoothing value to S. Higher values will create a smoother fit, but will interpolate less accurately.

’errorbar’

Uses error bar fitting algorithm. Best for noisy data or when the fitting accuracy is quite loose.

’biharm’

Set the basic function to the biharmonic spline. This is the default.

’triharm’

Set the basic function to the triharmonic spline.

’multi’

Set the basic function to the generalised multiquadric φ(r) = (r 2 + c 2 ) k .

’c’

Set the multiquadric constant to CONST.

’k’

Set multiquadric exponent to EXP/2. K must be odd and less than 6. The default is 1.

’degree’

Set the degree of the RBF polynomial to DEG. -1 for no polynomial, 0 for a constant, 1 for a linear, 2 for a quadratic.

’confidence’

Set the confidence that you have in the accuracy to CONF. CONF*length(P.Location) of the points must have the specified accuracy. The default is 1.0 (all points have the specified accuracy).

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’memlimit’

Set a soft limit on the amount of RAM to use to M megabytes.

May be exceeded if M is less than the core memory requirements. The default is half the available physical RAM.

’initial’

Specify an initial RBF solution. The benefit of using an RBF estimate depends how similar the estimate is to the final RBF.

’attr’

Fit to the fields associated with the NAME attribute instead of the default fields in the point list. The specified attribute must exist.

10.3.2 Point Eval P = FastRBF_PointEval(S,P) FastRBF_PointEval(S,P,ACC) FastRBF_PointEval(...,’gradient’) FastRBF_PointEval(...,’attr’, NAME) FastRBF_PointEval(...,’smooth’, W) Evaluate a 2D or 3D RBF on the points in a point list, with optional gradient evaluation.

The input point list P may also be a scan or mesh struct, and may contain attribute data. None of this information is used, but it will all be copied into the output struct. This allows you to add function values and gradients to an existing struct.

By default, the RBF values are written to the Value field of P, and Gradient field if the ’gradient’ option is given. The ’attr’ option may be used to write to the Value and Gradient fields of another, or a new attribute instead.

Argument

Description

S

Input RBF solution.

P

Input point list, scan or mesh. The Location field specifies the evaluation locations.

ACC

The evaluation accuracy. The default is S.DefaultEvalAcc, if the field exists. If the field does not exist, ACC must be specified.

’gradient’

Evaluate the RBF function gradient at each point as well as the value.

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’attr’

Write the RBF values to the NAME attribute instead of the default.

’smooth’

Applies a low pass filter of width W to the RBF before evaluating. In 3D, a smooth evaluation is only twice as long as an ordinary evaluation and independent of W. In 2D it is significantly slower with computation O( N 2 ) where N is the number of centres, but it is still independent of W.

10.3.3 Grid Eval G = FastRBF_GridEval(S, ...)

FastRBF_GridEval(S, ACC, ...)

FastRBF_GridEval(..., ’min’, MIN, ...)

FastRBF_GridEval(..., ’max’, MAX, ...)

FastRBF_GridEval(..., ’spacing’, SPACING) FastRBF_GridEval(..., ’size’, SIZE) FastRBF_GridEval(..., ’gradient’) FastRBF_PointEval(...,’smooth’, W) P = FastRBF_PointEval(...,’pointlist’) Evaluate a 2D or 3D RBF on a regular grid, with optional gradient evaluation.

There are four options for specifying the grid: ’min’, ’max’, ’spacing’ and ’size’. Any three of these fully specify the grid. Specifying all four causes an error. If the grid is under-specified then S.DataMin is used for the ’min’ argument. If the grid is still under-specified then S.DataMax is used for ’max’ argument. If any of MIN, MAX, SPACING or SIZE are scaler s, then the vector [s s] in 2D and [s s s] in 3D is used instead. See Section 9.3 for a description of the returned grid structure.

Argument

Description

S

The RBF solution to evaluate.

ACC

The evaluation accuracy. The default is S.DefaultEvalAcc, if the field exists. If the field does not exist, ACC must be specified.

’min’

Set the minimum corner of the evaluation grid to MIN.

’max’

Set the maximum corner of the evaluation grid to MAX.

’spacing’

Set the spacing between grid points to SPACING.

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’size’

Set the size of the evaluation grid to SIZE.

’gradient’

Evaluate the RBF gradient as well as the value at each grid point.

’smooth’

Applies a low pass filter of width W to the RBF before evaluating. In 3D, a smooth evaluation is only twice as long as an ordinary evaluation and independent of W. In 2D it is significantly slower with computation O( N 2 ) where N is the number of centres, but it is still independent of W.

’pointlist’

Return a pointlist struct instead of a grid struct. Grid structures are more compact, but the pointlist type is more flexible and suitable for export as text formatted data.

10.4

3D Surface Fitting

10.4.1 Mesh Checking X = FastRBF_CheckMesh(M) X = FastRBF_CheckMesh(M, PERPART) Checks that all the faces of a mesh M are consistently oriented. The face vertex order is validated via the connectivity (shared vertices) of the faces. Surface normals are not considered or altered. If the mesh consists of several unconnected parts then correct orientation between the parts is not assured. This utility can (and should) be used before calling NormalsFromMesh to ensure the generated normals all point outwards.

This routine tests that the mesh is manifold in the sense that each edge has at most 2 faces adjacent to it.

Note Note that this routine does not guarantee manifold-ness in the strictest sense, i.e., the mesh could still self-intersect.

Argument

Description

M

The mesh to check face consistency of.

X

The mesh property structure. The relevant information is stored in the Boolean fields: X.Consistent, X.Manifold and X.Closed.

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If PERPART is non-zero FastRBF_CheckMesh calculates the ’Consistent’, ’Manifold’ and ’Closed’ properties for each distinct (unconnected) sub-mesh. An array of mesh info structures is returned. If 0 (the default value) then a single structure is returned for the whole mesh, i.e., if one part of the mesh is open or non-manifold, but other parts are not, then the return structure will indicated that the net mesh data is open or non-manifold.

PERPART

10.4.2

Mesh Repair and Re-meshing

M = FastRBF_CheckMesh2Mesh(MESH,ACC,RES) Automatically fits an RBF to an existing (possibly partial) mesh, then isosurfaces that RBF to produce a manifold and evenly sampled output mesh. It applies FastRBF_CheckMesh and gives warnings if the raw mesh data contains errors. It does not correct these errors.

Argument

Description

M

The mesh to be fitted and reconstructed/repaired.

ACC

Fitting accuracy.

RES

Output mesh resolution.

10.4.3 P P P P P P P P S S S S M

= = = = = = = = = = = = =

Estimate Normals FastRBF_NormalsFromPoints(P) FastRBF_NormalsFromPoints(P, RADIUS) FastRBF_NormalsFromPoints(P, RADIUS, FACTOR) FastRBF_NormalsFromPoints(..., ’attr’, ’NAME’) FastRBF_NormalsFromSigns(P) FastRBF_NormalsFromSigns(P, RADIUS) FastRBF_NormalsFromSigns(P, RADIUS, FACTOR) FastRBF_NormalsFromSigns(..., ’attr’, ’NAME’) FastRBF_NormalsFromScan(S) FastRBF_NormalsFromScan(S, RADIUS) FastRBF_NormalsFromScan(S, RADIUS, FACTOR) FastRBF_NormalsFromScan(..., ’attr’, ’NAME’) FastRBF_NormalsFromMesh(M)

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M = FastRBF_NormalsFromMesh(..., ’attr’, ’NAME’) Estimate 3D surface normals using one of four methods. NormalsFromMesh sets the vertex normal to the average of the adjacent face normals.

NormalsFromSigns, NormalsFromScan and NormalsFromPoints estimate normals by fitting a least squares plane through a local neighbourhood around each point. NormalsFromScan utilises the position of the scanning device (line of sight) to resolve ambiguities in normal direction. NormalsFromSigns utilises an outward pointing vector stored in the gradient field of each point i.e.,P.Gradient. Note that this outward pointing vector need not be normal to the surface. NormalsFromPoints estimates normals without any additonal information. Note that in general this is a problem with ambiguities and the result generated may not always be the desired one. We recommend improving the robustness of normal estimation by adding additional information wherever possible.

The radius of the local neighbourhood sphere is controlled by the RADIUS argument. The default value RADIUS == 0 has a special function. When RADIUS == 0, at each point 4 neighbourhoods are calculated which contain 10, 30, 100 and 300 points. The neighbourhood which is most plane-like (has the largest plane-factor) is then used. This means that normals are found for most of the points with no knowledge about the scale of the data and no user intervention. However it is obvious that this method is about 4 times slower than using a fixed radius. Hence, if you are processing data from a single data source, experimenting with fixed RADIUS values to find one that works well with your data is advantageous.

The FACTOR argument specifies how plane-like the points in the spherical neighbourhood must be to accept a normal from it’s least squares plane. The plane factor is calculated as follows: PlaneFactor =

LineError / PlaneError .

PointError / LineError

where PointError = RMS of distances to best point, LineError = RMS of distances to best line, PlaneError = RMS of distances to best plane.

The FACTOR argument must be greater than 1. The default value of 1.8 is usually fine although 1.5 can work better for noisy data. The higher the value

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of FACTOR the flatter the local neighbourhoods are required to be and the more confident we are of the estimated plane.

Argument

Description

P, S, M

The surface data to estimate normals for.

RADIUS

The radius of the local patches for estimating surface normals. May be 0 to use the best plane from patches with 10, 30, 100 and 300 points. This is the default.

FACTOR

The minimum plane confidence factor (or plane-y-ness) required for the local patch for a normal to be accepted from it. Must be greater than 1. Default value is 1.8.

’attr’

Return the output vertex normals in the attr-array called NAME i.e., M.NAME.Gradient instead of the default attribute M.Gradient.

10.4.4 Density From Normals P = FastRBF_DensityFromNormals(Q, MIN, MAX) FastRBF_DensityFromNormals(Q, MAX) FastRBF_DensityFromNormals(..., ’fixedlen’) FastRBF_DensityFromNormals(..., ’onesided’) FastRBF_DensityFromNormals(..., ’increase’, INCR) FastRBF_DensityFromNormals(..., ’numgrid’, NUM) FastRBF_DensityFromNormals(..., ’attr’, NAME) Convert 3D surface points and normals to a density field. This creates new points inside and outside the surface by following the point normals. The Value field is filled with distance-to-surface data. Use the ’attr’ option to specify a particular attribute Value field. When an off-surface point is created it is assigned all the same attribute values as the surface point it was created from. Use the ’numgrid’ option to add a grid of points throughout the volume of interest. Grid point attributes are assigned the values of the closest point.

Argument

Description

Q

A point list, mesh or scan with surface normals.

MIN

The minimum projection distance. Default 0.

MAX

The maximum projection distance.

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’fixedlen’

Validate density values rather than changing the projection distance.

’onesided’

Generate new points on the outside of the surface only.

’increase’

Set the number of off-surface points to add, specified as a fraction of the number of input points, to INCR. The default value is 1, which doubles the number of points. The maximum value is 2, which triples the number of points.

’numgrid’

Generate approximately NUM additional off-surface points on a grid.

’attr’

Get input from, and put output into the fields of the attrarray called NAME (P.NAME.Gradient) rather than the default point list P.Gradient.

10.5 3D Isosurfacing 10.5.1 Isosurf M = FastRBF_Isosurf(S, RES) FastRBF_Isosurf(S, RES, ISOVALUE) FastRBF_Isosurf(S, RES, ISOVALUE, ACC) FastRBF_Isosurf(..., ’min’, MIN) FastRBF_Isosurf(..., ’max’, MAX) FastRBF_Isosurf(..., ’fit’, FIT) FastRBF_Isosurf(..., ’normals’) FastRBF_Isosurf(..., ’invert’) FastRBF_Isosurf(..., ’open’|’closeminus’|’closeplus’) FastRBF_Isosurf(..., ’optimal’|’plane’|’grid’) FastRBF_Isosurf(..., ’tri’|’quad’|’triquad’) FastRBF_Isosurf(..., ’orient’, T) FastRBF_Isosurf(..., ’up’, DIR) FastRBF_Isosurf(..., ’seeds’, P) FastRBF_Isosurf(..., ’seedlimit’, N) Extracts an isosurface from a 3D RBF solution.

The ’min’ and ’max’ options default to the bounding box of the input RBF, S. Individual elements in the MIN and MAX arguments can be omitted by putting NaN in the vector instead of a number. For example, passing [NaN NaN 1] for

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the MIN argument sets the minimum z value to 1, but uses the default values for x and y. Passing [NaN NaN NaN] is equivalent to omitting the option completely. Passing a scaler s is equivalent to passing [s s s].

By default the isosurfacer increases the sampling resolution slightly to get an exact fit to the specified bounding box. The ’fit’ parameter may be used to use the specified resolution in any direction. In this case the isosurface may extend slightly beyond the specified bounding box, so that an integer number of sampling intervals is achieved. The ’fit’ parameter is a string consisting of any combination of ’x’, ’y’ or ’z’. These signify exact clipping along the corresponding axes. For example, ’x’ causes the resolution in the x direction to be increased so that an exact fit to the bounding box is achieved in the x direction, and the specified resolution is used in the y and z directions.

The orientation of the sampling lattice, by default, corresponds to the data coordinate system implicit in the RBF data structure, S. This can be altered by specifying the orientation matrix T. The orientation matrix transforms a point in the RBF coordinate system into the lattice coordinate system. This 4x4 homogeneous transform may include an affine transform component and provides the user with a means of altering the relative sampling frequencies in the lattice directions as well as the lattice orientation. This will interact with the specified sampling resolution RES. A simpler way of specifying the orientation of the sampling lattice, if no scaling is desired, is to use the ’up’ option to specify the lattice ”up” direction (normal of the plane). DIR need not have unit length.

Note that valid seeds, for initialising surface-following, must lie within a sampling interval of the bounding box. ’min’ and ’max’ are specified in the RBF coordinate space. When the sampling lattice is rotated the effective bounding box in the new lattice coordinate system is chosen to just enclose that specified in the data coordinate system by min and max (Figure 10.1).

Argument

Description

S

The RBF solution to isosurface.

RES

Specifies the surface resolution. The default is 1 (one).

ISOVALUE

Specifies the function value of the isosurface. The default is zero.

ACC

Specifies the RBF evaluation accuracy. The default value is S.DefaultEvalAcc. If this field is missing then ACC must be present.

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Bounding box specified in data coordinates Effective bounding box

Lz

Lz Lattice transform

Lx

Ly Lx

z y

x Data coordinate system

z y

x

New lattice coordinate system Ly

Figure 10.1: When the sampling lattice is rotated relative to the RBF, the effective bounding box in the new lattice coordinate system is chosen to just enclose that specified in the data coordinate system by min and max.

’min’

Set the minimum corner of the evaluation box to MIN. The default is S.DataMin.

’max’

Set the maximum corner of the evaluation box to MAX. The default is S.DataMax.

’fit’

Tells the isosurfacer to adjust the sampling grid to fit exactly to the bounding box. The characters ’x’, ’y’ and ’z’ in FIT specify which axes to adjust along. The default is ’xyz’.

’normals’

Enable evaluation of the RBF gradient at each point to create surface normal vectors.

’invert’

Inverts the face orientation.

’open’

No boundary capping is performed. This is the default.

’closeminus’

Adds capping planes where the surface intersects the edges of the evaluation box. Assumes points outside the evaluation box have values less than the isosurface value.

’closeplus’

The same as ’closeminus’ but assumes points outside the evaluation box have values greater than the isosurface value.

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’orient’

Sets the evaluation lattice transformation matrix to the 4by-4 matrix T. This can be used with ’plane’ to set the orientation of the parallel planes.

’up’

Sets the evaluation lattice ”up” direction to DIR. This can be used with ’plane’ to set the orientation of the parallel planes.

’optimal’

Optimise the mesh faces. This is the default.

’plane’

Optimise the mesh faces, but force vertices to lie on parallel planes. The orientation of the plane defaults to the xy plane. This can be modified using the ’orient’ or ’up’ option.

’grid’

Do not optimise the mesh vertices. They will lie on the tetrahedral evaluation grid. This can be modified using the ’orient’ or ’up’ option.

’tri’

Construct the surface from triangular faces. This is the default.

’quad’

Construct the surface from quadrilateral faces.

’triquad’

Construct the surface from a combination of triangular and quadrilateral faces.

’seedlimit’

Use at most N successful surface seeds. The default is 1 (one). If the seedlimit is set to 0 then all available seeds are used.

’seeds’

Use the locations in P as surface seeds. The default is to use the centres from the RBF solution. P may be a point list or a 3 × N array.

10.5.2 Marching Cubes M = FASTRBF_MCUBES(GRID) FASTRBF_MCUBES(GRID, ISOVALUE) FASTRBF_MCUBES(...,’invert’|’noinvert’) FASTRBF_MCUBES(...,’open’|’closeminus’|’closeplus’) FASTRBF_MCUBES(...,’tri’|’triquad’) MCubes extracts an isosurface from a given grid of density values using the standard marching cubes algorithm. Using mcubes is recommended when

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isosurfacing density data since the 3D grid evaluation guarantees that all separate parts of the isosurface, greater than the sampling resolution, are extracted.

Also with general density data the desired isosurface may not be near any RBF centres and the seeding process of the isosurf command may fail.

Argument

Description

GRID

A grid of values to be isosurfaced.

ISOVALUE

Specifies isosurface function value. The default is zero.

’invert’

Inverts the face orientation.

’noinvert’

Does not invert the face orientation. The default.

’open’

No capping planes. This is the default.

’closeminus’

Adds capping planes where the surface intersects the edges of the evaluation box. Assumes points outside the evaluation box have values less than the isosurface value.

’closeplus’

The same as ’closeminus’ but assumes points outside the evaluation box have values greater than the isosurface value.

’tri’

Construct the surface from triangular faces. This is the default.

’triquad’

Construct the surface from a combination of triangular and quadrilateral faces.

10.5.3 Simplify N = FastRBF_Simplify(M, SOL, ERROR) FastRBF_Simplify(M, SOL, ERROR, ISOVALUE) FastRBF_Simplify(..., ’faces’, NF) FastRBF_Simplify(..., ’points’, NP) FastRBF_Simplify(..., ’angle’, A) FastRBF_Simplify(..., ’smooth’, W) FastRBF_Simplify(..., ’accuracy’, ACC) FastRBF_Simplify(..., ’inverted’) N = FastRBF_SimplifyRGB(M,SOL,ERROR,R,G,B,RGBSTEP) FastRBF_SimplifyRGB(M,SOL,ERROR,R,G,B,RGBSTEP,ISOVALUE) FastRBF_SimplifyRGB(..., ’smoothrgb’, V)

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FastRBF_SimplifyRGB(..., ’accuracyrgb’, ACCRGB) Simplifies an isosurface from a 3D RBF. This is a geometry preserving simplifier. Faces with bad aspect ratio are removed first. Simplification continues until one of the stop criteria options ’points’ or ’faces’ is met, or no more edges can be removed without violating the ERROR or ’angle’ conditions.

The ERROR argument prevents the simplifier from removing any faces that have or will cause a chordal distance to the isosurface of more than ERROR in a simplified face. The ’angle’ argument will prevent the simplifier creating faces which have a minimum internal vertex angle that is less than A degrees.

This controls the aspect ratio of the output faces. When ’points’ or ’faces’ is specified the faces with the least chordal error are removed first.

If severe simplification is desired it is best to do it in 2 or more steps. For example, if we have a mesh with 200 000 faces and we only want 2000, doing 2 calls with NF=20000 and then NF=2000 will produce a better result than just one call with NF=2000.

Argument

Description

M

The mesh to be simplified.

SOL

The RBF solution of which M is an isosurface.

ERROR

Sets the maximum allowed distance from a face centroid to the isosurface of the RBF solution of a split face.

R

(SimplifyRGB only) RBF solution interpolating red intensity over the surface.

G

(SimplifyRGB only) RBF solution interpolating green intensity over the surface.

B

(SimplifyRGB only) RBF solution interpolating blue intensity over the surface.

RGBSTEP

(SimplifyRGB only) Sets the maximum allowed variation in RGB values (calculated as a Euclidean distance) between vertices to s.

ISOVALUE

Sets the iso-value of the mesh. Default is 0 (zero).

’inverted’

Specifies that the face orientation has been inverted.

’points’

Sets the desired maximum number of vertices in the simplified mesh to NP. The default is no limit.

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’faces’

Sets the desired maximum number of faces in the simplified mesh to NF. The default is no limit.

’angle’

Sets the minimum vertex angle of the faces to A in degrees.

The default is 10.

’smooth’

Applies a low pass filter of width W to the RBF before evaluating. This must be the same value used when creating the passed isosurface.

’smoothrgb’

(SimplifyRGB only) Applies a low pass filter of width V to the colour RBFs before evaluating.

’accuracy’

Sets the RBF evaluation accuracy to ACC. The default value is SOL.DefaultEvalAcc. If this field is missing then ’accuracy’ must be present.

’accuracyrgb’

Sets the RBF evaluation accuracy to ACCRGB. The default value is R.DefaultEvalAcc. If this field is missing then ’accuracyrgb’ must be present.

10.6 Data Import and Export 10.6.1 Import X = FastRBF_Import(FILENAME) FastRBF_Import(..., ’ptx’|’pto’|’obj’|’stl’|’txt’) FastRBF_Import(..., ’format’, FMT) FastRBF_Import(..., ’comments’, STR) [X, RGB] = FastRBF_Import(...)

Loads external file formats into FastRBF data type. If no import type option is specified then the type will be determined from the file extension. RGB is only returned for the obj file type, supporting the non-standard extension of writing RGB values (on 0-255) on vertex lines after the vertex coordinates. In this case, the RGB values are also loaded into the attr-arrays Red, Green and Blue of X, and is equivalent to X = FastRBF_Import(..., ’obj’); RGB = [X.Red.Value; X.Green.Value; X.Blue.Value];

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Argument

Description

FILENAME

The name of the file to load.

0

ob j0 (or ext. .obj) Sets the import type to Alias|Wavefront mesh. X will be a mesh.

0

ptx0 (or ext. .ptx) Sets the import type to Cyra PTX. X will be a scan.

0

pto0 (or ext. .pto) Sets the import type to Maptek PTO. X will be a scan.

0

stl 0 (or ext. .stl) Sets the import type to Stereo-lithography binary. X will be a mesh.

0

txt0 (or ext. .txt) Sets the import type to ASCII text. X will be a point list.

’comments’

Lines starting with any character in the string STR are considered comments and are ignored.

’format’

Sets the format string for formatted text input to FMT. The string specifies the format for each line in the input file.

Required with ’txt’.

Text input files are interpreted according to the scanf-like format string fmt.

Three types of data field can be specified within the format string: required fields, optional attribute fields and skip fields.

The format for the required fields is: Field format: %datatype where datatype is one of the following: x Location[0] y Location[1] z Location[2] The fields %x and %y must occur in the format string. The dimension of the imported point list data, whether the data are 2D or 3D, is determined by the presence or absence of the %z field in the format string.

The format for optional fields is: Field format: %<attributename>datatype

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where datatype is one of the following: v Value a Accuracy u Upper (error-bar) l Lower (error-bar) dx Gradient[0] dy Gradient[1] dz Gradient[2] and attributename is the name of the attribute to which the data is assigned. If no attribute name is specified i.e., the format string is %datatype, then the data are read as the default value at the data location. Normally a Value field, %v, will be present in the format string, but it need not be when reading pointcloud surface data which may consist of just of the required data â&#x20AC;&#x201D; the location of the surface points.

All the above types are read as doubles.

Finally, the format for skipped fields is: Skip field format: %*[width]type where type is a scanf type, e.g. i, d, s, f, . . . , etc, and width is optionally used to specify a fixed number of characters to skip.

Some example format strings and correpsonding data lines are given below: Data: Format:

4.52e+04 3.63e+02 2.25e+04 23.0 %x %y %z %v\n

Data: Format:

4.52e+04 3.63e+02 2.25e+04 23.0 255 255 32 %x %y %z %v %<Red>v %<Green>v %<Blue>v\n

Data: Format:

4.52e+04 3.63e+02 2.25e+04 RBFsRus 23.0 0.5 0.5 0.707 %x %y %z %v %*s %dx %dy %dz\n

Care should be taken when specifying gradient fields (RBF data types dx, dy or dz) that all the components are assigned values as this is not enforced.

If multiple values are read to the same attribute, then the last entry read will overwrite the previous read values.

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10.6.2

Export

FastRBF_Export(X, FILENAME) FastRBF_Export(X, RGB, RGBMAX, FILENAME) FastRBF_Export(..., ’dxf’|’obj’|’stl’|’txt’|’vrml’) FastRBF_Export(..., ’format’, FMT) FastRBF_Export(..., ’precision’, N) FastRBF_Export(..., ’ascii’) Saves FastRBF data struct to third party file formats. If no export type option is specified then the type will be determined from the file extension.

Argument

Description

X

The data to export.

FILENAME

The name of the file to write to.

RGB

Specifies the RGB colour per vertex data, 3 × N. This argument is valid with VRML and Wavefront OBJ formats only. The Wavefront colour-per vertex format is not standard.

RGBMAX

Specifies the range of the RGB data. Usually 1 or 255.

0

0

dx f (or ext. .dxf) Sets the output file type to AutoDESK DXF. Use the precision option to set the number of decimal places.

0

ob j0 (or ext. .obj) Sets the export type to Alias|Wavefront OBJ.

0

stl 0 (or ext. .stl) Sets the export type to Stereo-lithography. Defaults to binary format. Use ’ascii’ to write ascii format.

0

vrml 0 (or ext. .wrl) Sets the export type to VRML.

0

txt0 (or ext. .txt) Sets the export type to formatted text.

’ascii’

Specifies ascii data format for STL type.

’precision’

Sets the number of decimal places in the DXF format to N. The default is to write full machine precision.

’format’

Sets the format string for formatted text output to FMT. The string specifies the format for each line in the output file.

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Functions

Text Export Format Specification The format specification for exporting text files contains printf-like format fields with additional syntax to specify the data to write to each line of the text file. The complete specification is: Export Format: %[([flags][width][.precision]type)][<attributename>]value Part

Description

flags

As for printf.

width

As for printf.

precision

As for printf.

type

One of g, f, E or e.

attributename

The name of the attribute, if none is given then the default is used.

value

Identifies the data that is to be formatted. One of: x Location[i][0] y Location[i][1] z Location[i][2] v Value[i] a Accuracy[i] a Upper[i] a Lower[i] dx Gradient[i][0] dy Gradient[i][1] dz Gradient[i][2]

An attribute name is not specified when formatting the location values x, y and z. If a specific attribute name is not given when formatting other values then the default value is used. The default printf format is (.10g). To write a value as an integer use (.0f). Other parts of the Format string are simply echoed to the output. As the file is opened in binary mode the end of line should be specified with \r\n for Windows and \n for U NIX.

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Some valid format strings are shown below: Format: Result:

%(.2e)x %(.2e)y %(.2e)z\n 4.52e+04 3.63e+02 2.25e+04

Format: Result:

rgb = %(.0f)<Red>v \t%(.0f)<Green>v \t%(.0f)<Blue>v)\n rgb = 2 120 34

Format: Result:

v %x %y %z \nvn %(.4f)dx %(.4f)dy %(.4f)dz\n v 1.2323456256 4446.5634454393 7.8934566432e-05 vn 0.8256 0.2008 0.5273

10.7

Load and Save FastRBF Files

10.7.1

Load

X = FastRBF_Load(FILENAME) Loads an A RANZ format file containing a FastRBF data type. If successful X be a point list, solution, grid, scan, or mesh.

Argument

Description

FILENAME

The name of the file to load.

10.7.2 Save FastRBF_Save(X, FILENAME) FASTRBF_SAVE(..., ’ascii’) Saves a FastRBF data struct X to an A RANZ format file.

Argument

Description

X

The data to export.

FILENAME

The name of the file to write to.

’ascii’

Specifies ascii format instead of binary.

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Chapter 11

File Formats The file types used by FastRBF are instances of a the A RANZ data file format.

In this chapter, we describe the general format first, then the specific types used by FastRBF.

11.1

A RANZ File Format

The files used by the FastRBF functions are in the A RANZ data file format. This format is designed for storing a set of large multidimensional arrays into a file in an ASCII or binary format.

The A RANZ file format starts with the header %ARANZ-1.0, where the two numbers are the major and minor versions numbers of the A RANZ format.

The only version at the moment is 1.0, but new versions may be added in the future.

The rest of the file is divided into sections, each of which starts with a string of the form [section-name]. The sections, in the order that they appear, are [comment] (optional), [index] and then either [binary] or [ascii].

11.1.1

Comment Section

The comment section starts with [comment] and continues until [index], the start of the index section. All the text in the comment section is ignored.

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11.1.2

Index Section

The index section describes the data stored in the file. It starts with the string [index] and ends with either [binary] or [ascii].

The data in an A RANZ file is divided into one or more ‘items’. Each item has an identifier, type and array size. The index section describes each item with a series of whitespace separated fields, terminated with a semicolon (;).

The fields are: Identifier is a string used to identify the item. It will start with a letter or underscore, followed by any number of letters, underscores or digits.

Type is the data type of the item: Integer, Double, Boolean or Character.

Array Sizes make up the rest of the list. This is a list of zero or more positive integers, where each is the size of the item’s data array in one dimension.

If there are no array sizes then the item is a single value.

A selection of example index entries is shown in Table 11.1.

Entry Points Double 3 20; Name Character 30; Grid Integer 10 10 10; Option Boolean;

Description List of 20 points in 3D. 30-character string.

10 × 10 × 10 grid of 1000 integers.

Single boolean value.

Table 11.1: Example index entries for the A RANZ file format.

11.1.3

Binary Section

The binary section stores the item data in binary form. It starts with [binary] and continues until the end of the file.

The first few bytes are used for check values, used when loading the file to ensure that the computer is interpreting the binary data correctly. These bytes are the integer 00f00fff16 , followed by the double-precision floating point number 1.23456789. An incorrect byte ordering in the file can be corrected but any other difference causes an error.

The rest of the file consists of the data arrays of each item, flattened and written to the file using the computer’s native sizes and formats (the boolean type is written as one byte for each element).

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File Formats

The order of flattening an array is such that with array of size N1 , N2 , N3 an item with zero based index (i1 , i2 , i3 ) has index iflat in the flat array, where iflat = i1 + i2 N1 + i3 N1 N2 .

11.1.4

(11.1)

ASCII Section

The ASCII section stores the item data in plain text form. It starts with [ascii] and continues to the end of the file.

The data arrays for each item are flattened, concatenated and written to the file. Elements are separated by whitespace (normally a line feed). The integer type is written as a decimal number. The double type is written in decimal scientific notation with 16 significant figures. The boolean type is written as TRUE or FALSE. Arrays of the character type are written as a double-quoted string, using the escape sequences described in Table 11.2.

Character Double quote Backslash Carriage return Line feed Form feed Tab Vertical tab Other non-printable

Sequence \" \\ \r \n \f \t \v \x. (x is the ASCII code, in decimal, for the character.)

Table 11.2: Escape sequences used in ASCII format character arrays.

11.1.5

Example File

The following is an example of an A RANZ format file, using ASCII encoding.

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%ARANZ-1.0 [comment] Example of the ARANZ file format [index] Foo Character 20; Bar Boolean; Wibble Double 3 20; Blarg Integer 5 5 5; [ascii] "string \0."

TRUE 3.000000000000000e+001 0.000000000000000e+000 0.000000000000000e+000 2.965925826289068e+001 0.000000000000000e+000 2.588190451025207e+000

11.2

FastRBF File Formats

Each of the file types used by FastRBF defines a set of item definitions for the A RANZ format (some items may be optional). The formats are described in the following sections. When sizes are specified, D is the number of dimensions (2 or 3) and Npoints should be read as “number of points”. If no sizes are given then the item is a single number.

Attribute fields Identifier Name Value Name Accuracy Name Upper Name Lower Name Gradient

Req.

Type Double Double Double Double Double

162

Size Npoints Npoints Npoints Npoints D × Npoints

File Formats

Attribute arrays may appear in Point List, Mesh and Scan files. They are stored in A RANZ format files by prefacing the field name with the attribute name. For example, if a point list had an attribute called Red, and the attribute had fields Value and Accuracy, then the items Red_Value and Red_Accuracy will appear in the point list file when saved. Similarly, if a mesh file has the item Intensity_Value, then the mesh will have an attribute array named Intensity with a Value field when loaded.

11.2.1

Point List

Identifier Location (default attribute fields) Value Accuracy Upper Lower Gradient (attribute arrays)

Req. ✓

Type Double

Size D × Npoints

Double Double Double Double Double

Npoints Npoints Npoints Npoints D × Npoints

The point list stores point locations, default attribute fields and any number of other attributes for a list of points in 2D or 3D. When present, the Value field stores the function value, the Accuracy stores the accuracy of that value and the Gradient stores the function gradient or a surface normal vector. In the context of an errorbar fit Upper stores the upper bound on a data value and Lower the lower bound on the value.

11.2.2

Mesh

Identifier Location (default attribute fields) Value Accuracy Gradient (attribute arrays) Tri Quad

Req. ✓

Type Double

Size 3 × Npoints

Double Double Double

Npoints Npoints 3 × Npoints

Integer Integer

3 × Ntri 4 × Ntri

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The mesh type is a specialisation of the 3D point list. It is used for storing surface meshes made out of triangles and/or quadrilaterals, defined as indexes into the point list (the first point is point zero). At least one of Tri or Quad is required.

11.2.3

Scan

Identifier Location (default attribute fields) Value Accuracy Gradient (attribute arrays) Origin Size

Req. ✓

✓ ✓

Type Double

Size 3 × Npoints

Double Double Double

Npoints Npoints 3 × Npoints

Double Integer

3 × Nscans Nscans

The scan type is also a specialisation of the 3D point list. It divides the points into Nscans parts and stores the location of a 3D surface scanner for each part.

The Size field stores the number of points in each part and Origin stores the scanner location.

11.2.4

Solution

Identifier AchievedAcc DefaultEvalAcc Centres Coeffs Indices Version DataMin DataMax FitType BasicFunc BasicFuncParam BasicFuncParam2 Rho Confidence

Req. ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Type Double Double Double Double Integer Character Double Double Integer Integer Double Integer Double Double

164

Size

D × Ncentres Ncoeffs Ncentres 32 D D

File Formats

Stores a 2D or 3D RBF solution. The relationship between Ncentres and Ncoeffs depends on the basic function used.

11.2.5

Grid

Identifier Min Max Spacing Value Gradient

Req. ✓ ✓ ✓ ✓

Type Double Double Double Double Double

Size D D D ( Nx × Ny ) or ( Nx × Ny × Nz ) (2 × Nx × Ny ) or (3 × Nx × Ny × Nz )

Grid stores the output of 2D or 3D grid evaluation. The number of dimensions of Value and Gradient differs in 2D and 3D. Like the point list, the function values goes in the Value item and the gradient in the Gradient item.

11.3

FastRBF File Viewer

fastrbffile FILENAME A stand-alone executable called fastrbffile is includded with the FastRBF distribution. This utility allows users under Windows to quickly view header information in a ARANZ format file (FILENAME), including mesh files, RBF solution files, pointlists and grids.

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166

Chapter 12

Troubleshooting 12.1 FastRBF License file not found FastRBF Toolbox inoperative with error messages relating to the license file FastRBF.lic.

Explanation

• FastRBF.lic is not available or has expired.

• FastRBF.lic corrupted (possibly interpreted as an ASCII file by a mail tool or Windows text editor — please treat the license as a Binary file.

• License installed in incorrect directory or environment variable/path not defined properly.

• License is node-locked to a particular machine.

Remedy

• Install lincense. This may require root or administrator priviledges.

• Installation usually involves copying FastRBF.lic to the appropriate installation directory. Under Windows ... this is likely to be something like: C:\Program Files... ... \FarField Technology\FastRBF v1.4\Command Line... ... or C:\Program Files\FarField Technology\FastRBF... v1.4\Matlab\toolbox\private. ...

• In Matlab add the toolbox to your path, i.e., addpath... ... ’C:\Program Files\FarField Technology\FastRBF...

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v1.4\Matlab’ • For CommandLine add the installation directory to your PATH environment variable.

• Under Linux the license file for the Matlab toolbox should ... be placed in the same directory as the FastRBF_MEXr13... .mexglx executable. This is stored in the toolbox subdirectory /private. This is likely to be something like, ... /usr/local/fastrbf/FastRBF_Matlab_Linux_v1.4.1/... toolbox/private. FastRBF will, however, first search for the license file specified by the enviornment variable FASTRBFLICENSE provided that both the variable, and the file it refers to, are present (you can define this variable in your .bashrc/.cshrc/.zshrc etc). FastRBF will lastly search for FastRBF.lic in the current directory where you are running Matlab.

• For the Linux CommandLine Toolbox, add the location of the executable to your PATH and set FASTRBFLICENSE to point to the location of the License file.

12.2

RBF Fitting

Using the reduce option does not reduce the number of centres.

Explanation

• If there are only a few hundred points this option is ignored.

• There was no redundant information in the data. That is, all the given points were required as RBF centres to achieve the desired accuracy. This is more likely to happen with true density data.

Remedy

None.

The returned RBF coefficient vector is all zeros.

Explanation All the interpolation values were smaller than the desired accuracy.

Remedy

Use a smaller accuracy.

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Troubleshooting

An evaluation routine returned all zeros.

Explanation Probably the RBF coefficients are all zero, see above.

Remedy

Fit the RBF again with a tighter accuracy.

Fitting accuracy achieved is not as expected Explanation

• The actual fit accuracy achieved at a point is determined by the method and the presence or absence of a global fitting accuracy and per-point bounds such as the Upper, Lower and Accuracy bounds.

• When a global accuracy and a per-point bound are both specified then for all fit methods the greater (i.e., easiest to meet) of the two is used at each point. This behaviour facilitates iterative fitting, i.e., quick initial but coarse fits.

• Only the errorbar fit uses the Upper and Lower data fields. If one of these is specified then it is treated as a symmetric bound applied at each point. If neither is available, but an accuracy per-point field is given, then this is treated as a symmetric bound. Otherwise the ... accuracy per-point field is ignored by the errorbar... fitter. If only a global accuracy is specified then it is applied at all data points as a symmetric bound.

Remedy

1. Check fitting method and present of per-point error bounds and accuracies as well as the global accuracy (if specified).

2. Specifying a global accuracy of zero forces FastRBF to use the per-point bounds.

Fitting reaches max iteration Explanation

• The more widely varied the density of data points, the less likely the fitter is going to succeed (due to worsening in matrix conditioning).

• Passing a large c parameter can also inhibit convergence.

• Highly tracked data can also not perform well.

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Remedy

1. Try applying unique to your data with a relative distance of 1e − 3 or similar. ...

2. If fitting with the multiquadric, try reducing the the c... parameter. A value roughly the same order of magnitude as the average distance between points should work best.

3. If your data is tracked in a uniform direction, scale it so that data points are uniformly spaced.

The progress meter sits in one place for a long time or goes up and down.

• All the fitting algorithms are iterative and it is hard to estimate the rate of convergence in advance.

Explanation

• The progress bar may appear to stall if a previous estimate of progress was too optimistic.

Remedy

None.

Fitting takes a very long time • Insufficient RAM can decrease performance. The RBF fitter requires a lot of memory, about 500Mb is desirable for 500 000 points.

Explanation

• Oversampled data is generally faster to fit since the inherint redundancy results in better convergence and ultimately fewer RBF centres for a given number of data points. The latter results in faster evaluation times during fitting.

• Non-uniformly sampled data (i.e., track data) generally takes longer to fit than uniformly sampled data.

• Specifying an overly tight fitting accuracy can dramatically increase fitting times.

• The errorbar algorithm can be slow when little centre reduction is achieved.

Remedy

1.

2.

3.

4.

Specify a looser fitting accuracy.

Try using the direct option.

Install more RAM. Buy a faster computer.

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Troubleshooting

Fitting reaches max iteration Explanation

• The more widely varied the density of data points, the less likely the fitter is going to succeed (due to worsening in matrix conditioning).

• Passing a large c parameter can also inhibit convergence.

• Highly tracked data can also not perform well.

Remedy

12.3

1. Try applying unique to your data with a relative distance of 1e − 3 or similar. ...

2. If fitting with the multiquadric, try reducing the the c... parameter. A value roughly the same order of magnitude as the average distance between points should work best.

3. If your data is tracked in a uniform direction, scale it so that data points are uniformly spaced.

Isosurfacing

Isosurface has undesirable blobs or pits.

Explanation

• Usually occurs during the DensityFromNormals step when reconstructing surface data. This step is not infallible and can sometimes create inconsistent distanceto-surface data.

• If the blob or pit consists of only one or 2 vertices it could be an aliasing problem (under-sampling).

Remedy

...

1. For surface reconstruction, repeat the DensityFromNormals... step with a shorter normal projection length.

2. Try using the smooth option to remove artifacts from high frequency noise.

No facets produced Explanation

• No valid seeds specified.

• Seed points are either all too far from the surface or outside the bounding box (or within a sampling interval of the bounding box).

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• The iso-value may not occur anywhere in the specified bounding box.

Remedy

1.

2.

3.

4.

Check bounding box dimensions — increase size.

Check the isosurface threshold.

Increase the seedlimit value.

Check number and location of seed points

Long initialisation time Explanation Large number of seeds specified, or all seeds specified (seedlimit = 0). Note that this may not necessarily result in overall processing times being significantly longer.

Remedy

1.

2.

3.

4.

Set a lower seed limit Spatially pre-sort the seeds Reduce the seed list Reduce evaluation accuracy

Too many facets Explanation Resolution parameter too small Remedy

Increase resolution

Slow performance • Algorithm initialised with too many seeds.

Explanation

• RBF gradient evaluation required by normals option can double surfacing time.

• Using the smooth option can increase surfacing time.

Remedy

1.

2.

3.

4.

5.

Set a lower seed limit Spatially pre-sort the seeds & reduce their number Do not specify normals Do not specify smooth Reduce evaluation accuracy

Small changes in surface detail or topology occur when varying the lattice orientation Explanation Changing the lattice orientation changes the physical locations where the function is evaluated. This may affect sur-

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Troubleshooting

face detail and topology at scales close to the sampling resolution.

Remedy

None.

Mesh contains errors, e.g., edges shared by more than 2 faces.

Explanation Finite precision in exported mesh or third party software has caused distinct vertices to become indistinct.

Remedy

1. Export mesh with higher precision 2. Choose a different export format.

3. Check the mesh with the unique function specifying a largish unique distance, (e.g., 0.001 relative distance).

Resolution appears to differ from that specified Explanation

• The default behaviour is to change the resolution slightly to exactly fit inside the specified bounding box.

• The RBF is sampled on a regular tetrahedral√lattice which means the sampling in the x-direction is 2 further than in the y and z directions.

• The resolution may be reduced if the bounding box z dimension divided by the resolution is > 232 .

• There may be an affine component in the lattice orientation matrix affecting the sampling frequency.

Remedy

1. Use the fit option to turn off resolution adjustment 2. Check z dimension of bounding box and resolution 3. Check lattice orientation matrix

Bounding box appears to differ from that specified Explanation

• The specified bounding box was invalid & default values were chosen • The bounding box is specified in the data coordinate system. The effective bounding box is the box in the lattice coordinate system which encloses it.

Remedy

1. Check bounding box arguments 2. Check lattice orientation matrix argument

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Unexpected surface artifacts Explanation Under-sampling often results in unexpected surface details or aliasing effects.

Remedy

Evaluate at a lower resolution value (higher detail)

Missing surface section Explanation Each separate sub-surface requires at least one seed Remedy

1. Increase the seedlimit.

2. Pre-sort the seeds for even spatial coverage.

3. Add appropriate seed(s) in vicinity of the missing surface 4. Make multiple calls to Isosurf with seed(s) in vicinity of the missing surface.

Slow fit with a small data set Explanation

• When a data set is small, FastRBF utilises a direct fit, even when a reduce fit is specified.

• For very small data sets below a few thousand points (< 2000), the benefits of the fast approximations that FastRBF uses are not so apparent and the set-up costs may even result in a fitting time slower than conventional methods.

Remedy

None.

Slow fit with a large data set Explanation The default fitting strategy is the direct fit. It utilises all data points as RBF centres. Consequently memory requirements can be larger and evaluations of intermediate solutions slower than when a centre reduction strategy such as ‘reduce’ or ‘errorbar’ are used. If redundancy exists in the data, relative to the fitting accuracy, then a centre reduction technique may be signifcantly faster. If there is little redunancy in the data, i.e., a reduction factor less than 3 is achievable, then a direct fit will actually be faster.

Remedy

1. Try a ‘reduce’ fit or an ’errorbar’ fit.

2. Large data sets always take some time.

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Appendix A

Direct methods for fitting and evaluating RBFs This appendix contains basic Matlab code to fit and evaluate an RBF to 3D data using the biharmonic basic function, φ(r) = |r|, and direct methods.

Essentially, the FastRBF Toolbox exhibits the same functionality as the following code except that it is very much faster, numerically more robust, and requires far less memory. These properties become obvious for larger data sets, where the number of data points typically exceeds 2000 (in 3D). Above a few thousand points, the solution and evaluation of an RBF by direct methods becomes impossible on most machines. The direct code is presented here for comparison and clarification of what the FastRBF Toolbox does.

A.1

Demonstration code for fitting and evaluating an RBF

% Demonstration file for a simple RBF biharmonic spline % fit in 3D % % n=100; cent=rand(100,3); f=rand(100,1); coeff=fitit(cent,f); fprintf(’Value to be interpolated %15.10e \n’,f(1));

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fprintf(’Value of RBF interpolant %15.10e \n’,... eval_direct(cent,coeff,cent(1,:)));

A.1.1

Direct fit of an RBF

Fitting an RBF involves setting up and solving the following linear system.

A P λ f = , PT 0 c 0 where Ai, j = Φ( xi − x j ), Pi, j = p j ( xi ),

i, j = 1, 2, . . . , N, i = 1, 2, . . . , N,

j = 1, 2, . . . , `,

and the pi ’s are the monomial basis for the polynomial space of p. function [coeff]=fitit(cent,f) % Function to find the radial basis function s consisting % of a linear plus a sum of shifts of \phi( | x | ) % interpolating to data % f_i at the point cent(i,:) for 1 \leq i \leq n. % % Syntax [coeff]=fitit(cent,f) % % Input % % cent n by dim array of centres % f n by 1 vector of values at centres % % Output % coeff (n+dim+1) vector of coefficients with the % coefficients of 1, x and y (2D) or 1, x, % y and z (3D) last. % A=direct(cent); [n dim] =size(cent); f=f(:); % Make sure its a column % add the zeros for the polynomial part % at the end of the column

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Direct methods for fitting and evaluating RBFs

f=[f ;zeros(dim+1,1)]; coeff=A\f;

A.1.2

Forming the interpolation matrix

function [A]=direct(cent) % % Form the (n+dim+1)*(n+dim+1) matrix corresponding to % RBF interpolation with basic function phi and linears. % % phi is assumed given as a function of r. It is coded in % the Matlab function phi.m % % Syntax [A]=direct(cent) % % Input % cent n*dim array coordinates of centres % % Output A (n+dim+1)* Symmetric matrix of the % (n+dim+1) linear system obtained if % we solve for the radial % basis function interpolant % directly. % % Write the matrix A in the form % % B P % A = % P^t O % % where P is the polynomial bit. % [n dim]=size(cent); A=zeros(n,n); for i=1:n for j=1:i r=norm(cent(i,:)-cent(j,:));

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temp=phi(r); A(i,j)=temp; A(j,i)=temp; end end % % Now the polynomial part % P=[ones(n,1) cent]; A = [ A P P’ zeros(dim+1,dim+1)];

A.1.3

Basic function

Here we consider the biharmonic basic function, φ(r) = |r|. function u=phi(r) % % Function phi of r % % Syntax [u] = phi(r) % % Remember if using something like the thin-plate spline % in 2D % u = r^2 * log(r) % you will need to test for r positive before taking % the log. u = r;

A.1.4

Direct evaluation

Evaluating the RBF s( x), where N

s ( x ) = p ( x ) + ∑ λi Φ ( x − xi ) . i =1

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Direct methods for fitting and evaluating RBFs

function s=eval_direct(cent,coeff,u) % % Function which directly evaluates the radial basis % function spline % % s(u) = linear polynomial + % \sum_i=1^n coeff(i) * phi(u - cent(i,:)) % % at the point u where the linear polynomial is given in % terms of its monomial coefficients. % % Syntax s=eval_direct(cent,coeff,u) % % Inputs % cent n by dim array The coordinates of the centres % coeff n+dim+1 vector Coefficients with the linear % polynomial part last. % u row vector Point at which to evaluate % % Output % s Value of the RBF at position u % % Code is written for clarity rather than Matlab % efficiency. [n dim] = size(cent); s=0; for j=1:n s = s + coeff(j)*phi( norm(u - cent(j,:)) ); end % Now the linear polynomial bit s=s+coeff(n+1); % The constant for i=1:dim s=s+coeff(i+n+1)*u(i); % The various components of u end

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Glossary Approximation Fitting a function which does not exactly reproduce the given data values but is ‘close’ to most of the data. Sec 1.1.3 p7.

Alt array An array of attribute values associated with a pointlist or a grid data type in FastRBF. Sec 5.1 p49.

Anti-aliasing Low pass filtering an RBF to avoid aliasing (under-sampling) artifacts during evaluation. See low pass filter. Sec 6.3 p74.

Basic Function A real-valued radially symmetric function φ(r), r ∈ R+ . Sec 1.2 p9.

Bi-harmonic spline (basic function) Also known as the thin plate spline, it minimises the energy in the 2nd derivative (Equation (1.5) in 3D). In 2D the biharmonic basic function is φ(r) = r2 ln r and in 3D φ(r) = r. Sec 1.2 p9.

Bounding box The region of interest over which an RBF or an iso-surface is evaluated. Sec 1.2 p9.

Centre reduction Fitting an RBF to the desired accuracy with fewer centres than the number of data points. Sec 1.3 p12.

Confidence A FastRBF parameter that specifies the fraction of fitting constraints (accuracies and error bounds) associated with a data set which must be achieved.

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Relaxing those fitting constraints which are most difficult to achieve can result speed increases and robustness to outliers. A typical value range is 0.98 to the default value, 1.0, which enforces all fitting constraints. Sec 1.1.4 p9.

Direct fit The standard (default) FastRBF method for fitting an RBF to a set of data where every data point contributes a centre to the RBF and the RBF interpolates the data. Sec 1.3 p12.

Direct methods Conventional solution of the RBF interpolation system of equations (Equation (1.3)) without using FastRBF approximations. Sec 1.1.2 p7, Appendix A p175.

Error-bar fit More correctly known as the smoothest restricted range fit, it finds the smoothest function which passes between the upper and lower bounds placed on a data point’s value. It can be shown that the smoothest function in this case is also an RBF. Only those bounds which constrain the RBF contribute centres to the RBF. This can result in significant data reduction. The maximum permitted error at occurs at those data points which contribute RBF centres when this method is used. Very effective for modelling many types of laser surface scan data. Sec 1.5 p15, Sec 6.4.2 p79, Sec 7.4 p96.

Evaluation The process of evaluating an RBF s( x) at the point x. Sec 1.2.2 p10.

Evaluation accuracy The precision with which the FastRBF evaluator determines the value of the RBF s( x), i.e., the maximum difference between the returned value and the true value of the RBF. Sec 1.1.3 p7.

Fitting The process of finding the coefficients λi in Equation (1.1) which determine an RBF s( x), i.e., solving the linear system of Equation (1.3). Sec 1.1 p6.

Fitting accuracy The maximum difference permitted between the value at a data point (interpolation node) and the value of a fitted RBF. Sec 1.1.3 p7.

Greedy algorithm The process by which centres are incrementally added to an RBF when fitting an RBF with the centre reduction (reduce) option. Sec 1.3 p12.

Low Pass Filtering (LPF) A smoothing technique applied during evaluation of an RBF which attenuates high frequencies in the RBF. Also referred to as an

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anti-aliasing filter. Sec 1.4 p14, Sec 6.3 p74, Sec 8.1.8 p113.

Implicit surface model Representing a surface as the level set (iso-surface) of a 3D function. Sec 2 p21, Capt. 7 p87.

Interpolation Fitting a function which reproduces the given data values exactly.

Interpolation node A data point which an RBF must pass through.

Iso-surfacing The process of extracting a polygonal representation of the level set of a function in R3 , that is the set of points where the function has constant value. Sec 3 p29, Capt. 8 p103.

Marching Cubes (mcubes) An alternative method to the FastRBF surface-following isosurfacer that requires evaluation of an RBF on a full grid.

Sec 8.1.7 p112.

Multiquadric (basic function) The multiquadric basic function is equivalent to convolving the appropriate bi-harmonic basic function with a smoothing kernel. Consequently, the multiquadric is used in FastRBF to implement low pass filtering during RBF evaluations. Generalised form: Ď&#x2020;(r) = (r2 + c2)k in 2D and (r2 + c2)k k odd in 3D. Sec 1.2 p9, Sec 1.2 p9.

Nyquist sampling limit The Nyquist sampling criterion says that aliasing artifacts will be avoided if a signal is sampled at a rate at least twice the highest frequency present in the signal. This is relevant when evaluating an RBF at a particular sampling interval/resolution. Sec 1.4 p14, Sec 6.3 p74.

Pointlist The basic FastRBF data type: an array of points where a point consists of a location and associated attributes. Sec 5.1 p49, Capt. 9 p125.

Polyharmonic spline The family of energy minimising splines that includes the thin plate spline and the bi-harmonic and tri-harmonic basic functions used in FastRBF. Sec 1.1 p6.

RBF â&#x20AC;&#x201C; Radial Basis Function a weighted sum of a shifted, radially symmetric basic function (Equation (1.1)). Sec 1.1 p6.

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RBF centre An RBF consists of a weighted sum of basic functions shifted from the origin to lie at various locations. These locations are called RBF centres.

Sec 1.1 p6.

Seed points The points from which surface-following is initiated when the FastRBF iso-surfacer is called. Sec 3.1 p30, Sec 8.1.5 p108.

Smoothest interpolant A function which passes through the given data points and minimises some energy functional, e.g., the thin plate spline minimises the integral of the 2nd derivative (Equation (1.5)). Sec 1.2.3 p11.

Smoothest restricted range fit see error-bar fit.

Spline smoothing A traditional method of fitting an RBF to noisy data which achieves a smoother function by approximating the given data values. Sec 1.5 p15, Sec 1.5.3 p17.

Surface-following The method which the FastRBF isosurfacer uses to efficiently extract iso-surfaces. Recommended for generating meshes from RBF implicit surface models. Sec 3 p29, Sec 8.1 p103.

Thin plate spline A name given to the biharmonic family of splines which minimise the energy functional Equation (1.5), the integral of the 2nd derivative.

Sec 1.2 p9, Sec 1.2.3 p11.

Tri-harmonic spline (basic function) Minimises the energy in the 3rd derivative. In 2D the triharmonic basic function is Ď&#x2020;(r) = r4 ln r in 2D and Ď&#x2020;(r) = r3 in 3D. Sec 1.2 p9.

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