Curvature for Fourier Heat Transformation Method: A Shape based Object Recognition

Studies in System Science (SSS) Volume 2, 2014 www.as‐se.org/sss

Curvature for Fourier Heat Transformation Method: A Shape based Object Recognition M. Radhika Mani1, Dr. G.P.S. Varma2, Dr. Potukichi DM3, Dr. Ch. Satyanarayana*4 1

Pragati Engg. College, Surampalem, A.P., India‐533437

2

SRKR Engg. College, Bhimavaram, A.P., India‐534204

3

JNT University Kakinada, Kakinada, A.P., India‐533003

4

JNT University Kakinada, Kakinada, A.P., India‐533003

1

radhika_madireddy@yahoo.com; 2gpsvarma@yahoo.com; 3potukuchidm@yahoo.com; chsatyanarayana@yahoo.com

*4

Abstract In Shape based Object Recognition processes, heat transformations are widely used. Elegance for the computation of it with Fourier transformation is presented. Generic involvement of Cartesian coordinates for the computation is discussed. A novel method that utilizes the features of curvature of the object in the computation of heat transformation, CFHT is proposed. An invariant feature vector is generated with the help of Fourier transformation. New distance measure is used for the computation of shape similarity. The proposed CFHT method is testified over MPEG CE‐1 set B database. The results and the analysis infer relative efficiency for the present method than those with normal Cartesian coordinate method.

Keywords Polar Coordinates; Distance Measure; Heat Kernel; Feature Vector; Fourier Transform

Introduction Shape based methods is widely used in the image processing and pattern recognition process recently [1]. The shape based approach consists of different stages viz., shape representation, shape description and shape toning. The shape representation stage identifies the effective representative points of an object. There are two categories of shape representation methods viz., boundary based and region based methods [2, 3]. Moment Invariants [4], Zernike Moments [5, 6] , Krawtchouk moments [7], Chebyshev moments [8, 9] represent some other types of popular moments used for region based descriptors. However, the region based shape representation methods involves the procedures as the medial axis transform [10, 11, 12], grid method [13], and generic Fourier transform [14] etc.

Recently, the inelastic deformations that involved with the shape of an object are described by geodesic geometry [15, 16, 17] , inner distances with dynamic programming [18, 19], Laplacian transforms [20, 21] and heat diffusion [22, 23]. Currently, the heat flow equation [24, 25] is successfully applied in different fields. The diffusion process is tested in a more generalized perspective which can be connected to other analogus processes. Nevertheless, Fourier transformation can efficiently describe [26] the intrinsic features of the given object. Based on the shape representation points, shape signature [26, 27] would be constructed and it can be used as an input to the Fourier transformation to generate the shape characteristic feature vector.

In the wake of the necessity (i) to conceive the fineness of shape information mediated by details of curvature information and (ii) overcome the problem of mis coverage finer detail information of sharp bends; the existing heat equation based methods needs a more efficient, but simpler method which includes the curvature of the object.

Hence, the authors intend to propose a novel curvature based Fourier Heat Transformation method for Shape based Object Recognition. The paper is organized in 3‐sections. Introduction to heat equation based shape description methods is briefed in Section‐I. The details of methodology implemented presently and the relevance is detailed in section‐II. Results of implementing the novel algorithm designed by curvature approach to the standard databases and their analysis with relevant discussion is presented in Section‐III. Methodology An innovative Curvature based Fourier Heat Transformation (CFHT) method for object recognition process is presently proposed. The proposal consists of 4 successive steps viz., (i) Shape representation with contour (ii) CFHT construction (iii) Shape description by using Fourier Transform and (iv) Shape similarity and ranking, respectively.

The first step involves the contour representation of the shape surface. Surface is sampled by a finite number of

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representation points by Equal Arc Length sampling (EAL) [26] method. Let f(x) be a continuous signal which decays sufficiently fast at infinity so that its Fourier transform exists. Then, the solution to the one‐dimensional heat equation satisfying the initial condition u(x, 0) = f(x) is shown by (1). u ( x, t ) 



 K ( x, y; t ) f ( y)dy (1)



Where, f(y) is the Euclidean distance of a point in polar coordinate and k(x,y;t) is defined by (2) K ( x, y; t ) 

1

 x  y / 4t e     (2) 2

4 t

Where, x and y are representing the Cartesian coordinates of a point and t represents time.

In the second step, the proposed CFHT method uses f(y) as the curvature of the object. During the third stage, the Fourier transformation is used to derive the feature vector in the proposed system. The Fourier transformation is shown by (3).

FDn



1 N

  - j 2 nt    N  (3)

N 1

 s (t )  e

t 0

In order to improve the quality of proposed method, the proposed signature feature vector is further augmented by global descriptors (GD). The GD feature vector {S, C, A} contains the measures of solidity, circularity and aspect ratio.

The last step of object recognition involves shape toning process, which is comprised of the similarity measurement of test object feature vector with that of the training sample feature vector. Generally, distance measures are used to evaluate the similarity. Presently, a new distance measure D is proposed as shown in Equation (4).

M

eTE TR

i 0

TEi  TRi 6

D(TE , TR )  

i

i

(4)

Where TE represents the test shape feature vector, TR represents the trained shape feature vector and M represents the length of the feature vector. Among various performance measues, the precision and recall are considered to be prevalently adopted measures; Further, they can quantify the similarity measurement. The Precision (P) and Recall (R) measures are defined by: P  R 

x (5) y

x (6) groupsize

where x denotes the true recognition results, y denotes the total recognized result and groupsize denotes the maximum true recognition result. The Average Precision for Low Recall (APLR) denotes the average precision for recalls less than or equal to 50. In the contrary, the Average Precision for High Recall (APHR) represents the average precision for recalls greater than 50. The proposed shape descriptor is compared with the four standard descriptors viz., Angular Radial Transform Descriptor (ARTD) [28], Moment Invariant Descriptor (MID) [28], Zernike Moment Descriptor (ZMD) [29] and Curvature‐Scale‐Space‐Descriptor (CSSD) [29] to estimate the efficiency. The performance for the present curvature based technique is compared with the feature sizes of 35 for ARTD (n<3, m<12), 6 for MID, and 34 for ZMD (order ranges from 2 to 10). But, the CSSD feature size is varied from one image to another, because of the apparent variation in number of the peaks.

Results and Discussion The proposed Curvature based Fourier Heat Transformation (CFHT) is applied on the MPEG CE‐1 Set B database [30].

During the first stage, the contour of the object shape is extracted and by using EAL method. The number of representative points is fixed to 128. Then, during the second stage, the CFHT signature is constructed with consideration of the angular or curvature measurement of each of the representative point of the shape. In the third stage, the Fourier transformation is applied on the proposed signature and the first six coefficients is used to form the feature vector of the proposed descriptor. The proposed method feature vector is extended by the GD features.

During the final stage, the proposed new distance measure is used for shape similarity and ranking. The proposed

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descriptor performance is measured by APLR and APHR values.

The performance measure of proposed method is compared to different popular methods by Table‐ 1. From the table 1, it is clear that the CFHT+GD is giving better performance, when compared with other standard descriptors such as ARTD, MID, ZMD, and CSSSD. The CFHT+GD descriptor is found to yield increased performance measure for the APHR than APLR. The trends of Precision‐Recall (PR) for these 5 descriptors evaluated for set B are presented in Figure‐1. Figure‐1 reveals that all the six descriptors are yield considerable enhancement to the precision measure on the Set B database, than especially correspond to the low recalls. However, at high recalls, the present descriptor is found to result for an improved precision measure when compared with other standard descriptors. It is also noticed that the proposed descriptor is found to increase the precision measure marginally at low recalls i.e. for less than or equal to 50. The proposed descriptor has a significant increase in the precision measure at high recalls i.e. for greater than 50.

TABLE 1 APLR AND APHR RESULTS OF VARIOUS DESCRIPTORS USING SET B DATABASE.

Descriptor

APHR

APLR

CFHT+GD

83.43

46.04

Average 64.74

ARTD

82.10

45.69

63.90

MID

79.54

44.50

62.02

ZMD

82.56

45.62

64.09

CSSD

78.61

41.81

60.21

FIG. 1 PR GRAPH FOR VARIOUS DESCRIPTORS CORRESPONDING TO SET B DATABASE

For the Set‐B database, the accuracy for the recognized results corresponding to the top 20 images is illustrated in Figure 2. Among various standard descriptors, the ZMD seems to be superior to other descriptors. Hence, it is imperative that the recognition result of proposed descriptor is compared with the ZMD result. The Figure 2(a) illustrates the results corresponding to Teddy18 image from Set Bwhich used as the query image. The Figure 2(b) potrays the recognition result of ZMD and Figure 2(c) provides the recognition result of proposed descriptor. The recognition rate for dissimilar images in recognition result of ZMD is reduced in the recognition result of proposed descriptor.

(a) (b) (c)

FIG. 2 RETRIEVAL RESULTS OF TEDDY18 QUERY IMAGE FROM SET B DATABASE (a) QUERY IMAGE (b) ZMD RECOGNITION RESULTS (c) PROPOSED CFHT+GD RECOGNITION RESULTS.

Conclusion 

A simple Curvature based Fourier Heat Transfer method involving polar coordinate transformation of its surface can be effective tool for the shape representation and its description.

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

Inclusion of angular variables for the object recognition technique enhances the performance.

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