3D Fourier transforms

Page 1

2144

J. Opt. Soc. Am. A / Vol. 27, No. 10 / October 2010

Natalie Baddour

Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates Natalie Baddour Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada (nbaddour@uottawa.ca) Received July 12, 2010; accepted August 12, 2010; posted August 12, 2010 (Doc. ID 131538); published September 13, 2010 For functions that are best described with spherical coordinates, the three-dimensional Fourier transform can be written in spherical coordinates as a combination of spherical Hankel transforms and spherical harmonic series. However, to be as useful as its Cartesian counterpart, a spherical version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the spherical version of the standard Fourier operation toolset. In particular, convolution in various forms is discussed in detail as this has important consequences for filtering. It is shown that standard multiplication and convolution rules do apply as long as the correct definition of convolution is applied. © 2010 Optical Society of America OCIS codes: 070.6020, 070.4790, 350.6980, 100.6950.

1. INTRODUCTION The Fourier transform has proved to be a powerful tool in many diverse disciplines and indispensable to signal processing. One of its powerful features is it can easily be extended to n dimensions. The strength of the Fourier transform is that it is accompanied by a toolset of operational properties that simplify the calculation of more complicated transforms through the use of these standard rules, turning otherwise complex calculations into those that can be done via a look-up table and a core set of transforms. Specifically, the standard Fourier toolset consists of results for translation (spatial shift), multiplication, and convolution, along with the basic transforms of the Dirac-delta function and complex exponential. This basic toolset of operational rules is well known for the regular Fourier transform in single and multiple dimensions [1,2]. The convolution rules are particularly important as they form the basis of most filtering operations. As is also known, the Fourier transform in three dimensions can be developed in terms of spherical polar coordinates [3], most usefully when the function being transformed has some underlying spherical symmetry. For example, this has seen application in the field of photoacoustics [4] and some attempts to translate ideas from continuous to discrete domain [5]. The primary inspiration behind this paper actually lies in the Fourier diffraction theorem [6] of acoustic tomography, versions of which also appear in other imaging modalities [7]. The Fourier diffraction theorem relates the Fourier transform of the forward scattered acoustic field to the value of the Fourier transform of the object on a circular [two-dimensional (2D)] or a spherical [three-dimensional (3D)] arc. The fact that the scattered field is related to the Fourier transform on an arc is one of the primary motivators to switch the formulation of tomographic problems to curvilinear coor1084-7529/10/102144-12/$15.00

dinates. The development of this operational toolset was thus motivated by the desire to write various (acoustic, thermal, and photoacoustic) tomographic problems in curvilinear coordinates, while no Fourier toolset to aid in this formulation could be found in the literature. A complete interpretation of the standard Fourier operational toolset in terms of 2D polar coordinates has been developed in [8], and the equivalent toolset for the 3D transform in spherical coordinates is still missing from the literature. Thus, the aim of this paper is to develop this missing toolset for the 3D Fourier transform in spherical coordinates. Some results are already known, but the results on shift, multiplication, and in particular convolution are incomplete. Therefore, we feel that it is worthwhile to present a complete, detailed, and unified account of the 3D curvilinear toolset for archival purposes, describing the mathematical foundation underlying all results. What is to the author’s knowledge of particular novelty in this paper is the treatment of shift, multiplication, and convolution. It is known that 3D Fourier transforms for radially (spherically) symmetric functions can be interpreted in terms of a (zeroth order) spherical Hankel transform. It is also known that Hankel transforms do not have a multiplication/convolution rule, a rule which has found so much use in the Cartesian version of the transform. In this paper, the multiplication/convolution rule is treated in detail for the curvilinear version of the transform, and in particular it is shown that the spherical Hankel transform does obey a multiplication/convolution rule—once the proper interpretation of convolution is applied. This paper carefully considers the definition of convolution and derives the correct interpretation of this in terms of the curvilinear coordinates so that the standard multiplication/convolution rule is again applicable. Con© 2010 Optical Society of America


Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.
3D Fourier transforms by Natalie Baddour - Issuu