IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 31, NO. 9, SEPTEMBER 2009 1715 Rotational Invariance Based on Fourier coordinates. It would be of great advantage if the image can be Analysis in Polar and Spherical Coordinates decomposed into wave-like basic patterns that have clear radial and angular structures. Ideally, this decomposition should be an extension of the normal Fourier analysis and can, therefore, be Qing Wang, Olaf Ronneberger, and called Fourier analysis in the corresponding coordinates. To fulfill Hans Burkhardt, Member, IEEE these requirements, the basis functions should take the separation- of-variable form Abstract—In this paper, polar and spherical Fourier analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the RðrÞÈð’Þð2Þ eigenfunctions being separable in the corresponding coordinates. The proposed for 2D and transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures. The theory is compactly presented with an RðrÞÂð#ÞÈð’Þ¼RðrÞ ð#; ’Þð3Þ emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and the normal Fourier transform is for 3D, where ðr; ’Þ and ðr; #; ’Þ are the polar and spherical explored. As examples of applications, rotation-invariant descriptors based on coordinates, respectively. They should also be the eigenfunctions polar and spherical Fourier coefficients are tested on pattern classification of the Laplacian so that they represent wave-like patterns and that problems. the associated transform is closely related to the normal Fourier transform. The concrete form of the angular and radial parts of the Index Terms—Invariants, Fourier analysis, radial transform, multidimensional. basis functions will be investigated and elaborated in the coming Ç sections but will be briefly introduced below in order to show previous work related to them. 1INTRODUCTION For polar coordinates, the angular part of a basis function is NOTHING needs to be said about the importance of Fourier given by transform in image processing and pattern recognition. Usually, 1 Fourier transform is formulated in Cartesian coordinates, where a Èð’Þ¼pffiffiffiffiffiffi eim’; ð4Þ separable basis function in 3D space without normalization is 2 given by where m is an integer. The associated transform in angular coordinate is just the normal 1D Fourier transform. For spherical eikÁr ¼ eikxxeikyyeikz z; ð1Þ coordinates, the angular part is a spherical harmonic where ðx; y; zÞ are coordinates of the position r and ðkx;ky;kzÞ are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi components of the wave vector k along the corresponding axis. 2l þ 1 ðl À mÞ! im’ ð#; ’Þ¼Ylmð#; ’Þ¼ Plmðcos #Þe ; ð5Þ The basis function (1) represents a plane wave, which is a periodic 4 ðl þ mÞ! pattern. Fourier analysis of an image is, therefore, the decomposi- P l m tion of the image into the basic patterns represented by (1). where lm is an associated Legendre polynomial and and are The Laplacian is an important operator in mathematics and integers, l 0 and jmjl. The corresponding transform is called physics. Its eigenvalue problem gives the time-independent wave Spherical Harmonic (SH) transform and has been widely used in equation. In Cartesian coordinates, the operator is written as representation and registration of 3D shapes [6], [7], [8]. The angular parts of the transforms in 2D and 3D are, therefore, @2 @2 @2 very familiar. Not so well known are the transforms in the radial r2 ¼r2 þr2 þr2 ¼ þ þ : x y z @x2 @y2 @z2 direction. The radial basis function is a Bessel function JmðkrÞ for polar coordinates and a spherical Bessel function jlðkrÞ for spherical Equation (1) is an eigenfunction of the Laplacian and is separable coordinates. In both cases, the parameter k can take either in Cartesian coordinates. continuous or discrete values, depending on whether the region is When defined on the whole space, functions given in (1) are infinite or finite. For functions defined on ð0; 1Þ, the transform with mutually orthogonal for different k; wave vectors take contin- JmðkrÞ as integral kernel and r as weight is known as the Hankel uous values and it is said that one has a continuous spectrum. transform. For functions defined on a finite interval, with zero-value Over finite regions, the mutual orthogonality generally does not boundary condition for the basis functions, one gets the Fourier- hold. To get an orthogonal basis, k can only take values from a Bessel series [1]. Although the theory on Fourier-Bessel series has discrete set and the spectrum becomes discrete. The continuous long been available, it, mainly, has applications in physics-related Fourier transform reduces to Fourier series or to the discrete areas [16], [17]. Zana and Cesar’s work [10] and a few references Fourier transform. therein are the only ones we can find that employ Fourier-Bessel For objects with certain rotational symmetry, it is more effective series expansion for 2D image analysis. Methods based on Zernike for them to be investigated in polar (2D) or spherical (3D) moments are, on the other hand, much more popular in applications where we believe the Fourier-Bessel expansion also fits. The SH transform works on the spherical surface. To describe . The authors are with the Computer Science Department and the Centre for Biological Signalling Studies, University of Freiburg, Chair of Pattern 3D volume data, one can use a collection of SH features calculated on Recognition and Image Processing, Georges-Koehler-Allee Geb. 052, concentric spherical surfaces of different radii, as suggested in [7]. D-79110 Freiburg, Germany. This approach treats each spherical surface as independent to one E-mail: {qwang, ronneber, hans.burkhardt}@informatik.uni-freiburg.de. another; therefore, it cannot describe the radial structures effec- Manuscript received 27 Sept. 2007; revised 10 June 2008; accepted 12 Jan. tively. This observation motivated the whole work presented here. 2009; published online 23 Jan. 2009. In this paper, the operations that transform a function into the Recommended for acceptance by W. Fo¨rstner. coefficients of the basis functions, given in (2) and (3) and For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number described above, will simply be called polar and spherical Fourier TPAMI-2007-09-0646. transform, respectively. It should be noted though that, in the Digital Object Identifier no. 10.1109/TPAMI.2009.29. literature, the former often refers to the normal Fourier transform 0162-8828/09/$25.00 ß 2009 IEEE Published by the IEEE Computer Society 1716 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 31, NO. 9, SEPTEMBER 2009 with wave vectors expressed in polar coordinates [14] and the The solution to (9) is simply latter often refers to the SH transform [15]. 1 im’ Due to the extreme importance of the Laplacian in physics, the Èmð’Þ¼pffiffiffiffiffiffi e ; ð11Þ expansion of functions with respect to its eigenfunctions is 2 naturally not new there (e.g., [18], [19]). The idea that these with m being an integer. eigenfunctions can be used as basis functions for analyzing 2D or 3D images is unfamiliar to the pattern recognition society. There 2.1.2 Radial Basis Functions also lacks a simple and systematic presentation of the expansion The general nonsingular solution to (10) is from the point of view of signal analysis. Therefore, although part RðrÞ¼J ðkrÞ; ð12Þ of the derivation is scattered in books like [1], we rederive the basis m functions to emphasize the analogy to the normal Fourier trans- where Jm is the mth order Bessel function. Bessel functions satisfy form. Employment of the Sturm-Liouville theory makes this the orthogonality relation analogy clearer and the derivation more compact. Z 1 1 The proposed polar and spherical Fourier transforms are Jmðk1rÞJmðk2rÞrdr ¼ ðk1 À k2Þ; ð13Þ connected with the normal Fourier transform by the Laplacian. 0 k1 We investigate the relation between them so that one can under- just like the complex exponential functions satisfy stand the proposed transforms from another point of view. It is Z 1 Âà found that the relation also provides computational convenience. ik1x ik2x à e e dx ¼ 2ðk1 À k2Þ: ð14Þ An advantage of the polar and the spherical Fourier transforms is 1 that rotation-invariant descriptors can be very easily extracted from Actually, J ðkrÞ with k 0 forms an orthogonal basis for functions the transform coefficients. We will show how to do this. m defined on ð0; 1Þ. Section 2 deals with the polar Fourier transform. Besides For the normal Fourier transform, an infinite space corre- presentation of the theory, issues about calculation of the sponds to a continuous spectrum and a finite space to a discrete coefficients are discussed. A short comparison between polar spectrum, which is selected by proper boundary conditions. The Fourier basis functions and Zernike functions is made at the end. same also applies to the radial basis functions (12). Over the finite Parallel to Section 2, the theory for the spherical Fourier interval ð0;aÞ, the orthogonality relation, like in (13), generally transform is summarized in Section 3. As examples for the does not hold any more, instead, application of the theory, rotation-invariant descriptors based on Z the polar and spherical Fourier coefficients are applied to object a J k r J k r rdr classification tasks in Section 4. At the end, conclusion and mð 1 Þ mð 2 Þ 0 ÂÃð15Þ outlook are given. a 0 0 ¼ 2 2 k2Jmðk1aÞJmðk2aÞk1Jmðk2aÞJmðk1aÞ : To avoid the ideas being hidden by too much mathematics, the k1 À k2 derivation and theories are only outlined in this paper. Those who are interested in more details are referred to [23].