Fourier Analysis in Polar and Spherical Coordinates

1 ALBERT-LUDWIGS-UNIVERSIT AT FREIBURGINSTITUT F UR INFORMATIKL ehrstuhl f ur Mustererkennung und BildverarbeitungFourier Analysis in Polar and SphericalCoordinatesInternal Report 1/08 Qing Wang, Olaf Ronneberger, Hans BurkhardtFourier Analysis in Polar and SphericalCoordinatesQing Wang, Olaf Ronneberger, Hans BurkhardtAbstractIn this paper, Polar and Spherical Fourier Analysis are defined as thedecomposition of a function in terms of eigenfunctions of the Laplacianwith the eigenfunctions being separable in the corresponding eigenfunction represents a basic pattern with the wavenumber in-dicating the scale. The proposed transforms provide an effective radialdecomposition in addition to the well-known angular decomposition. Thederivation of the basis functions is compactly presented with an emphasison the analogy to the normal Fourier transform. The relation betweenthe Polar or Spherical Fourier transform and normal Fourier transform isexplored.

2 Possible applications of the proposed transforms are IntroductionFourier transform is very important in image processing and pattern recognitionboth as a theory and as a tool. Usually it is formulated in Cartesian Coordinates ,where a separable basis function in 3D space without normalization iseik r=eikxxeikyyeikzz(1)where (x, y, z) are Coordinates of the positionrandkx,ky,kzare componentsof the wave vectorkalong the corresponding axis. The basis function (1)represents a plane wave. Fourier Analysis is therefore the decomposition of afunction into plane waves. As the basis function is separable inx,yandz, Thedecomposition can be understood as being made up of three decompositions (for3D).The Laplacian is an important operator in mathematics and physics. Itseigenvalue problem gives the time-independent wave equation. In Cartesiancoordinates the operator is written as 2= 2x+ 2y+ 2z= 2 x2+ 2 y2+ 2 3D space.

3 (1) is an eigenfunction of the Laplacian and is separable in Carte-sian defined on the whole space, functions given in (1) are mutually or-thogonal for differentk; wave vectors take continuous values and it is said thatone has a continuous spectrum. Over finite regions, the mutual orthogonalitygenerally does not hold. To get an orthogonal basis,kcan only take values from1a discrete set and the spectrum becomes discrete. The continuous Fourier trans-form reduced to Fourier series expansion (with continuous spatial Coordinates )or to the discrete Fourier transform (with discrete spatial Coordinates ).For objects with certain rotational symmetry, it is more effective for them to beinvestigated in Polar (2D) or Spherical (3D) Coordinates . It would be of greatadvantage if the image can be decomposed into wave-like basic patterns thathave simple radial and angular structures, so that the decomposition is madeup of radial and angular decompositions.

4 Ideally this decomposition should bean extension of the normal Fourier Analysis and can therefore be called Fourieranalysis in the corresponding Coordinates . To fulfill these requirements, thebasis functions should take the separation-of-variable form:R(r) ( )(2)for 2D andR(r) ( ) ( ) =R(r) ( , )(3)for 3D where (r, ) and (r, , ) are the Polar and Spherical Coordinates respec-tively. They should also be the eigenfunctions of the Laplacian so that theyrepresent wave-like patterns and that the associated transform is closely relatedto the normal Fourier transform. The concrete form of the angular and radialparts of the basis functions will be investigated and elaborated in the comingsections but will be briefly introduced below in order to show previous workrelated to Polar Coordinates , as will be shown in the next section, the angular partof a basis function is simply ( ) =1 2 eim (4)wheremis an integer, which is a natural result of the single-value requirement: ( ) = ( + 2 ), a special kind of boundary condition.

5 The associated trans-form in angular coordinate is nothing else but the normal 1D Fourier Spherical Coordinates , the angular part of a basis function is a Spherical har-monic ( , ) =Ylm( , ) =s2l+ 14 (l m)!(l+m)!Plm(cos )eim (5)wherePlmis an associated Legendre polynomial andlandmare integers,l 0and|m| l. It also satisfies the single-value requirement. The correspondingtransform is called Spherical Harmonic (SH) transform and has been widelyused in representation and registration of 3D shapes [8 10].The angular parts of the transforms in 2D and 3D are therefore very so well-known are the transforms in the radial direction. The radial basisfunction is a Bessel functionJm(kr) for Polar Coordinates and a Spherical Besselfunctionjl(kr) for Spherical Coordinates . In both cases, The parameterkcantake either continuous or discrete values, depending on whether the region isinfinite or finite.

6 For functions defined on (0, ), the transform withJm(kr) asintegral kernel andras weight is known as the Hankel transform. For functions2defined on a finite interval, with zero-value boundary condition for the basisfunctions, one gets the Fourier -Bessel series [1]. Although the theory on Fourier -Bessel series has long been available, it mainly has applications in physics-relatedareas [18, 19]. [12] and a few references therein are the only we can find thatemploy Fourier -Bessel series expansion for 2D image Analysis . Methods basedon zernike moments are on the other hand much more popular in applicationswhere we believe the Fourier -Bessel expansion also fits. The zernike polynomialsare a set of orthogonal polynomials defined on a unit disk, which have the sameangular part as (4).The SH transform works on the Spherical surface. When it is used for 3 Dvolume data, the SH features (extracted from SH coefficients) can be calculatedon concentric Spherical surfaces of different radii and be collected to describean object, as suggested in [9].

7 This approach treats each Spherical surface asindependent to one another and has a good localization nature. it fails to de-scribe the relation of angular properties of different radius as a whole, thereforecannot represent the radial structures effectively. The consideration of how todescribe the radial variation of the SH coefficients actually motivated the wholework presented this paper, the operations that transform a function into the coefficients ofthe basis functions given in (2) and (3) and described above will simply be calledpolarandspherical Fourier transformrespectively. It should be noted thoughthat in the literature, the former often refers to the normal Fourier transformwith wave vectorskexpressed in Polar Coordinates (k, k) [16] and the latteroften refers to the SH transform [17].Due to the extreme importance of the Laplacian in physics, the expansionof functions with respect to its eigenfunctions is naturally not new there.

8 Forexample, in [20] and [21], the eigenfunctions of the Laplacian are used for expan-sion of sought wave functions. The idea that these eigenfunctions can be usedas basis functions for analyzing 2D or 3D images is unfamiliar to the patternrecognition society. There also lacks a simple and systematic presentation of theexpansion from the point of view of signal Analysis . Therefore, although partsof the derivation are scattered in books like [1], we rederive the basis functionsto emphasize the analogy to the normal Fourier transform. Employment ofthe Sturm-Liouville theory makes this analogy clearer and the derivation proposed Polar and Spherical Fourier transforms are connected with thenormal Fourier transform by the Laplacian. We investigate the relations betweenthem so that one can understand the proposed transforms more completely anddeeply. It is found that the relations also provide computational advantage of the proposed transforms is that when a function is rotatedaround the origin, the change of its transform coefficients can be relativelysimply expressed in terms of the rotation parameters.

9 This property can, onthe one hand, be used to estimate rotation parameters, on the other hand, beused to extract rotation-invariant descriptors. We will show how to do 2 deals with the Polar Fourier transform. Besides presentation ofthe theory, issues about calculation of the coefficients are discussed. A shortcomparison between Polar Fourier basis functions and zernike functions is madeat the end. Parallel to section 2, the theory for the Spherical Fourier transformis given in section 3. In section 4 we investigate the possible applications of the3polar and Spherical Fourier transforms. At the end, conclusion and outlook Polar Fourier Basis Helmholtz Equation and Angular Basis FunctionsAs a direct extension from the Cartesian case, we begin with the eigenfunctionsof the Laplacian, whose expression in Polar Coordinates is given by: 2= 2r+1r2 2 (6)where 2r=1r r r r (7)and 2 = 2 2.

10 (8)are the radial and angular parts. The eigenvalue problem can be written as 2r (r, ) +1r2 2 (r, ) +k2 (r, ) = 0,(9)which is the Helmholtz differential equation in Polar Coordinates . We requirethatk2 0 as with negativek2, the radial functions are exponentially growingor decaying, which are not interesting for our purpose. It will be shown later thatsuch a requirement does not prevent the eigenfunctions from forming a simplicity, it is further required thatk 0. Substituting the separation-of-variable form (r, ) =R(r) ( ) into (9), one gets 2 2 +m2 = 0(10)1r r r r R+ k2 m2r2 R= 0.(11)The solution to (10) is simply m( ) =1 2 eim (12)withmbeing an Radial Basis FunctionsThe general solution to (11) isR(r) =AJm(kr) +BYm(kr)(13)4whereJmandYmare them-th order Bessel functions and Neumann functionsrespectively [1];AandBare constant multipliers. A nonsingular requirementofRat the origin leavesR(r) =Jm(kr)(14)asYmis singular at the origin.