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 RðrÞð’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 RðrÞð#Þð’Þ¼RðrÞð#; ’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 ð’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 eikr 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 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 mð’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 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 ÂÃ found that the relation also provides computational convenience. ik1x ik2x e e dx ¼ 2ðk1 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 k2 derivation and theories are only outlined in this paper. Those who are interested in more details are referred to [23]. Boundary conditions need to be imposed on RðrÞ¼JmðkrÞ to select a set of k values so that the corresponding functions are mutually orthogonal. According to the Sturm-Liouville (S-L) theory [2], for 2POLAR FOURIER TRANSFORM all the nonnegative k satisfying the boundary condition 2.1 Basis Functions RðaÞ cos aR0ðaÞ sin ¼ 0;2½0;Þ; ð16Þ 2.1.1 Helmholtz Equation and Angular Basis Functions and leaving RðrÞ¼J ðkrÞ as nonzero functions, the eigenfunc- As an extension from the Cartesian case, we begin with the m tions fRðrÞ¼J ðkrÞg form an orthogonal basis on ð0;aÞ. eigenfunctions of the Laplacian, whose expression in polar m With x ¼ ka and RðrÞ¼J ðkrÞ, (16) becomes coordinates is given by m 0 JmðxÞ cos xJmðxÞ sin ¼ 0: ð17Þ 2 2 1 2 r ¼rr þ 2 r’; ð6Þ r Suppose ðxm1

  1 2 qffiffiffiffiffiffiffiffiffiffiffi 1 @ @ 2 m RnmðrÞ¼ JmðknmrÞ: ð21Þ r R þ k R ¼ 0: ð10Þ ðmÞ r @r @r r2 Nn IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 31, NO. 9, SEPTEMBER 2009 1717

The set of functions fgRnmjn ¼ 1; 2; ... forms an orthonormal basis with weight r on ð0;aÞ. A function fðrÞ defined on this interval can be expanded as Z X1 a fðrÞ¼ fðÞRnmðÞd RnmðrÞ: ð22Þ n¼1 0 In (17), can take any values in ½0;Þ. Two cases are especially interesting: Zero-value boundary condition. With sin ¼ 0, (17) reduces to

JmðxÞ¼0 ð23Þ

(note that x ¼ ka). xmn should be the positive zeros of JmðxÞ. Under this condition,

a2 NðmÞ ¼ J2 ðx Þ; ð24Þ n 2 mþ1 mn and the right-hand side of (22) is usually known as the mth order Fourier-Bessel series of fðrÞ. Derivative boundary condition. With cos ¼ 0, (17) becomes

0 JmðxÞ¼0: ð25Þ 0 xmn should be the positive zeros of JmðxÞ except for one special case: x01 ¼ 0. This case is not explicitly covered in [1]. One can 0 verify that x ¼ 0 is a solution to J0ðxÞ¼0 and J0ð0 r=aÞ¼1 is nonzero and is, indeed, an eigenfunction of the S-L system. Under the derivative boundary condition, we have  2 2 ðmÞ a m 2 Nn ¼ 1 2 JmðxmnÞ; ð26Þ 2 xmn ð0Þ 2 with the special case N1 ¼ a =2. Different boundary conditions lead to different spectra of the system. The choice should depend on the problems under investigation. To give an impression how the radial functions look like, we show the first few of them for m ¼ 2 with zero and derivative boundary conditions in Figs. 1a and 1b. It is intuitive to choose the zero boundary condition when the images tend to be zero at r ¼ a and the derivative condition when the images tend to be constant in radial direction near r ¼ a. Often, it is necessary to Fig. 1. The first few radial basis functions for 2D with m ¼ 2 and a ¼ 1: (a) Rnm with zero boundary condition, (b) R with derivative boundary condition, and do some experiments to find the better choice. nm e (c) normalized radial Zernike function Znm. The number beside each curve is The asymptotic behavior of the Bessel functions [1] reveals the the value of n. 2 1 wave-like property of the radial function: for knmr jm 4 j, 2  Those familiar with quantum mechanics will recognize knm as the 1ffiffiffi m RnmðrÞp cos knmr : ð27Þ energy level (except for a constant factor) of the system. The basis r 2 4 functions nmðr; ’Þ with the lowest energy levels are shown in R ðrÞ approaches a cosine function with the amplitude decreas- nm pffiffiffi Fig. 2. One can find that the higher the energy level, the finer the ing as fast as 1= r. There is a phase shift of ðm=2 þ =4Þ, which structures. Therefore, for image analysis, the value of k is an corresponds to a “delay” of the function to take the wave-like form indication of the scale of the basic patterns, which is consistent near the origin (see Fig. 6a). with the normal Fourier transform. 2.1.3 Basis Functions 2.2 Expansion and Rotation-Invariant Descriptors The basis function for the polar Fourier transform comprises the A 2D function fðr; ’Þ defined on the whole space can be expanded radial and the angular parts. Consequently, for the transform with respect to k;m, as defined in (28): defined on the whole space, the basis function is given by Z 1 X1 pffiffiffi fðr; ’Þ¼ P ðr; ’Þk dk; ð31Þ r; ’ kJ kr ’ 28 k;m k;m k;mð Þ¼ mð Þ mð ÞðÞ 0 m¼1 with k taking continuous nonnegative values. For the transform where defined on the finite region r

Fig. 3. Isolines of knm.

on the radial basis function show that it is enough to have Mr akmax= þ 25 to ensure that (19) holds with a relative error about or less than 1 percent. Mapping of the image is a process of interpolating and sampling. It must be handled carefully to avoid aliasing. If the resolution in Ipolar is coarser than in the original image I, one can either first smooth I then perform the mapping, or alternatively, first map I to polar coordinates with proper Fig. 2. Basic patterns represented by nm with zero boundary condition. Shown resolutions followed by smoothing and downscaling in r or ’. are the real parts of the functions. ðn; mÞ pairs are given under each pattern. The Which approach to take depends on the aspect ratio of I . patterns are listed in the increasing order of the value of knm. polar 2.3 Relation to Normal Fourier Transform in 2D finite region. A function fðr; ’Þ, defined on r

Fig. 4. Illustration of the relation of the polar Fourier coefficients and the normal k0 Jm ðk0 aÞ Fig. 5. j 2 2 j as a function of k0 for n ¼ 8;m¼ 5, and a ¼ 32, where k knm Fourier coefficients Ck;’k . (a) When the space is infinite, Pk;m is the Fourier 0 knm ¼ 0:994. coefficient of Ck;’k , with ’k as the variable. (b) When the space is finite, Pnm is the weighted sum of the Fourier coefficients on different circles. With zero boundary condition, the weight function is proportional to (44). which has its maximum absolute value at k0 ¼ knm. According to the asymptotic behavior of Bessel functions, it arrives to its first We first expand a plane wave in f nmg on the disk. As shown zeros approximately at jk0 knmj¼=a and will oscillatingly in (35), a plane wave can be expanded in k;m, which, in turn, can decrease on both sides. Fig. 5 shows the absolute value of (44) be expanded easily in fnmg on the disk. We have multiplied by k0, which comes from the fact that the number of Z pixels at radius k is approximately proportional to k. X pffiffiffiffiffiffi a ikr m f g e ¼ i 2 RnmðÞJmðkÞd For completeness, if nm is determined with the derivative n;m 0 ð38Þ boundary condition, we can get eim’k ðr; ’Þ pffiffiffi nm 2 ak X kJ0 ka n m nm mð Þ imk Pnm ¼ð1Þ i pffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e Ck : ð45Þ r

The function fðr; ’Þ can as well be expanded in fnmg. Let the im’ Vnmðr; ’Þ¼ZnmðrÞe ; ð46Þ coefficient for nm be Pnm. With the expansion (39), one can get pffiffiffi X where m is any integer, n 0 is an integer and is the order of the 2 J ðk aÞ n m m 0 im’k n jmj n jmj Pnm ¼ð1Þ i pffiffiffiffi knm e 0 Ck ð41Þ polynomial, , and is even. The angular part is the A k2 k2 0 k0 0 nm same as that of (29). The radial Zernike function Znm is a polynomial in r: for the zero boundary condition. We rewrite the main parts of Pk;m njmj and Pnm from (37) and (41) for comparison, X2 s ðn sÞ! n2s Z ZnmðrÞ¼ ð1Þ r : ð47Þ nþjmj njmj 0 0 im’ 0 s¼0 s! s ! s ! k 0 2 2 Pk;m dk ðk kÞ e Ck ;’k0 ; ð42Þ It has ðn jmjÞ=2 zeros between 0 and 1. The orthogonality relation X of the radial functions is given by Jmðk0aÞ im’ P e k0 C : 43 nm 2 2 k0 ð Þ Z k0 knm 1 k0 1 ZnmðrÞZn0mðrÞrdr¼ nn0 : ð48Þ When the space becomes finite, the integral over the wave vector is 0 2n þ 2 replaced by a summation and the sharp function of the For comparison, we define the normalized radial function as wavenumber ðk0 kÞ is replaced by a more spreading one (see ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e p Fig. 4b for an illustration) Znm ¼ 2n þ 2Znm: ð49Þ The first few normalized radial functions for m ¼ 2 are shown Jmðk0aÞ e 2 2 ; ð44Þ in Fig. 1c. The typical form of Znm with relatively large m and n is k0 knm shown in Fig. 6 together with Rnm for comparison. One can find 1720 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 31, NO. 9, SEPTEMBER 2009

X1 X1 Xl fðr; #; ’Þ¼ Snlmnlmðr; #; ’Þ; ð56Þ n¼1 l¼0 m¼l where Z Z Z a 2 2 Snlm ¼ fðr; #; ’Þnlmðr; #; ’Þr sin #drd#d’ ð57Þ 0 0 0 are the spherical Fourier coefficients (S stands for Spherical). Extracting rotational invariants of objects in 3D is slightly more complicated than in 2D as there are two angular coordinates now.

It is well known that Ylm with l m l span a subspace Yl that is invariant with respect to the rotation group. For a function defined Fig. 6. (a) Rð11Þ8 as defined in (21) with derivative boundary condition for a ¼ 1 and e on the spherical surface and having expansion (b) Zð28Þ8 as defined in (49). Both have 10 zeros on ð0; 1Þ.

e X1 Xl that Znm has also a wave-like form. Like Rnm, it has also a “delay” fð#; ’Þ¼ hlmYlmð#; ’Þ; ð58Þ near the origin for the “wave” to begin. Unlike R , the amplitude l¼0 m¼l nm qPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the “wave” does not decrease monotonically; instead, the l 2 the magnitude of its projection onto Yl, i.e., m¼l jhlmj remains “wavelength” decreases with r. unchanged under rotation and is a rotational invariant (will be called a SH descriptor) of the function. In [7], it is suggested to use 3SPHERICAL FOURIER TRANSFORM SH descriptors on spherical surfaces of a set of radii to describe volume data. These descriptors will be called radius-wise SH The theory on the spherical Fourier transform is only summarized Descriptors (rwSHD) in this paper. It is worth noting that a rwSHD here. Interested readers are referred to [23] for more details. should be scaled with the square of its radius to account for the The angular part of the basis function is a spherical harmonic change of the area of the spherical surface with the radius. It is Ylmð#; ’Þ, as defined in (5). The corresponding radial part is straightforward to know that, for a function defined on a solid

RðrÞjlðkrÞ; ð50Þ sphere, its projection onto the subspace spanned by nlm ¼ RnlYlm with l m l has also a fixed magnitude under rotation as all where jl is the so-called spherical Bessel function of order l and is related to the ordinary Bessel function by the variance of the basis function nlm under rotation is captured rffiffiffiffiffiffi by its angular part. That is to say, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u jlðxÞ¼ Jlþ1ðxÞ: ð51Þ 2x 2 u Xl t 2 jSnlmj ð59Þ The normalized basis function for spherical Fourier transform m¼l defined on the whole space is for any n and l is a rotation-invariant property of that function. We rffiffiffi 2 call (59) a Spherical Fourier Descriptor (SFD). ðr; #; ’Þ¼ kj ðkrÞY ð#; ’Þ; ð52Þ k;l;m l lm The relation between the spherical Fourier transform and the normal Fourier transform can be derived similarly as in the 2D case. where k takes continuous values. When defined on a solid sphere Any function fðr; #; ’Þ defined on the whole space can be expanded of radius a, the normalized basis function is 3 ðp1ffiffiffiffiÞ eikr in either plane waves 2 or in spherical waves k;l;m: qffiffiffiffiffiffiffiffiffi1 nlmðr; #; ’Þ¼ jlðknlrÞYlmð#; ’Þ; ð53Þ fðr; #; ’Þ ðlÞ Z Z Z  Nn 1 2 3 1ffiffiffiffiffiffi ikr 2 ¼ Ck;#k;’k p e k sin #k dk d#k d’k n k NðlÞ 0 0 0 2 where is a positive integer; nl and n are determined from the Z 1X1 Xl S-L boundary conditions. With the zero-value boundary condi- 2 ¼ Sk;l;mk;l;mðr; #; ’Þk dk; tion, let ðxl1

Fig. 8. Recognition rates on 3D data with nearest neighbor classifier based on L1-norm distance.

111 512 512 in spherical coordinates ðr; #; ’Þ. The sizes for # and ’ are chosen to be the same due to the requirement of S2Kit,a software package for fast SH transform [26]. Due to the existence of fast programs for Fourier transform [25] and SH transform [26], the angular-part transformation can be done first. The radial part needs only to be performed for the kept terms. We use the Gnu Scientific Library [27] for Bessel functions and keep a lookup table for the radial functions. 4.1 2D Experiments Fig. 7. Recognition rates on 2D data with (a) nearest-neighbor classifiers based on We test PFDs with the zero and the derivative boundary L1-norm distance and (b) support vector machines. conditions. They will be called PFD-0 (0 for zero) and PFD-d X 1ffiffiffiffi ikr (d for derivative) respectively. For comparison, Zernike descrip- fðr; #; ’Þ¼ Ck0 p e V tors are also calculated, which are the magnitude of the Xk0 coefficients in normalized Zernike functions. The truncation of ¼ Snlmnlmðr; #; ’Þ: nlm the polar Fourier expansion is according to the energy level and the truncation of the Zernike expansion is according to the order The coefficients have the following relationship if knm are selected with the zero boundary condition: of the polynomials. We test on descriptor sets of different sizes. "#rffiffiffiffiffi Each set is normalized to have a sum of 1. As a note for the X À Á m 170 n 57 n l 2a knljlðk0aÞ interested readers, we have max ¼ and max ¼ for 4,096 Snlm ¼ð1Þ i 4 Y #k ;’k Ck : V k2 k2 lm 0 0 0 PFD-0 features. k0 0 nm We first perform leave-one-out tests using nearest neighbor classifiers based on the L1-norm distance. As Fig. 7a shows, PFD-0 4EXPERIMENTS performs best for most cases, but the difference of performance is not very distinct. We further employ support vector machines [24] In this paper, we conduct experiments on the application of PFD for classification. The 10-fold cross-validation results are given in and SFD to pattern classification problems. The aim here is to Fig. 7b. Although the Zernike descriptors catch up at large numbers compare their performance with other features like Zernike descriptors or rwSHD. We shall calculate descriptors directly from of descriptors, they are worse when the number is low. The the gray-value images instead of trying to get a higher classifica- recognition rate of PFD-0 reaches its highest with 1,024 descriptors tion rate by combination with other techniques that are irrelevant and is “saturated” after that. The other two sets of descriptors to the comparison, such as, texture decomposition or image reach a slightly lower “saturation” level with about 2,070 or warping according to the object shape. more descriptors. PFD-0, therefore, describes this data set more The data set for the experiments is 3D confocal laser scanning efficiently for the considered classification problem. microscopic images of pollen grains. It contains 387 images of single pollen grains from 26 species, with 13-15 objects for each species. 4.2 3D Experiments The class of every pollen grain is known. The size of each 3D image For 3D, we test SFDs with the zero boundary condition against is 128 128 128, with the same resolution in each direction. rwSHDs. These two kinds of descriptors correspond to the same Essentially, they are the same as used in [20]. We have only applied angular transformation and differ in how radial patterns are new segmentation and downscaling methods on the raw data here. represented. As the recognition rates of rwSHDs can vary with the As pollen grains usually have near-spherical shapes and show number of sampling points nr in radius, we compare SFDs and certain rotational symmetry, they make up a good data set for rwSHDs for the same SH band limits L (l is limited to l

5CONCLUSION AND OUTLOOK [12] W.-Y. Kim and Y.-S. Kim, “Robust Rotation Angle Estimator,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 8, pp. 768-773, Aug. We propose to use the eigenfunctions of the Laplacian that are 1999. [13] E.M. Arvacheh and H.R. Tizhoosh, “Pattern Analysis Using Zernike separable in polar and spherical coordinates as basis functions for Moments,” Proc. IEEE Instrumentation and Measurement Technology Conf., Image analysis. This idea puts the proposed polar and spherical vol. 2, pp. 1574-1578, 2005. [14] A. Averbuch, R.R. Coifman, D.L. Donoho, M. Elad, and M. Israeli, Fourier transform and the normal Fourier transform into the “Accurate and Fast Discrete Polar Fourier Transform,” Proc. Conf. Record same framework and ensures close resemblance and relation 37th Asilomar Conf. Signals, Systems, and Computers, vol. 2, pp. 1933-1937, between them. 2003. [15] A. Makadia, L. Sorgi, and K. Daniilidis, “Rotation Estimation from Rotation-invariant descriptors based on polar and spherical Spherical Images,” Proc. 17th Int’l Conf. Pattern Recognition, vol. 3, Fourier coefficients have been tested in object classification tasks. pp. 590-593, 2004. For 2D data, it is shown that PFDs generally perform better than [16] R. Skomski, J.P. Liu, and D.J. Sellmyer, “Quasicoherent Nucleation Mode in Two-Phase Nanomagnets,” Physical Rev. B, vol. 60, pp. 7359-7365, 1999. Zernike descriptors for the selected data set, leaving PFDs also as [17] B. Pons, “Ability of Monocentric Close-Coupling Expansions to Describe candidates for applications where Zernike descriptors are used. Ionization in Atomic Collisions,” Physical Rev. A, vol. 63, p. 012704, 2000. [18] R. Bisseling and R. Kosloff, “The Fast Hankel Transform as a Tool in the For 3D data, SFDs perform better than rwSHDs with the same Solution of the Time Dependent Schro¨dinger Equation,” J. Computational SH band limits, showing that the radial structure is effectively Physics, vol. 59, no. 1, pp. 136-151, 1985. coded into the spherical Fourier coefficients. Also, it should be [19] D. Lemoine, “The Discrete Bessel Transform Algorithm,” J. Chemical Physics, vol. 101, no. 5, pp. 3936-3944, 1994. emphasized that the radial transform is automatically a multiscale [20] O. Ronneberger, E. Schultz, and H. Burkhardt, “Automated Pollen approach that decomposes the pattern into different scales, just like Recognition Using 3D Volume Images from Fluorescence Microscopy,” Aerobiologia, vol. 18, pp. 107-115, 2002. the normal Fourier transform does. [21] H. Burkhardt, Transformationen zur lageinvarianten Merkmalgewinnung, In the experiments, the features are calculated for the whole Ersch. als Fortschrittbericht (Reihe 10, Nr. 7) der VDI-Zeitschriften. objects. They can also be calculated pointwise to detect local VDI-Verlag, 1979. [22] O. Ronneberger, J. Fehr, and H. Burkhardt, “Voxel-Wise Gray Scale structures. In that case, no mapping to the corresponding Invariants for Simultaneous Segmentation and Classification,” Proc. 27th coordinates is necessary. The coefficients can be calculated by DAGM Symp., pp. 85-92, 2005. [23] Q. Wang, O. Ronneberger, and H. Burkhardt, “Fourier Analysis in Polar convolution, which can be done fast with the help of FFT [22]. and Spherical Coordinates,” internal report, http://lmb.informatik.uni- Although the experiments are only performed on problems where freiburg.de/papers/, 2008. rotational invariance is important, the polar and spherical Fourier [24] C.C. Chang and C.J. Lin, “LIBSVM—A Library for Support Vector Machines,” http://www.csie.ntu.edu.tw/cjlin/libsvm/, 2009. analysis can also be applied to other problems like registration, [25] FFTW Home Page, http://www.fftw.org/, 2009. where the orientation of the patterns is the essential information. [26] Fast Spherical Harmonic Transforms, http://www.cs.dartmouth.edu/ geelong/sphere/, 2009. For this kind of application, phases of the coefficients shall be kept [27] The Gnu Scientific Library, http://www.gnu.org/software/gsl/, 2009. as they carry the information of the angular positions. In this paper, we are mainly concerned with the numerical precision of the calculations. The radial transform is calculated for . For more information on this or any other computing topic, please visit our every coefficient independently, which is surely not an efficient Digital Library at www.computer.org/publications/dlib. way. Whether fast algorithms exist for radial transforms is a question to be answered.

ACKNOWLEDGMENTS This study was supported by the Excellence Initiative of the German Federal and State Governments (EXC 294).

REFERENCES [1] N.N. Lebedev, Special Functions and Their Applications (translated from Russian by R.A. Silverman), chapters 5, 6, 8. Dover, 1972. [2] W. Kaplan, Advanced Calculus, fourth ed., pp. 696-698. Addison-Wesley, 1991. [3] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, seventh ed., pp. 523-525. Cambridge, 1989. [4] E.W. Weisstein, “Jacobi-Anger Expansion,” From MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/Jacobi-AngerExpansion. html, 2009. [5] K.E. Schmidt, “The Expansion of a Plane Wave,” http://fermi.la.asu.edu/ PHY577/notes/plane.pdf, 2009. [6] S. Erturk and T.J. Dennis, “3D Model Representation Using Spherical Harmonics,” Electronics Letters, vol. 33, no. 11, pp. 951-952, 1997. [7] M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz, “Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors,” Proc. Symp. Geometry Processing, pp. 167-175, 2003. [8] H. Huang, L. Shen, R. Zhang, F. Makedon, A. Saykin, and J. Pearlman, “A Novel Surface Registration Algorithm with Medical Modeling Applica- tions,” IEEE Trans. Information Technology in Biomedicine, vol. 11, no. 4, pp. 474-482, July 2007. [9] C.-H. Teh and R.T. Chin, “On Image Analysis by the Methods of Moments,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 4, pp. 496-513, July 1988. [10] Y. Zana and R.M. Cesar Jr., “Face Recognition Based on Polar Frequency Features,” ACM Trans. Applied Perception, vol. 3, no. 1, pp. 62-82, 2006. [11] A. Khotanzad and Y.H. Hong, “Invariant Image Recognition by Zernike Moments,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 5, pp. 489-497, May 1990.