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Convolution on the N-Sphere with Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE

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Convolution on the N-Sphere with Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 1157 Convolution on the n-Sphere With Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE Abstract—In this paper, we derive an explicit form of the convo- emphasis on wavelet transform in [8]–[12]. Computation of the lution theorem for functions on an -sphere. Our motivation comes Fourier transform and convolution on groups is studied within from the design of a probability density estimator for -dimen- the theory of noncommutative harmonic analysis. Examples sional random vectors. We propose a probability density function (pdf) estimation method that uses the derived convolution result of applications of noncommutative harmonic analysis in engi- on . Random samples are mapped onto the -sphere and esti- neering are analysis of the motion of a rigid body, workspace mation is performed in the new domain by convolving the samples generation in robotics, template matching in image processing, with the smoothing kernel density. The convolution is carried out tomography, etc. A comprehensive list with accompanying in the spectral domain. Samples are mapped between the -sphere theory and explanations is given in [13]. and the -dimensional Euclidean space by the generalized stereo- graphic projection. We apply the proposed model to several syn- Statistics of random vectors whose realizations are observed thetic and real-world data sets and discuss the results. along manifolds embedded in Euclidean spaces are commonly termed directional statistics. An excellent review may be found Index Terms—Convolution, density estimation, hypersphere, hy- perspherical harmonics, -sphere, rotations, spherical harmonics. in [14]. It is of interest to develop tools for the directional sta- tistics in analogy with the ordinary Euclidean. In particular, one can address a probability density function (pdf) estimation I. INTRODUCTION problem from the observations on a manifold. For the ubiquitous XTENSION of signal processing methodologies to spherical case, a common domain example is the earth surface. E noneuclidean spaces is a natural way of dealing with Typical pdf estimation methods are Parzen windowing [15], also directional data. A common example is a 2-sphere and many called kernel density estimation, and mixture models, with the Euclidean paradigms that have found their spherical analogue. most popular Gaussian mixture model (GMM) [16]–[18]. Ap- The most basic is the notion of Fourier transform that on the plication of the GMM to source coding is discussed in [19]. A sphere corresponds to the expansion of functions into series discussion of Parzen windowing on manifolds is available in of familiar spherical harmonics. Vast amount of literature is [20]. available on such expansions, mostly from quantum mechanics In this paper, we discuss convolution on a sphere in an -di- and mathematical physics [1]. It is known from group repre- mensional space regarded as a homogeneous space sentation theory that the Fourier transform can be defined on of the group of rotations of the -dimensional space . any compact Lie group and consequently on homogeneous A sphere in the -dimensional space is a manifold of dimen- spaces of these groups [2]. An example is the group of rota- sion , hence the seemingly odd notation. We derive the ex- tions of a 3-D space about the origin and the 2-sphere plicit expression for convolution in the frequency domain. Our as a corresponding homogeneous space. Application of motivation comes from the design of a probability density es- signal processing methodologies to different manifolds is timator for data of arbitrary dimension, more generally from facilitated by the discovery of efficient algorithms, analogous source coding. to the fast Fourier transform (FFT). Efficient algorithms have We propose a pdf estimation model that maps -di- been designed and implemented for Fourier transform on the mensional samples onto the . The involved stereographic 2-sphere [3], [4] and on the rotation group [5]. Some mapping is beneficial for a class of bell-shaped densities as dis- FFT generalization ideas are surveyed in [6]. The generalized cussed in Section IV-D. For the discussion of benefits of trans- concept of convolution on groups is intimately related to the formations in density estimation, see [21]. The pdf estimation concept of filtering on homogeneous spaces. Some insight is performed in the new domain using the kernel density esti- into spherical filtering can be found in [7] and with particular mation technique. This corresponds to putting the smoothing kernel function at the place of each sample and summing their contributions. Since the described technique can be regarded as Manuscript received April 28, 2009; accepted August 17, 2009. First pub- a convolution between the symmetric kernel and the data set, we lished September 29, 2009; current version published February 10, 2010. This can use the derived convolution result to realize the estimation work was supported by the Ministry of Science, Education and Sports of Croatia in the frequency domain. The spatial estimate can be efficiently under Grants MZOS 0036054 and 036-0362214-1987. The associate editor co- computed from the spectral model using the fast inverse spher- ordinating the review of this manuscript and approving it for publication was Dr. Kainam Thomas Wong. ical Fourier transform [3], [4], [6]. The authors are with the Department of Electronic Systems and In- Modern work on kernel density estimation deals with depen- formation Processing, Faculty of Electrical Engineering and Computing, dent random variables. It has been proposed by several authors University of Zagreb, 10000 Zagreb, Croatia (e-mail: [email protected]; [email protected]). [22]–[27] that the convolution formula can be used to improve Digital Object Identifier 10.1109/TSP.2009.2033329 the asymptotic properties of such estimators. The main idea in 1053-587X/$26.00 © 2010 IEEE 1158 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 this setting is to improve the estimator by using the structure of In the space of square integrable functions on , we can the model (i.e., by convolution). Note that we use the convolu- choose a particular orthonormal basis called hyperspher- tion only to compute the estimate, not to handle the dependency. ical harmonics [2], [33]. Thus, for ,wehave The pdf estimation is important in source coding where it is interesting to have compact representations of the data that (3) enable adaptation of a quantizer to the changing statistics of the source through a narrow side channel. Another class of pdf estimators are those based on expansions where , and are projections of of the pdf into series of orthogonal functions. If the smoothing onto the basis functions kernel in the proposed approach degenerates to a Dirac delta function, an orthogonal series estimator is obtained that corre- (4) sponds to Fourier expansions. An excellent general treatment of the density estimation theory along with treatment of orthog- In (3) and (4), and index the basis functions. A compound onal series estimators can be found in [28]. and cases index is a sequence of integers such that are treated in [29]. In this context, the expansion coefficients, . For functions that i.e., Fourier transform of the pdf estimate, may be regarded as are not bandlimited, . In practice, we have to limit model parameters. On the sphere, this corresponds to spherical the range of . That limit corresponds to the spectral bandwidth. harmonic expansions. Density estimation from samples on Intuitively, one can think of as of a master index that deter- using Fourier transform is treated in [30]. Deconvolution den- mines some global level of detail modeled by corresponding sity estimation is examined in [31] for the case of observations basis functions and as a way of counting all the basis func- on corrupted by random rotations and in [32] for observa- tions at the th detail level. A rotation invariant area element on tions on . is given in the introduced coordinates as This paper is organized as follows. In Section II, some background on hyperspherical harmonics and rotations in (5) is given. In Section III, we discuss convolution on and how it can be applied to density estimation. We derive the where normalization is chosen such that the measure of the exact expression for convolution transform in Appendix II. In whole sphere is equal to 1. Note that in the remainder of this Section IV, we lay out the details of the proposed estimation paper the hat notation will be used for the Fourier transform algorithm on . Section V gives some results on the perfor- of the function while tilde notation for the estimate of the mance of the estimator. Experiments that verify the sanity of the model are described in Section VI. density . Therefore, implies the spectral representation of the estimate. Furthermore, we will use the Greek letter to de- II. DEFINITIONS AND PREREQUISITES note points on when spherical polar parameterization is By a unit sphere in we refer to a set of points whose Eu- assumed. clidean distance from the origin is equal to unity Analysis in the sense of (3) is a Fourier series for functions from . In the 2-sphere case, functions coin- (1) cide with familiar spherical harmonics . An explicit expression for hyperspherical harmonics is given in As already mentioned, the superscript is due to the fact Appendix I. that a sphere in as defined by (1) is a manifold of dimension Rotations of , or equivalently, rotations in , form a . may be parameterized by a set of hyperspherical group whose elements are real orthogonal matrices polar coordinates. If are Cartesian coordinates of unit determinant. may be parameterized by in , then we define the angles and the radius real numbers. Natural choice are the Euler angles. It can be as shown that any rotation may be written as a product of three matrices, where , , , and represents the matrix of a rotation about the -axis while is the rotation about the -axis.
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