Convolution on the -Sphere With Application to PDF Modeling
ABSTRACT In this paper, we derive an explicit form of the convolution theorem for functions on an n -sphere. Our motivation comes from the design of a probability density estimator for n -dimensional random vectors. We propose a probability density function (pdf) estimation method that uses the derived convolution result on Sn. Random samples are mapped onto the n -sphere and estimation is performed in the new domain by convolving the samples with the smoothing kernel density. The convolution is carried out in the spectral domain. Samples are mapped between the n -sphere and the n -dimensional Euclidean space by the generalized stereographic projection. We apply the proposed model to several synthetic and real-world data sets and discuss the results.
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ABSTRACT: This paper examines filtering on a sphere, by first examining the roles of spherical harmonic magnitude and phase. We show that phase is more important than magnitude in determining the structure of a spherical function. We examine the properties of linear phase shifts in the spherical harmonic domain, which suggest a mechanism for constructing finite-impulse-response (FIR) filters. We show that those filters have desirable properties, such as being associative, mapping spherical functions to spherical functions, allowing directional filtering, and being defined by relatively simple equations. We provide examples of the filters for both spherical and manifold data.IEEE Transactions on Signal Processing 08/2012; 60(12). · 2.81 Impact Factor
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ABSTRACT: We propose a simple model for approximate scaling of spherical functions in the Fourier domain. The proposed scaling model is analogous to the scaling property of the classical Euclidean Fourier transform. Spherical scaling is used for example in spherical wavelet transform and filter banks or illumination in computer graphics. Since the function that requires scaling is often represented in the Fourier domain, our method is of significant interest. Furthermore, we extend the result to higher-dimensional spheres. We show how this model follows naturally from consideration of a hypothetical continuous spectrum. Experiments confirm the applicability of the proposed method for several signal classes. The proposed algorithm is compared to an existing linear operator formulation.IEEE Transactions on Signal Processing 12/2010; · 2.81 Impact Factor