Convolution on the n-Sphere With Application to PDF Modeling

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

Efficient Approximate Scaling of Spherical Functions in the Fourier Domain With Generalization to Hyperspheres

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