Spatio-spectral analysis on the unit sphere

This thesis is focussed on the development of new signal processing techniques to analyse signals defined on the sphere. Analysis and processing of signals defined on the sphere find applications in various fields of science and engineering, such as cosmology, geophysics and medical imaging. The objective to develop new signal processing methods is served by formulating, extending and tailoring existing Euclidean domain signal processing theories in ways that they become suitable for analysis of signals defined on the sphere. The first part of this thesis develops a new type of convolution between two signals on the sphere. This is the first type of convolution on the sphere which is commutative. Two other advantages, in comparison with existing definitions in the literature, are that the new convolution admits anisotropic filters and signals and the domain of the output remains on the sphere. The spectral analysis of the convolution is provided and a fast algorithm for efficient computation of convolution output is developed. The second part of the thesis is focused on the development of signal processing techniques to analyse signals on the sphere in joint spatio-spectral~(spatial-spectral) domain. A transform analogous to short-time Fourier transform(STFT) in time-frequency analysis is formulated for signals defined on the sphere, in order to devise a spatio-spectral representation of a signal. The proposed transform is referred as the spatially localized spherical harmonic transform~(SLSHT) and is defined as windowed spherical harmonic transform, resulting in the SLSHT distribution. The properties of the SLSHT distribution and its analysis in the spherical harmonic domain are also provided. Furthermore, examples are provided to demonstrate the capability of SLSHT to reveal spatially localized spectral contents in a signal that were not obtainable from traditional spherical harmonics analysis. With the consideration that data-sets on the sphere can be of considerable size and the SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented. A general framework for spatially-varying spectral filtering of signals defined on the unit sphere is also developed, as an analogy to joint time-frequency filtering. For spatio-spectral filtering, the spherical signals are first mapped from the spatial domain into a joint spatio-spectral domain using SLSHT, where a spatio-spectral signal transformation or modification is introduced. Next, a suitable scheme to transform the modified signal from the spatio-spectral domain back to an admissible signal in the spatial domain using the least squares approach is proposed. It is shown that the overall action of the SLSHT and spatio-spectral signal modification can be described through a single transformation matrix, which is useful in practice. Finally, two specific and useful instances of spatially-varying spectral filtering are presented, defined through multiplicative and convolutive modification of the SLSHT distribution. The proposed framework enables filtering or modification in the spatio-spectral domain which cannot be carried out in either the spatial or spectral domain

Ambiguity Function and Wigner Distribution on the Sphere

The ambiguity function and the Wigner distribution are fundamental tools in the time-frequency analysis. In this paper, we present an analog of the ambiguity function and the Wigner distribution for signals on the sphere. First, we formulate the ambiguity function for signals on the sphere which represents the signals in joint spatio-spectral domain and derive an inversion operation to obtain the signal from its ambiguity function. Next, we formulate the Wigner distribution for azimuthally symmetric signals on the sphere as a two dimensional spherical harmonics transform of the ambiguity function. We provide the matrix formulation of the Wigner distribution and discuss some of its useful properties. Finally, we illustrate the use of Wigner distribution for spatial and/or spectral localization of a signal in joint spatio-spectral domain. The obtained results provide the first step in designing more sophisticated transforms on the sphere