Applied Sampling and Reconstruction of Signals on the Sphere

Abstract

Signals collected with spherical geometry appear in a large number and diverse range of real-world applications in various disciplines, such as geophysics, medical imaging, computer graphics, wireless communications and acoustics. This thesis focuses on the development of signal processing techniques for sampling and reconstruction of signals on the sphere. The objective of developing new spherical signal measurement and reconstruction techniques is driven by meeting the practical requirements of applications where signals are inherently defined on the sphere. The first part of this thesis develops novel sampling schemes on the sphere for the applications of measuring and reconstructing the head-related transfer function (HRTF) in acoustics and the diffusion signal in diffusion magnetic resonance imaging (dMRI). In contrast to existing sampling schemes in the literature, the developed schemes attain the optimal number of samples, equal to the degrees of freedom in spherical harmonic space, while enabling computation of the spherical harmonic transform (SHT) to near machine precision accuracy. In addition, the proposed sampling schemes have computationally efficient algorithms for the computation of the SHT and are designed to satisfy all the practical requirements of their applications. The proposed scheme for HRTF measurement and reconstruction has non-dense sampling near the poles and is iso-latitude, allowing for the measurements to be more easily obtained. Furthermore, the proposed scheme can be configured as fully hierarchical which enables the HRTF to be analyzed at all frequencies in the audible range using the same arrangement of speakers (or microphones). Numerical experiments and simulations are conducted to demonstrate the reconstruction accuracy of the proposed sampling scheme. Two sampling schemes are proposed for the measurement and reconstruction of the diffusion signal in dMRI which exploit the antipodal symmetry of the diffusion signal in the spatial and spectral domains respectively to attain an optimal number of samples. In addition, the reconstruction accuracy of both schemes does not change significantly with rotation. The spectral antipodal sampling scheme has slightly greater reconstruction accuracy and smaller storage requirements, while the spatial antipodal sampling scheme is more similar to the schemes used by the dMRI community. The second part of the thesis develops signal processing techniques for signal reconstruction in the Slepian basis. Slepian functions must be computed numerically, and therefore inexactly, for arbitrary regions on the sphere. An analytical formulation is developed for the Slepian spatial-spectral concentration problem on the sphere for a limited colatitude-longitude spatial region which enables exact computation. The formulation is extended for an arbitrary region comprised of a union of rotated limited colatitude-longitude subregions. In addition, a computationally efficient algorithm for the implementation of the proposed analytical formulation is developed. Computational complexity analysis is performed and examples provided to illustrate the use of the algorithm in applications. A new method for the computation of Slepian functions on the sphere for an arbitrary spatial region is proposed. Computing Slepian functions with large band-limits is infeasible using the conventional method. The proposed method enables faster computation and has smaller memory storage requirements while maintaining accurate computation, enabling the computation of Slepian functions with larger band-limits.