Irregular Sampling for Multidimensional Polar Processing of Integral Transforms and Prolate Spheroidal Wave Functions

Analyzing physical phenomena using a computer inevitably requires to bridge between the continuous nature of the physical phenomena and the discrete nature of computers. This bridging is known as sampling. The continuous signal is sampled such that it is represented by a discrete set of samples. The sampling scheme used to discretize the signal is often irregular. This may be due to either the physical constraints or due to computational considerations, as irregular sampling is usually superior to regular sampling. Despite the fact that irregular sampling is situated in the core of many scientific applications, there are very few efficient numerical tools that allow robust processing of irregularly sampled data. In this thesis we present a coherent related family of theories that enable to process irregularly sampled data. We show the relation between irregular sampling and discrete integral transforms, demonstrate the application of irregular sampling to image processing problems, and derive approximation algorithms that are based on unequally spaced samples. We consider two sampling methodologies. One is based on sampling the Fourier domain and the other is based on sampling the signal space. For the first approach, we describe 2D and 3D irregular sampling geometries of the frequency domain, derive efficient numerical algorithms that implement them, prove their correctness, and provide theory and algorithms that invert them. We also show that these sampling geometries are closely related to discrete integral transforms. For the second approach, we derive 1D and 2D sampling schemes that are