Irregular sampling for multidimensional polar processing of integral transforms

We survey a family of theories that enable to process polar data via integral transforms. We show the relation between irregular sampling and discrete integral transforms, demonstrate the application of irregular (polar) sampling to image processing problems, and derive approximation algorithms that are based on unequally spaced samples. It is based on sampling the Fourier domain. 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. The proposed underlying methodology bridges via sampling between the continuous nature of the physical phenomena and the discrete nature world. 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.