Polar, Spiral, and Generalized Sampling and Interpolation

Abstract

In this chapter we cohect a number of results related to polar, spiral, and generahzed non-uniform sampling. In most cases we shall not give proofs since those are often lengthy and the proofs are given in other places. A number of polar sampling theorems and their proofs are given in [858]. However, these are of limited application since they involve sampling the function at zeros of Bessel functions. A notable exception is the uniform sampling theorem which has apphcations in computerized tomography (CT). Another result with application in medical imaging is reconstruction from samples along spiral scans. This result was motivated by and has application in magnetic resonance imaging (MRI). A generahzed reconstruction formula from non-uniform samples is given in Section 6.4. The derivation of this result is based on the powerful theory of projections onto convex sets (POCS) also known as the theory of convex projections. Other results on non-uniform sampling theory are given by Marvasti in a recent book [628] as well as in a chapter by him in this book.