Analytic continuation and incomplete data tomography.

Abstract

A unique feature of medical imaging is that the object to be imaged has a compact support. In mathematics, the Fourier transform of a function that has a compact support is an entire function. In theory, an entire function can be uniquely determined by its values in a small region, using, for example, power series expansions. Power series expansions require evaluation of all orders of derivatives of a function, which is an impossible task if the function is discretely sampled. In this paper, we propose an alternative method to perform analytic continuation of an entire function, by using the Nyquist-Shannon sampling theorem. The proposed method involves solving a system of linear equations and does not require evaluation of derivatives of the function. Noiseless data computer simulations are presented. Analytic continuation turns out to be extremely ill-conditioned.