Shape Reconstruction with A Priori Knowledge Based on Integral Invariants

Abstract

We investigate the applicability of integral invariants as geometrical shape descriptors in the context of ill-posed inverse problems. We propose the use of a Tikhonov functional, where the penalty term is based on the difference of integral invariants. As a case example, we consider the problem of inverting the Radon transform of an object with only limited angular data available. We approximate the ill-posed operator equation by a minimization problem involving a Tikhonov functional and show existence of minimizers of the functional. Because of its nondifferentiability, we derive for the numerical minimization smooth approximations, which converge in the sense of $\Gamma$-limits.