Optimal resolution in maximum entropy image reconstruction from projections with multigrid acceleration

Mark A. Limber, Thomas A. Manteuffel and Stephen F. McCormick

Computational Mathematics Group, University of Colorado at Denver, Denver, CO

David S. Sholl

Program in Applied Mathematics, University of Colorado at Boulder, Boulder, CO


Abstract

We consider the problem of image reconstruction from a finite number of projections over the space L^1(\Omega), where \Omega is a compact subset of \Reals^2. We prove that, given a discretization of the projection space, the function that generates the correct projection data and maximizes the Boltzmann-Shannon entropy is piecewise constant on a certain discretization of \Omega, which we call the optimal grid. It is on this grid that one obtains the maximum resolution given the problem setup. The size of this grid grows very quickly as the number of projections and number of cells per projection grows, indicating fast computational methods are essential to make its use feasible.

We use a Fenchel duality formulation of the problem to keep the number of variables small while still using the optimal discretization, and propose a multilevel scheme to improve convergence of a simple cyclic maximization scheme applied to the dual problem.