Reconstructing Binary Polygonal Objects from Projections: A Statistical View

Abstract

In many applications of tomography, the fundamental quantities of interest in an image are geometric ones. In these instances, pixel-based signal processing and reconstruction is at best inefficient, and, at worst, nonrobust in its use of the available tomographic data. Classical reconstruction techniques such as filtered back-projection tend to produce spurious features when data is sparse and noisy; these "ghosts" further complicate the process of extracting what is often a limited number of rather simple geometric features. In this paper, we present a framework that, in its most general form, is a statistically optimal technique for the extraction of specific geometric features of objects directly from the noisy projection data. We focus on the tomographic reconstruction of binary polygonal objects from sparse and noisy data. In our setting, the tomographic reconstruction problem is essentially formulated as a (finite-dimensional) parameter estimation problem. In particular, the vertices of binary polygons are used as their defining parameters. Under the assumption that the projection data are corrupted by Gaussian white noise, we use the maximum likelihood (ML) criterion, when the number of parameters is assumed known, and the minimum description length (MDL) criterion for reconstruction when the number of parameters is not known. The resulting optimization problems are nonlinear and thus are plagued by numerous extraneous local extrema, making their solution far from trivial. In particular, proper initialization of any iterative technique is essential for good performance. To this end, we provide a novel method to construct a reliable yet simple initial guess for the solution. This procedure is based on the estimated moments of the object, which may be conveniently obtained directly from the noisy projection data.