Statistical approaches to the tomographic reconstruction of finitelyparameterized geometric objects,

摘要

Abstract: Tomographic reconstruction in two dimensions is concerned with the reconstruction of a positive, bounded function f(x, y) and its compact domain of support approximately icron from noisy and possibly sparse samples of its radon- transform projections, g(t, approximately icron@). If the pair (f, approximately icron@) is referred to as an object, a finitely parameterized object is one in which both f(x, y) and approximately icron are determined uniquely by a finite number of parameters. For instance, a binary N-sided polygonal object in the plane is uniquely specified by exactly 2N parameters which may be the vertices, normals to the sides, etc. In this work we study the optimal reconstruction of finitely parameterized objects from noisy projections. In specific, we focus our study on the optimal reconstruction of binary polygonal objects from noisy projections. We show that when the projections are corrupted by Gaussian white noise, the optimal maximum likelihood (ML) solution to the reconstruction problem is the solution to a nonlinear optimization problem. This optimization problem is formulated over a parameter space which is a finite dimensional Euclidean space. We also demonstrate that in general, the moments of an object can be estimated directly from the projection data and that using these estimated moments, a good initial guess for the numerical solution to the nonlinear optimization problem may be constructed. Finally, we study the performance of the proposed algorithms from both statistical and computational viewpoints. !11