Reconstructing parameterized objects from projections : a statistical view

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, non-robust 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; and these "ghosts" further complicate the process of extracting what is often a limited number of rather simple geometric features. In addition, even if our interest is not primarily geometric, such a perspective can provide a rational framework for focusing information in those cases where the quality or quantity of the available data will not support the generation of a dense pixel-based field. 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 or objects directly from the noisy projection data. We present an approach that is applicable to the reconstruction of any finite parameterization of an object, but in specific, 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. The parameters to be estimated correspond to features of the underlying object. 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 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. f(XA,y) g(t,O) 0 6 w d, dy cos(O) sin(O) ir V i j m n O Y Vml log E vmap d VMDL N / f S k XI A R h T Vref L C Vinit det k tan I A U £ E z I P * …