Reconstructing Binary Polygonal Objects from Projections: A Statistical View

Abstract

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. AbstractIn 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; and 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.2

Model-based Tomographic Reconstruction Literature Search

“…Follow on work by Sauer and Liu [4], Wang, et al [5], and Karl, et al [6] generalized this to multiple ellipses and three-dimensional ellipsoids. Milanfor, et al [7,8] applied the maximum likelihood method to parametric models of polygons. The elliptical object model was extended to cylinders by Bresler [9,10] and Fessler [11], who also used both minimum mean-square error (MMSE) and maximum a posteriori (MAP) approaches to estimating the object parameters.…”

Polygonal and Polyhedral Contour Reconstruction in Computed Tomography

“…1) Maximum Likelihood: First, ML estimation was used to reconstruct a polygon with a very low number of vertices [8]. Considering the assumptions on the errors of the data model , the ML estimator is equal to the leastsquare estimator (6) where denotes the th data value for projection .…”

“…On the other hand, the direct use of deformable contours in image reconstruction has been very limited because of the nonlinearity of the direct model, relating the surface parameters and the projection values. For those applications, explicit contour models are mostly parametric, and often involve a very few parameters [5]- [8].…”

Contour-based models for 3D binary reconstruction in X-ray tomography

“…These contours have mainly been used in image segmentation [14]. Considering a super-ellipsoid as a star-shaped contour, and denoting its central point, this surface is defined by: ¢ from the data using explicit relations between the moments of the projections and those of the object [3,15]. Then, we process the development (12) in the basis formed by these axes, where the third coordinate corresponds to the greatest moment of inertia.…”

“…In particular, we discuss first order spline models which yield polyhedral shapes [3,8,7]. In recent works [9,7], we have proposed regularization-based methods to estimate the vertices coordinates of a polyhedron directly from the projections.…”