Statistical approaches to the tomographic reconstruction of finitely parameterized geometric objects

Peyman Milanfar; William Clement Karl; Alan S. Willsky

doi:10.1117/12.130884

16 December 1992 Statistical approaches to the tomographic reconstruction of finitely parameterized geometric objects

Peyman Milanfar, William Clement Karl, Alan S. Willsky

Proceedings Volume 1766, Neural and Stochastic Methods in Image and Signal Processing; (1992) https://doi.org/10.1117/12.130884
Event: San Diego '92, 1992, San Diego, CA, United States

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 (Omicron) from noisy and possibly sparse samples of its radon-transform projections, g(t, (Omicron) ). If the pair (f, (Omicron) ) is referred to as an object, a finitely parameterized object is one in which both f(x, y) and (Omicron) 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.

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Peyman Milanfar, William Clement Karl, and Alan S. Willsky "Statistical approaches to the tomographic reconstruction of finitely parameterized geometric objects", Proc. SPIE 1766, Neural and Stochastic Methods in Image and Signal Processing, (16 December 1992); https://doi.org/10.1117/12.130884

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KEYWORDS

Signal to noise ratio

Reconstruction algorithms

Binary data

Signal processing

Stochastic processes

Image processing

Tomography

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