New Advances in Bayesian Calculation for Linear and Nonlinear Inverse Problems

The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires the minimization of a compound criterion which, in general, has two parts: a data fitting part and a prior part. In many situations the criterion to be minimized becomes multimodal. The cost of the Simulated Annealing (SA) based techniques is in general huge for inverse problems. Recently a deterministic optimization technique, based on Graduated Non Convexity (GNC), have been proposed to overcome this difficulty. The objective of this paper is to show two specific implementations of this technique for the following situations: -- Linear inverse problems where the solution is modeled as a piecewise continuous function. The non convexity of the criterion is then due to the special choice of the prior; -- A nonlinear inverse problem which arises in inverse scattering where the non convexity of the criterion is due to the likelihood part. Keywords: Inverse problems, Regularization, Bayesian calculation, Global optimization, Graduated Non Convexity.Comment: Presented at MaxEnt96. Appeared in Proceedings of the Maximum Entropy Conference, Berg-en-Dal, South Africa, M. Sears, V. Nedeljkovic, N.E. Pendock and S. Sibisi (Ed.), pp 92-10

A Bayesian Approach to Shape Reconstruction of a Compact Object from a Few Number of Projections

Image reconstruction in X ray tomography consists in determining an object from its projections. In many applications such as non destructive testing, we look for an image who has a constant value inside a region (default) and another constant value outside that region (homogeneous region surrounding the default). The image reconstruction problem becomes then the determination of the shape of that region. In this work we model the object (the default region) as a polygonal disc and propose a new method for the estimation of the coordinates of its vertices directly from a very limited number of its projections. Keywords: Computed Imaging, Tomography, Shape reconstruction, Non destructive testing, Regularization, Bayesian estimation, Deformable contours.Comment: Presented at MaxEnt96. Appeared in Proceedings of the Maximum Entropy Conference, Berg-en-Dal, South Africa, M. Sears, V. Nedeljkovic, N.E. Pendock and S. Sibisi (Ed.), pp 68-7

Shape reconstruction in X-ray tomography from a small number of projections using deformable models

X-ray tomographic image reconstruction consists of determining an object function from its projections. In many applications such as non-destructive testing, we look for a fault region (air) in a homogeneous, known background (metal). The image reconstruction problem then becomes the determination of the shape of the default region. Two approaches can be used: modeling the image as a binary Markov random field and estimating the pixels of the image, or modeling the shape of the fault and estimating it directly from the projections. In this work we model the fault shape by a deformable polygonal disc or a deformable polyhedral volume and propose a new method for directly estimating the coordinates of its vertices from a very limited number of its projections. The basic idea is not new, but in other competing methods, in general, the fault shape is modeled by a small number of parameters (polygonal shapes with very small number of vertices, snakes and deformable templates) and these parameters are estimated either by least squares or by maximum likelihood methods. We propose modeling the shape of the fault region by a polygon with a large number of vertices, allowing modeling of nearly any shape and estimation of its vertices' coordinates directly from the projections by defining the solution as the minimizer of an appropriate regularized criterion. This formulation can also be interpreted as a maximum a posteriori (MAP) estimate in a Bayesian estimation framework. To optimize this criterion we use either a simulated annealing or a special purpose deterministic algorithm based on iterated conditional modes (ICM). The simulated results are very encouraging, especially when the number and the angles of projections are very limited.Comment: Presented at MaxEnt97. Appeared in Maximum Entropy and Bayesian Methods, G.J. Erickson, J.T. Rychert and C.R. Smith (Ed.), Kluwer Academic Publishers (http://www.wkap.nl/prod/b/0-7923-5047-2