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 nondestructive 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 re­ gion. 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 an­ nealing or a special purpose deterministic algorithm based on iterated conditional modes (leM). The simulated results are very encouraging, especially when the number and the angles of projections are very limited.