Binary Tomography Using Gibbs Priors

Abstract

The problem of reconstructing a binary image (usually an image in the plane and not necessarily on a Cartesian grid) from a few projections translates into the problem of solving a system of equations,which is very underdetermined and leads in general to a large class of solutions. It is desirable to limit the class of possible solutions, by using appropriate prior information, to only those which are reasonably typical of the class of images which contains the unknown image that we wish to reconstruct. One may indeed pose the following hypothesis: if the image is a typical member of a class of images having a certain distribution, then by using this information we can limit the class of possible solutions to only those which are close to the given unknown image. This hypothesis is experimentally validated for the specific case of a class of binary images defined on the hexagonal grid, where the probability of the occurrence of a particular image of the class is determined by a Gibbs distribution and reconstruction is to be done from the three natural projections. Another case for which the hypothesis is tested is reconstruction, from the three projections, of semiconductor surface phantoms defined on the square grid. The time-consuming nature of the stochastic reconstruction algorithm is ameliorated by a preprocessing step that discovers image locations at which the value is the same in all images having the given projections; this reduces the search space considerably. We discuss, in particular, a linear-programming approach to finding such “invariant” locations.