A deterministic algorithm for edge-preserving computed imaging using Legendre transform

Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. We first propose sufficient conditions for the design of such an edge-preserving regularization. Using the Legendre transform, it is then possible to introduce an auxiliary variable which role is twofold. Firstly, it marks the discontinuities and ensures their preservation from smoothing. Secondly, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic algorithm, based on alternate minimizations over the image and the auxiliary variable. We apply this algorithm to the problem of SPECT reconstruction.