An Iterative Method for Edge-Preserving MAP Estimation When Data-Noise Is Poisson

Abstract

In numerous applications of image processing, e.g., astronomical and medical imaging, data-noise is well modeled by a Poisson distribution. This motivates the use of the negative-log Poisson likelihood function for data fitting. However, difficulties arise when this likelihood is used due to the fact that it is nonquadratic and requires a nonnegativity constraint when minimized. Moreover, for ill-posed problems, which are our interest here, regularization is required. In this paper we present an edge-preserving, quadratic regularization function. The resulting regularized, negative-log Poisson likelihood minimization problem is solved with an iterative method that we present here. We prove that the method is convergent when applied to the problem of interest. Finally, in order to test the methodology, we apply the algorithm and edge-preserving regularization scheme to synthetically generated data.