Multivariate Myriad Filters based on Parameter Estimation of Student-$t$ Distributions

The contribution of this paper is twofold: First, we prove existence and uniqueness of the weighted maximum likelihood estimator of the multivariate Student-$t$ distribution and propose an efficient algorithm for its computation that we call generalized multivariate myriad filter (GMMF). Second, we use the GMMF in a nonlocal framework for the denoising of images corrupted by different kinds of noise. The resulting method is very flexible and can handle very heavy-tailed noise such as Cauchy noise, but also also Gaussian or wrapped Cauchy noise.