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

The contribution of this paper is twofold: First, we propose an efficient algorithm for the computation of the (weighted) maximum likelihood estimators for the parameters of the multivariate Student-$t$ distribution, which we call generalized multivariate myriad filter. Second, we use the generalized multivariate myriad filter in a nonlocal framework for the denoising of images corrupted by different kinds of noise. The resulting method is very flexible and can handle heavy-tailed noise such as Cauchy noise, but on the other extreme also Gaussian noise is covered. Furthermore, we detail how a special case of our approach can be used for the robust denoising of periodic data, in particular for images corrupted by wrapped Cauchy noise on the circle $\mathbb S^1$.