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

TL;DR

This paper introduces a new algorithm for estimating parameters of multivariate Student-$t$ distributions and applies it to develop a flexible image denoising method capable of handling various noise types, including heavy-tailed distributions.

Contribution

We propose an efficient algorithm for maximum likelihood estimation of multivariate Student-$t$ parameters and integrate it into a nonlocal denoising framework adaptable to different noise models.

Findings

The generalized multivariate myriad filter effectively denoises images with heavy-tailed noise.

The method handles both Gaussian and Cauchy noise robustly.

Application to circular data shows promising results for wrapped Cauchy noise.

Abstract

The contribution of this study 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, as well as the other extreme, namely Gaussian noise. Furthermore, we detail how the limiting case $\nu \rightarrow 0$ of the projected normal distribution in two dimensions can be used for the robust denoising of periodic data, in particular for images with circular data corrupted by wrapped Cauchy noise.