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

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 ν → 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. distribution we propose to start with the family of more general Student-t distributions, which possesses an additional degree of freedom parameter ν > 0 that allows to control the robustness of the resulting filter. While the Cauchy distribution is obtained as the special case ν = 1, the Student-t distribution converges for ν → ∞ to the normal distribution, so that in the limit also mean filters are covered.The multivariate Student t-distribution is frequently used in statistics [22], whereas the multivariate Cauchy distribution is far less common and in contrast to the one-dimensional case usually not considered separately from the Student-t distribution. The parameter(s) of a multivariate Student t-distribution are usually estimated via the Maximum Likelihood (ML) method in combination with the EM algorithm [5,7,8,10,34]. The EM algorithm for the Student-t distribution has been derived, e.g. in [23], For an overview of estimation methods for the multivariate Student t-distribution, in particular the EM algorithm and its variants, we refer to [37] and the references therein.Recently, the Student-t distribution and the closely related Student-t mixture models (SMM) have found interesting applications in various image processing tasks. One of the first papers which suggested a variational approach for denoising of images corrupted by Cauchy noise was [1]. In [24], the authors proposed a unified framework for images corrupted by white noise that can handle (range constrained) Cauchy noise as well. Other recent approaches that consider also the task of deblurring include [11,54]. Concerning mixture models, in [48] it has been shown that Student-t mixture models are superior to Gaussian mixture models for modeling image patches and the authors proposed an application in image compression. Further applications include robust image segmentation [2,38,45] as well as robust registration [16,55]. In both cases, the SMM is estimated using the EM algorithm derived in [39].In this paper, we propose an application of the Student-t distribution to robust denoising of images corrupted by different kinds of noise. The initial motivation for this work were the recent papers [26,35,43] for Cauchy noise removal. In [35,43] the authors propos...

Alternatives to the EM Algorithm for ML-Estimation of Location, Scatter Matrix and Degree of Freedom of the Student-$t$ Distribution

“…We restrict our attention to the case µ = 0. For an approach how to extend the results to arbitrary µ for fixed ν we refer to [13]. For fixed ν > 0, it is known that there exists a unique solution of (3) and for ν = 0 that there exists solutions of (3) which differ only by a multiplicative positive constant, see, e.g.…”

“…For fixed ν > 0, it is known that there exists a unique solution of (3) and for ν = 0 that there exists solutions of (3) which differ only by a multiplicative positive constant, see, e.g. [13]. In contrast, if we do not fix ν, we have roughly to distinguish between the two cases that the samples tend to come from a Gaussian distribution or not.…”

“…Assume that either {λ r1 : r ∈ N} is unbounded or that {λ rd : r ∈ N} has zero as a cluster point. Then, we know by [13,Theorem 4.3] that there exists a subsequence of (Σ r ) r , which we again denote by (Σ r ) r , such that for any fixed ν > 0 it holds…”

“…In [24] it has been shown that Student-t mixture models are superior to Gaussian mixture models for modeling image patches and the authors proposed an application in image compression. Image denoising based on Student-t models was addressed in [13] and image deblurring in [5,26]. Further applications include robust image segmentation [3,18,22] as well as robust registration [6,27].…”

“…For denoising images corrupted by additive Cauchy noise, a similar approach was addressed in [13] based on ML estimation for the family of Student-t distributions, of which the Cauchy distribution forms a special case. The authors call this approach generalized multivariate myriad filter.…”