On Robust Estimation and smoothing with Spatial and Tonal Kernels

Abstract

This paper deals with establishing relations between a number of widely-used nonlinear filters for digital image processing. We cover robust statistical estimation with (local) M-estimators, local mode filtering in image or histogram space, bilateral filtering, nonlinear diffusion, and regularisation approaches. Although these methods originate in different mathematical theories, we show that their implementation reveals a highly similar structure. We demonstrate that all these methods can be cast into a unified framework of functional minimisation combining nonlocal data and nonlocal smoothness terms. This unification contributes to a better understanding of the individual methods, and it opens the way to new techniques combining the advantages of known filters.

Automated Diffeomorphic Registration of Anatomical Structures with Rigid Parts: Application to Dynamic Cervical MRI

“…Solving this minimisation problem is known as local M-smoothing [15], due to the use of a robust error norm and of spatial neighbourhoods to design C; C can be minimised using gradient descent, where each transformation can be estimated independently of the others. Using a particular adaptive, data-dependent step size leads to an easy-to-interpret update formula for each log S i [15]:…”

Rotationally invariant similarity measures for nonlocal image denoising

“…It belongs to the class of adaptive averaging filters, like the Yaroslavsky [41] or the bilateral filter [3,35,38]. In this context, Mrázek et al [32] and Pizarro et al [33] have presented unified frameworks that include many denoising methods like bilateral filtering or M-smoothers [10,39] as special cases. The difference between NL means and previous approaches is the way of calculating the weights for the averaging process with the consideration of neighbourhood information.…”

Generalised Nonlocal Image Smoothing

“…There exist numerous approaches to image smoothing emerging from statistical methods, information theory, transforms in the frequency domain, partial differential equations (PDEs) and variational methods [82,1,84,18]. Establishing equivalences and relations between the different approaches has been focus of intense research in recent years [6,31,32,56,67,68,77,71,85]. Mrázek et al [56] pointed out the relations between several nonlinear smoothing methods such as M-estimators [23,85], bilateral filtering [75], diffusion filters [61,81], and regularisation/Bayesian techniques [7,35,57,85].…”

“…Establishing equivalences and relations between the different approaches has been focus of intense research in recent years [6,31,32,56,67,68,77,71,85]. Mrázek et al [56] pointed out the relations between several nonlinear smoothing methods such as M-estimators [23,85], bilateral filtering [75], diffusion filters [61,81], and regularisation/Bayesian techniques [7,35,57,85]. Although these methods seem very different at the first glance and originate in different mathematical theories, Mrázek et al showed that they lead to highly similar discrete algorithms, and that all these methods can be cast in a single unified framework of discrete regularisation theory.…”

“…This method has inspired the appearance of numerous so-called patch-based approaches for image smoothing, deblurring, segmentation, inpainting, super-resolution, and texture synthesis, among others. In this paper we propose the Generalised NDS (GNDS) framework for image smoothing as an extension of the NDS model of Mrázek et al [56]. Instead of penalising deviations from similarity considering only single pixel differences, as in the NDS model, we introduce a discrete variational approach with nonlocal constraints that penalise general dissimilarity measures defined on image patches.…”