Gaussian Priors for Image Denoising

Abstract

This chapter is dedicated to the study of Gaussian priors for patch-based image denoising. In the last twelve years, patch priors have been widely used for image restoration. In a Bayesian framework, such priors on patches can be used for instance to estimate a clean patch from its noisy version, via classical estimators such as the conditional expectation or the maximum a posteriori. As we will recall, in the case of Gaussian white noise, simply assuming Gaussian (or Mixture of Gaussians) priors on patches leads to very simple closed-form expressions for some of these estimators. Nevertheless, the convenience of such models should not prevail over their relevance. For this reason, we also discuss how these models represent patches and what kind of information they encode. The end of the chapter focuses on the different ways in which these models can be learned on real data. This stage is particularly challenging because of the curse of dimensionality. Through these different questions, we compare and connect several denoising methods using this framework.

Generalized Wasserstein barycenters between probability measures living on different subspaces

“…From this follows that the right hand-side of ( 12) is greater than or equal to the right hand-side of (11). Since the other inequality is obvious, this completes the proof of (12). Now, let us prove (11).…”

“…A concrete application where a distribution must be reconstructed from a set of marginals appears in image processing with patch-based aggregation [28]. Patches are small overlapping image pieces and it is usual to infer stochastic models (for example a Gaussian or GMM distribution) on these patches [12]. Typically, each of these models is a distribution in R 9 (for 3 × 3 patches).…”

Research Trends and Case Studies of Deep Learning Applications in Geo-electric and Electromagnetic Surveys

“…여기서, 시간영역 전자탐사(transient/ time-domain EM, TEM)과 시간영역 유도분극과 같은 탐 사에서의 잡음 제거는 비유일해를 가진 불량조건 문제에서 수치 저하(forward degraded) 과정을 기반으로 진행할 수 있다 (Chen et al, 2020). 즉, 베이지안(Bayesian) 관점에서 사후 확률 분포(posteriori distribution)로서의 해인  는 일 반적으로 최대 사후 확률(maximum a posteriori, MAP) 문 제를 해결함으로써 다음과 같이 계산할 수 있다 (Zhang et al, 2017b;Kataoka and Yasuda, 2019;Delon and Houdard, 2018).…”

PatchNR: Learning from Small Data by Patch Normalizing Flow Regularization

“…Denoising methods based on the comparison of similar patches provided state-of-the-art methods [9,12,39,40] for a long time. Recently, the approximation of patch distributions of images was successfully exploited in certain papers [3,13,22,24,25,56]. In particular, the authors of [71] proposed the negative log likelihood of all patches of an image as a regularizer, where the underlying patch distribution was assumed to follow a Gaussian mixture model (GMM) which parameters were learned from few clean images.…”