Patch-Based Image Restoration Using Expectation Propagation

Dan Yao,Stephen Mclaughlin,Yoann Altmann

doi:10.1137/21m1427541

Abstract

This paper presents a new Expectation Propagation (EP) framework for image restoration using patch-based prior distributions. While Monte Carlo techniques are classically used to sample from intractable posterior distributions, they can suffer from scalability issues in high-dimensional inference problems such as image restoration. To address this issue, EP is used here to approximate the posterior distributions using products of multivariate Gaussian densities. Moreover, imposing structural constraints on the covariance matrices of these densities allows for greater scalability and distributed computation. While the method is naturally suited to handle additive Gaussian observation noise, it can also be extended to non-Gaussian noise. Experiments conducted for denoising, inpainting, and deconvolution problems with Gaussian and Poisson noise illustrate the potential benefits of such a flexible approximate Bayesian method for uncertainty quantification in imaging problems, at a reduced computational cost compared to sampling techniques.

Highlights

1.1 Problem FormulationIn this paper, we address the problem of image restoration which consists of estimating an unknown image from its degraded observation, e.g., noisy, blurry, or missing pixels
We proposed a new Expectation Propagation (EP) framework for image restoration using patch-based priors
In a similar fashion to Variational Bayes (VB), EP approximates the posterior distribution of interest by a simpler distribution whose moments are tractable

Summary

Introduction

We address the problem of image restoration which consists of estimating an unknown image from its degraded observation, e.g., noisy, blurry, or missing pixels. The (vectorized) observed image y ∈ RN is modeled as a known linear transformation of the unknown image of interest x ∈ RN. The linear transformation is denoted by Hx and the matrix H ∈ RN×N is either diagonal (e.g., as in denoising and inpainting problems) or diagonalizable in a transformed domain (e.g., as in deconvolution problems). The observation noise can be either Gaussian, independently and identically distributed (i.i.d.) or more complex, e.g., non-Gaussian. Adopting a Bayesian approach, the observation model described by the likelihood function fy|x(y|Hx), and a prior distribution fx(x|θ) parameterized by θ and based on image patches, are adopted to perform image restoration.

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