A Bayesian Hyperprior Approach for Joint Image Denoising and Interpolation, With an Application to HDR Imaging

Abstract

Abstract-Recently, impressive denoising results have been achieved by Bayesian approaches which assume Gaussian models for the image patches. This improvement in performance can be attributed to the use of per-patch models. Unfortunately such an approach is particularly unstable for most inverse problems beyond denoising. In this work, we propose the use of a hyperprior to model image patches, in order to stabilize the estimation procedure. There are two main advantages to the proposed restoration scheme: Firstly it is adapted to diagonal degradation matrices, and in particular to missing data problems (e.g. inpainting of missing pixels or zooming). Secondly it can deal with signal dependent noise models, particularly suited to digital cameras. As such, the scheme is especially adapted to computational photography. In order to illustrate this point, we provide an application to high dynamic range imaging from a single image taken with a modified sensor, which shows the effectiveness of the proposed scheme.Index Terms-Non-local patch-based restoration, Bayesian restoration, Maximum a Posteriori, Gaussian Mixture Models, hyper-prior, conjugate distributions, high dynamic range imaging, single shot HDR, hierarchical models.

Video Denoising via Empirical Bayesian Estimation of Space-Time Patches

“…As in [14,15,32,39] we perform two stages. The first stage computes a basic estimate u (1) . The second stage computes the final estimate u (2) using the basic estimate as an oracle: patches from the basic estimate are used to compute the patch distances and to estimate the mean and covariance matrix of the a priori distribution.…”

“…However, the potential of these techniques for video restoration remains mostly unexplored. 1 We refer to our method as an empirical Bayesian approach because although it is based on a Bayesian model for patches, the parameters of the prior are determined from the data using frequentist estimators. The empirical (or data-driven) Bayesian approach is a usual choice when the parameters of the prior are unknown, but it is a departure from a strict Bayesian methodology which would require a hyperprior to estimate the parameters (see [1] for a closely related method considering a Bayesian estimation of the parameters of the prior).…”

“…Extending these approaches to video is challenging because of the high variability of spatio-temporal patches. Thus we shall stick to the idea of using video redundancy to build a local Gaussian model for each patch from its most similar patches as in [1,32,61]. Such an approach is applied in [8] to video denoising, but restricted to 2D patches.…”

Gaussian Priors for Image Denoising

“…Among these methods, one of the most popular remains the Non-Local Means [5], which sees similar patches as independent realizations of the same distribution and averages these repeated structures to reduce noise variance. If numerous approaches have built on the same core ideas since 2004, the recent and most convincing approaches in patch-based denoising rely on a Bayesian reformulation of the denoising problem, using local or global statistical priors for the distribution of each patch [12,24,23,20,1,11]. Under the white Gaussian noise model (2), the conditional distribution of a noisy patch y knowing its original version x (we omit the index i in the following) can be written…”

“…On the other hand, under a Gaussian prior, the conditional expectation, Wiener estimator and MAP coincide, as we will see in Section 3. For these reasons, these priors are favored in most recent works on patch-based image denoising [6,12,1]. A slightly more involved prior used in the literature is the Gaussian Mixture Model (GMM) [24,19,23,20,11].…”