An Interpretation of Regularization by Denoising and its Application with the Back-Projected Fidelity Term

TL;DR

This paper provides a new interpretation of Regularization by Denoising (RED) as a gradient of a prior function evaluated at a denoised image, and introduces combining RED with a Back-Projection fidelity term to improve image recovery tasks.

Contribution

It offers a theoretical interpretation of RED as a prior gradient at denoised points and proposes integrating RED with Back-Projection fidelity for enhanced image reconstruction.

Findings

RED can be viewed as a prior gradient at a denoised image

Combining RED with Back-Projection improves deblurring and super-resolution

Back-Projection outperforms Least Squares in this context

Abstract

The vast majority of image recovery tasks are ill-posed problems. As such, methods that are based on optimization use cost functions that consist of both fidelity and prior (regularization) terms. A recent line of works imposes the prior by the Regularization by Denoising (RED) approach, which exploits the good performance of existing image denoising engines. Yet, the relation of RED to explicit prior terms is still not well understood, as previous work requires too strong assumptions on the denoisers. In this paper, we make two contributions. First, we show that the RED gradient can be seen as a (sub)gradient of a prior function--but taken at a denoised version of the point. As RED is typically applied with a relatively small noise level, this interpretation indicates a similarity between RED and traditional gradients. This leads to our second contribution: We propose to combine RED with the Back-Projection (BP) fidelity term rather than the common Least Squares (LS) term that is used in previous works. We show that the advantages of BP over LS for image deblurring and super-resolution, which have been demonstrated for traditional gradients, carry on to the RED approach.