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

arXiv: https://arxiv.org/abs/2101.11599 · HTML: https://mlsanity.com/paper/2101.11599/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 22 references that are in the map.
- First posted on arXiv 27 Jan 2021.

## Builds on (most cited first, 15 of 15)

- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [DeepRED: Deep Image Prior Powered by RED](https://mlsanity.com/paper/1903.10176/) (2019)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [From Denoising to Compressed Sensing](https://mlsanity.com/paper/1406.4175/) (2014)
- [Plug-and-Play Unplugged: Optimization Free Reconstruction using Consensus Equilibrium](https://mlsanity.com/paper/1705.08983/) (2017)
- [Regularization by Denoising via Fixed-Point Projection (RED-PRO)](https://mlsanity.com/paper/2008.00226/) (2020)
- [Deep Mean-Shift Priors for Image Restoration](https://mlsanity.com/paper/1709.03749/) (2017)
- [Block Coordinate Regularization by Denoising](https://mlsanity.com/paper/1905.05113/) (2019)
- [Super-Resolution via Image-Adapted Denoising CNNs: Incorporating External and Internal Learning](https://mlsanity.com/paper/1811.12866/) (2018)
- [Back-Projection based Fidelity Term for Ill-Posed Linear Inverse Problems](https://mlsanity.com/paper/1906.06794/) (2019)
- [BP-DIP: A Backprojection based Deep Image Prior](https://mlsanity.com/paper/2003.05417/) (2020)
- [Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors](https://mlsanity.com/paper/2010.01446/) (2020)
- [On the Convergence Rate of Projected Gradient Descent for a Back-Projection based Objective](https://mlsanity.com/paper/2005.00959/) (2020)

## Abstract (arXiv)

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.