# Back-Projection based Fidelity Term for Ill-Posed Linear Inverse Problems

arXiv: https://arxiv.org/abs/1906.06794 · HTML: https://mlsanity.com/paper/1906.06794/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2019.

## Built on by (most cited first, 11 of 11)

- [Image Restoration by Denoising Diffusion Models with Iteratively Preconditioned Guidance](https://mlsanity.com/paper/2312.16519/) (2023)
- [Zero-Shot Image Restoration Using Few-Step Guidance of Consistency Models (and Beyond)](https://mlsanity.com/paper/2412.20596/) (2024)
- [Deep Internal Learning: Deep Learning from a Single Input](https://mlsanity.com/paper/2312.07425/) (2023)
- [Image Restoration by Deep Projected GSURE](https://mlsanity.com/paper/2102.02485/) (2021)
- [A Comparison of Image Denoising Methods](https://mlsanity.com/paper/2304.08990/) (2023)
- [Deep Unfolding of the DBFB Algorithm with Application to ROI CT Imaging with Limited Angular Density](https://mlsanity.com/paper/2209.13264/) (2022)
- [PnP-CM: Consistency Models as Plug-and-Play Priors for Inverse Problems](https://mlsanity.com/paper/2509.22736/) (2025)
- [An unsupervised deep learning framework for medical image denoising](https://mlsanity.com/paper/2103.06575/) (2021)
- [Bi-level Guided Diffusion Models for Zero-Shot Medical Imaging Inverse Problems](https://mlsanity.com/paper/2404.03706/) (2024)
- [An Interpretation of Regularization by Denoising and its Application with the Back-Projected Fidelity Term](https://mlsanity.com/paper/2101.11599/) (2021)
- [MrSARP: A Hierarchical Deep Generative Prior for SAR Image Super-resolution](https://mlsanity.com/paper/2212.00069/) (2022)

## Abstract (arXiv)

Ill-posed linear inverse problems appear in many image processing applications, such as deblurring, super-resolution and compressed sensing. Many restoration strategies involve minimizing a cost function, which is composed of fidelity and prior terms, balanced by a regularization parameter. While a vast amount of research has been focused on different prior models, the fidelity term is almost always chosen to be the least squares (LS) objective, that encourages fitting the linearly transformed optimization variable to the observations. In this paper, we examine a different fidelity term, which has been implicitly used by the recently proposed iterative denoising and backward projections (IDBP) framework. This term encourages agreement between the projection of the optimization variable onto the row space of the linear operator and the pseudo-inverse of the linear operator ("back-projection") applied on the observations. We analytically examine the difference between the two fidelity terms for Tikhonov regularization and identify cases (such as a badly conditioned linear operator) where the new term has an advantage over the standard LS one. Moreover, we demonstrate empirically that the behavior of the two induced cost functions for sophisticated convex and non-convex priors, such as total-variation, BM3D, and deep generative models, correlates with the obtained theoretical analysis.