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

## TL;DR

This paper introduces a novel fidelity term based on back-projection for ill-posed linear inverse problems, demonstrating its advantages over traditional least squares in certain conditions and validating its effectiveness with various priors.

## Contribution

The paper proposes a new back-projection based fidelity term for inverse problems, providing theoretical analysis and empirical validation against standard least squares methods.

## Key findings

- The back-projection fidelity term performs better with badly conditioned operators.
- Theoretical analysis shows advantages of the new term in specific scenarios.
- Empirical results confirm improved performance with complex priors.

## Abstract

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.