A Projectional Ansatz to Reconstruction

TL;DR

This paper introduces a projectional approach for inverse problems that integrates learned and non-learned priors while ensuring data consistency, using a novel neural architecture and connections to existing methods like RED and DIP.

Contribution

The paper develops a projectional method combining Plug-and-Play priors and end-to-end training with a new neural architecture for customized priors, advancing inverse problem solutions.

Findings

PM can incorporate various priors while guaranteeing data consistency.

The approach explains RED as an approximation of PM.

It helps prevent degradation in DIP reconstructions.

Abstract

Recently the field of inverse problems has seen a growing usage of mathematically only partially understood learned and non-learned priors. Based on first principles, we develop a projectional approach to inverse problems that addresses the incorporation of these priors, while still guaranteeing data consistency. We implement this projectional method (PM) on the one hand via very general Plug-and-Play priors and on the other hand, via an end-to-end training approach. To this end, we introduce a novel alternating neural architecture, allowing for the incorporation of highly customized priors from data in a principled manner. We also show how the recent success of Regularization by Denoising (RED) can, at least to some extent, be explained as an approximation of the PM. Furthermore, we demonstrate how the idea can be applied to stop the degradation of Deep Image Prior (DIP) reconstructions over time.