A Projectional Ansatz to Reconstruction
This work addresses a foundational problem in inverse problems for researchers and practitioners, offering a principled framework that is incremental but clarifies and enhances prior methods.
The paper tackles the challenge of incorporating learned and non-learned priors into inverse problems while ensuring data consistency, resulting in a projectional method that explains existing techniques like Regularization by Denoising and improves Deep Image Prior reconstructions.
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.