A Projectional Ansatz to Reconstruction
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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 which 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.
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