Regularization by Denoising: Clarifications and New Interpretations
This work addresses theoretical inconsistencies in a popular image-recovery framework, providing clarifications and new interpretations that could impact researchers and practitioners in computational imaging and machine learning, though it is incremental in refining existing methods.
The paper challenges the theoretical justification of the Regularization by Denoising (RED) framework by showing that its explicit regularization claims rely on unrealistic symmetry assumptions not met by practical denoisers, and it proposes a new Score-Matching by Denoising (SMD) framework with connections to kernel density estimation and new algorithms offering acceleration and convergence guarantees.
Regularization by Denoising (RED), as recently proposed by Romano, Elad, and Milanfar, is powerful image-recovery framework that aims to minimize an explicit regularization objective constructed from a plug-in image-denoising function. Experimental evidence suggests that the RED algorithms are state-of-the-art. We claim, however, that explicit regularization does not explain the RED algorithms. In particular, we show that many of the expressions in the paper by Romano et al. hold only when the denoiser has a symmetric Jacobian, and we demonstrate that such symmetry does not occur with practical denoisers such as non-local means, BM3D, TNRD, and DnCNN. To explain the RED algorithms, we propose a new framework called Score-Matching by Denoising (SMD), which aims to match a "score" (i.e., the gradient of a log-prior). We then show tight connections between SMD, kernel density estimation, and constrained minimum mean-squared error denoising. Furthermore, we interpret the RED algorithms from Romano et al. and propose new algorithms with acceleration and convergence guarantees. Finally, we show that the RED algorithms seek a consensus equilibrium solution, which facilitates a comparison to plug-and-play ADMM.