Regularization by Denoising: Clarifications and New Interpretations
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Regularization by Denoising (RED), as recently proposed by Romano, Elad, and Milanfar, is powerful new image-recovery framework that aims to construct an explicit regularization objective from a plug-in image-denoising function. Evidence suggests that the RED algorithms are, indeed, state-of-the-art. However, a closer inspection suggests that explicit regularization may not explain the workings of these algorithms. In this paper, we clarify that the expressions in 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. Going further, we prove that non-symmetric denoising functions cannot be modeled by any explicit regularizer. In light of these results, there remains the question of how to justify the good-performing RED algorithms for practical denoisers. In response, we propose a new framework called "score-matching by denoising" (SMD), which aims to match the score (i.e., the gradient of the log-prior) instead of designing the regularizer itself. We also show tight connections between SMD, kernel density estimation, and constrained minimum mean-squared error denoising. Finally, we provide new interpretations of the RED algorithms proposed by Romano et al., and we propose novel RED algorithms with faster convergence.
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