  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.
