  The vast majority of image recovery tasks are ill-posed problems. As such,
methods that are based on optimization use cost functions that consist of both
fidelity and prior (regularization) terms. A recent line of works imposes the
prior by the Regularization by Denoising (RED) approach, which exploits the
good performance of existing image denoising engines. Yet, the relation of RED
to explicit prior terms is still not well understood, as previous work requires
too strong assumptions on the denoisers. In this paper, we make two
contributions. First, we show that the RED gradient can be seen as a
(sub)gradient of a prior function--but taken at a denoised version of the
point. As RED is typically applied with a relatively small noise level, this
interpretation indicates a similarity between RED and traditional gradients.
This leads to our second contribution: We propose to combine RED with the
Back-Projection (BP) fidelity term rather than the common Least Squares (LS)
term that is used in previous works. We show that the advantages of BP over LS
for image deblurring and super-resolution, which have been demonstrated for
traditional gradients, carry on to the RED approach.
