  Removal of noise from an image is an extensively studied problem in image
processing. Indeed, the recent advent of sophisticated and highly effective
denoising algorithms lead some to believe that existing methods are touching
the ceiling in terms of noise removal performance. Can we leverage this
impressive achievement to treat other tasks in image processing? Recent work
has answered this question positively, in the form of the Plug-and-Play Prior
($P^3$) method, showing that any inverse problem can be handled by sequentially
applying image denoising steps. This relies heavily on the ADMM optimization
technique in order to obtain this chained denoising interpretation.
  Is this the only way in which tasks in image processing can exploit the image
denoising engine? In this paper we provide an alternative, more powerful and
more flexible framework for achieving the same goal. As opposed to the $P^3$
method, we offer Regularization by Denoising (RED): using the denoising engine
in defining the regularization of the inverse problem. We propose an explicit
image-adaptive Laplacian-based regularization functional, making the overall
objective functional clearer and better defined. With a complete flexibility to
choose the iterative optimization procedure for minimizing the above
functional, RED is capable of incorporating any image denoising algorithm,
treat general inverse problems very effectively, and is guaranteed to converge
to the globally optimal result. We test this approach and demonstrate
state-of-the-art results in the image deblurring and super-resolution problems.
