  Prior probability models are a fundamental component of many image processing
problems, but density estimation is notoriously difficult for high-dimensional
signals such as photographic images. Deep neural networks have provided
state-of-the-art solutions for problems such as denoising, which implicitly
rely on a prior probability model of natural images. Here, we develop a robust
and general methodology for making use of this implicit prior. We rely on a
statistical result due to Miyasawa (1961), who showed that the least-squares
solution for removing additive Gaussian noise can be written directly in terms
of the gradient of the log of the noisy signal density. We use this fact to
develop a stochastic coarse-to-fine gradient ascent procedure for drawing
high-probability samples from the implicit prior embedded within a CNN trained
to perform blind (i.e., with unknown noise level) least-squares denoising. A
generalization of this algorithm to constrained sampling provides a method for
using the implicit prior to solve any linear inverse problem, with no
additional training. We demonstrate this general form of transfer learning in
multiple applications, using the same algorithm to produce state-of-the-art
levels of unsupervised performance for deblurring, super-resolution,
inpainting, and compressive sensing.
