  Deep neural network based methods are the state of the art in various image
restoration problems. Standard supervised learning frameworks require a set of
noisy measurement and clean image pairs for which a distance between the output
of the restoration model and the ground truth, clean images is minimized. The
ground truth images, however, are often unavailable or very expensive to
acquire in real-world applications. We circumvent this problem by proposing a
class of structured denoisers that can be decomposed as the sum of a nonlinear
image-dependent mapping, a linear noise-dependent term and a small residual
term. We show that these denoisers can be trained with only noisy images under
the condition that the noise has zero mean and known variance. The exact
distribution of the noise, however, is not assumed to be known. We show the
superiority of our approach for image denoising, and demonstrate its extension
to solving other restoration problems such as blind deblurring where the ground
truth is not available. Our method outperforms some recent unsupervised and
self-supervised deep denoising models that do not require clean images for
their training. For blind deblurring problems, the method, using only one noisy
and blurry observation per image, reaches a quality not far away from its fully
supervised counterparts on a benchmark dataset.
