  We present a method for supervised learning of sparsity-promoting
regularizers for image denoising. Sparsity-promoting regularization is a key
ingredient in solving modern image reconstruction problems; however, the
operators underlying these regularizers are usually either designed by hand or
learned from data in an unsupervised way. The recent success of supervised
learning (mainly convolutional neural networks) in solving image reconstruction
problems suggests that it could be a fruitful approach to designing
regularizers. As a first experiment in this direction, we propose to denoise
images using a variational formulation with a parametric, sparsity-promoting
regularizer, where the parameters of the regularizer are learned to minimize
the mean squared error of reconstructions on a training set of (ground truth
image, measurement) pairs. Training involves solving a challenging bilievel
optimization problem; we derive an expression for the gradient of the training
loss using Karush-Kuhn-Tucker conditions and provide an accompanying gradient
descent algorithm to minimize it. Our experiments on a simple synthetic,
denoising problem show that the proposed method can learn an operator that
outperforms well-known regularizers (total variation, DCT-sparsity, and
unsupervised dictionary learning) and collaborative filtering. While the
approach we present is specific to denoising, we believe that it can be adapted
to the whole class of inverse problems with linear measurement models, giving
it applicability to a wide range of image reconstruction problems.
