Provably Convergent Algorithms for Denoiser-Driven Image Regularization

Abstract

Some fundamental reconstruction tasks in image processing can be posed as an inverse problem where we are required to invert a given forward model. For example, in deblurring and superresolution, the ground-truth image needs to be estimated from blurred and low-resolution images, whereas in CT and MR imaging, a high-resolution image must be reconstructed from a few linear measurements. Such inverse problems are invariably ill-posed—they exhibit non-unique solutions and the process of direct inversion is unstable. Some form of image model (or prior) on the ground truth is required to regularize the inversion process. For example, a classical solution involves minimizing f + g , where the loss term f is derived from the forward model and the regularizer g is used to constrain the search space. The challenge is to come up with a formula for g that can yield good image reconstructions. This has been the center of research activity in image reconstruction for the last few decades. “Regularization using denoising" is a recent breakthrough in which a powerful denoiser is used for regularization purposes, instead of having to specify some hand-crafted g (but the loss f is still used). This has been empirically shown to yield significantly better results than staple f + g minimization. In fact, the results are generally comparable and often superior to state-of-the-art deep learning methods. In this thesis, we consider two such popular models for image regularization—Plug-and-Play (PnP) and Regularization by Denoising (RED). In particular, we focus on the convergence aspect of these iterative algorithms which is not well understood even for simple denoisers. This is important since the lack of convergence guarantee can result in spurious reconstructions in imaging applications. The contributions of the thesis in this regard are as follows. PnP with linear denoisers: We show that for a class of non-symmetric linear denoisers that includes kernel denoisers such as nonlocal means, one can associate a convex regularizer g with the denoiser. More precisely, we show that any such linear denoiser can be expressed as the proximal operator of a convex function, provided we work with a non-standard inner product (instead of the Euclidean inner product). In particular, the regularizer is quadratic, but unlike classical quadratic regularizers, the quadratic form is derived from the observed data. A direct implication of this observation is that (a simple variant of) the PnP algorithm based on this linear denoiser amounts to solving an optimization problem of the form f + g , though it was not originally conceived this way. Consequently, if f is convex, both objective and iterate convergence are guaranteed for the PnP algorithm. Apart from the convergence guarantee, we go on to show that this observation has algorithmic value as well. For example, in the case of linear inverse problems such as superresolution, deblurring and inpainting (where f is quadratic), we can reduce the problem of minimizing f + g to a linear system. In particular, we show how using Krylov solvers we can solve this system efficiently in just few iterations. Surprisingly, the reconstructions are found to be comparable with state-of-theart deep learning methods. To the best of our knowledge, the possibility of achieving near state-of-the-art image reconstructions using a linear solver has not been demonstrated before. PnP and RED with learning-based denoisers: In general, state-of-the-art PnP and RED algorithms rely on trained CNN denoisers such as DnCNN. Unlike linear denoisers, it is difficult to place PnP and RED algorithms within an optimization framework in the case of CNN denoisers. Nonetheless, we can still try to understand the convergence of the sequence of iterates generated by these algorithms. For convex loss f , we show that this question can be resolved using the theory of monotone operators — the denoiser being averaged (a subclass of nonexpansive operators) is sufficient for iterate convergence of PnP and RED. Using numerical examples, we show that existing CNN denoisers are not nonexpansive and can cause PnP and RED algorithms to diverge. Can we train denoisers that are provably nonexpansive? Unfortunately, this is computationally challenging—simply checking nonexpansivity of a CNN is known to be intractable. As a result, existing algorithms for training nonexpansive CNNs either cannot guarantee nonexpansivity or are computation intensive. We show that this problem can be solved by moving away from CNN denoisers to unfolded deep denoisers. In particular, we are able to construct unfolded networks that are efficiently trainable and come with convergence guarantees for PnP and RED algorithms, and whose regularization capacity can be matched withCNNdenoisers. Presumably, we are the first to propose a simple framework for training provably averaged (contractive) denoisers using unfolding networks. We provide numerical results to validate our theoretical results and compare our algorithms with state-of-the-art regularization techniques. We also point out some future research directions stemming from the thesis.