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PhD Thesis Colloquium
April 28, 2022 @ 11:00 am - 12:00 pm UTC+0
Name of Student: Ruturaj Gavaskar.
Guide: Prof. Kunal Narayan Chaudhuryhttps://tinyurl.com/bdfardzzDate: April 28, Thursday. Time: 11-12 am. Venue: MS Teams (online). Link:
Title: On plug-and-play regularization using linear denoisers.Abstract: The problem of inverting a given measurement model comes up in several computational imaging applications. For example, in CT and MRI, we are required to reconstruct a high-resolution image from incomplete noisy measurements, whereas in superresolution and deblurring, we try to infer the ground-truth from low-resolution or blurred images. While several forms of regularization and associated optimization methods have been proposed in the imaging literature of the last few decades, the use of denoisers (aka denoising priors) for image regularization is a relatively recent phenomenon. This has partly been triggered by the advances in image denoising in the last 20 years, leading to the development of powerful image denoisers. In this thesis, we look at a recent protocol called Plug-and-Play (PnP) method, where powerful image denoisers such as BM3D and DnCNN are deployed within iterative algorithms for image regularization. Surprisingly, the reconstructed images are of high quality and competitive with state-of-the-art methods. Following this, researchers have tried explaining why plugging a denoiser within an inversion algorithm should work in the first place, why it produces high-quality images, and whether the final reconstruction is optimal in some sense. We have tried answering some of these questions in this thesis. At a high level, the contributions of the thesis are as follows. Based on the theory of proximal operators, we prove that a PnP algorithm in fact minimizes a convex objective function provided the plugged denoiser belongs to a broad class L of linear filters. In particular, L has a simple characterization and includes kernel and GMM denoisers. That we are able to characterize the reconstruction (for class L denoisers) as the solution of a convex optimization problem helps in settling some of the above questions. For example, this allows us to establish iterate convergence for PnP regularization. Obtaining such a guarantee for complex nonlinear denoisers such as BM3D and neural denoisers is nontrivial. As a more profound application, we are able to provide guarantees on signal recovery for the compressed sensing problem. More precisely, under certain verifiable assumptions, we are able to prove that a signal can be recovered exactly (resp. stably) with high probability from random clean (resp. noisy) measurements using PnP regularization. To the best of our knowledge, this is the first such result where the underlying assumptions are verifiable. We will present and discuss these and other theoretical findings in greater detail during the colloquium. We will also present numerical results to validate our findings.