PhD Thesis Colloquium

Name of Student: Ruturaj Gavaskar.

Guide: Prof. Kunal Narayan Chaudhury

Date:  April 28, Thursday.               

Time: 11-12 am.

Venue: MS Teams (online).

Link: https://tinyurl.com/bdfardzz

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.