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1. Block Coordinate Plug-and-Play Methods for Blind Inverse Problems

   Gan, Weijie ; Shoushtari, Shirin ; Hu, Yuyang ; Liu, Jiaming ; An, Hongyu ; Kamilov, Ulugbek S. ( December 2023 , Conference on Neural Information Processing Systems (NeurIPS))

   Plug-and-play (PnP) prior is a well-known class of methods for solving imaging inverse problems by computing fixed-points of operators combining physical measurement models and learned image denoisers. While PnP methods have been extensively used for image recovery with known measurement operators, there is little work on PnP for solving blind inverse problems. We address this gap by presenting a new block-coordinate PnP (BC-PnP) method that efficiently solves this joint estimation problem by introducing learned denoisers as priors on both the unknown image and the unknown measurement operator. We present a new convergence theory for BC-PnP compatible with blind inverse problems by considering nonconvex data-fidelity terms and expansive denoisers. Our theory analyzes the convergence of BC-PnP to a stationary point of an implicit function associated with an approximate minimum mean-squared error (MMSE) denoiser. We numerically validate our method on two blind inverse problems: automatic coil sensitivity estimation in magnetic resonance imaging (MRI) and blind image deblurring. Our results show that BC-PnP provides an efficient and principled framework for using denoisers as PnP priors for jointly estimating measurement operators and images. 

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2. Regularized Fourier Ptychography using an Online Plug-and-Play Algorithm

   https://doi.org/doi:10.1109/icassp.2019.8683057  

   Sun, Y. ; Xu, S. ; Tian, L. ; Wohlberg, B. ; Kamilov, U. ( May 2019 , ICASSP proceedings)

   The plug-and-play priors (PnP) framework has been recently shown to achieve state-of-the-art results in regularized image reconstruction by leveraging a sophisticated denoiser within an iterative algorithm. In this paper, we propose a new online PnP algorithm for Fourier ptychographic microscopy (FPM) based on the accelerated proximal gradient method (APGM). Specifically, the proposed algorithm uses only a subset of measurements, which makes it scalable to a large set of measurements. We validate the algorithm by showing that it can lead to significant performance gains on both simulated and experimental data. 

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3. Online Regularization by Denoising with Applications to Phase Retrieval

   https://doi.org/10.1109/ICCVW.2019.00482  

   Wu, Zihui ; Sun, Yu ; Liu, Jiaming ; Kamilov, Ulugbek ( October 2019 , 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW))

   Regularization by denoising (RED) is a powerful framework for solving imaging inverse problems. Most RED algorithms are iterative batch procedures, which limits their applicability to very large datasets. In this paper, we address this limitation by introducing a novel online RED (On-RED) algorithm, which processes a small subset of the data at a time. We establish the theoretical convergence of On-RED in convex settings and empirically discuss its effectiveness in non-convex ones by illustrating its applicability to phase retrieval. Our results suggest that On-RED is an effective alternative to the traditional RED algorithms when dealing with large datasets. 

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4. Block Coordinate Regularization by Denoising

   Sun, Yu ; Liu, Jiaming ; Kamilov, Ulugbek S ( January 2019 , Advances in neural information processing systems)

   We consider the problem of estimating a vector from its noisy measurements using a prior specified only through a denoising function. Recent work on plug- and-play priors (PnP) and regularization-by-denoising (RED) has shown the state- of-the-art performance of estimators under such priors in a range of imaging tasks. In this work, we develop a new block coordinate RED algorithm that decomposes a large-scale estimation problem into a sequence of updates over a small subset of the unknown variables. We theoretically analyze the convergence of the algorithm and discuss its relationship to the traditional proximal optimization. Our analysis complements and extends recent theoretical results for RED-based estimation methods. We numerically validate our method using several denoiser priors, including those based on convolutional neural network (CNN) denoisers. 

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5. Convergence of Nonconvex PNP-ADMM with MMSE Denoisers

   https://doi.org/10.1109/CAMSAP58249.2023.10403463  

   Park, Chicago ; Shoushtari, Shirin ; Gan, Weijie ; Kamilov, Ulugbek S. ( December 2023 , IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing)

   Plug-and-Play Alternating Direction Method of Multipliers (PnP-ADMM) is a widely-used algorithm for solving inverse problems by integrating physical measurement models and convolutional neural network (CNN) priors. PnP-ADMM has been theoretically proven to converge for convex data-fidelity terms and nonexpansive CNNs. It has however been observed that PnP-ADMM often empirically converges even for expansive CNNs. This paper presents a theoretical explanation for the observed stability of PnP-ADMM based on the interpretation of the CNN prior as a minimum mean-squared error (MMSE) denoiser. Our explanation parallels a similar argument recently made for the iterative shrinkage/thresholding algorithm variant of PnP (PnP-ISTA) and relies on the connection between MMSE denoisers and proximal operators. We also numerically evaluate the performance gap between PnP-ADMM using a nonexpan-sive DnCNN denoiser and expansive DRUNet denoiser, thus motivating the use of expansive CNNs. 

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