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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. Block Matrix and Tensor Randomized Kaczmarz Methods for Linear Feasibility Problems

   https://doi.org/10.1007/s44007-025-00163-z  

   Zhang, Minxin; Haddock, Jamie; Needell, Deanna (May 2025, La Matematica)

   Abstract The randomized Kaczmarz methods are a popular and effective family of iterative methods for solving large-scale linear systems of equations, which have also been applied to linear feasibility problems. In this work, we propose a new block variant of the randomized Kaczmarz method, B-MRK, for solving linear feasibility problems defined by matrices. We show that B-MRK converges linearly in expectation to the feasible region. Furthermore, we extend the method to solve tensor linear feasibility problems defined under the tensor t-product. A tensor randomized Kaczmarz (TRK) method, TRK-L, is proposed for solving linear feasibility problems that involve mixed equality and inequality constraints. Additionally, we introduce another TRK method, TRK-LB, specifically tailored for cases where the feasible region is defined by linear equality constraints coupled with bound constraints on the variables. We show that both of the TRK methods converge linearly in expectation to the feasible region. Moreover, the effectiveness of our methods is demonstrated through numerical experiments on various Gaussian random data and applications in image deblurring. 

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3. A new nonstationary preconditioned iterative method for linear discrete ill‐posed problems with application to image deblurring

   https://doi.org/10.1002/nla.2353  

   Buccini, Alessandro; Donatelli, Marco; Reichel, Lothar; Zhang, Wei‐Hong (December 2020, Numerical Linear Algebra with Applications)

   Abstract Discrete ill‐posed inverse problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods aim at reducing this sensitivity. This article considers an iterative regularization method, based on iterated Tikhonov regularization, that was proposed in M. Donatelli and M. Hanke, Fast nonstationary preconditioned iterative methods for ill‐posed problems, with application to image deblurring,Inverse Problems, 29 (2013), Art. 095008, 16 pages. In this method, the exact operator is approximated by an operator that is easier to work with. However, the convergence theory requires the approximating operator to be spectrally equivalent to the original operator. This condition is rarely satisfied in practice. Nevertheless, this iterative method determines accurate image restorations in many situations. We propose a modification of the iterative method, that relaxes the demand of spectral equivalence to a requirement that is easier to satisfy. We show that, although the modified method is not an iterative regularization method, it maintains one of the most important theoretical properties for this kind of methods, namely monotonic decrease of the reconstruction error. Several computed experiments show the good performances of the proposed method. 

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4. Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems

   Hurault, Samuel; Kamilov, Ulugbek S.; Leclaire, Arthur; Papadakis, Nicolas (December 2023, Conference on Neural Information Processing Systems (NeurIPS))

   Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using the Bregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance. 

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5. Random mesh projectors for inverse problems

   Konik Kothari, Sidharth Gupta (May 2019, 7th International Conference on Learning Representations, ICLR 2019)

   We propose a new learning-based approach to solve ill-posed inverse problems in imaging. We address the case where ground truth training samples are rare and the problem is severely ill-posed-both because of the underlying physics and because we can only get few measurements. This setting is common in geophysical imaging and remote sensing. We show that in this case the common approach to directly learn the mapping from the measured data to the reconstruction becomes unstable. Instead, we propose to first learn an ensemble of simpler mappings from the data to projections of the unknown image into random piecewise-constant subspaces. We then combine the projections to form a final reconstruction by solving a deconvolution-like problem. We show experimentally that the proposed method is more robust to measurement noise and corruptions not seen during training than a directly learned inverse. 

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