The Anscombe transform offers an approximate conversion of a Poisson random variable into a Gaussian one. This transform is important and appealing, as it is easy to compute, and becomes handy in various inverse problems with Poisson noise contamination. Solution to such problems can be done by first applying the Anscombe transform, then applying a Gaussian-noise-oriented restoration algorithm of choice, and finally applying an inverse Anscombe transform. The appeal in this approach is due to the abundance of high-performance restoration algorithms designed for white additive Gaussian noise (we will refer to these hereafter as "Gaussian-solvers"). This process is known to work well for high SNR images, where the Anscombe transform provides a rather accurate approximation. When the noise level is high, the above path loses much of its effectiveness, and the common practice is to replace it with a direct treatment of the Poisson distribution. Naturally, with this we lose the ability to leverage on vastly available Gaussian-solvers.
In this work we suggest a novel method for coupling Gaussian denoising algorithms to Poisson noisy inverse problems, which is based on a general approach termed "Plug-and-Play". Deploying the Plug-and-Play approach to such problems leads to an iterative scheme that repeats several key steps: 1) A convex programming task of simple form that can be easily treated; 2) A powerful Gaussian denoising algorithm of choice; and 3) A simple update step.
Such a modular method, just like the Anscombe transform, enables other developers to plug their own Gaussian denoising algorithms to our scheme in an easy way. While the proposed method bares some similarity to the Anscombe operation, it is in fact based on a different mathematical basis, which holds true for all SNR ranges.
In this work we propose an efficient stochastic plug-and-play (PnP) algorithm for imaging inverse problems. The PnP stochastic gradient descent methods have been recently proposed and shown improved performance in some imaging applications over standard deterministic PnP methods. However, current stochastic PnP methods need to frequently compute the image denoisers which can be computationally expensive. To overcome this limitation, we propose a new stochastic PnP-ADMM method which is based on introducing stochastic gradient descent inner-loops within an inexact ADMM framework. We provide the …
Single-Shot Plug-and-Play Methods for Inverse Problems
University of Cambridge（剑桥大学）；Hong Kong Polytechnic University（香港理工大学）；The University of Hong Kong（香港大学）；Lingnan University（岭南大学）；Tsinghua University（清华大学）；City University of Hong Kong（香港城市大学）
Beijing Institute of Technology（北京理工大学）；University of Cambridge（剑桥大学）
Plug-and-play (PnP) is a non-convex framework that combines ADMM or other proximal algorithms with advanced denoiser priors. Recently, PnP has achieved great empirical success, especially with the integration of deep learning-based denoisers. However, a key problem of PnP based approaches is that they require manual parameter tweaking. It is necessary to obtain high-quality results across the high discrepancy in terms of imaging conditions and varying scene content. In this work, we present a tuning-free PnP proximal algorithm, which can automatically determine the internal parameters includin…
Instituto de Telecomunicações（电信研究所）；Instituto Superior Técnico（高等技术学院）
The recently proposed plug-and-play (PnP) framework allows leveraging recent developments in image denoising to tackle other, more involved, imaging inverse problems. In a PnP method, a black-box denoiser is plugged into an iterative algorithm, taking the place of a formal denoising step that corresponds to the proximity operator of some convex regularizer. While this approach offers flexibility and excellent performance, convergence of the resulting algorithm may be hard to analyze, as most state-of-the-art denoisers lack an explicit underlying objective function. In this paper, we propose a …
We introduce a novel energy formulation for Plug- and-Play (PnP) image recovery. Traditional PnP methods that use a convolutional neural network (CNN) do not have an energy based formulation. The primary focus of this work is to introduce an energy-based PnP formulation, which relies on a CNN that learns the log of the image prior from training data. The score function is evaluated as the gradient of the energy model, which resembles a UNET with shared encoder and decoder weights. The proposed score function is thus constrained to a conservative vector field, which is the key difference with c…
Plug-and-Play (PnP) methods solve ill-posed inverse problems through iterative proximal algorithms by replacing a proximal operator by a denoising operation. When applied with deep neural network denoisers, these methods have shown state-of-the-art visual performance for image restoration problems. However, their theoretical convergence analysis is still incomplete. Most of the existing convergence results consider nonexpansive denoisers, which is non-realistic, or limit their analysis to strongly convex data-fidelity terms in the inverse problem to solve. Recently, it was proposed to train th…
Deep Equilibrium models for Poisson Imaging Inverse problems via Mirror Descent
MaLGa Center, DIBRIS, University of Genoa（马尔加中心，热拉尔迪布里斯大学）；MaLGa Center, DIIMA, University of Genoa（马尔加中心，迪玛大学）；CNRS & Université Côte d’Azur, Laboratoire J. A. Dieudonné（法国国家科学研究中心与科蒂迪昂大学，J. A. Dieudonné实验室）；MMS, Istituto Italiano di Tecnologia (IIT)（MMS，意大利技术研究院（IIT））
Alternating direction method of multiplier (ADMM) is a widely used algorithm for solving constrained optimization problems in image restoration. Among many useful features, one critical feature of the ADMM algorithm is its modular structure which allows one to plug in any off-the-shelf image denoising algorithm for a subproblem in the ADMM algorithm. Because of the plug-in nature, this type of ADMM algorithms is coined the name "Plug-and-Play ADMM". Plug-and-Play ADMM has demonstrated promising empirical results in a number of recent papers. However, it is unclear under what conditions and by …
Beijing Institute of Technology（北京理工大学）；University of Cambridge（剑桥大学）；Shanghai Jiao Tong University（上海交通大学）；Beijing Normal University（北京师范大学）
Plug-and-Play (PnP) is a non-convex optimization framework that combines proximal algorithms, for example, the alternating direction method of multipliers (ADMM), with advanced denoising priors. Over the past few years, great empirical success has been obtained by PnP algorithms, especially for the ones that integrate deep learning-based denoisers. However, a key challenge of PnP approaches is the need for manual parameter tweaking as it is essential to obtain high-quality results across the high discrepancy in imaging conditions and varying scene content. In this work, we present a class of t…
We present a novel algorithm for blind denoising of images corrupted by mixed impulse, Poisson, and Gaussian noises. The algorithm starts by applying the Anscombe variance-stabilizing transformation to convert the Poisson into white Gaussian noise. Then it applies a combinatorial optimization technique to denoise the mixed impulse Gaussian noise using proximal algorithms. The result is then processed by the inverse Anscombe transform. We compare our algorithm to state of the art methods on standard images, and show its superior performance in various noise conditions.
Plug-and-Play optimization recently emerged as a powerful technique for solving inverse problems by plugging a denoiser into a classical optimization algorithm. The denoiser accounts for the regularization and therefore implicitly determines the prior knowledge on the data, hence replacing typical handcrafted priors. In this paper, we extend the concept of plug-and-play optimization to use denoisers that can be parameterized for non-constant noise variance. In that aim, we introduce a preconditioning of the ADMM algorithm, which mathematically justifies the use of such an adjustable denoiser. …
Plug-and-Play methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser. Although Plug-and-Play methods can lead to tremendous visual performance for various image problems, the few existing convergence guarantees are based on unrealistic (or suboptimal) hypotheses on the denoiser, or limited to strongly convex data terms. In this work, we propose a new type of Plug-and-Play methods, based on half-quadratic splitting, for which the denoiser is realized as a gradient descent step on a functional parameterized by a deep…
Inverse problem regularization with hierarchical variational autoencoders
Univ. Bordeaux（波尔多大学）；CNRS（法国国家科学研究中心）；Bordeaux INP（波尔多国立综合理工学院）；IMB（数学与计算机科学研究所）；Ubisoft La Forge（育碧拉福奇工作室）；Université Paris Cité（巴黎城市大学）；MAP5（数学与应用概率科学研究所）
Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems
School of Mathematical Sciences, East China Normal University（东华大学数学系）；School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education), Shanghai Key Laboratory of PMMP, East China Normal University（东华大学数学系）；Institute for Advanced Study, Beijing Normal-Hong Kong Baptist University（北京师范大学-香港 Baptist大学高级研究院）；School of Mathematics and Statistics, Guangzhou Nanfang College（广州南方学院数学系）
The Ohio State University（俄亥俄州立大学）；Purdue University（普渡大学）
Magnetic Resonance Imaging (MRI) is a non-invasive diagnostic tool that provides excellent soft-tissue contrast without the use of ionizing radiation. Compared to other clinical imaging modalities (e.g., CT or ultrasound), however, the data acquisition process for MRI is inherently slow, which motivates undersampling and thus drives the need for accurate, efficient reconstruction methods from undersampled datasets. In this article, we describe the use of "plug-and-play" (PnP) algorithms for MRI image recovery. We first describe the linearly approximated inverse problem encountered in MRI. Then…
You KAN Do It in a Single Shot: Plug-and-Play Methods with Single-Instance Priors
Department of Applied Mathematics and Theoretical Physics（应用数学与理论物理系）；University of Cambridge（剑桥大学）；Yau Mathematical Sciences Center（叶中伟数学科学中心）；Tsinghua University（清华大学）
提出 KAN-PnP 优化框架，将 KAN 作为去噪器融入即插即用范式解决单实例先验逆问题，证明了其 Lipschitz 连续性与收敛性，并在超分辨率等任务上以更少迭代实现高精度。
Washington University in St. Louis（圣路易斯华盛顿大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Plug-and-play priors (PnP) is a powerful framework for regularizing imaging inverse problems by using advanced denoisers within an iterative algorithm. Recent experimental evidence suggests that PnP algorithms achieve state-of-the-art performance in a range of imaging applications. In this paper, we introduce a new online PnP algorithm based on the iterative shrinkage/thresholding algorithm (ISTA). The proposed algorithm uses only a subset of measurements at every iteration, which makes it scalable to very large datasets. We present a new theoretical convergence analysis, for both batch and on…
Removing optical and atmospheric blur from galaxy images significantly improves galaxy shape measurements for weak gravitational lensing and galaxy evolution studies. This ill-posed linear inverse problem is usually solved with deconvolution algorithms enhanced by regularisation priors or deep learning. We introduce a so-called "physics-informed deep learning" approach to the Point Spread Function (PSF) deconvolution problem in galaxy surveys. We apply algorithm unrolling and the Plug-and-Play technique to the Alternating Direction Method of Multipliers (ADMM), in which a neural network learns…
Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problems
Institut Denis Poisson（德尼·泊松研究所）；CNRS（法国国家科学研究中心）；University of Tours（图尔大学）；Univ Rouen Normandie（鲁昂诺曼底大学）；INSA Rouen Normandie（鲁昂诺曼底国立应用科学学院）；Normandie Univ（诺曼底大学）；LITIS（信息、信号与图像处理实验室）；University of Toulouse（图卢兹大学）；IRIT（图卢兹计算机研究所）；INP-ENSEEIHT（图卢兹国立综合理工学院-高等电力、电子、计算机、水力与电信学院）；CEA（法国原子能和替代能源委员会）；University of Paris-Saclay（巴黎萨克雷大学）
Plug-and-play (PnP) method is a recent paradigm for image regularization, where the proximal operator (associated with some given regularizer) in an iterative algorithm is replaced with a powerful denoiser. Algorithmically, this involves repeated inversion (of the forward model) and denoising until convergence. Remarkably, PnP regularization produces promising results for several restoration applications. However, a fundamental question in this regard is the theoretical convergence of the PnP iterations, since the algorithm is not strictly derived from an optimization framework. This question …
ETH Zürich（苏黎世联邦理工学院）；KU Leuven（鲁汶大学）；Harbin Institute of Technology（哈尔滨工业大学）；The Hong Kong Polytechnic University（香港理工大学）
Recent works on plug-and-play image restoration have shown that a denoiser can implicitly serve as the image prior for model-based methods to solve many inverse problems. Such a property induces considerable advantages for plug-and-play image restoration (e.g., integrating the flexibility of model-based method and effectiveness of learning-based methods) when the denoiser is discriminatively learned via deep convolutional neural network (CNN) with large modeling capacity. However, while deeper and larger CNN models are rapidly gaining popularity, existing plug-and-play image restoration hinder…
In this work, we propose Regularization-by-Equivariance (REV), a novel structure-adaptive regularization scheme for solving imaging inverse problems under incomplete measurements. This regularization scheme utilizes the equivariant structure in the physics of the measurements -- which is prevalent in many inverse problems such as tomographic image reconstruction -- to mitigate the ill-poseness of the inverse problem. Our proposed scheme can be applied in a plug-and-play manner alongside with any classic first-order optimization algorithm such as the accelerated gradient descent/FISTA for simpl…
Washington University in St. Louis（圣路易斯华盛顿大学）；University of California, Riverside（加州大学河滨分校）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
The plug-and-play priors (PnP) and regularization by denoising (RED) methods have become widely used for solving inverse problems by leveraging pre-trained deep denoisers as image priors. While the empirical imaging performance and the theoretical convergence properties of these algorithms have been widely investigated, their recovery properties have not previously been theoretically analyzed. We address this gap by showing how to establish theoretical recovery guarantees for PnP/RED by assuming that the solution of these methods lies near the fixed-points of a deep neural network. We also pre…
