Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser
Center for Data Science（数据科学中心）；Howard Hughes Medical Institute（霍华德·休斯医学研究所）；New York University（纽约大学）；Center for Neural Science（神经科学中心）；Courant Inst. of Mathematical Sciences（柯朗数学科学研究所）
Zahra Kadkhodaie, Eero P. Simoncelli
备注：
19 pages, 12 figures. Changes: more detailed description of relationships to previous literature, including empirical comparisons for super-resolution, debarring, and compressive sensing
发表信息：
updated version published in Proc. Neural Information Processing Systems (NeurIPS) 2021
Prior probability models are a fundamental component of many image processing problems, but density estimation is notoriously difficult for high-dimensional signals such as photographic images. Deep neural networks have provided state-of-the-art solutions for problems such as denoising, which implicitly rely on a prior probability model of natural images. Here, we develop a robust and general methodology for making use of this implicit prior. We rely on a statistical result due to Miyasawa (1961), who showed that the least-squares solution for removing additive Gaussian noise can be written directly in terms of the gradient of the log of the noisy signal density. We use this fact to develop a stochastic coarse-to-fine gradient ascent procedure for drawing high-probability samples from the implicit prior embedded within a CNN trained to perform blind (i.e., with unknown noise level) least-squares denoising. A generalization of this algorithm to constrained sampling provides a method for using the implicit prior to solve any linear inverse problem, with no additional training. We demonstrate this general form of transfer learning in multiple applications, using the same algorithm to produce state-of-the-art levels of unsupervised performance for deblurring, super-resolution, inpainting, and compressive sensing.
Harbin Institute of Technology（哈尔滨工业大学）；The Hong Kong Polytechnic University（香港理工大学）
Model-based optimization methods and discriminative learning methods have been the two dominant strategies for solving various inverse problems in low-level vision. Typically, those two kinds of methods have their respective merits and drawbacks, e.g., model-based optimization methods are flexible for handling different inverse problems but are usually time-consuming with sophisticated priors for the purpose of good performance; in the meanwhile, discriminative learning methods have fast testing speed but their application range is greatly restricted by the specialized task. Recent works have …
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. …
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…
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…
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…
Washington University in St. Louis（圣路易斯华盛顿大学）；Department of Electrical and Systems Engineering（电气与系统工程系）；Department of Computer Science and Engineering（计算机科学与工程系）
Regularization by denoising (RED) is a widely-used framework for solving inverse problems by leveraging image denoisers as image priors. Recent work has reported the state-of-the-art performance of RED in a number of imaging applications using pre-trained deep neural nets as denoisers. Despite the recent progress, the stable convergence of RED algorithms remains an open problem. The existing RED theory only guarantees stability for convex data-fidelity terms and nonexpansive denoisers. This work addresses this issue by developing a new monotone RED (MRED) algorithm, whose convergence does not …
Technion – Israel Institute of Technology（以色列理工学院）
Image denoising (removal of additive white Gaussian noise from an image) is one of the oldest and most studied problems in image processing. An extensive work over several decades has led to thousands of papers on this subject, and to many well-performing algorithms for this task. Indeed, 10 years ago, these achievements have led some researchers to suspect that "Denoising is Dead", in the sense that all that can be achieved in this domain has already been obtained. However, this turned out to be far from the truth, with the penetration of deep learning (DL) into image processing. The era of D…
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 …
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）
Nanyang Technological University（南洋理工大学）；Michigan State University（密歇根州立大学）
Image restoration schemes based on the pre-trained deep models have received great attention due to their unique flexibility for solving various inverse problems. In particular, the Plug-and-Play (PnP) framework is a popular and powerful tool that can integrate an off-the-shelf deep denoiser for different image restoration tasks with known observation models. However, obtaining the observation model that exactly matches the actual one can be challenging in practice. Thus, the PnP schemes with conventional deep denoisers may fail to generate satisfying results in some real-world image restorati…
Harbin Institute of Technology（哈尔滨工业大学）；The Hong Kong Polytechnic University（香港理工大学）；Graz University of Technology（格拉茨技术大学）；Xi’an Jiaotong University（西安交通大学）
Discriminative model learning for image denoising has been recently attracting considerable attentions due to its favorable denoising performance. In this paper, we take one step forward by investigating the construction of feed-forward denoising convolutional neural networks (DnCNNs) to embrace the progress in very deep architecture, learning algorithm, and regularization method into image denoising. Specifically, residual learning and batch normalization are utilized to speed up the training process as well as boost the denoising performance. Different from the existing discriminative denois…
End-to-end deep neural networks (DNNs) have become the state-of-the-art (SOTA) for solving inverse problems. Despite their outstanding performance, during deployment, such networks are sensitive to minor variations in the testing pipeline and often fail to reconstruct small but important details, a feature critical in medical imaging, astronomy, or defence. Such instabilities in DNNs can be explained by the fact that they ignore the forward measurement model during deployment, and thus fail to enforce consistency between their output and the input measurements. To overcome this, we propose a f…
In this work we introduce a novel stochastic algorithm dubbed SNIPS, which draws samples from the posterior distribution of any linear inverse problem, where the observation is assumed to be contaminated by additive white Gaussian noise. Our solution incorporates ideas from Langevin dynamics and Newton's method, and exploits a pre-trained minimum mean squared error (MMSE) Gaussian denoiser. The proposed approach relies on an intricate derivation of the posterior score function that includes a singular value decomposition (SVD) of the degradation operator, in order to obtain a tractable iterati…
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。
Technion Israel Institute of Technology（以色列理工学院）；Google Research（谷歌研究院）
Inverse problems in image processing are typically cast as optimization tasks, consisting of data-fidelity and stabilizing regularization terms. A recent regularization strategy of great interest utilizes the power of denoising engines. Two such methods are the Plug-and-Play Prior (PnP) and Regularization by Denoising (RED). While both have shown state-of-the-art results in various recovery tasks, their theoretical justification is incomplete. In this paper, we aim to bridge between RED and PnP, enriching the understanding of both frameworks. Towards that end, we reformulate RED as a convex op…
Deep-learning-based hyperspectral image (HSI) restoration methods have gained great popularity for their remarkable performance but often demand expensive network retraining whenever the specifics of task changes. In this paper, we propose to restore HSIs in a unified approach with an effective plug-and-play method, which can jointly retain the flexibility of optimization-based methods and utilize the powerful representation capability of deep neural networks. Specifically, we first develop a new deep HSI denoiser leveraging gated recurrent convolution units, short- and long-term skip connecti…
Nanyang Technological University（南洋理工大学）；Delft University of Technology（代尔夫特理工大学）
State-of-the-art image denoisers exploit various types of deep neural networks via deterministic training. Alternatively, very recent works utilize deep reinforcement learning for restoring images with diverse or unknown corruptions. Though deep reinforcement learning can generate effective policy networks for operator selection or architecture search in image restoration, how it is connected to the classic deterministic training in solving inverse problems remains unclear. In this work, we propose a novel image denoising scheme via Residual Recovery using Reinforcement Learning, dubbed R3L. W…
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）
State-of-the-art algorithms for imaging inverse problems (namely deblurring and reconstruction) are typically iterative, involving a denoising operation as one of its steps. Using a state-of-the-art denoising method in this context is not trivial, and is the focus of current work. Recently, we have proposed to use a class-adapted denoiser (patch-based using Gaussian mixture models) in a so-called plug-and-play scheme, wherein a state-of-the-art denoiser is plugged into an iterative algorithm, leading to results that outperform the best general-purpose algorithms, when applied to an image of a …
College of Computer Science Chongqing University（重庆大学计算机学院）
Image denoising is a classical problem in low level computer vision. Model-based optimization methods and deep learning approaches have been the two main strategies for solving the problem. Model-based optimization methods are flexible for handling different inverse problems but are usually time-consuming. In contrast, deep learning methods have fast testing speed but the performance of these CNNs is still inferior. To address this issue, here we propose a novel deep residual learning model that combines the dilated residual convolution and multi-scale convolution groups. Due to the complex pa…
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）
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）；Department of Computer Science and Engineering, Washington University in St. Louis（圣路易斯华盛顿大学计算机科学与工程系）；Department of Electrical and Systems Engineering, Washington University in St. Louis（圣路易斯华盛顿大学电气与系统工程系）
Regularization by denoising (RED) is a recently developed framework for solving inverse problems by integrating advanced denoisers as image priors. Recent work has shown its state-of-the-art performance when combined with pre-trained deep denoisers. However, current RED algorithms are inadequate for parallel processing on multicore systems. We address this issue by proposing a new asynchronous RED (ASYNC-RED) algorithm that enables asynchronous parallel processing of data, making it significantly faster than its serial counterparts for large-scale inverse problems. The computational complexity…
Principled MAP estimation for inverse problems: bridging the gap between convergence and performance
ENS de Lyon（里昂高等师范学校）；CNRS（法国国家科学研究中心）；Université Claude Bernard Lyon 1（里昂第一大学）；Inria（法国国家信息与自动化研究所）；Univ. Bordeaux（波尔多大学）；Bordeaux INP（波尔多国立理工学院）；IMB（波尔多数学研究所）
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…
Ulsan National Institute of Science and Technology（蔚山科学技术院）
Compressive sensing is a method to recover the original image from undersampled measurements. In order to overcome the ill-posedness of this inverse problem, image priors are used such as sparsity in the wavelet domain, minimum total-variation, or self-similarity. Recently, deep learning based compressive image recovery methods have been proposed and have yielded state-of-the-art performances. They used deep learning based data-driven approaches instead of hand-crafted image priors to solve the ill-posed inverse problem with undersampled data. Ironically, training deep neural networks for them…
It's well-known that inverse problems are ill-posed and to solve them meaningfully, one has to employ regularization methods. Traditionally, popular regularization methods are the penalized Variational approaches. In recent years, the classical regularization approaches have been outclassed by the so-called plug-and-play (PnP) algorithms, which copy the proximal gradient minimization processes, such as ADMM or FISTA, but with any general denoiser. However, unlike the traditional proximal gradient methods, the theoretical underpinnings, convergence, and stability results have been insufficient …
Deep model-based architectures (DMBAs) are widely used in imaging inverse problems to integrate physical measurement models and learned image priors. Plug-and-play priors (PnP) and deep equilibrium models (DEQ) are two DMBA frameworks that have received significant attention. The key difference between the two is that the image prior in DEQ is trained by using a specific measurement model, while that in PnP is trained as a general image denoiser. This difference is behind a common assumption that PnP is more robust to changes in the measurement models compared to DEQ. This paper investigates t…
The Ohio State University（俄亥俄州立大学）；The University of Maryland（马里兰大学）
To solve inverse problems, plug-and-play (PnP) methods replace the proximal step in a convex optimization algorithm with a call to an application-specific denoiser, often implemented using a deep neural network (DNN). Although such methods yield accurate solutions, they can be improved. For example, denoisers are usually designed/trained to remove white Gaussian noise, but the denoiser input error in PnP algorithms is usually far from white or Gaussian. Approximate message passing (AMP) methods provide white and Gaussian denoiser input error, but only when the forward operator is sufficiently …
Ulm University（乌尔姆大学）；University College London（伦敦大学学院）
Scanning Transmission Electron Microscopes (STEMs) acquire 2D images of a 3D sample on the scale of individual cell components. Unfortunately, these 2D images can be too noisy to be fused into a useful 3D structure and facilitating good denoisers is challenging due to the lack of clean-noisy pairs. Additionally, representing a detailed 3D structure can be difficult even for clean data when using regular 3D grids. Addressing these two limitations, we suggest a differentiable image formation model for STEM, allowing to learn a joint model of 2D sensor noise in STEM together with an implicit 3D m…
Technion（以色列理工学院）；Google Research and Machine Intelligence（谷歌研究与机器学习智能部门）
Inverse problems in imaging are extensively studied, with a variety of strategies, tools, and theory that have been accumulated over the years. Recently, this field has been immensely influenced by the emergence of deep-learning techniques. One such contribution, which is the focus of this paper, is the Deep Image Prior (DIP) work by Ulyanov, Vedaldi, and Lempitsky (2018). DIP offers a new approach towards the regularization of inverse problems, obtained by forcing the recovered image to be synthesized from a given deep architecture. While DIP has been shown to be quite an effective unsupervis…
