图像去噪器可通过即插即用（Plug-and-Play, PnP）架构应用于广泛的图像复原问题。本文提出一种通用框架，通过展开以图拉普拉斯正则化器（Graph Laplacian Regularizer, GLR）作为信号先验的最大后验（Maximum A Posteriori, MAP）问题的解，构建可解释的基于图的深度去噪器（Graph-based Deep Denoiser, GDD）。利用近期一项定理——在温和条件下，任意（伪）线性去噪器$\boldsymbol Ψ$都可映射到采用GLR正则化的MAP去噪问题的解，我们首先通过$\boldsymbol Ψ^{-1}$的截断泰勒级数展开（Truncated Taylor Series Expansion, TSE）初始化图拉普拉斯矩阵$\boldsymbol L$。随后，我们将共轭梯度（Conjugate Gradient, CG）算法的迭代展开为一系列神经层，构成前馈网络来求解MAP线性系统，该网络便于参数调优。最终得到的GDD网络具备“图可解释性”，参数量低，且由于$\boldsymbol L$源自已知的高性能去噪器$\boldsymbol Ψ$，初始化十分简便。实验结果表明，GDD在图像去噪性能上与现有方法相当，但参数量远少于对比方法，且对协变量偏移的鲁棒性更强。
英文摘要：
An image denoiser can be used for a wide range of restoration problems via the Plug-and-Play (PnP) architecture. In this paper, we propose a general framework to build an interpretable graph-based deep denoiser (GDD) by unrolling a solution to a maximum a posteriori (MAP) problem equipped with a graph Laplacian regularizer (GLR) as signal prior. Leveraging a recent theorem showing that any (pseudo-)linear denoiser $\boldsymbol Ψ$, under mild conditions, can be mapped to a solution of a MAP denoising problem regularized using GLR, we first initialize a graph Laplacian matrix $\mathbf L$ via truncated Taylor Series Expansion (TSE) of $\boldsymbol Ψ^{-1}$. Then, we compute the MAP linear system solution by unrolling iterations of the conjugate gradient (CG) algorithm into a sequence of neural layers as a feed-forward network -- one that is amenable to parameter tuning. The resulting GDD network is "graph-interpretable", low in parameter count, and easy to initialize thanks to $\mathbf L$ derived from a known well-performing denoiser $\boldsymbol Ψ$. Experimental results show that GDD achieves competitive image denoising performance compared to competitors, but employing far fewer parameters, and is more robust to covariate shift.
In inverse problems we aim to reconstruct some underlying signal of interest from potentially corrupted and often ill-posed measurements. Classical optimization-based techniques proceed by optimizing a data consistency metric together with a regularizer. Current state-of-the-art machine learning approaches draw inspiration from such techniques by unrolling the iterative updates for an optimization-based solver and then learning a regularizer from data. This loop unrolling (LU) method has shown tremendous success, but often requires a deep model for the best performance leading to high memory c…
The Plug-and-Play (PnP) framework makes it possible to integrate advanced image denoising priors into optimization algorithms, to efficiently solve a variety of image restoration tasks generally formulated as Maximum A Posteriori (MAP) estimation problems. The Plug-and-Play alternating direction method of multipliers (ADMM) and the Regularization by Denoising (RED) algorithms are two examples of such methods that made a breakthrough in image restoration. However, while the former method only applies to proximal algorithms, it has recently been shown that there exists no regularization that exp…
Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms
Laboratoire Interdisciplinaire des Sciences du Numériques, Inria, Université Paris-Saclay, CNRS, Gif-sur-Yvette, France（跨学科数字科学实验室，法国国家信息与自动化研究所，巴黎-萨克雷大学，法国国家科学研究中心，法国吉夫-sur-伊夫特）；Inria, Université Paris-Saclay, CEA, Palaiseau, France（法国国家信息与自动化研究所，巴黎-萨克雷大学，法国原子能委员会，法国帕莱索）；School of Mathematics and Computer Sciences and School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK（数学与计算机科学学院和工程与物理科学学院，赫里奥特-瓦特大学，英国爱丁堡）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（数学科学麦克斯韦研究所，英国爱丁堡）
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 …
A Deep Unrolling Model with Hybrid Optimization Structure for Hyperspectral Image Deconvolution
Department of Computer Engineering and Informatics, Patras University（计算机工程与信息学院，帕特拉大学）；Department Digital Media and Communication, Ionian University（数字媒体与通信系，伊奥尼亚大学）
In plug-and-play image restoration, the regularization is performed using powerful denoisers such as nonlocal means (NLM) or BM3D. This is done within the framework of alternating direction method of multipliers (ADMM), where the regularization step is formally replaced by an off-the-shelf denoiser. Each plug-and-play iteration involves the inversion of the forward model followed by a denoising step. In this paper, we present a couple of ideas for improving the efficiency of the inversion and denoising steps. First, we propose to use linearized ADMM, which generally allows us to perform the in…
In plug-and-play (PnP) regularization, the knowledge of the forward model is combined with a powerful denoiser to obtain state-of-the-art image reconstructions. This is typically done by taking a proximal algorithm such as FISTA or ADMM, and formally replacing the proximal map associated with a regularizer by nonlocal means, BM3D or a CNN denoiser. Each iterate of the resulting PnP algorithm involves some kind of inversion of the forward model followed by denoiser-induced regularization. A natural question in this regard is that of optimality, namely, do the PnP iterations minimize some f+g, w…
A standard model for image reconstruction involves the minimization of a data-fidelity term along with a regularizer, where the optimization is performed using proximal algorithms such as ISTA and ADMM. In plug-and-play (PnP) regularization, the proximal operator (associated with the regularizer) in ISTA and ADMM is replaced by a powerful image denoiser. Although PnP regularization works surprisingly well in practice, its theoretical convergence -- whether convergence of the PnP iterates is guaranteed and if they minimize some objective function -- is not completely understood even for simple …
Deep Unfolded BM3D: Unrolling Non-local Collaborative Filtering into a Trainable Neural Network
Electronics and Communication Engineering Department, Istanbul Technical University, Istanbul, Turkey（伊斯坦布尔技术大学电子与通信工程系）；Electrical and Electronics Engineering Department, Istanbul University, Istanbul, Turkey（伊斯坦布尔大学电气与电子工程系）
City University of Hong Kong（香港城市大学）；The Chinese University of Hong Kong（香港中文大学）；Hong Kong Centre for Cerebro-Cardiovascular Health Engineering（香港脑心血管健康工程中心）；University of Hong Kong（香港大学）
Spherical image processing has been widely applied in many important fields, such as omnidirectional vision for autonomous cars, global climate modelling, and medical imaging. It is non-trivial to extend an algorithm developed for flat images to the spherical ones. In this work, we focus on the challenging task of spherical image inpainting with deep learning-based regularizer. Instead of a naive application of existing models for planar images, we employ a fast directional spherical Haar framelet transform and develop a novel optimization framework based on a sparsity assumption of the framel…
Elasticity image, visualizing the quantitative map of tissue stiffness, can be reconstructed by solving an inverse problem. Classical methods for magnetic resonance elastography (MRE) try to solve a regularized optimization problem comprising a deterministic physical model and a prior constraint as data-fidelity term and regularization term, respectively. For improving the elasticity reconstructions, appropriate prior about the underlying elasticity distribution is required which is not unique. This article proposes an infused approach for MRE reconstruction by integrating the statistical repr…
School of Marine Science and Technology, Northwestern Polytechnical University, China（西北工业大学海洋科学与技术学院）；State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, China（北京交通大学轨道交通控制与安全国家重点实验室）
Nowadays, nonnegative matrix factorization (NMF) based methods have been widely applied to blind spectral unmixing. Introducing proper regularizers to NMF is crucial for mathematically constraining the solutions and physically exploiting spectral and spatial properties of images. Generally, properly handcrafting regularizers and solving the associated complex optimization problem are non-trivial tasks. In our work, we propose an NMF based unmixing framework which jointly uses a handcrafting regularizer and a learnt regularizer from data. we plug learnt priors of abundances where the associated…
College of Computer Science, Sichuan University（四川大学计算机学院）；Laboratory of Image Science and Technology, Southeast University（东南大学图像科学与技术实验室）；Beijing Engineering Research Center of Radiographic Techniques and Equipment, Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所北京市射线成像技术与装备工程研究中心）；School of Electrical Engineering Information, Sichuan University（四川大学电气信息学院）；Department of Radiology, West China Hospital of Sichuan University（四川大学华西医院放射科）
Compressed sensing (CS) computed tomography has been proven to be important for several clinical applications, such as sparse-view computed tomography (CT), digital tomosynthesis and interior tomography. Traditional compressed sensing focuses on the design of handcrafted prior regularizers, which are usually image-dependent and time-consuming. Inspired by recently proposed deep learning-based CT reconstruction models, we extend the state-of-the-art LEARN model to a dual-domain version, dubbed LEARN++. Different from existing iteration unrolling methods, which only involve projection data in th…
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 have been the penalized Variational approaches. In recent years, the classical regularized-reconstruction approaches have been outclassed by the (deep-learning-based) learned reconstruction algorithms. However, unlike the traditional regularization methods, the theoretical underpinnings, such as stability and regularization, have been insufficient for such learned reconstruction algorithms. Hence, the results obtained from su…
There has been tremendous research on the design of image regularizers over the years, from simple Tikhonov and Laplacian to sophisticated sparsity and CNN-based regularizers. Coupled with a model-based loss function, these are typically used for image reconstruction within an optimization framework. The technical challenge is to develop a regularizer that can accurately model realistic images and be optimized efficiently along with the loss function. Motivated by the recent plug-and-play paradigm for image regularization, we construct a quadratic regularizer whose reconstruction capability is…
Regularization by denoising (RED) is an image reconstruction framework that uses an image denoiser as a prior. Recent work has shown the state-of-the-art performance of RED with learned denoisers corresponding to pre-trained convolutional neural nets (CNNs). In this work, we propose to broaden the current denoiser-centric view of RED by considering priors corresponding to networks trained for more general artifact-removal. The key benefit of the proposed family of algorithms, called regularization by artifact-removal (RARE), is that it can leverage priors learned on datasets containing only un…
Ultrasound elasticity images which enable the visualization of quantitative maps of tissue stiffness can be reconstructed by solving an inverse problem. Classical model-based approaches for ultrasound elastography use deterministic finite element methods (FEMs) to incorporate the governing physical laws resulting in poor performance in noisy conditions. Moreover, these approaches utilize fixed regularizers for various tissue patterns while appropriate data-adaptive priors might be required for capturing the complex spatial elasticity distribution. In this regard, we propose a joint model-based…
Spectral unmixing is a widely used technique in hyperspectral image processing and analysis. It aims to separate mixed pixels into the component materials and their corresponding abundances. Early solutions to spectral unmixing are performed independently on each pixel. Nowadays, investigating proper priors into the unmixing problem has been popular as it can significantly enhance the unmixing performance. However, it is non-trivial to handcraft a powerful regularizer, and complex regularizers may introduce extra difficulties in solving optimization problems in which they are involved. To addr…
Physics-guided deep learning (PG-DL) via algorithm unrolling has received significant interest for improved image reconstruction, including MRI applications. These methods unroll an iterative optimization algorithm into a series of regularizer and data consistency units. The unrolled networks are typically trained end-to-end using a supervised approach. Current supervised PG-DL approaches use all of the available sub-sampled measurements in their data consistency units. Thus, the network learns to fit the rest of the measurements. In this study, we propose to improve the performance and robust…
SIAT（中国科学院深圳先进技术研究院）；National Innovation Center for Advanced Medical Devices（国家高性能医疗器械创新中心）；Shanghai Jiaotong University（上海交通大学）；Technical University of Munich（慕尼黑工业大学）
Recently, model-driven deep learning unrolls a certain iterative algorithm of a regularization model into a cascade network by replacing the first-order information (i.e., (sub)gradient or proximal operator) of the regularizer with a network module, which appears more explainable and predictable compared to common data-driven networks. Conversely, in theory, there is not necessarily such a functional regularizer whose first-order information matches the replaced network module, which means the network output may not be covered by the original regularization model. Moreover, up to now, there is…
Harvard Medical School（哈佛医学院）；Boston Children’s Hospital（波士顿儿童医院）
Fast data acquisition in Magnetic Resonance Imaging (MRI) is vastly in demand and scan time directly depends on the number of acquired k-space samples. Conventional MRI reconstruction methods for fast MRI acquisition mostly relied on different regularizers which represent analytical models of sparsity. However, recent data-driven methods based on deep learning has resulted in promising improvements in image reconstruction algorithms. In this paper, we propose a deep plug-and-play prior framework for parallel MRI reconstruction problems which utilize a deep neural network (DNN) as an advanced d…
University of Electronic Science and Technology of China（电子科技大学）；Southwestern University of Finance and Economics（西南财经大学）；Xi’an Jiaotong University（西安交通大学）
Multi-dimensional images, such as color images and multi-spectral images, are highly correlated and contain abundant spatial and spectral information. However, real-world multi-dimensional images are usually corrupted by missing entries. By integrating deterministic low-rankness prior to the data-driven deep prior, we suggest a novel regularized tensor completion model for multi-dimensional image completion. In the objective function, we adopt the newly emerged tensor nuclear norm (TNN) to characterize the global low-rankness prior of the multi-dimensional images. We also formulate an implicit…
