It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low-rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorization component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. However, low rank is not adequate to reflect the intrinsic characteristics of a natural image. Thus, algorithms with only the constraint of low rank are insufficient to perform image restoration well. In this work, we propose a Regularized Deep Matrix Factorized (RDMF) model for image restoration, which utilizes the implicit bias of the low rank of deep neural networks and the explicit bias of total variation. We demonstrate the effectiveness of our RDMF model with extensive experiments, in which our method surpasses the state of art models in common examples, especially for the restoration from very few observations. Our work sheds light on a more general framework for solving other inverse problems by combining the implicit bias of deep learning with explicit regularization.
In the past decade, sparsity-driven regularization has led to significant improvements in image reconstruction. Traditional regularizers, such as total variation (TV), rely on analytical models of sparsity. However, increasingly the field is moving towards trainable models, inspired from deep learning. Deep image prior (DIP) is a recent regularization framework that uses a convolutional neural network (CNN) architecture without data-driven training. This paper extends the DIP framework by combining it with the traditional TV regularization. We show that the inclusion of TV leads to considerabl…
College of Medicine and Biological Information Engineering, Northeastern University（东北大学医学与生物信息工程学院）；Key Laboratory of Intelligent Computing in Medical Image, Ministry of Education（教育部医学影像智能计算重点实验室）；Department of Electrical and Computer Engineering, Stevens Institute of Technology（史蒂文斯理工学院电气与计算机工程系）
The iterative weighted shrinkage-thresholding algorithm (IWSTA) has shown superiority to the classic unweighted iterative shrinkage-thresholding algorithm (ISTA) for solving linear inverse problems, which address the attributes differently. This paper proposes a new entropy regularized IWSTA (ERIWSTA) that adds an entropy regularizer to the cost function to measure the uncertainty of the weights to stimulate attributes to participate in problem solving. Then, the weights are solved with a Lagrange multiplier method to obtain a simple iterative update. The weights can be explained as the probab…
INRIA（法国国家信息与自动化研究所）；Département d’informatique de l’ENS（巴黎高等师范学院计算机系）；ENS（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；PSL University（巴黎文理研究大学）
We present a novel approach to image restoration that leverages ideas from localized structured prediction and non-linear multi-task learning. We optimize a penalized energy function regularized by a sum of terms measuring the distance between patches to be restored and clean patches from an external database gathered beforehand. The resulting estimator comes with strong statistical guarantees leveraging local dependency properties of overlapping patches. We derive the corresponding algorithms for energies based on the mean-squared and Euclidean norm errors. Finally, we demonstrate the practic…
Northwest A\&F University, College of Science, Yangling 712100, P.R. China
Several bandwise total variation (TV) regularized low-rank (LR)-based models have been proposed to remove mixed noise in hyperspectral images (HSIs). Conventionally, the rank of LR matrix is approximated using nuclear norm (NN). The NN is defined by adding all singular values together, which is essentially a $L_1$-norm of the singular values. It results in non-negligible approximation errors and thus the resulting matrix estimator can be significantly biased. Moreover, these bandwise TV-based methods exploit the spatial information in a separate manner. To cope with these problems, we propose …
Chinese University of Hongkong（香港中文大学）；University of Electronic Science and Technology of China（电子科技大学）
Hyperspectral images (HSIs) are often corrupted by a mixture of several types of noise during the acquisition process, e.g., Gaussian noise, impulse noise, dead lines, stripes, and many others. Such complex noise could degrade the quality of the acquired HSIs, limiting the precision of the subsequent processing. In this paper, we present a novel tensor-based HSI restoration approach by fully identifying the intrinsic structures of the clean HSI part and the mixed noise part respectively. Specifically, for the clean HSI part, we use tensor Tucker decomposition to describe the global correlation…
University at Albany, SUNY（纽约州立大学奥尔巴尼分校）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Several important classes of images such as text, barcode and pattern images have the property that pixels can only take a distinct subset of values. This knowledge can benefit the restoration of such images, but it has not been widely considered in current restoration methods. In this work, we describe an effective and efficient approach to incorporate the knowledge of distinct pixel values of the pristine images into the general regularized least squares restoration framework. We introduce a new regularizer that attains zero at the designated pixel values and becomes a quadratic penalty func…
This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample s…
University of Electronic Science and Technology of China（中国电子科技大学）
Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing and medical imaging, and this kind of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being non-convex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the sin…
Institute of Optics and Electronics, Chinese Academy of Sciences（中国科学院光电技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Image quality is the basis of image communication and understanding tasks. Due to the blur and noise effects caused by imaging, transmission and other processes, the image quality is degraded. Blind image restoration is widely used to improve image quality, where the main goal is to faithfully estimate the blur kernel and the latent sharp image. In this study, based on experimental observation and research, an adaptively sparse regularized minimization method is originally proposed. The high-order gradients combine with low-order ones to form a hybrid regularization term, and an adaptive opera…
Restoration of digital images from their degraded measurements has always been a problem of great theoretical and practical importance in numerous applications of imaging sciences. A specific solution to the problem of image restoration is generally determined by the nature of degradation phenomenon as well as by the statistical properties of measurement noises. The present study is concerned with the case in which the images of interest are corrupted by convolutional blurs and Poisson noises. To deal with such problems, there exists a range of solution methods which are based on the principle…
Energy-oriented Diffusion Bridge for Image Restoration with Foundational Diffusion Models
Department of Computer Science, City University of Hong Kong（香港城市大学计算机科学系）；Department of Computer Science, City University of Hong Kong (Dongguan)（香港城市大学（东莞）计算机科学系）
The Chinese University of Hong Kong（香港中文大学）；Nanyang Technological University（南洋理工大学）；The University of Hong Kong（香港大学）
Learning a good image prior is a long-term goal for image restoration and manipulation. While existing methods like deep image prior (DIP) capture low-level image statistics, there are still gaps toward an image prior that captures rich image semantics including color, spatial coherence, textures, and high-level concepts. This work presents an effective way to exploit the image prior captured by a generative adversarial network (GAN) trained on large-scale natural images. As shown in Fig.1, the deep generative prior (DGP) provides compelling results to restore missing semantics, e.g., color, p…
Shandong University of Traditional Chinese Medicine（山东中医药大学）；Nankai University（南开大学）；ShanghaiTech University（上海科技大学）；Taishan University（泰山学院）；United Imaging Co.（联影公司）
Real-world face super-resolution (SR) is a highly ill-posed image restoration task. The fully-cycled Cycle-GAN architecture is widely employed to achieve promising performance on face SR, but prone to produce artifacts upon challenging cases in real-world scenarios, since joint participation in the same degradation branch will impact final performance due to huge domain gap between real-world and synthetic LR ones obtained by generators. To better exploit the powerful generative capability of GAN for real-world face SR, in this paper, we establish two independent degradation branches in the fo…
The split Bregman (SB) method [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323-43] is a fast splitting-based algorithm that solves image reconstruction problems with general l1, e.g., total-variation (TV) and compressed sensing (CS), regularizations by introducing a single variable split to decouple the data-fitting term and the regularization term, yielding simple subproblems that are separable (or partially separable) and easy to minimize. Several convergence proofs have been proposed, and these proofs either impose a "full column rank" assumption to the split or assume ex…
Many tasks in image processing can be tackled by modeling an appropriate data fidelity term $Φ: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$ and then solve one of the regularized minimization problems \begin{align*} &{}(P_{1,τ}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \big\{ Φ(x) \;{\rm s.t.}\; Ψ(x) \leq τ\big\} \\ &{}(P_{2,λ}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \{ Φ(x) + λΨ(x) \}, \; λ> 0 \end{align*} with some function $Ψ: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}$ and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}&…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
The cross-entropy loss commonly used in deep learning is closely related to the defining properties of optimal representations, but does not enforce some of the key properties. We show that this can be solved by adding a regularization term, which is in turn related to injecting multiplicative noise in the activations of a Deep Neural Network, a special case of which is the common practice of dropout. We show that our regularized loss function can be efficiently minimized using Information Dropout, a generalization of dropout rooted in information theoretic principles that automatically adapts…
University of Maryland, College Park（马里兰大学帕克分校）；Cornell University（康奈尔大学）
Conventional algorithms for sparse signal recovery and sparse representation rely on $l_1$-norm regularized variational methods. However, when applied to the reconstruction of $\textit{sparse images}$, i.e., images where only a few pixels are non-zero, simple $l_1$-norm-based methods ignore potential correlations in the support between adjacent pixels. In a number of applications, one is interested in images that are not only sparse, but also have a support with smooth (or contiguous) boundaries. Existing algorithms that take into account such a support structure mostly rely on non-convex meth…
University of Bristol（布里斯托大学）；New York University（纽约大学）
The L1 norm regularized least squares method is often used for finding sparse approximate solutions and is widely used in 1-D signal restoration. Basis pursuit denoising (BPD) performs noise reduction in this way. However, the shortcoming of using L1 norm regularization is the underestimation of the true solution. Recently, a class of non-convex penalties have been proposed to improve this situation. This kind of penalty function is non-convex itself, but preserves the convexity property of the whole cost function. This approach has been confirmed to offer good performance in 1-D signal denois…
EquiReg: Equivariance Regularized Diffusion for Inverse Problems
University of Alberta（阿尔伯塔大学）；Alberta Machine, Intelligence Institute (Amii)（阿尔伯塔机器智能研究所）；Neuroscience and Mental Health Institute, University of Alberta（阿尔伯塔大学神经科学与心理健康研究所）；New York University（纽约大学）；California Institute of Technology（加州理工学院）；University of Oxford（牛津大学）
University Paris-Est（巴黎东大学）；IBM Research China（IBM中国研究院）；CentraleSupelec（中央理工-高等电力学院）；University Paris-Saclay（巴黎萨克雷大学）
In this paper, a methodology is investigated for signal recovery in the presence of non-Gaussian noise. In contrast with regularized minimization approaches often adopted in the literature, in our algorithm the regularization parameter is reliably estimated from the observations. As the posterior density of the unknown parameters is analytically intractable, the estimation problem is derived in a variational Bayesian framework where the goal is to provide a good approximation to the posterior distribution in order to compute posterior mean estimates. Moreover, a majorization technique is emplo…
