Ting-Zhu Huang

Xian-Ming Gu, Ting-Zhu Huang, Cui-Cui Ji et al.

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Grünwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $O(N^3)$ to $O(N\log N)$ in each iterative step, where $N$ is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.

Yong-Liang Zhao, Ting-Zhu Huang, Xian-Ming Gu et al.

In this paper, we consider a fast and second-order implicit difference method for approximation of a class of time-space fractional variable coefficients advection-diffusion equation. To begin with, we construct an implicit difference scheme, based on $L2-1_σ$ formula [A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, \emph{J. Comput. Phys.}, 280 (2015)] for the temporal discretization and weighted and shifted Grünwald method for the spatial discretization. Then, unconditional stability of the implicit difference scheme is proved, and we theoretically and numerically show that it converges in the $L_2$-norm with the optimal order $\mathcal{O}(τ^2 + h^2)$ with time step $τ$ and mesh size $h$. Secondly, three fast Krylov subspace solvers with suitable circulant preconditioners are designed to solve the discretized linear systems with the Toeplitz matrix. In each iterative step, these methods reduce the memory requirement of the resulting linear equations from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $\mathcal{O}(N^3)$ to $\mathcal{O}(N \log N)$, where $N$ is the number of grid nodes. Finally, numerical experiments are carried out to demonstrate that these methods are more practical than the traditional direct solvers of the implicit difference methods, in terms of memory requirement and computational cost.

Jin-Ju Wang, Nicolas Dobigeon, Marie Chabert et al.

In the context of Earth observation, change detection boils down to comparing images acquired at different times by sensors of possibly different spatial and/or spectral resolutions or different modalities (e.g., optical or radar). Even when considering only optical images, this task has proven to be challenging as soon as the sensors differ by their spatial and/or spectral resolutions. This paper proposes a novel unsupervised change detection method dedicated to images acquired by such so-called heterogeneous optical sensors. It capitalizes on recent advances which formulate the change detection task into a robust fusion framework. Adopting this formulation, the work reported in this paper shows that any off-the-shelf network trained beforehand to fuse optical images of different spatial and/or spectral resolutions can be easily complemented with a network of the same architecture and embedded into an adversarial framework to perform change detection. A comparison with state-of-the-art change detection methods demonstrates the versatility and the effectiveness of the proposed approach.

Xian-Ming Gu, Bruno Carpentieri, Ting-Zhu Huang et al.

In the present study, we establish two new block variants of the Conjugate Orthogonal Conjugate Gradient (COCG) and the Conjugate A-Orthogonal Conjugate Residual (COCR) Krylov subspace methods for solving complex symmetric linear systems with multiple right hand sides. The proposed Block iterative solvers can fully exploit the complex symmetry property of coefficient matrix of the linear system. We report on extensive numerical experiments to show the favourable convergence properties of our newly developed Block algorithms for solving realistic electromagnetic simulations.

Huan-Yan Jian, Ting-Zhu Huang, Xi-Le Zhao et al.

The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in $L_2$-norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods.

Wei-Hao Wu, Ting-Zhu Huang, Xi-Le Zhao et al.

Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation ability and data adaptation ability, resulting in unsatisfactory performance. To address these challenges, we introduce the neural basis function by leveraging an untrained neural network as the basis function. Equipped with the proposed neural basis function, we suggest the neural approximation (NeuApprox) paradigm for multivariate function approximation. Specifically, the underlying multivariate function behind the multi-dimensional data is decomposed into a sum of block terms. The clear physically-interpreted block term is the product of expressive neural basis functions and their corresponding learnable coefficients, which allows us to faithfully capture distinct components of the underlying data and also flexibly adapt to new data by readily fine-tuning the neural basis functions. Attributed to the elaborately designed block terms, the suggested NeuApprox enjoys strong approximation ability and flexible data adaptation ability over the hand-crafted basis function-based methods. We also theoretically prove that NeuApprox can approximate any multivariate continuous function to arbitrary accuracy. Extensive experiments on diverse multi-dimensional datasets (including multispectral images, light field data, videos, traffic data, and point cloud data) demonstrate the promising performance of NeuApprox in terms of both approximation capability and adaptability.

Xiao Wu, Ting-Zhu Huang, Liang-Jian Deng et al.

Medical decision-making often involves integrating knowledge from multiple clinical specialties, typically achieved through multidisciplinary teams. Inspired by this collaborative process, recent work has leveraged large language models (LLMs) in multi-agent collaboration frameworks to emulate expert teamwork. While these approaches improve reasoning through agent interaction, they are limited by static, pre-assigned roles, which hinder adaptability and dynamic knowledge integration. To address these limitations, we propose KAMAC, a Knowledge-driven Adaptive Multi-Agent Collaboration framework that enables LLM agents to dynamically form and expand expert teams based on the evolving diagnostic context. KAMAC begins with one or more expert agents and then conducts a knowledge-driven discussion to identify and fill knowledge gaps by recruiting additional specialists as needed. This supports flexible, scalable collaboration in complex clinical scenarios, with decisions finalized through reviewing updated agent comments. Experiments on two real-world medical benchmarks demonstrate that KAMAC significantly outperforms both single-agent and advanced multi-agent methods, particularly in complex clinical scenarios (i.e., cancer prognosis) requiring dynamic, cross-specialty expertise. Our code is publicly available at: https://github.com/XiaoXiao-Woo/KAMAC.

Tai-Xiang Jiang, Ting-Zhu Huang, Xi-Le Zhao et al.

Rain streak removal is an important issue in outdoor vision systems and has recently been investigated extensively. In this paper, we propose a novel video rain streak removal approach FastDeRain, which fully considers the discriminative characteristics of rain streaks and the clean video in the gradient domain. Specifically, on the one hand, rain streaks are sparse and smooth along the direction of the raindrops, whereas on the other hand, clean videos exhibit piecewise smoothness along the rain-perpendicular direction and continuity along the temporal direction. Theses smoothness and continuity results in the sparse distribution in the different directional gradient domain, respectively. Thus, we minimize 1) the $\ell_1$ norm to enhance the sparsity of the underlying rain streaks, 2) two $\ell_1$ norm of unidirectional Total Variation (TV) regularizers to guarantee the anisotropic spatial smoothness, and 3) an $\ell_1$ norm of the time-directional difference operator to characterize the temporal continuity. A split augmented Lagrangian shrinkage algorithm (SALSA) based algorithm is designed to solve the proposed minimization model. Experiments conducted on synthetic and real data demonstrate the effectiveness and efficiency of the proposed method. According to comprehensive quantitative performance measures, our approach outperforms other state-of-the-art methods especially on account of the running time. The code of FastDeRain can be downloaded at https://github.com/TaiXiangJiang/FastDeRain.

Yu-Bang Zheng, Xi-Le Zhao, Junhua Zeng et al.

Tensor network (TN) representation is a powerful technique for computer vision and machine learning. TN structure search (TN-SS) aims to search for a customized structure to achieve a compact representation, which is a challenging NP-hard problem. Recent "sampling-evaluation"-based methods require sampling an extensive collection of structures and evaluating them one by one, resulting in prohibitively high computational costs. To address this issue, we propose a novel TN paradigm, named SVD-inspired TN decomposition (SVDinsTN), which allows us to efficiently solve the TN-SS problem from a regularized modeling perspective, eliminating the repeated structure evaluations. To be specific, by inserting a diagonal factor for each edge of the fully-connected TN, SVDinsTN allows us to calculate TN cores and diagonal factors simultaneously, with the factor sparsity revealing a compact TN structure. In theory, we prove a convergence guarantee for the proposed method. Experimental results demonstrate that the proposed method achieves approximately 100 to 1000 times acceleration compared to the state-of-the-art TN-SS methods while maintaining a comparable level of representation ability.

Tian-Jing Zhang, Liang-Jian Deng, Ting-Zhu Huang et al.

Pansharpening refers to the fusion of a panchromatic image with a high spatial resolution and a multispectral image with a low spatial resolution, aiming to obtain a high spatial resolution multispectral image. In this paper, we propose a novel deep neural network architecture with level-domain based loss function for pansharpening by taking into account the following double-type structures, \emph{i.e.,} double-level, double-branch, and double-direction, called as triple-double network (TDNet). By using the structure of TDNet, the spatial details of the panchromatic image can be fully exploited and utilized to progressively inject into the low spatial resolution multispectral image, thus yielding the high spatial resolution output. The specific network design is motivated by the physical formula of the traditional multi-resolution analysis (MRA) methods. Hence, an effective MRA fusion module is also integrated into the TDNet. Besides, we adopt a few ResNet blocks and some multi-scale convolution kernels to deepen and widen the network to effectively enhance the feature extraction and the robustness of the proposed TDNet. Extensive experiments on reduced- and full-resolution datasets acquired by WorldView-3, QuickBird, and GaoFen-2 sensors demonstrate the superiority of the proposed TDNet compared with some recent state-of-the-art pansharpening approaches. An ablation study has also corroborated the effectiveness of the proposed approach.

Ben-Zheng Li, Xi-Le Zhao, Teng-Yu Ji et al.

The linear transform-based tensor nuclear norm (TNN) methods have recently obtained promising results for tensor completion. The main idea of this type of methods is exploiting the low-rank structure of frontal slices of the targeted tensor under the linear transform along the third mode. However, the low-rankness of frontal slices is not significant under linear transforms family. To better pursue the low-rank approximation, we propose a nonlinear transform-based TNN (NTTNN). More concretely, the proposed nonlinear transform is a composite transform consisting of the linear semi-orthogonal transform along the third mode and the element-wise nonlinear transform on frontal slices of the tensor under the linear semi-orthogonal transform, which are indispensable and complementary in the composite transform to fully exploit the underlying low-rankness. Based on the suggested low-rankness metric, i.e., NTTNN, we propose a low-rank tensor completion (LRTC) model. To tackle the resulting nonlinear and nonconvex optimization model, we elaborately design the proximal alternating minimization (PAM) algorithm and establish the theoretical convergence guarantee of the PAM algorithm. Extensive experimental results on hyperspectral images, multispectral images, and videos show that the our method outperforms linear transform-based state-of-the-art LRTC methods qualitatively and quantitatively.

Yun-Yang Liu, Xi-Le Zhao, Guang-Jing Song et al.

The robust tensor completion (RTC) problem, which aims to reconstruct a low-rank tensor from partially observed tensor contaminated by a sparse tensor, has received increasing attention. In this paper, by leveraging the superior expression of the fully-connected tensor network (FCTN) decomposition, we propose a $\textbf{FCTN}$-based $\textbf{r}$obust $\textbf{c}$onvex optimization model (RC-FCTN) for the RTC problem. Then, we rigorously establish the exact recovery guarantee for the RC-FCTN. For solving the constrained optimization model RC-FCTN, we develop an alternating direction method of multipliers (ADMM)-based algorithm, which enjoys the global convergence guarantee. Moreover, we suggest a $\textbf{FCTN}$-based $\textbf{r}$obust $\textbf{n}$on$\textbf{c}$onvex optimization model (RNC-FCTN) for the RTC problem. A proximal alternating minimization (PAM)-based algorithm is developed to solve the proposed RNC-FCTN. Meanwhile, we theoretically derive the convergence of the PAM-based algorithm. Comprehensive numerical experiments in several applications, such as video completion and video background subtraction, demonstrate that proposed methods are superior to several state-of-the-art methods.

Jin-Fan Hu, Ting-Zhu Huang, Liang-Jian Deng

Hyperspectral image has become increasingly crucial due to its abundant spectral information. However, It has poor spatial resolution with the limitation of the current imaging mechanism. Nowadays, many convolutional neural networks have been proposed for the hyperspectral image super-resolution problem. However, convolutional neural network (CNN) based methods only consider the local information instead of the global one with the limited kernel size of receptive field in the convolution operation. In this paper, we design a network based on the transformer for fusing the low-resolution hyperspectral images and high-resolution multispectral images to obtain the high-resolution hyperspectral images. Thanks to the representing ability of the transformer, our approach is able to explore the intrinsic relationships of features globally. Furthermore, considering the LR-HSIs hold the main spectral structure, the network focuses on the spatial detail estimation releasing from the burden of reconstructing the whole data. It reduces the mapping space of the proposed network, which enhances the final performance. Various experiments and quality indexes show our approach's superiority compared with other state-of-the-art methods.

Jin-Fan Hu, Ting-Zhu Huang, Liang-Jian Deng et al.

Hyperspectral images are of crucial importance in order to better understand features of different materials. To reach this goal, they leverage on a high number of spectral bands. However, this interesting characteristic is often paid by a reduced spatial resolution compared with traditional multispectral image systems. In order to alleviate this issue, in this work, we propose a simple and efficient architecture for deep convolutional neural networks to fuse a low-resolution hyperspectral image (LR-HSI) and a high-resolution multispectral image (HR-MSI), yielding a high-resolution hyperspectral image (HR-HSI). The network is designed to preserve both spatial and spectral information thanks to an architecture from two folds: one is to utilize the HR-HSI at a different scale to get an output with a satisfied spectral preservation; another one is to apply concepts of multi-resolution analysis to extract high-frequency information, aiming to output high quality spatial details. Finally, a plain mean squared error loss function is used to measure the performance during the training. Extensive experiments demonstrate that the proposed network architecture achieves best performance (both qualitatively and quantitatively) compared with recent state-of-the-art hyperspectral image super-resolution approaches. Moreover, other significant advantages can be pointed out by the use of the proposed approach, such as, a better network generalization ability, a limited computational burden, and a robustness with respect to the number of training samples.

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao et al.

In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on the convex relaxation by penalizing the weighted sum of nuclear norm of TR unfolding matrices. However, this method treats each singular value equally and neglects their physical meanings, which usually leads to suboptimal solutions in practice. In this paper, we propose to use the logdet-based function as a nonconvex smooth relaxation of the TR rank for tensor completion, which can more accurately approximate the TR rank and better promote the low-rankness of the solution. To solve the proposed nonconvex model efficiently, we develop an alternating direction method of multipliers algorithm and theoretically prove that, under some mild assumptions, our algorithm converges to a stationary point. Extensive experiments on color images, multispectral images, and color videos demonstrate that the proposed method outperforms several state-of-the-art competitors in both visual and quantitative comparison. Key words: nonconvex optimization, tensor ring rank, logdet function, tensor completion, alternating direction method of multipliers.

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao et al.

The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($\textrm{order} >3$). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with \emph{ket augmentation}, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completion by simultaneously exploring the spatial, temporal/spectral, and nonlocal redundancy in visual data. More precisely, the TT rank minimization is performed on a formed higher-order tensor called group by stacking similar cubes, which naturally and fully takes advantage of the ability of TT rank for high-order tensors. Moreover, the perturbation analysis for the TT low-rankness of each group is established. We develop the alternating direction method of multipliers tailored for the specific structure to solve the proposed model. Extensive experiments demonstrate that the proposed method is superior to several existing state-of-the-art methods in terms of both qualitative and quantitative measures.

Tai-Xiang Jiang, Michael K. Ng, Xi-Le Zhao et al.

The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods.

Yu-Bang Zheng, Ting-Zhu Huang, Xi-Le Zhao et al.

As low-rank modeling has achieved great success in tensor recovery, many research efforts devote to defining the tensor rank. Among them, the recent popular tensor tubal rank, defined based on the tensor singular value decomposition (t-SVD), obtains promising results. However, the framework of the t-SVD and the tensor tubal rank are applicable only to three-way tensors and lack of flexibility to handle different correlations along different modes. To tackle these two issues, we define a new tensor unfolding operator, named mode-$k_1k_2$ tensor unfolding, as the process of lexicographically stacking the mode-$k_1k_2$ slices of an $N$-way tensor into a three-way tensor, which is a three-way extension of the well-known mode-$k$ tensor matricization. Based on it, we define a novel tensor rank, the tensor $N$-tubal rank, as a vector whose elements contain the tubal rank of all mode-$k_1k_2$ unfolding tensors, to depict the correlations along different modes. To efficiently minimize the proposed $N$-tubal rank, we establish its convex relaxation: the weighted sum of tensor nuclear norm (WSTNN). Then, we apply WSTNN to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). The corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient alternating direction method of multipliers (ADMM)-based algorithms are developed to solve the proposed models. Numerical experiments demonstrate that the proposed models significantly outperform the compared ones.

Ye-Tao Wang, Xi-Le Zhao, Tai-Xiang Jiang et al.

Rain streak removal is an important issue and has recently been investigated extensively. Existing methods, especially the newly emerged deep learning methods, could remove the rain streaks well in many cases. However the essential factor in the generative procedure of the rain streaks, i.e., the motion blur, which leads to the line pattern appearances, were neglected by the deep learning rain streaks approaches and this resulted in over-derain or under-derain results. In this paper, we propose a novel rain streak removal approach using a kernel guided convolutional neural network (KGCNN), achieving the state-of-the-art performance with simple network architectures. We first model the rain streak interference with its motion blur mechanism. Then, our framework starts with learning the motion blur kernel, which is determined by two factors including angle and length, by a plain neural network, denoted as parameter net, from a patch of the texture component. Then, after a dimensionality stretching operation, the learned motion blur kernel is stretched into a degradation map with the same spatial size as the rainy patch. The stretched degradation map together with the texture patch is subsequently input into a derain convolutional network, which is a typical ResNet architecture and trained to output the rain streaks with the guidance of the learned motion blur kernel. Experiments conducted on extensive synthetic and real data demonstrate the effectiveness of the proposed method, which preserves the texture and the contrast while removing the rain streaks.

Tai-Xiang Jiang, Ting-Zhu Huang, Xi-Le Zhao et al.

In this paper, we investigate tensor recovery problems within the tensor singular value decomposition (t-SVD) framework. We propose the partial sum of the tubal nuclear norm (PSTNN) of a tensor. The PSTNN is a surrogate of the tensor tubal multi-rank. We build two PSTNN-based minimization models for two typical tensor recovery problems, i.e., the tensor completion and the tensor principal component analysis. We give two algorithms based on the alternating direction method of multipliers (ADMM) to solve proposed PSTNN-based tensor recovery models. Experimental results on the synthetic data and real-world data reveal the superior of the proposed PSTNN.

Xian-Ming Gu, Ting-Zhu Huang, Guojian Yin et al.

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method [M. Heyouni, Méthode de Hessenberg Généralisée et Applications, Ph.D. Thesis, Université des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough CPU time elapsed to converge than the earlier established restarted shifted FOM, weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recent popular applications of handling the time fractional differential equations.

Liang-Jian Deng, Weihong Guo, Ting-Zhu Huang

Image super-resolution is a process to enhance image resolution. It is widely used in medical imaging, satellite imaging, target recognition, etc. In this paper, we conduct continuous modeling and assume that the unknown image intensity function is defined on a continuous domain and belongs to a space with a redundant basis. We propose a new iterative model for single image super-resolution based on an observation: an image is consisted of smooth components and non-smooth components, and we use two classes of approximated Heaviside functions (AHFs) to represent them respectively. Due to sparsity of the non-smooth components, a $L_{1}$ model is employed. In addition, we apply the proposed iterative model to image patches to reduce computation and storage. Comparisons with some existing competitive methods show the effectiveness of the proposed method.

Yuantao Cai, Marco Donatelli, Davide Bianchi et al.

Thresholding iterative methods are recently successfully applied to image deblurring problems. In this paper, we investigate the modified linearized Bregman algorithm (MLBA) used in image deblurring problems, with a proper treatment of the boundary artifacts. We consider two standard approaches: the imposition of boundary conditions and the use of the rectangular blurring matrix. The fast convergence of the MLBA depends on a regularizing preconditioner that could be computationally expensive and hence it is usually chosen as a block circulant circulant block (BCCB) matrix, diagonalized by discrete Fourier transform. We show that the standard approach based on the BCCB preconditioner may provide low quality restored images and we propose different preconditioning strategies, that improve the quality of the restoration and save some computational cost at the same time. Motivated by a recent nonstationary preconditioned iteration, we propose a new algorithm that combines such method with the MLBA.We prove that it is a regularizing and convergent method. A variant with a stationary preconditioner is also considered. Finally, a large number of numerical experiments shows that our methods provide accurate and fast restorations, when compared with the state of the art.

Gang Liu, Ting-Zhu Huang, Xiao-Guang Lv et al.

The least-square regression problems or inverse problems have been widely studied in many fields such as compressive sensing, signal processing, and image processing. To solve this kind of ill-posed problems, a regularization term (i.e., regularizer) should be introduced, under the assumption that the solutions have some specific properties, such as sparsity and group sparsity. Widely used regularizers include the $\ell_1$ norm, total variation (TV) semi-norm, and so on. Recently, a new regularization term with overlapping group sparsity has been considered. Majorization minimization iteration method or variable duplication methods are often applied to solve them. However, there have been no direct methods for solve the relevant problems because of the difficulty of overlapping. In this paper, we proposed new explicit shrinkage formulas for one class of these relevant problems, whose regularization terms have translation invariant overlapping groups. Moreover, we apply our results in TV deblurring and denoising with overlapping group sparsity. We use alternating direction method of multipliers to iterate solve it. Numerical results also verify the validity and effectiveness of our new explicit shrinkage formulas.

Gang Liu, Ting-Zhu Huang, Jun Liu et al.

The total variation (TV) regularization method is an effective method for image deblurring in preserving edges. However, the TV based solutions usually have some staircase effects. In this paper, in order to alleviate the staircase effect, we propose a new model for restoring blurred images with impulse noise. The model consists of an $\ell_1$-fidelity term and a TV with overlapping group sparsity (OGS) regularization term. Moreover, we impose a box constraint to the proposed model for getting more accurate solutions. An efficient and effective algorithm is proposed to solve the model under the framework of the alternating direction method of multipliers (ADMM). We use an inner loop which is nested inside the majorization minimization (MM) iteration for the subproblem of the proposed method. Compared with other methods, numerical results illustrate that the proposed method, can significantly improve the restoration quality, both in avoiding staircase effects and in terms of peak signal-to-noise ratio (PSNR) and relative error (ReE).

Jun Liu, Ting-Zhu Huang, Ivan W. Selesnick et al.

Image restoration is one of the most fundamental issues in imaging science. Total variation (TV) regularization is widely used in image restoration problems for its capability to preserve edges. In the literature, however, it is also well known for producing staircase-like artifacts. Usually, the high-order total variation (HTV) regularizer is an good option except its over-smoothing property. In this work, we study a minimization problem where the objective includes an usual $l_2$ data-fidelity term and an overlapping group sparsity total variation regularizer which can avoid staircase effect and allow edges preserving in the restored image. We also proposed a fast algorithm for solving the corresponding minimization problem and compare our method with the state-of-the-art TV based methods and HTV based method. The numerical experiments illustrate the efficiency and effectiveness of the proposed method in terms of PSNR, relative error and computing time.