Non-Convex Weighted Lp Nuclear Norm based ADMM Framework for Image Restoration

Since the matrix formed by nonlocal similar patches in a natural image is of low rank, the nuclear norm minimization (NNM) has been widely used in various image processing studies. Nonetheless, nuclear norm based convex surrogate of the rank function usually over-shrinks the rank components and makes different components equally, and thus may produce a result far from the optimum. To alleviate the above-mentioned limitations of the nuclear norm, in this paper we propose a new method for image restoration via the non-convex weighted Lp nuclear norm minimization (NCW-NNM), which is able to more accurately enforce the image structural sparsity and self-similarity simultaneously. To make the proposed model tractable and robust, the alternative direction multiplier method (ADMM) is adopted to solve the associated non-convex minimization problem. Experimental results on various types of image restoration problems, including image deblurring, image inpainting and image compressive sensing (CS) recovery, demonstrate that the proposed method outperforms many current state-of-the-art methods in both the objective and the perceptual qualities.Comment: arXiv admin note: text overlap with arXiv:1611.0898

From Group Sparse Coding to Rank Minimization: A Novel Denoising Model for Low-level Image Restoration

Recently, low-rank matrix recovery theory has been emerging as a significant progress for various image processing problems. Meanwhile, the group sparse coding (GSC) theory has led to great successes in image restoration (IR) problem with each group contains low-rank property. In this paper, we propose a novel low-rank minimization based denoising model for IR tasks under the perspective of GSC, an important connection between our denoising model and rank minimization problem has been put forward. To overcome the bias problem caused by convex nuclear norm minimization (NNM) for rank approximation, a more generalized and flexible rank relaxation function is employed, namely weighted nonconvex relaxation. Accordingly, an efficient iteratively-reweighted algorithm is proposed to handle the resulting minimization problem combing with the popular L_(1/2) and L_(2/3) thresholding operators. Finally, our proposed denoising model is applied to IR problems via an alternating direction method of multipliers (ADMM) strategy. Typical IR experiments on image compressive sensing (CS), inpainting, deblurring and impulsive noise removal demonstrate that our proposed method can achieve significantly higher PSNR/FSIM values than many relevant state-of-the-art methods.Comment: Accepted by Signal Processin

Edge-adaptive l2 regularization image reconstruction from non-uniform Fourier data

Total variation regularization based on the l1 norm is ubiquitous in image reconstruction. However, the resulting reconstructions are not always as sparse in the edge domain as desired. Iteratively reweighted methods provide some improvement in accuracy, but at the cost of extended runtime. In this paper we examine these methods for the case of data acquired as non-uniform Fourier samples. We then develop a non-iterative weighted regularization method that uses a pre-processing edge detection to find exactly where the sparsity should be in the edge domain. We show that its performance in terms of both accuracy and speed has the potential to outperform reweighted TV regularization methods

Group-based Sparse Representation for Image Compressive Sensing Reconstruction with Non-Convex Regularization

Patch-based sparse representation modeling has shown great potential in image compressive sensing (CS) reconstruction. However, this model usually suffers from some limits, such as dictionary learning with great computational complexity, neglecting the relationship among similar patches. In this paper, a group-based sparse representation method with non-convex regularization (GSR-NCR) for image CS reconstruction is proposed. In GSR-NCR, the local sparsity and nonlocal self-similarity of images is simultaneously considered in a unified framework. Different from the previous methods based on sparsity-promoting convex regularization, we extend the non-convex weighted Lp (0 < p < 1) penalty function on group sparse coefficients of the data matrix, rather than conventional L1-based regularization. To reduce the computational complexity, instead of learning the dictionary with a high computational complexity from natural images, we learn the principle component analysis (PCA) based dictionary for each group. Moreover, to make the proposed scheme tractable and robust, we have developed an efficient iterative shrinkage/thresholding algorithm to solve the non-convex optimization problem. Experimental results demonstrate that the proposed method outperforms many state-of-the-art techniques for image CS reconstruction

Nonconvex Nonsmooth Low-Rank Minimization for Generalized Image Compressed Sensing via Group Sparse Representation

Group sparse representation (GSR) based method has led to great successes in various image recovery tasks, which can be converted into a low-rank matrix minimization problem. As a widely used surrogate function of low-rank, the nuclear norm based convex surrogate usually leads to over-shrinking problem, since the standard soft-thresholding operator shrinks all singular values equally. To improve traditional sparse representation based image compressive sensing (CS) performance, we propose a generalized CS framework based on GSR model, which leads to a nonconvex nonsmooth low-rank minimization problem. The popular L_2-norm and M-estimator are employed for standard image CS and robust CS problem to fit the data respectively. For the better approximation of the rank of group-matrix, a family of nuclear norms are employed to address the over-shrinking problem. Moreover, we also propose a flexible and effective iteratively-weighting strategy to control the weighting and contribution of each singular value. Then we develop an iteratively reweighted nuclear norm algorithm for our generalized framework via an alternating direction method of multipliers framework, namely, GSR-AIR. Experimental results demonstrate that our proposed CS framework can achieve favorable reconstruction performance compared with current state-of-the-art methods and the robust CS framework can suppress the outliers effectively.Comment: This paper has been submitted to the Journal of the Franklin Institute. arXiv admin note: substantial text overlap with arXiv:1903.0978

Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset

Recent research on problem formulations based on decomposition into low-rank plus sparse matrices shows a suitable framework to separate moving objects from the background. The most representative problem formulation is the Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit (PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix. However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal of these similar problem formulations is to obtain explicitly or implicitly a decomposition into low-rank matrix plus additive matrices. In this context, this work aims to initiate a rigorous and comprehensive review of the similar problem formulations in robust subspace learning and tracking based on decomposition into low-rank plus additive matrices for testing and ranking existing algorithms for background/foreground separation. For this, we first provide a preliminary review of the recent developments in the different problem formulations which allows us to define a unified view that we called Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine carefully each method in each robust subspace learning/tracking frameworks with their decomposition, their loss functions, their optimization problem and their solvers. Furthermore, we investigate if incremental algorithms and real-time implementations can be achieved for background/foreground separation. Finally, experimental results on a large-scale dataset called Background Models Challenge (BMC 2012) show the comparative performance of 32 different robust subspace learning/tracking methods.Comment: 121 pages, 5 figures, submitted to Computer Science Review. arXiv admin note: text overlap with arXiv:1312.7167, arXiv:1109.6297, arXiv:1207.3438, arXiv:1105.2126, arXiv:1404.7592, arXiv:1210.0805, arXiv:1403.8067 by other authors, Computer Science Review, November 201

Non-Convex Weighted Lp Minimization based Group Sparse Representation Framework for Image Denoising

Nonlocal image representation or group sparsity has attracted considerable interest in various low-level vision tasks and has led to several state-of-the-art image denoising techniques, such as BM3D, LSSC. In the past, convex optimization with sparsity-promoting convex regularization was usually regarded as a standard scheme for estimating sparse signals in noise. However, using convex regularization can not still obtain the correct sparsity solution under some practical problems including image inverse problems. In this paper we propose a non-convex weighted $\ell_p$ minimization based group sparse representation (GSR) framework for image denoising. To make the proposed scheme tractable and robust, the generalized soft-thresholding (GST) algorithm is adopted to solve the non-convex $\ell_p$ minimization problem. In addition, to improve the accuracy of the nonlocal similar patches selection, an adaptive patch search (APS) scheme is proposed. Experimental results have demonstrated that the proposed approach not only outperforms many state-of-the-art denoising methods such as BM3D and WNNM, but also results in a competitive speed

A Survey on Nonconvex Regularization Based Sparse and Low-Rank Recovery in Signal Processing, Statistics, and Machine Learning

In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based sparse and low-rank recovery is of considerable interest and it in fact is a main driver of the recent progress in nonconvex and nonsmooth optimization. This paper gives an overview of this topic in various fields in signal processing, statistics and machine learning, including compressive sensing (CS), sparse regression and variable selection, sparse signals separation, sparse principal component analysis (PCA), large covariance and inverse covariance matrices estimation, matrix completion, and robust PCA. We present recent developments of nonconvex regularization based sparse and low-rank recovery in these fields, addressing the issues of penalty selection, applications and the convergence of nonconvex algorithms. Code is available at https://github.com/FWen/ncreg.git.Comment: 22 page

Compressive Sensing of Color Images Using Nonlocal Higher Order Dictionary

This paper addresses an ill-posed problem of recovering a color image from its compressively sensed measurement data. Differently from the typical 1D vector-based approach of the state-of-the-art methods, we exploit the nonlocal similarities inherently existing in images by treating each patch of a color image as a 3D tensor consisting of not only horizontal and vertical but also spectral dimensions. A group of nonlocal similar patches form a 4D tensor for which a nonlocal higher order dictionary is learned via higher order singular value decomposition. The multiple sub-dictionaries contained in the higher order dictionary decorrelate the group in each corresponding dimension, thus help the detail of color images to be reconstructed better. Furthermore, we promote sparsity of the final solution using a sparsity regularization based on a weight tensor. It can distinguish those coefficients of the sparse representation generated by the higher order dictionary which are expected to have large magnitude from the others in the optimization. Accordingly, in the iterative solution, it acts like a weighting process which is designed by approximating the minimum mean squared error filter for more faithful recovery. Experimental results confirm improvement by the proposed method over the state-of-the-art ones.Comment: 13 pages, 10 figure

Robust and Low-Rank Representation for Fast Face Identification with Occlusions

In this paper we propose an iterative method to address the face identification problem with block occlusions. Our approach utilizes a robust representation based on two characteristics in order to model contiguous errors (e.g., block occlusion) effectively. The first fits to the errors a distribution described by a tailored loss function. The second describes the error image as having a specific structure (resulting in low-rank in comparison to image size). We will show that this joint characterization is effective for describing errors with spatial continuity. Our approach is computationally efficient due to the utilization of the Alternating Direction Method of Multipliers (ADMM). A special case of our fast iterative algorithm leads to the robust representation method which is normally used to handle non-contiguous errors (e.g., pixel corruption). Extensive results on representative face databases (in constrained and unconstrained environments) document the effectiveness of our method over existing robust representation methods with respect to both identification rates and computational time. Code is available at Github, where you can find implementations of the F-LR-IRNNLS and F-IRNNLS (fast version of the RRC) : https://github.com/miliadis/FIRCComment: IEEE Transactions on Image Processing (TIP), 201