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Multi-dimensional images, such as color images and multi-spectral images (MSIs), 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 multi-dimensional images. We also formulate an implicit regularizer by plugging a denoising neural network (termed as deep denoiser), which is convinced to express the deep image prior learned from a large number of natural images. The resulting model can be solved by the alternating directional method of multipliers algorithm under the plug-and-play (PnP) framework. Experimental results on color images, videos, and MSIs demonstrate that the proposed method can recover both the global structure and fine details very well and achieve superior performance over competing methods in terms of quality metrics and visual effects. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel model for remote sensing images destriping, which includes the Schatten 1 ∕ 2 -norm and the unidirectional first-order and high-order total variation regularization. The main idea is that the stripe layer is low-rank, and the desired image possesses smoothness across stripes. Therefore, we use the Schatten 1 ∕ 2 -norm regularization to depict the low-rankness of stripes, and use the unidirectional total variation and the unidirectional high-order total variation to guarantee the smoothness of the underlying image. We develop the alternating direction method of multipliers algorithm to solve the proposed model. Extensive experiments on synthetic and real data are reported to show the superiority of the proposed method over state-of-the-art methods in terms of both quantitative and qualitative assessments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a novel model to restore an image corrupted by blur and Cauchy noise. The model is composed of a data fidelity term and two regularization terms including total variation and high-order total variation. Total variation provides well-preserved edge features, but suffers from staircase effects in smooth regions, whereas high-order total variation can alleviate staircase effects. Moreover, we introduce a strategy for adaptively selecting regularization parameters. We develop an efficient alternating minimization algorithm for solving the proposed model. Numerical examples suggest that the proposed method has the advantages of better preserving edges and reducing staircase effects. [ABSTRACT FROM AUTHOR]

In this paper, we address the total variation (TV)-based nonlinear image restoration problems. In nonlinear image restoration problems, an original image is corrupted by a spatiallyinvariant blur, the build-in nonlinearity in imaging system, and the additive Gaussian white noise. We study the objective function consisting of the nonlinear least squares data-fitting term and the TV regularization term of the restored image. By making use of the structure of the objective function, an efficient alternating direction method of multipliers can be developed for solving the proposed model. The convergence of the numerical scheme is also studied. Numerical examples, including nonlinear image restoration and high-dynamic range imaging are reported to demonstrate the effectiveness of the proposed model and the efficiency of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

Recently, tensor train rank, defined by a well-balanced matricization scheme, has been shown the powerful capacity to capture the hidden correlations among different modes of a tensor, leading to great success in tensor completion problem. Most of the high-dimensional data in the real world are more likely to be grossly corrupted with sparse noise. In this paper, based on tensor train rank, we consider a new model for tensor robust principal component analysis which aims to recover a low-rank tensor corrupted by sparse noise. The alternating direction method of multipliers algorithm is developed to solve the proposed model. A tensor augmentation tool called ket augmentation is used to convert lower-order tensors to higher-order tensors to enhance the performance of our method. Experiments of simulated data show the superiority of the proposed method in terms of PSNR and SSIM values. Moreover, experiments of the real rain streaks removal and the real stripe noise removal also illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

• Adopt the nonlocal self-similarity prior for video rain streaks removal. • Utilize the tensor nuclear norm to regularize the low-rankness of similar-patch-formed tensors. • Develop an efficient alternating direction method of multipliers to tackle the proposed model. Outdoor videos captured in rainy weather may be significantly corrupted by the undesired rain streaks, which severely affect the video processing tasks in outdoor computer vision systems. In this paper, we propose a tensor-based video rain streaks removal method using the nonlocal low-rank regularization. Specifically, we first divide videos into overlapped spatial–temporal patches. Then for each patch, we group its nonlocal similar spatial–temporal patches to form a third-order tensor. To model the clean videos, we characterize the wealth redundancy by adopting the tensor nuclear norm to regularize the low-rankness of the third-order tensors formed by similar spatial–temporal patches of clean videos. We also consider the piecewise smoothness and the temporal continuity of clean videos and utilize the unidirectional total variation to enhance the smoothness and continuity. Moreover, as rain streaks are sparse and smooth along the rain direction, we model the rain streaks by employing an ℓ 1 norm and the unidirectional total variation penalty to boost the sparsity and directional smoothness, respectively. We develop an efficient alternating direction method of multipliers to solve the proposed model. Experimental results on both synthetic and real rainy videos show that our method outperforms the state-of-the-art methods quantitatively and qualitatively. [ABSTRACT FROM AUTHOR]

Highlights • We propose a nonconvex variational model for Retinex. • An efficient method of multipliers to tackle the proposed model is developed. • Numerical results show the method is comparable to the state-of-the-art methods. Abstract Retinex aims at estimating real reflectance images by removing the effect of illumination. We propose a nonconvex variational model for Retinex with novel priors for reflectance and illumination. Based on the statistics of the gradients of reflectance and illumination, we use the hyper-Laplacian prior to characterize the gradients of reflectance, and the hybrid hyper-Laplacian and Tikhonov prior to characterize the gradients of illumination. An efficient alternating direction method of multipliers (ADMM) is developed to solve the proposed model. Extensive numerical experiments show that the proposed method is comparable to the state-of-the-art methods quantitatively and qualitatively. [ABSTRACT FROM AUTHOR]

Rain removal from a single image is an important issue in the fields of outdoor vision. Rain, a kind of bad weather that is often seen, usually causes complex local intensity changes in images and has negative impact on vision performance. Many existing rain removal approaches have been proposed recently, such as some dictionary learning-based methods and layer decomposition-based methods. Although these methods can improve the visibility of rain images, they fail to consider the intrinsic directional and structural information of rain streaks, thus usually leave undesired rain streaks or change the background intensity of rain-free region significantly. In the paper, we propose a simple but efficient method to remove rain streaks from a single rainy image. The proposed method formulates a global sparse model that involves three sparse terms by considering the intrinsic directional and structural knowledge of rain streaks, as well as the property of image background information. We employ alternating direction method of multipliers (ADMM) to solve the proposed convex model which guarantees the global optimal solution. Results on a variety of synthetic and real rainy images demonstrate that the proposed method outperforms two recent state-of-the-art rain removal methods. Moreover, the proposed method needs no training and requires much less computation significantly. [ABSTRACT FROM AUTHOR]

Low-rankness has been widely exploited for the tensor completion problem. Recent advances have suggested that the tensor nuclear norm often leads to a promising approximation for the tensor rank. It treats the singular values equally to pursue the convexity of the objective function, while the singular values for the practical images have clear physical meanings with different importance and should be treated differently. In this paper, we propose a non-convex logDet function as a smooth approximation for tensor rank instead of the convex tensor nuclear norm and introduce it into the low-rank tensor completion problem. An alternating direction method of multiplier (ADMM)-based method is developed to solve the problem. Experimental results have shown that the proposed method can significantly outperform existing state-of-the-art nuclear norm-based methods for tensor completion. [ABSTRACT FROM AUTHOR]

• We propose a new low-rank sparsity relaxation by imposing the logdet function on tensor ring (TR) unfolding matrices. • We propose an efficient ADMM algorithm to solve the LogTR model with convergence analysis. • Extensive experiments show the effectiveness of the proposed LogTR in the tensor completion problem. 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. To alleviate this weakness, this paper proposes an enhanced low-rank sparsity measure, which can more accurately approximate the TR rank and better promote the low-rankness of the solution. In specific, we apply the logdet function onto TR unfolding matrices to shrink less the larger singular values while shrink more the smaller ones. 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. [ABSTRACT FROM AUTHOR]

In many sparsity-based image processing problems, compared with the convex ℓ 1 norm approximation of the nonconvex ℓ 0 quasi-norm, one can often preserve the structures better by taking full advantage of the nonconvex ℓ p quasi-norm (0 ≤ p < 1). In this paper, we propose a nonconvex ℓ p quasi-norm approximation in the total generalized variation (TGV)-shearlet regularization for image reconstruction. By introducing some auxiliary variables, the nonconvex nonsmooth objective function can be solved by an efficient alternating direction method of multipliers with convergence analysis. Especially, we use a generalized iterated shrinkage operator to deal with the ℓ p quasi-norm subproblem, which is easy to implement. Extensive experimental results show clearly that the proposed nonconvex sparsity approximation outperforms some state-of-the-art algorithms in both the visual and quantitative measures for different sampling ratios and noise levels. [ABSTRACT FROM AUTHOR]

Stripe noise degradation is a common phenomenon in remote sensing image, which largely affects the visual quality and brings great difficulty for subsequent processing. In contrast to existing stripe noise removal (destriping) models in which the reconstruction is performed to directly estimate the clean image from the striped one, the proposed model achieves the destriping by estimating the stripe component firstly. Since the stripe component possesses column sparse structure, the group sparsity is employed in this study. In addition, difference-based constraints are used to describe the direction information of the stripes. Then, we build a novel convex optimization model which consists of a unidirectional total variation term, a group sparsity term and a gradient domain fidelity term solved by an efficient alternating direction method of multiplier. Compared with the state-of-the-art methods, experiment results on simulated and real data are reported to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Although the extensive research on Gaussian noise removal, few works consider the Cauchy noise removal problem. In this paper, we propose a novel group-based low-rank method for Cauchy noise removal. By exploiting the nonlocal self-similarity of natural images, we consider a group of similar patches as an approximate low-rank matrix, and formulate the denoising of each group as a low-rank matrix recovery problem. Meanwhile, we develop the alternating direction method of multipliers algorithm to solve the proposed nonconvex model with guaranteed convergence. Experiments illustrate that our method has superior performance over the state-of-the-art methods in terms of both visual and quantitative measures. [ABSTRACT FROM AUTHOR]