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The nonconvex optimization method has attracted increasing attention due to its excellent ability of promoting sparsity in signal processing, image restoration, and machine learning. In this paper, we consider a new minimization method L1 -ß Lq ((ß, q)-[0, 1]×[1, 8)\(1, 1)) and its applications in signal recovery and image reconstruction because L1 -ß Lq minimization provides an effective way to solve the q-ratio sparsity minimization model. Our main contributions are to establish a convex hull decomposition for L1 -ß Lq and investigate RIP-based conditions for stable signal recovery and image reconstruction by L1 -ß Lq minimization. For one-dimensional signal recovery, our derived RIP condition extends existing results. For two-dimensional image recovery under L1 -ß Lq minimization of image gradients, we provide the error estimate of the resulting optimal solutions in terms of sparsity and noise level, which is missing in the literature. Numerical results of the limited angle problem in computed tomography imaging and image deblurring are presented to validate the efficiency and superiority of the proposed minimization method among the state-of-art image recovery methods. [ABSTRACT FROM AUTHOR]

The image nonlocal self-similarity (NSS) prior refers to the fact that a local patch often has many nonlocal similar patches to it across the image and has been widely applied in many recently proposed machining learning algorithms for image processing. However, there is no theoretical analysis on its working principle in the literature. In this paper, we discover a potential causality between NSS and low-rank property of color images, which is also available to grey images. A new patch group based NSS prior scheme is proposed to learn explicit NSS models of natural color images. The numerical low-rank property of patched matrices is also rigorously proved. The NSS-based QMC algorithm computes an optimal low-rank approximation to the high-rank color image, resulting in high PSNR and SSIM measures and particularly the better visual quality. A new tensor NSS-based QMC method is also presented to solve the color video inpainting problem based on quaternion tensor representation. The numerical experiments on color images and videos indicate the advantages of NSS-based QMC over the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Transformed L1 (TL1) regularization has been shown to have comparable signal recovery capability with L1 L2 regularization and L1/L2 regularization, regardless of whether the measurement matrix satisfies the restricted isometry property (RIP). In the spirit of the TL1 method, we introduce a transformed total variation (TTV) minimization model to investigate robust image recovery from a certain number of noisy measurements by the proposed TTV minimization model in this paper. An optimal error bound, up to a logarithmic factor, of robust image recovery from compressed measurements via the TTV minimization model is established, and the RIP based condition is improved compared with total variation (TV) minimization. Numerical results of image reconstruction demonstrate our theoretical results and illustrate the efficiency of the TTV minimization model among state-of-the-art methods. Empirically, the error bound between the reconstructed image and the original image is shown to be better than that produced by TV minimization. [ABSTRACT FROM AUTHOR]

Heterogeneous domain adaptation (HDA) is a challenging problem because of the different feature representations in the source and target domains. Most HDA methods search for mapping matrices from the source and target domains to discover latent features for learning. However, these methods barely consider the reconstruction error to measure the information loss during the mapping procedure. In this paper, we propose to jointly capture the information and match the source and target domain distributions in the latent feature space. In the learning model, we propose to minimize the reconstruction loss between the original and reconstructed representations to preserve information during transformation and reduce the Maximum Mean Discrepancy between the source and target domains to align their distributions. The resulting minimization problem involves two projection variables with orthogonal constraints that can be solved by the generalized gradient flow method, which can preserve orthogonal constraints in the computational procedure. We conduct extensive experiments on several image classification datasets to demonstrate that the effectiveness and efficiency of the proposed method are better than those of state-of-the-art HDA methods. [ABSTRACT FROM AUTHOR]

Color image restoration is one of the important tasks in color image processing. Covariant differentiation has been applied to handle vector bundles arising from color images in the red, green, blue (RGB) color space. However, there are strong correlations among these three channels, and color image regularization in RGB color space may not be effective enough. The main aim of this paper is to study vector bundles of color images in saturation-value color space and to develop color image regularization models based on vector bundles in saturation-value color space. We develop the saturation-value metric of a vector bundle of R5-valued functions, and we generalize the vectorial total variation and the vector bundle-valued total variation in saturation-value color space based on the saturation-value metric via the transformation between RGB color space and saturation-value color space. We then develop a saturation-value total variation regularization on vector bundles. We study color image restoration models by using such total variation, and show numerical examples that the proposed color image restoration model outperforms existing methods in terms of visual quality, peak signal-to-noise ratio, and structural similarity. [ABSTRACT FROM AUTHOR]

Non-Lipschitz regularization has got much attention in image restoration with additive noise removal recently, which can preserve neat edges in the restored image. In this paper, we consider a class of minimization problems with gradient compounded non-Lipschitz regularization applied to non-additive noise removal, with Poisson and multiplicative one as examples. The existence of a solution of the general model is discussed. We also extend the recent iterative support shrinkage strategy to give an algorithm to minimize it, where the subproblem at each iteration is allowed to be solved inexactly. Moreover, this paper is the first one to give the subdifferential of the gradient compounded non-Lipschitz regularization term, based on which we are able to establish the global convergence of the iterative sequence to a stationary point of the original objective function. This is, to our best knowledge, stronger than all the convergence results for gradient compounded non-Lipschitz minimization problems in the current published literature. Numerical experiments show that our proposed method performs well. [ABSTRACT FROM AUTHOR]

Although image denoising as a basic task of image restoration has been widely studied in the past decades, there are not many studies on mixed noise denoising. In this paper, we propose two structured dictionary learning models to recover images corrupted by mixed Gaussian and impulse noise. These two models can be merged as $\ell _{p}$ -norm fidelity plus $\ell _{q}$ -norm regularization. The fidelity term is used to fit image patches and the regularization term is employed for sparse coding. Particularly, we utilize proximal (and proximal linearized) alternating minimization methods as the main solvers to deal with these two models. We remove the Gaussian noise under the assumption that the uncorrupted image can be approximated with a linear representation under an appropriate orthogonal basis. We use different ways to remove impulse noise for these two models. The experimental results are reported to compare the existing methods and demonstrate the performance of the proposed denoising model is better than the other existing methods in terms of some quality assessment metrics. [ABSTRACT FROM AUTHOR]

In this paper, we study ℓ1 − αℓ2 (0 < α ⩽ 1) minimization methods for signal and image reconstruction with impulsive noise removal. The data fitting term is based on ℓ1 fidelity between the reconstruction output and the observational data, and the regularization term is based on ℓ1 − αℓ2 nonconvex minimization of the reconstruction output or its total variation. Theoretically, we show that under the generalized restricted isometry property that the underlying signal or image can be recovered exactly. Numerical algorithms are also developed to solve the resulting optimization problems. Experimental results have shown that the proposed models and algorithms can recover signal or images under impulsive noise degradation, and their performance is better than that of the existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose and develop a novel nonlocal variational technique based on structural similarity (SS) information for image restoration problems. In the literature, patches extracted from images are compared according to their pixel values, and then nonlocal filtering can be employed for image restoration. The disadvantage of this approach is that intensity-based patch distance may not be effective in image restoration, especially for images containing texture or structural information. The main aim of this paper is to propose using SS between image patches to develop nonlocal regularization models. In particular, two types of nonlocal regularizing functions are studied: an SS-based nonlocal quadratic function (SS-NLH1) and an SS-based nonlocal total variation function (SS-NLTV) for regularization of image restoration problems. Moreover, we employ iterative algorithms to solve these SS-NLH1 and SS-NLTV variational models numerically and discuss the convergence of these algorithms. The experimental results are presented to demonstrate the effectiveness of the proposed models. [ABSTRACT FROM AUTHOR]

Color image restoration is one of the important tasks in color image processing. Total variation regularizaton was proposed and employed for the recovery of edges in a grayscale image. In the literature, there are several methods for extension of total variation regularization for color images, for example, based on color channel coupling and tensor regularization. The main contribution of this paper is to propose and develop a new saturation-value (SV) color total variation regularization in the hue, saturation, amd value color space instead of in the original red, green, and blue color space. The development of this SV total variation can be studied via the representation of color images in the quaternion framework for color edge detection. We will investigate the properties of the SV total variation regularization and the resulting optimization model for color image restoration. Numerical examples are presented to demonstrate that the performance of the new SV total variation is better than that of existing color image total variation methods in terms of some criteria such as PSNR, SSIM, and S-CIELAB error. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a nonconvex data fitting term and a total variation regularization term for image restoration with impulse noise removal. The proposed model is different from existing image restoration models where the data fitting term is based on the ℓ1- or ℓ2-norm, and the regularization term is based on the total variation, ℓ1-norm, or some nonconvex functions. Theoretically, we analyze the properties of minimizers of the proposed objective function with the nonconvex data fitting term and the total variation regularization term. We show that minimizers can preserve piecewise constant regions or match with the data points perfectly. This property is particularly useful for impulse noise removal. The proposed image restoration model can be solved by the proximal linearized minimization algorithm, and the global convergence of the iterative algorithm can also be established according to Kurdyka–Lojasiewicz property. The performance of the proposed model is tested for image restoration with salt-and-pepper impulse noise or random-valued impulse noise. We demonstrate that the restored images by the proposed Nonconvex-TV model are better (in terms of PSNR and visual quality) than those by the other existing data fitting plus regularization models, including ℓ1 plus total variation (L1TV) and ℓ1 plus nonconvex (L1Nonconvex) methods. [ABSTRACT FROM AUTHOR]

Missing information, such as dead pixel values and cloud effects, is very common image quality degradation problems in remote sensing. Missing information can reduce the accuracy of the subsequent image processing, in applications such as classification, unmixing, and target detection, and even the quantitative retrieval process. The main aim of this paper is to study an adaptive weighted tensor completion (AWTC) method for the recovery of remote sensing images with missing data. Our idea is to collectively make use of the spatial, spectral, and temporal information to build a new weighted tensor low-rank regularization model for recovering the missing data. In the model, the weights are determined adaptively by considering the contribution of the spatial, spectral, and temporal information in each dimension. Experimental results based on both simulated and real data sets are presented to verify that the proposed method can recover missing data, and its performance is found to be better than the other tested methods. In the simulated experiments, the peak signal-to-noise ratio is improved by more than 3 dB, compared with the original tensor completion model. In the real data experiments, the proposed AWTC model can better recover the dead line problem in Aqua Moderate Resolution Imaging Spectroradiometer band 6 and the scan-line corrector-off problem in enhanced thematic mapper plus images, with the smallest spectral distortion. [ABSTRACT FROM PUBLISHER]

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]

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]

• SVTV regularization term is applied to model the target color image in HSV color space. • The fidelity term is well-adapted to multiplicative noise. • The existence and uniqueness of the minimizer of the proposed minimization problem is shown. • The convergence of an implicit scheme of the associated evolution problem is given. In this paper, we propose and develop a novel Saturation-Value Total Variation (SVTV) model for multiplicative noise and blur removal of color images. In the proposed model, SVTV regularization term is applied to model the target color image in HSV color space instead of RGB color space, and the fidelity term is well-adapted to multiplicative noise. We investigate into the existence and uniqueness of the minimizer of the proposed minimization problem. We study and show the convergence of an implicit scheme of the associated evolution problem for the numerical solution of the proposed SVTV model. Numerical examples are presented to demonstrate the performance of the proposed SVTV model is significantly better than that of other testing methods in terms of some criteria such as PSNR, SSIM and S-CIELAB color error. [ABSTRACT FROM AUTHOR]

In this paper, we make use of the fact that the matrix u is (approximately) low-rank in image inpainting, and the corresponding gradient transform matrices D x u , D y u are sparse in image reconstruction and restoration. Therefore we consider that these gradient matrices D x u , D y u also are (approximately) low-rank, and also verify it by numerical test and theoretical analysis. We propose a model called compressive total variation (CTV) to characterize the sparsity and low-rank prior knowledge of an image. In order to solve the proposed model, we design a concrete algorithm with provably convergence, which is based on inertial proximal ADMM. The performance of the proposed model is tested for magnetic resonance imaging (MRI) reconstruction, image denoising and image deblurring. The proposed method not only recovers edges of the image but also preserves fine details of the image. And our model is much better than the existing regularization models based on the TGV, Shearlet-TGV, ℓ 1 − α ℓ 2 TV and BM3D in test for images with piecewise constant regions. And it visibly improves the performances of TV, ℓ 1 − α ℓ 2 TV and TGV, and is comparable to Shearlet-TGV in test for natural images. [ABSTRACT FROM AUTHOR]