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Abstract Image restoration with impulse noise is an important task in image processing. Taking into account the statistical distribution of impulse noise, the ℓ1‐norm data fidelity and total variation (ℓ1TV) model has been widely used in this area. However, the ℓ1TV model usually performs worse when the noise level is high. To overcome this drawback, several nonconvex models have been proposed. In this paper, an efficient iterative algorithm is proposed to solve nonconvex models arising in impulse noise. Compared to existing algorithms, the proposed algorithm is a completely explicit algorithm in which every subproblem has a closed‐form solution. The key idea is to transform the original nonconvex models into an equivalent constrained minimization problem with two separable objective functions, where one is differentiable but nonconvex. As a consequence, the proximal linearized alternating direction method of multipliers is employed to solve it. Extensive numerical experiments are presented to demonstrate the efficiency and effectiveness of the proposed algorithm.

Super-resolution (SR) aims to recover a high-resolution image from a single or multiple low-resolution images, compensating for the limitations of satellite sensor imaging. Deep convolutional neural networks have made great achievement in remote sensing image SR. In this article, we propose a novel gradient prior dilated convolutional network (GPDCN) for remote sensing image SR, which obtains contextual spatial connections and alleviates structural distortions. The GPDCN comprises a multiscale feature extraction network and a feature reconstruction network. The former employs a double-path dilated residual block with dilation convolution to increase a receptive field, a global self-attention module to detect long-range reliance among image patches, and a gradient propagation network to extract high-level gradient information. The latter uses the mixed high-order attention module to reconstruct the feature by collecting the high-order characteristics of multiple frequency bands. Experiments with the Massachusetts_Roads and 3K VEHICLE_SR datasets demonstrate that the GPDCN outperforms recent techniques concerning both quantitative and qualitative measures.

Researchers have resorted to model quantization to compress and accelerate graph neural networks (GNNs). Nevertheless, several challenges remain: (1) quantization functions overlook outliers in the distribution, leading to increased quantization errors; (2) the reliance on full-precision teacher models results in higher computational and memory overhead. To address these issues, this study introduces a novel framework called quantized graph neural networks for image classification (QGNN-IC), which incorporates a novel quantization function, Pauta quantization (PQ), and two innovative self-distillation methods, attention quantization distillation (AQD) and stochastic quantization distillation (SQD). Specifically, PQ utilizes the statistical characteristics of distribution to effectively eliminate outliers, thereby promoting fine-grained quantization and reducing quantization errors. AQD enhances the semantic information extraction capability by learning from beneficial channels via attention. SQD enhances the quantization robustness through stochastic quantization. AQD and SQD significantly improve the performance of the quantized model with minimal overhead. Extensive experiments show that QGNN-IC not only surpasses existing state-of-the-art quantization methods but also demonstrates robust generalizability.

PurposeDuring neoadjuvant chemotherapy (NACT), breast tumor morphological and vascular characteristics are usually changed. This study aimed to evaluate the tumor shrinkage pattern and response to NACT by preoperative multiparametric magnetic resonance imaging (MRI), including dynamic contrast-enhanced MRI (DCE-MRI), diffuse weighted imaging (DWI) and T2 weighted imaging (T2WI).MethodIn this retrospective analysis, female patients with unilateral unifocal primary breast cancer were included for predicting tumor pathologic/clinical response to NACT (n=216, development set, n=151 and validation set, n=65) and for discriminating the tumor concentric shrinkage (CS) pattern from the others (n=193; development set, n=135 and validation set, n=58). Radiomic features (n=102) of first-order statistical, morphological and textural features were calculated on tumors from the multiparametric MRI. Single- and multiparametric image-based features were assessed separately and were further combined to feed into a random forest-based predictive model. The predictive model was trained in the testing set and assessed on the testing dataset with an area under the curve (AUC). Molecular subtype information and radiomic features were fused to enhance the predictive performance.ResultsThe DCE-MRI-based model showed higher performance (AUCs of 0.919, 0.830 and 0.825 for tumor pathologic response, clinical response and tumor shrinkage patterns, respectively) than either the T2WI or the ADC image-based model. An increased prediction performance was achieved by a model with multiparametric MRI radiomic feature fusion.ConclusionsAll these results demonstrated that multiparametric MRI features and their information fusion could be of important clinical value for the preoperative prediction of treatment response and shrinkage pattern.

Internal Solitary Waves (ISWs) play a pivotal role in transporting energy and matter within the ocean and also pose substantial risks to ocean engineering, navigation, and underwater communication systems. Consequently, measures need to be adopted to alleviate their negative effects and minimize linked risks. An effective method entails extracting ISW positions from Synthetic Aperture Radar (SAR) data for precise trajectory prediction and efficient avoidance strategies. However, manual extraction of ISWs from SAR data is time-consuming and prone to inaccuracies. Hence, it is imperative to develop a high-precision, rapid, and automated ISW-extraction algorithm. In this paper, we introduce Middle Transformer U2-net (MTU2-net), an innovative model that integrates a distinctive loss function and Transformer to improve the accuracy of ISWs’ extraction. The novel loss function enhances the model’s capacity to extract bow waves, whereas the Transformer ensures coherence in ISW’s patterns. By conducting experiments involving 762 image scenes, incorporating ISWs, from the South China Sea, we established a standardized dataset. The Mean Intersection over Union (MIoU) achieved on this dataset was 71.57%, surpassing the performance of other compared methods. The experimental outcomes showcase the remarkable performance of our proposed model in precisely extracting bow wave attributes from SAR data.

The restoration of images corrupted by blurring and structured noise has attracted growing attention in the domains of image processing and computer vision. However, many works only focus on the restoration of the images degraded by blurring and additive structured noise or multiplicative structured noise separately. It is still a challenge and an open problem to restore degraded images with blurring and multiplicative structured noise, simultaneously. In this paper, based on the total variation (TV), the statistical property of the Gamma noise and the maximum a posteriori (MAP) estimator, we obtain a convex variational model to recover blurred images with multiplicative structured noise. Especially, to get this convex model, we reformulate the prior assumption of the images degradation model by division instead of multiplication. For solving this convex model, an effective alternating direction method of multipliers (ADMM) is employed. Numerical experiments are presented to illustrate the effectiveness and efficiency of the proposed model.

This paper investigates the convergence properties and applications of the three-operator splitting method, also known as the Davis-Yin splitting (DYS) method, integrated with extrapolation and plug-and-play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possibly nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward-backward splitting and extrapolated Douglas-Rachford splitting methods. To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka-Łojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization step is replaced by a gradient step-based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image superresolution problems, where our numerical results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images. [ABSTRACT FROM AUTHOR]

The Mumford-Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford-Shah model, which seeks a smoothed approximation solution to the Mumford-Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford-Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods. [ABSTRACT FROM AUTHOR]

Low-light image enhancement plays an important role in computer vision applications, which is a fundamental low-level task and can affect high-level computer vision tasks. To solve this ill-posed problem, a lot of methods have been proposed to enhance low-light images. However, their performance degrades significantly under nonuniform lighting conditions. Due to the rapid variation of illuminance in different regions in natural images, it is challenging to enhance low-light parts and retain normal-light parts simultaneously in the same image. Commonly, either the low-light parts are underenhanced or the normal-light parts are overenhanced, accompanied by color distortion and artifacts. To overcome this problem, we propose a simple and effective Retinex-based model with reflectance map reweighting for images under nonuniform lighting conditions. An alternating proximal gradient (APG) algorithm is proposed to solve the proposed model, in which the illumination map, the reflectance map, and the weighting map are updated iteratively. To make our model applicable to a wide range of light conditions, we design an initialization scheme for the weighting map. A theoretical analysis of the existence of the solution to our model and the convergence of the APG algorithm are also established. A series of experiments on real-world low-light images are conducted, which demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Spherical image processing has been widely applied in many important fields, such as omnidirectional vision for autonomous cars, global climate modeling, and medical imaging. It is nontrivial to extend an algorithm developed for flat images to the spherical ones. In this work, we focus on the challenging task of spherical image inpainting with a deep learning-based regularizer. Instead of a naive application of existing models for planar images, we employ a fast directional spherical Haar framelet transform and develop a novel optimization framework based on a sparsity assumption of the framelet transform. Furthermore, by employing progressive encoder-decoder architecture, a new and better-performed deep CNN denoiser is carefully designed and works as an implicit regularizer. Finally, we use a plug-and-play method to handle the proposed optimization model, which can be implemented efficiently by training the CNN denoiser prior. Numerical experiments are conducted and show that the proposed algorithms can greatly recover damaged spherical images and achieve the best performance over purely using a deep learning denoiser and a plug-and-play model. [ABSTRACT FROM AUTHOR]

Image deblurring under Rician noise has attracted considerable attention in imaging science. Frequently appearing in medical imaging, Rician noise leads to an interesting nonconvex optimization problem, termed as the MAP-Rician model, which is based on the Maximum a Posteriori (MAP) estimation approach. As the MAP-Rician model is deeply rooted in Bayesian analysis, we want to understand its mathematical analysis carefully. Moreover, one needs to properly select a suitable algorithm for tackling this nonconvex problem to get the best performance. This paper investigates both issues. Indeed, we first present a theoretical result about the existence of a minimizer for the MAP-Rician model under mild conditions. Next, we aim to adopt an efficient boosted difference of convex functions algorithm (BDCA) to handle this challenging problem. Basically, BDCA combines the classical difference of convex functions algorithm (DCA) with a backtracking line search, which utilizes the point generated by DCA to define a search direction. In particular, we apply a smoothing scheme to handle the nonsmooth total variation (TV) regularization term in the discrete MAP-Rician model. Theoretically, using the Kurdyka--Lojasiewicz (KL) property, the convergence of the numerical algorithm can be guaranteed. We also prove that the sequence generated by the proposed algorithm converges to a stationary point with the objective function values decreasing monotonically. Numerical simulations are then reported to clearly illustrate that our BDCA approach outperforms some state-of-the-art methods for both medical and natural images in terms of image recovery capability and CPU-time cost. [ABSTRACT FROM AUTHOR]

In ptychography experiments, redundant scanning is usually required to guarantee the stable recovery, such that a huge number of frames are generated, and thus it poses a great demand on parallel computing to solve this large-scale inverse problem. In this paper, we propose the overlapping domain decomposition methods to solve the nonconvex optimization problem in ptychographic imaging. They decouple the problem defined on the whole domain into subproblems only defined on the subdomains with synchronizing information in the overlapping regions of these subdomains, thus leading to highly parallel algorithms with good load balance. More specifically, for the nonblind recovery (with known probe in advance), by enforcing the continuity of the overlapping regions for the image (sample), the nonlinear optimization model is established based on a novel smooth-truncated amplitude-Gaussian metric (ST-AGM). Such a metric allows for fast calculation of the proximal mapping with closed form, and meanwhile provides the possibility for the convergence guarantee of the first-order nonconvex optimization algorithm due to its Lipschitz smoothness. Then the alternating direction method of multipliers is utilized to generate an efficient overlapping domain decomposition based ptychography algorithm (OD2P) for the two-subdomain domain decomposition (DD), where all subproblems can be computed with closed-form solutions. Due to the Lipschitz continuity for the gradient of the objective function with ST-AGM, the convergence of the proposed OD2P is derived under mild conditions. Moreover, it is extended to more general cases including multiple-subdomain DD and blind recovery. Numerical experiments are further conducted to show the performance of proposed algorithms, demonstrating good convergence speed and robustness to the noise. Especially, we report the virtual wall-clock time of proposed algorithm up to 10 processors, which shows potential for upcoming massively parallel computations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new three-stage segmentation framework based on a convex variant of the Mumford--Shah model and the intensity inhomogeneity information of an image. The first stage in our framework is to perform a dimension lifting method. An intensity inhomogeneity image is added as an additional channel, which results in a vector-valued image. In the second stage, a convex variant of the Mumford--Shah model is applied to each channel of the vector-valued image to obtain a smooth approximation. We use the semi--proximal alternating direction method of multipliers (sPADMM) to solve this model and prove that the sPADMM for solving this convex model has Q-linear convergence rate. In the last stage, we apply a thresholding method to the smoothed vector-valued image to get the final segmentation. Experiments demonstrate clearly that the proposed methods can provide more accurate segmentation results in comparison with five state-of-the-art methods including a deep learning approach. [ABSTRACT FROM AUTHOR]

The piecewise constant Mumford{Shah (PCMS) model and the Rudin{Osher{Fatemi (ROF) model are two important variational models in image segmentation and image restoration, respectively. In this paper, we explore a linkage between these models. We prove that for the twophase segmentation problem a partial minimizer of the PCMS model can be obtained by thresholding the minimizer of the ROF model. A similar linkage is still valid for multiphase segmentation under specific assumptions. Thus it opens a new segmentation paradigm: image segmentation can be done via image restoration plus thresholding. This new paradigm, which circumvents the innate nonconvex property of the PCMS model, therefore, improves the segmentation performance in both eficiency (much faster than state-of-the-art methods based on the PCMS model, particularly when the phase number is high)the and effectiveness (producing segmentation results with better quality) due to the exibility of the ROF model in tackling degraded images, such as noisy images, blurry images, or images with information loss. As a by-product of the new paradigm, we derive a novel segmentation method, called thresholded-ROF (T-ROF) method, to illustrate the virtue of managing image segmentation through image restoration techniques. The convergence of the T-ROF method is proved, and elaborate experimental results and comparisons are presented. [ABSTRACT FROM AUTHOR]

We reformulate the original phase retrieval problem into two variational models (with and without regularization), both containing a globally Lipschitz differentiable term. These two models can be efficiently solved via the proposed Partially Preconditioned Proximal Alternating Linearized Mini- mization (P3ALM) for masked Fourier measurements. Thanks to the Lipschitz differentiable term, we prove the global convergence of P3ALM for solving the nonconvex phase retrieval problems. Extensive experiments are conducted to show the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

The phase retrieval problem has drawn considerable attention, as many optical detection devices can only measure magnitudes of the Fourier transform of the underlying object (signal or image). This paper addresses the phase retrieval problem from incomplete data, where only partial magnitudes of Fourier transform are obtained. In particular, we consider structured illuminated patterns in holography and find that noninteger values used in designing such patterns often yield better reconstruction than the conventional integer-valued ones. Furthermore, we demonstrate theoretically and numerically that three diffracted sets of (complete) magnitude data are sufficient to recover the object. To compensate for incomplete information, we incorporate a total variation regularization a priori to guarantee that the reconstructed image satisfies some desirable properties. The proposed model can be solved efficiently by an alternative directional multiplier method with provable convergence. Numerical experiments validate the theoretical finding and demonstrate the effectiveness of the proposed method in recovering objects from noisy and incomplete data. [ABSTRACT FROM AUTHOR]

In this paper, we focus on variational approaches for image inpainting in transform domain and propose two new algorithms, iterative coupled transform domain inpainting (ICTDI) and iterative decoupled transform domain inpainting. In the derivation of ICTDI, we use operator splitting and the quadratic penalty technique to get a new approximate problem of the basic model. By the alternating minimization method, the approximate problem can be decomposed as three relatively simple subproblems with closed-form solutions. However, ICTDI is not efficient when some adaptive regularization operator is used, such as the learned BM3D frame. To overcome this drawback, with some modifications, we decouple our framework into three relatively independent parts: denoising, linear combination in the transform domain, and linear combination in the image domain. Therefore, we can use any existing denoising method in the denoising step. We consider three choices for regularization operators in our approach: gradient operator, tight framelet transform, and learned BM3D frame. The numerical experiments and comparisons on various images demonstrate the effectiveness of the proposed methods. The convergence of the numerical algorithms is proved under some assumptions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study image enhancement by using sparse and redundant representations of the reflectance component in the Retinex model over a learned dictionary. This approach is different from existing variational methods, and the advantage of this approach is that the reflectance component in the Retinex model can be represented with more details by the dictionary. A variational method based on the dynamic dictionaries is adopted here, where it changes with respect to iterations of the enhancement algorithm. Numerical examples are also reported to demonstrate that the proposed methods can provide better visual quality of the enhanced high-contrast images than the other variational methods, i.e., revealing more details in the low-light part. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new fast variational approach to dehaze and denoise simultaneously. The proposed method first estimates a transmission map using a windows adaptive method based on the celebrated dark channel prior. This transmission map can significantly reduce the edge artifact in the resulting image and enhance the estimation precision. The transmission map is then converted to a depth map, with which the new variational model can be built to seek the final haze- and noise-free image. The existence and uniqueness of a minimizer of the proposed variational model is further discussed. A numerical procedure based on the Chambolle-Pock algorithm is given, and the convergence of the algorithm is ensured. Extensive experimental results on real scenes demonstrate that our method can restore vivid and contrastive haze- and noise-free images effectively. [ABSTRACT FROM AUTHOR]

In this paper, a two-stage method for segmenting blurry images in the presence of Poisson or multiplicative Gamma noise is proposed. The method is inspired by a previous work on two-stage segmentation and the usage of an I-divergence term to handle the noise. The first stage of our method is to find a smooth solution u to a convex variant of the Mumford-Shah model where the ℓ2 data-fidelity term is replaced by an I-divergence term. A primal-dual algorithm is adopted to efficiently solve the minimization problem. We prove the convergence of the algorithm and the uniqueness of the solution u. Once u is obtained, in the second stage, the segmentation is done by thresholding u into different phases. The thresholds can be given by the users or can be obtained automatically by using any clustering method. In our method, we can obtain any K-phase segmentation (K ≥ 2) by choosing (K - 1) thresholds after u is found. Changing K or the thresholds does not require u to be recomputed. Experimental results show that our two-stage method performs better than many standard two-phase or multiphase segmentation methods for very general images, including antimass, tubular, magnetic resonance imaging, and low-light images. [ABSTRACT FROM AUTHOR]

In this paper, we study the image recovery problem where the observed image is simultaneously corrupted by blur and impulse noise. Our proposed patch-based model contains three terms: the sparse representation prior, the total variation regularization, and the data-fidelity term. We are interested in the two-phase approach. The first phase is to identify the possible impulse noise positions; the second phase is to recover the image via the patch-based model using noise position information. An alternating minimization method is then applied to solve the model. This approach works extremely well for image deblurring under salt-and-pepper noise. However, as the detection for random-valued noise is usually unreliable, extra work is then needed. Indeed, to get better recovery results for the latter case, we combine the two separate phases to simultaneously detect the random-valued noise positions and to recover the image. The numerical experiments clearly demonstrate the super performance of the proposed methods. [ABSTRACT FROM AUTHOR]

The restoration of images corrupted by blurring and multiplicative noise is a challenging problem in applied mathematics that has attracted much attention in recent years. In this article, we propose a two-step approach to solve the problem of restoring images degraded by multiplicative noise and blurring, where the multiplicative noise is first reduced by nonlocal filters and then a convex variational model is adopted to obtain the final restored images. The variational model of the second step is composed of an L1-L2 data-fidelity term and a total variation regularization term. The alternating direction method (ADM) is utilized to solve this variational problem, and we also prove that the ADM algorithm converges at least linearly. Experimental results show that the proposed two-step approach performs better than the existing methods for restoring images with multiplicative noise and blurring, both in the quality of the restored images and the convergence speed of the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a new variational model for restoring blurred images with multiplicative noise is proposed. Based on the statistical property of the noise, a quadratic penalty function technique is utilized in order to obtain a strictly convex model under a mild condition, which guarantees the uniqueness of the solution and the stabilization of the algorithm. For solving the new convex variational model, a primal-dual algorithm is proposed, and its convergence is studied. The paper ends with a report on numerical tests for the simultaneous deblurring and denoising of images subject to multiplicative noise. A comparison with other methods is provided as well. [ABSTRACT FROM AUTHOR]

The Mumford-Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a two- stage segmentation method based on the Mumford-Shah model. The first stage of our method is to find a smooth solution g to a convex variant of the Mumford-Shah model. Once g is obtained, then in the second stage the segmentation is done by thresholding g into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, g can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle- Pock method. We prove that our method is convergent and that the solution g is always unique. In our method, there is no need to specify the number of segments K (K ≥ 2) before finding g. We can obtain any K-phase segmentations by choosing (K - 1) thresholds after g is found in the first stage, and in the second stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our two-stage method performs better than many standard two-phase or multiphase segmentation methods for very general images, including antimass, tubular, MRI, noisy, and blurry images. [ABSTRACT FROM AUTHOR]

The goal of this paper is to provide a theoretical study of a total variation (TV) dictionary model. Based on the properties of convex analysis and bounded variation functions, the existence of solutions of the TV dictionary model is proved. We then show that the dual form of the model can be given by the minimization of the sum of the l¹-norm of the dual solution and the Bregman distance between the curvature of the primal solution and the subdifferential of TV norm of the dual solution. This theoretical result suggests that the dictionary must represent sparsely the curvatures of solution image in order to obtain a better denoising performance. [ABSTRACT FROM AUTHOR]

Recently, a significant improvement of the well-known least significant bit (LSB) matching steganography has been proposed, reducing the changes to the cover image for the same amount of embedded secret data. When the embedding rate is 1, this method decreases the expected number of modification per pixel (ENMPP) from 0.5 to 0.375. In this letter, we propose the so-called generalized LSB matching (G-LSB-M) scheme, which generalizes this method and LSB matching. The lower bound of ENMPP for G-LSB-M is investigated, and a construction of G-LSB-M is presented by using the sum and difference covering set of finite cyclic group. Compared with the previous works, we show that the suitable G-LSB-M can further reduce the ENMPP and lead to more secure steganographic schemes. Experimental results illustrate clearly the better resistance to steganalysis of G-LSB-M. [ABSTRACT FROM AUTHOR]

Image sharpening can highlight fine details in images but with a tendency of amplifying noise. This paper proposes a novel idea of incorporating an image sharpening operator into a framelet-based model for image deblurring. The proposed model is convex and hence it can be solved efficiently by the semi-proximal alternating direction method of multipliers (sPADMM) with guaranteed linear rate convergence, which covers the classical ADMM. The experimental results on different blurring kernels and Gaussian noise levels show that the proposed approach outperforms the state-of-the-art methods in terms of PSNR, SSIM, relative error, and visual quality. [ABSTRACT FROM AUTHOR]

Blind image deconvolution is the process of estimating both the original image and the blur kernel from the degraded image with only partial or no information about degradation and the imaging system. It is a bilinear ill-posed inverse problem corresponding to the direct problem of convolution. Regularization methods are used to handle the ill-posedness of blind deconvolution and get meaningful solutions. In this paper, we investigate a convex regularized inverse filtering method for blind deconvolution of images. We assume that the support region of the blur object is known, as has been done in a few existing works. By studying the inverse filters of signal and image restoration problems, we observe the oscillation structure of the inverse filters. Inspired by the oscillation structure of the inverse filters, we propose to use the star norm to regularize the inverse filter. Meanwhile, we use the total variation to regularize the resulting image obtained by convolving the inverse filter with the degraded image. The proposed minimization model is shown to be convex. We employ the first-order primal-dual method for the solution of the proposed minimization model. Numerical examples for blind image restoration are given to show that the proposed method outperforms some existing methods in terms of peak signal-to-noise ratio (PSNR), structural similarity (SSIM), visual quality and time consumption. [ABSTRACT FROM AUTHOR]

In this work, we consider a new fractional derivative with nonsingular kernel introduced by Caputo-Fabrizio (CF) and propose a finite difference method for computing the CF fractional derivatives. Based on an iterative technique, we can reduce the computational complexity from O(J²N) to O(JN), and the corresponding storage will be cut down from O(JN) to O(N), which makes the computation much more efficient. Besides, by adopting piece-wise Lagrange polynomials of degrees 1, 2, and 3, we derive the second, third, and fourth order discretization formulas respectively. The error analysis and numerical experiments are carefully provided for the validation of the accuracy and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, we study dictionary learning (DL) approach to identify the representation of low-dimensional subspaces from high-dimensional and nonnegative data. Such representation can be used to provide an affinity matrix among different subspaces for data clustering. The main contribution of this paper is to consider both nonnegativity and sparsity constraints together in DL such that data can be represented effectively by nonnegative and sparse coding coefficients and nonnegative dictionary bases. In the algorithm, we employ the proximal point technique for the resulting DL and sparsity optimization problem. We make use of coding coefficients to perform spectral clustering (SC) for data partitioning. Extensive experiments on real-world high-dimensional and nonnegative data sets, including text, microarray, and image data demonstrate that the proposed method can discover their subspace structures. Experimental results also show that our algorithm is computationally efficient and effective for obtaining high SC performance and interpreting the clustering results compared with the other testing methods.