Reweighted low-rank matrix recovery and its application in image restoration

Hyperspectral remote sensing image destriping via spectral-spatial factorization

Hyperspectral images (HSIs) are gradually playing an important role in many fields because of their ability to obtain spectral information. However, sensor response differences and other reasons may lead to the generation of stripe noise in HSIs, which will greatly degrade the image quality. To solve the problem of HSIs destriping, a new iterative method via spectral-spatial factorization is proposed. We first rearrange the HSI data to get a new two-dimensional matrix. Then the original noise-free HSI is decomposed into a spectral information matrix and a spatial information matrix. The sparsity of stripe noise, the group sparsity of spatial information matrix, the smoothness of spectral information matrix can be used to achieve sufficient removal of stripe noise while effectively retaining spectral information and spatial details of the original HSI. Numerical tests on simulated datasets show that our method achieves an average PSNR growth above 4dB and a better SSIM result. The proposed method also obtains good results when processing real datasets polluted by Gaussian noise and stripe noise.

Joint constraints of guided filtering based confidence and nonlocal sparse tensor for color polarization super-resolution imaging

This paper introduces a camera-array-based super-resolution color polarization imaging system designed to simultaneously capture color and polarization information of a scene in a single shot. Existing snapshot color polarization imaging has a complex structure and limited generalizability, which are overcome by the proposed system. In addition, a novel reconstruction algorithm is designed to exploit the complementarity and correlation between the twelve channels in acquired color polarization images for simultaneous super-resolution (SR) imaging and denoising. We propose a confidence-guided SR reconstruction algorithm based on guided filtering to enhance the constraint capability of the observed data. Additionally, by introducing adaptive parameters, we effectively balance the data fidelity constraint and the regularization constraint of nonlocal sparse tensor. Simulations were conducted to compare the proposed system with a color polarization camera. The results show that color polarization images generated by the proposed system and algorithm outperform those obtained from the color polarization camera and the state-of-the-art color polarization demosaicking algorithms. Moreover, the proposed algorithm also outperforms state-of-the-art SR algorithms based on deep learning. To evaluate the applicability of the proposed imaging system and reconstruction algorithm in practice, a prototype was constructed for color polarization image acquisition. Compared with conventional acquisition, the proposed solution demonstrates a significant improvement in the reconstructed color polarization images.

Multi-View Enhanced Tensor Nuclear Norm and Local Constraint Model for Cancer Clustering and Feature Gene Selection

The analysis of cancer data from multi-omics can effectively promote cancer research. The main focus of this article is to cluster cancer samples and identify feature genes to reveal the correlation between cancers and genes, with the primary approach being the analysis of multi-view cancer omics data. Our proposed solution, the Multi-View Enhanced Tensor Nuclear Norm and Local Constraint (MVET-LC) model, aims to utilize the consistency and complementarity of omics data to support biological research. The model is designed to maximize the utilization of multi-view data and incorporates a nuclear norm and local constraint to achieve this goal. The first step involves introducing the concept of enhanced partial sum of tensor nuclear norm, which significantly enhances the flexibility of the tensor nuclear norm. After that, we incorporate total variation regularization into the MVET-LC model to further augment its performance. It enables MVET-LC to make use of the relationship between tensor data structures and sparse data while paying attention to the feature details of the tensor data. To tackle the iterative optimization problem of MVET-LC, the alternating direction method of multipliers is utilized. Through experimental validation, it is demonstrated that our proposed model outperforms other comparison models.

Spectral Clustering Super-Resolution Imaging Based on Multispectral Camera Array

Although multispectral and hyperspectral imaging acquisitions are applied in numerous fields, the existing spectral imaging systems suffer from either low temporal or spatial resolution. In this study, a new multispectral imaging system-camera array based multispectral super resolution imaging system (CAMSRIS) is proposed that can simultaneously achieve multispectral imaging with high temporal and spatial resolutions. The proposed registration algorithm is used to align pairs of different peripheral and central view images. A novel, super-resolution, spectral-clustering-based image reconstruction algorithm was developed for the proposed CAMSRIS to improve the spatial resolution of the acquired images and preserve the exact spectral information without introducing false information. The reconstructed results showed that the spatial and spectral quality and operational efficiency of the proposed system are better than those of a multispectral filter array (MSFA) based on different multispectral datasets. The PSNR of the multispectral super-resolution images obtained by the proposed method were respectively higher by 2.03 and 1.93 dB than those of GAP-TV and DeSCI, and the execution time was significantly shortened by approximately 54.55 s and 9820.19 s when the CAMSI dataset was used. The feasibility of the proposed system was verified in practical applications based on different scenes captured by the self-built system.

Tensor Cascaded-Rank Minimization in Subspace: A Unified Regime for Hyperspectral Image Low-Level Vision

Low-rank tensor representation philosophy has enjoyed a reputation in many hyperspectral image (HSI) low-level vision applications, but previous studies often failed to comprehensively exploit the low-rank nature of HSI along different modes in low-dimensional subspace, and unsurprisingly handled only one specific task. To address these challenges, in this paper, we figured out that in addition to the spatial correlation, the spectral dependency of HSI also implicitly exists in the coefficient tensor of its subspace, this crucial dependency that was not fully utilized by previous studies yet can be effectively exploited in a cascaded manner. This led us to propose a unified subspace low-rank learning regime with a new tensor cascaded rank minimization, named STCR, to fully couple the low-rankness of HSI in different domains for various low-level vision tasks. Technically, the high-dimensional HSI was first projected into a low-dimensional tensor subspace, then a novel tensor low-cascaded-rank decomposition was designed to collapse the constructed tensor into three core tensors in succession to more thoroughly exploit the correlations in spatial, nonlocal, and spectral modes of the coefficient tensor. Next, difference continuity-regularization was introduced to learn a basis that more closely approximates the HSI's endmembers. The proposed regime realizes a comprehensive delineation of the self-portrait of HSI tensor. Extensive evaluations conducted with dozens of state-of-the-art (SOTA) baselines on eight datasets verified that the proposed regime is highly effective and robust to typical HSI low-level vision tasks, including denoising, compressive sensing reconstruction, inpainting, and destriping. The source code of our method is released at https://github.com/CX-He/STCR.git.

Structural Smoothing Low-Rank Matrix Restoration Based on Sparse Coding and Dual-Weighted Model

Group sparse coding (GSC) uses the non-local similarity of images as constraints, which can fully exploit the structure and group sparse features of images. However, it only imposes the sparsity on the group coefficients, which limits the effectiveness of reconstructing real images. Low-rank regularized group sparse coding (LR-GSC) reduces this gap by imposing low-rankness on the group sparse coefficients. However, due to the use of non-local similarity, the edges and details of the images are over-smoothed, resulting in the blocking artifact of the images. In this paper, we propose a low-rank matrix restoration model based on sparse coding and dual weighting. In addition, total variation (TV) regularization is integrated into the proposed model to maintain local structure smoothness and edge features. Finally, to solve the problem of the proposed optimization, an optimization method is developed based on the alternating direction method. Extensive experimental results show that the proposed SDWLR-GSC algorithm outperforms state-of-the-art algorithms for image restoration when the images have large and sparse noise, such as salt and pepper noise.

Deep Unfolding of Iteratively Reweighted ADMM for Wireless RF Sensing

We address the detection of material defects, which are inside a layered material structure using compressive sensing-based multiple-input and multiple-output (MIMO) wireless radar. Here, strong clutter due to the reflection of the layered structure's surface often makes the detection of the defects challenging. Thus, sophisticated signal separation methods are required for improved defect detection. In many scenarios, the number of defects that we are interested in is limited, and the signaling response of the layered structure can be modeled as a low-rank structure. Therefore, we propose joint rank and sparsity minimization for defect detection. In particular, we propose a non-convex approach based on the iteratively reweighted nuclear and ℓ1-norm (a double-reweighted approach) to obtain a higher accuracy compared to the conventional nuclear norm and ℓ1-norm minimization. To this end, an iterative algorithm is designed to estimate the low-rank and sparse contributions. Further, we propose deep learning-based parameter tuning of the algorithm (i.e., algorithm unfolding) to improve the accuracy and the speed of convergence of the algorithm. Our numerical results show that the proposed approach outperforms the conventional approaches in terms of mean squared errors of the recovered low-rank and sparse components and the speed of convergence.

Recursive Sample Scaling Low-Rank Representation

The low-rank representation (LRR) method has recently gained enormous popularity due to its robust approach in solving the subspace segmentation problem, particularly those concerning corrupted data. In this paper, the recursive sample scaling low-rank representation (RSS-LRR) method is proposed. The advantage of RSS-LRR over traditional LRR is that a cosine scaling factor is further introduced, which imposes a penalty on each sample to minimize noise and outlier influence better. Specifically, the cosine scaling factor is a similarity measure learned to extract each sample’s relationship with the low-rank representation’s principal components in the feature space. In order words, the smaller the angle between an individual data sample and the low-rank representation’s principal components, the more likely it is that the data sample is clean. Thus, the proposed method can then effectively obtain a good low-rank representation influenced mainly by clean data. Several experiments are performed with varying levels of corruption on ORL, CMU PIE, COIL20, COIL100, and LFW in order to evaluate RSS-LRR’s effectiveness over state-of-the-art low-rank methods. The experimental results show that RSS-LRR consistently performs better than the compared methods in image clustering and classification tasks.

Clutter Suppression for Indoor Self-Localization Systems by Iteratively Reweighted Low-Rank Plus Sparse Recovery

Self-localization based on passive RFID-based has many potential applications. One of the main challenges it faces is the suppression of the reflected signals from unwanted objects (i.e., clutter). Typically, the clutter echoes are much stronger than the backscattered signals of the passive tag landmarks used in such scenarios. Therefore, successful tag detection can be very challenging. We consider two types of tags, namely low-Q and high-Q tags. The high-Q tag features a sparse frequency response, whereas the low-Q tag presents a broad frequency response. Further, the clutter usually showcases a short-lived response. In this work, we propose an iterative algorithm based on a low-rank plus sparse recovery approach (RPCA) to mitigate clutter and retrieve the landmark response. In addition to that, we compare the proposed approach with the well-known time-gating technique. It turns out that RPCA outperforms significantly time-gating for low-Q tags, achieving clutter suppression and tag identification when clutter encroaches on the time-gating window span, whereas it also increases the backscattered power at resonance by approximately 12 dB at 80 cm for high-Q tags. Altogether, RPCA seems a promising approach to improve the identification of passive indoor self-localization tag landmarks.

Removing Adversarial Noise via Low-Rank Completion of High-Sensitivity Points

Deep neural networks are fragile under adversarial attacks. In this work, we propose to develop a new defense method based on image restoration to remove adversarial attack noise. Using the gradient information back-propagated over the network to the input image, we identify high-sensitivity keypoints which have significant contributions to the image classification performance. We then partition the image pixels into the two groups: high-sensitivity and low-sensitivity points. For low-sensitivity pixels, we use a total variation (TV) norm-based image smoothing method to remove adversarial attack noise. For those high-sensitivity keypoints, we develop a structure-preserving low-rank image completion method. Based on matrix analysis and optimization, we derive an iterative solution for this optimization problem. Our extensive experimental results on the CIFAR-10, SVHN, and Tiny-ImageNet datasets have demonstrated that our method significantly outperforms other defense methods which are based on image de-noising or restoration, especially under powerful adversarial attacks.

Joint Embedding Learning and Low-Rank Approximation: A Framework for Incomplete Multiview Learning

In real-world applications, not all instances in the multiview data are fully represented. To deal with incomplete data, incomplete multiview learning (IML) rises. In this article, we propose the joint embedding learning and low-rank approximation (JELLA) framework for IML. The JELLA framework approximates the incomplete data by a set of low-rank matrices and learns a full and common embedding by linear transformation. Several existing IML methods can be unified as special cases of the framework. More interestingly, some linear transformation-based complete multiview methods can be adapted to IML directly with the guidance of the framework. Thus, the JELLA framework improves the efficiency of processing incomplete multiview data, and bridges the gap between complete multiview learning and IML. Moreover, the JELLA framework can provide guidance for developing new algorithms. For illustration, within the framework, we propose the IML with the block-diagonal representation (IML-BDR) method. Assuming that the sampled examples have an approximate linear subspace structure, IML-BDR uses the block-diagonal structure prior to learning the full embedding, which would lead to more correct clustering. A convergent alternating iterative algorithm with the successive over-relaxation optimization technique is devised for optimization. The experimental results on various datasets demonstrate the effectiveness of IML-BDR.

Low-Rank Matrix Recovery from Noise via an MDL Framework-Based Atomic Norm

The recovery of the underlying low-rank structure of clean data corrupted with sparse noise/outliers is attracting increasing interest. However, in many low-level vision problems, the exact target rank of the underlying structure and the particular locations and values of the sparse outliers are not known. Thus, the conventional methods cannot separate the low-rank and sparse components completely, especially in the case of gross outliers or deficient observations. Therefore, in this study, we employ the minimum description length (MDL) principle and atomic norm for low-rank matrix recovery to overcome these limitations. First, we employ the atomic norm to find all the candidate atoms of low-rank and sparse terms, and then we minimize the description length of the model in order to select the appropriate atoms of low-rank and the sparse matrices, respectively. Our experimental analyses show that the proposed approach can obtain a higher success rate than the state-of-the-art methods, even when the number of observations is limited or the corruption ratio is high. Experimental results utilizing synthetic data and real sensing applications (high dynamic range imaging, background modeling, removing noise and shadows) demonstrate the effectiveness, robustness and efficiency of the proposed method.

Infrared Small Target Detection via Non-Convex Tensor Rank Surrogate Joint Local Contrast Energy

Small target detection is a crucial technique that restricts the performance of many infrared imaging systems. In this paper, a novel detection model of infrared small target via non-convex tensor rank surrogate joint local contrast energy (NTRS) is proposed. To improve the latest infrared patch-tensor (IPT) model, a non-convex tensor rank surrogate merging tensor nuclear norm (TNN) and the Laplace function, is utilized for low rank background patch-tensor constraint, which has a useful property of adaptively allocating weight for every singular value and can better approximate l 0 -norm. Considering that the local prior map can be equivalent to the saliency map, we introduce a local contrast energy feature into IPT detection framework to weight target tensor, which can efficiently suppress the background and preserve the target simultaneously. Besides, to remove the structured edges more thoroughly, we suggest an additional structured sparse regularization term using the l 1 , 1 , 2 -norm of third-order tensor. To solve the proposed model, a high-efficiency optimization way based on alternating direction method of multipliers with the fast computing of tensor singular value decomposition is designed. Finally, an adaptive threshold is utilized to extract real targets of the reconstructed target image. A series of experimental results show that the proposed method has robust detection performance and outperforms the other advanced methods.

Hyperspectral Images Denoising via Nonconvex Regularized Low-Rank and Sparse Matrix Decomposition

Hyperspectral images (HSIs) are often degraded by a mixture of various types of noise during the imaging process, including Gaussian noise, impulse noise, and stripes. Such complex noise could plague the subsequent HSIs processing. Generally, most HSI denoising methods formulate sparsity optimization problems with convex norm constraints, which over-penalize large entries of vectors, and may result in a biased solution. In this paper, a nonconvex regularized low-rank and sparse matrix decomposition (NonRLRS) method is proposed for HSI denoising, which can simultaneously remove the Gaussian noise, impulse noise, dead lines, and stripes. The NonRLRS aims to decompose the degraded HSI, expressed in a matrix form, into low-rank and sparse components with a robust formulation. To enhance the sparsity in both the intrinsic low-rank structure and the sparse corruptions, a novel nonconvex regularizer named as normalized ε -penalty, is presented, which can adaptively shrink each entry. In addition, an effective algorithm based on the majorization minimization (MM) is developed to solve the resulting nonconvex optimization problem. Specifically, the MM algorithm first substitutes the nonconvex objective function with the surrogate upper-bound in each iteration, and then minimizes the constructed surrogate function, which enables the nonconvex problem to be solved in the framework of reweighted technique. Experimental results on both simulated and real data demonstrate the effectiveness of the proposed method.

Tensor Completion via Nonlocal Low-Rank Regularization

Tensor completion (TC), aiming to recover original high-order data from its degraded observations, has recently drawn much attention in hyperspectral images (HSIs) domain. Generally, the widely used TC methods formulate the rank minimization problem with a convex trace norm penalty, which shrinks all singular values equally, and may generate a much biased solution. Besides, these TC methods assume the whole high-order data is of low-rank, which may fail to recover the detail information in high-order data with diverse and complex structures. In this paper, a novel nonlocal low-rank regularization-based TC (NLRR-TC) method is proposed for HSIs, which includes two main steps. In the first step, an initial completion result is generated by the proposed low-rank regularization-based TC (LRR-TC) model, which combines the logarithm of the determinant with the tensor trace norm. This model can more effectively approximate the tensor rank, since the logarithm function values can be adaptively tuned for each input. In the second step, the nonlocal spatial-spectral similarity is integrated into the LRR-TC model, to obtain the final completion result. Specifically, the initial completion result is first divided into groups of nonlocal similar cubes (each group forms a 3-D tensor), and then the LRR-TC is applied to each group. Since similar cubes within each group contain similar structures, each 3-D tensor should have low-rank property, and thus further improves the completion result. Experimental results demonstrate that the proposed NLRR-TC method outperforms state-of-the-art HSIs completion techniques.

Infrared Small Target Detection Based on Partial Sum of the Tensor Nuclear Norm

Excellent performance, real time and strong robustness are three vital requirements for infrared small target detection. Unfortunately, many current state-of-the-art methods merely achieve one of the expectations when coping with highly complex scenes. In fact, a common problem is that real-time processing and great detection ability are difficult to coordinate. Therefore, to address this issue, a robust infrared patch-tensor model for detecting an infrared small target is proposed in this paper. On the basis of infrared patch-tensor (IPT) model, a novel nonconvex low-rank constraint named partial sum of tensor nuclear norm (PSTNN) joint weighted l1 norm was employed to efficiently suppress the background and preserve the target. Due to the deficiency of RIPT which would over-shrink the target with the possibility of disappearing, an improved local prior map simultaneously encoded with target-related and background-related information was introduced into the model. With the help of a reweighted scheme for enhancing the sparsity and high-efficiency version of tensor singular value decomposition (t-SVD), the total algorithm complexity and computation time can be reduced dramatically. Then, the decomposition of the target and background is transformed into a tensor robust principle component analysis problem (TRPCA), which can be efficiently solved by alternating direction method of multipliers (ADMM). A series of experiments substantiate the superiority of the proposed method beyond state-of-the-art baselines.

Hyperspectral Image Restoration under Complex Multi-Band Noises

Hyperspectral images (HSIs) are always corrupted by complicated forms of noise during the acquisition process, such as Gaussian noise, impulse noise, stripes, deadlines and so on. Specifically, different bands of the practical HSIs generally contain different noises of evidently distinct type and extent. While current HSI restoration methods give less consideration to such band-noise-distinctness issues, this study elaborately constructs a new HSI restoration technique, aimed at more faithfully and comprehensively taking such noise characteristics into account. Particularly, through a two-level hierarchical Dirichlet process (HDP) to model the HSI noise structure, the noise of each band is depicted by a Dirichlet process Gaussian mixture model (DP-GMM), in which its complexity can be flexibly adapted in an automatic manner. Besides, the DP-GMM of each band comes from a higher level DP-GMM that relates the noise of different bands. The variational Bayes algorithm is also designed to solve this model, and closed-form updating equations for all involved parameters are deduced. The experiment indicates that, in terms of the mean peak signal-to-noise ratio (MPSNR), the proposed method is on average 1 dB higher compared with the existing state-of-the-art methods, as well as performing better in terms of the mean structural similarity index (MSSIM) and Erreur Relative Globale Adimensionnelle de Synthèse (ERGAS).

Discriminative Structured Dictionary Learning on Grassmann Manifolds and Its Application on Image Restoration

Image restoration is a difficult and challenging problem in various imaging applications. However, despite of the benefits of a single overcomplete dictionary, there are still several challenges for capturing the geometric structure of image of interest. To more accurately represent the local structures of the underlying signals, we propose a new problem formulation for sparse representation with block-orthogonal constraint. There are three contributions. First, a framework for discriminative structured dictionary learning is proposed, which leads to a smooth manifold structure and quotient search spaces. Second, an alternating minimization scheme is proposed after taking both the cost function and the constraints into account. This is achieved by iteratively alternating between updating the block structure of the dictionary defined on Grassmann manifold and sparsifying the dictionary atoms automatically. Third, Riemannian conjugate gradient is considered to track local subspaces efficiently with a convergence guarantee. Extensive experiments on various datasets demonstrate that the proposed method outperforms the state-of-the-art methods on the removal of mixed Gaussian-impulse noise.

Rank-One Matrix Completion With Automatic Rank Estimation via L1-Norm Regularization

Completing a matrix from a small subset of its entries, i.e., matrix completion is a challenging problem arising from many real-world applications, such as machine learning and computer vision. One popular approach to solve the matrix completion problem is based on low-rank decomposition/factorization. Low-rank matrix decomposition-based methods often require a prespecified rank, which is difficult to determine in practice. In this paper, we propose a novel low-rank decomposition-based matrix completion method with automatic rank estimation. Our method is based on rank-one approximation, where a matrix is represented as a weighted summation of a set of rank-one matrices. To automatically determine the rank of an incomplete matrix, we impose L1-norm regularization on the weight vector and simultaneously minimize the reconstruction error. After obtaining the rank, we further remove the L1-norm regularizer and refine recovery results. With a correctly estimated rank, we can obtain the optimal solution under certain conditions. Experimental results on both synthetic and real-world data demonstrate that the proposed method not only has good performance in rank estimation, but also achieves better recovery accuracy than competing methods.

Reweighted Low-Rank Matrix Analysis With Structural Smoothness for Image Denoising

In this paper, we develop a new low-rank matrix recovery algorithm for image denoising. We incorporate the total variation (TV) norm and the pixel range constraint into the existing reweighted low-rank matrix analysis to achieve structural smoothness and to significantly improve quality in the recovered image. Our proposed mathematical formulation of the low-rank matrix recovery problem combines the nuclear norm, TV norm, and norm, thereby allowing us to exploit the low-rank property of natural images, enhance the structural smoothness, and detect and remove large sparse noise. Using the iterative alternating direction and fast gradient projection methods, we develop an algorithm to solve the proposed challenging non-convex optimization problem. We conduct extensive performance evaluations on single-image denoising, hyper-spectral image denoising, and video background modeling from corrupted images. Our experimental results demonstrate that the proposed method outperforms the state-of-the-art low-rank matrix recovery methods, particularly for large random noise. For example, when the density of random sparse noise is 30%, for single-image denoising, our proposed method is able to improve the quality of the restored image by up to 4.21 dB over existing methods.

Edge-Preserving Depth Map Upsampling by Joint Trilateral Filter

Compared to the color images, their associated depth images captured by the RGB-D sensors are typically with lower resolution. The task of depth map super-resolution (SR) aims at increasing the resolution of the range data by utilizing the high-resolution (HR) color image, while the details of the depth information are to be properly preserved. In this paper, we present a joint trilateral filtering (JTF) algorithm for depth image SR. The proposed JTF first observes context information from the HR color image. In addition to the extracted spatial and range information of local pixels, our JTF further integrates local gradient information of the depth image, which allows the prediction and refinement of HR depth image outputs without artifacts like textural copies or edge discontinuities. Quantitative and qualitative experimental results demonstrate the effectiveness and robustness of our approach over prior depth map upsampling works.

Constrained Low-Rank Learning Using Least Squares-Based Regularization

Low-rank learning has attracted much attention recently due to its efficacy in a rich variety of real-world tasks, e.g., subspace segmentation and image categorization. Most low-rank methods are incapable of capturing low-dimensional subspace for supervised learning tasks, e.g., classification and regression. This paper aims to learn both the discriminant low-rank representation (LRR) and the robust projecting subspace in a supervised manner. To achieve this goal, we cast the problem into a constrained rank minimization framework by adopting the least squares regularization. Naturally, the data label structure tends to resemble that of the corresponding low-dimensional representation, which is derived from the robust subspace projection of clean data by low-rank learning. Moreover, the low-dimensional representation of original data can be paired with some informative structure by imposing an appropriate constraint, e.g., Laplacian regularizer. Therefore, we propose a novel constrained LRR method. The objective function is formulated as a constrained nuclear norm minimization problem, which can be solved by the inexact augmented Lagrange multiplier algorithm. Extensive experiments on image classification, human pose estimation, and robust face recovery have confirmed the superiority of our method.

Constructing the L2-Graph for Robust Subspace Learning and Subspace Clustering

Under the framework of graph-based learning, the key to robust subspace clustering and subspace learning is to obtain a good similarity graph that eliminates the effects of errors and retains only connections between the data points from the same subspace (i.e., intrasubspace data points). Recent works achieve good performance by modeling errors into their objective functions to remove the errors from the inputs. However, these approaches face the limitations that the structure of errors should be known prior and a complex convex problem must be solved. In this paper, we present a novel method to eliminate the effects of the errors from the projection space (representation) rather than from the input space. We first prove that l₁ -, l₂ -, l_∞ -, and nuclear-norm-based linear projection spaces share the property of intrasubspace projection dominance, i.e., the coefficients over intrasubspace data points are larger than those over intersubspace data points. Based on this property, we introduce a method to construct a sparse similarity graph, called L2-graph. The subspace clustering and subspace learning algorithms are developed upon L2-graph. We conduct comprehensive experiment on subspace learning, image clustering, and motion segmentation and consider several quantitative benchmarks classification/clustering accuracy, normalized mutual information, and running time. Results show that L2-graph outperforms many state-of-the-art methods in our experiments, including L1-graph, low rank representation (LRR), and latent LRR, least square regression, sparse subspace clustering, and locally linear representation.

Reconstruction of Structurally-Incomplete Matrices With Reweighted Low-Rank and Sparsity Priors

Most matrix reconstruction methods assume that missing entries randomly distribute in the incomplete matrix, and the low-rank prior or its variants are used to well pose the problem. However, in practical applications, missing entries are structurally rather than randomly distributed, and cannot be handled by the rank minimization prior individually. To remedy this, this paper introduces new matrix reconstruction models using double priors on the latent matrix, named Reweighted Low-rank and Sparsity Priors (ReLaSP). In the proposed ReLaSP models, the matrix is regularized by a low-rank prior to exploit the inter-column and inter-row correlations, and its columns (rows) are regularized by a sparsity prior under a dictionary to exploit intra-column (-row) correlations. Both the low-rank and sparse priors are reweighted on the fly to promote low-rankness and sparsity, respectively. Numerical algorithms to solve our ReLaSP models are derived via the alternating direction method under the augmented Lagrangian multiplier framework. Results on synthetic data, image restoration tasks, and seismic data interpolation show that the proposed ReLaSP models are quite effective in recovering matrices degraded by highly structural missing and various types of noise, complementing the classic matrix reconstruction models that handle random missing only.

Weighted Joint Sparse Representation for Removing Mixed Noise in Image

Joint sparse representation (JSR) has shown great potential in various image processing and computer vision tasks. Nevertheless, the conventional JSR is fragile to outliers. In this paper, we propose a weighted JSR (WJSR) model to simultaneously encode a set of data samples that are drawn from the same subspace but corrupted with noise and outliers. Our model is desirable to exploit the common information shared by these data samples while reducing the influence of outliers. To solve the WJSR model, we further introduce a greedy algorithm called weighted simultaneous orthogonal matching pursuit to efficiently approximate the global optimal solution. Then, we apply the WJSR for mixed noise removal by jointly coding the grouped nonlocal similar image patches. The denoising performance is further improved by incorporating it with the global prior and the sparse errors into a unified framework. Experimental results show that our denoising method is superior to several state-of-the-art mixed noise removal methods.

Edge-Aware Filtering with Local Polynomial Approximation and Rectangle-Based Weighting

This paper presents a novel method for performing guided image filtering using local polynomial approximation (LPA) with range guidance. In our method, the LPA is introduced into a multipoint framework for reliable model regression and better preservation on image spatial variation which usually contains the essential information in the input image. In addition, we develop a weighting scheme which has the spatial flexibility during the filtering process. All components in our method are efficiently implemented and a constant computation complexity is achieved. Compared with conventional filtering methods, our method provides clearer boundaries and performs especially better in recovering spatial variation from noisy images. We conduct a number of experiments for different applications: depth image upsampling, joint image denoising, details enhancement, and image abstraction. Both quantitative and qualitative comparisons demonstrate that our method outperforms state-of-the-art methods.