Low-rank Regularization Research Articles

Blind Unmixing of Hyperspectral Images Based on L 1 Norm and Tucker Tensor Decomposition

Most of the traditional hyperspectral unmixing methods are based on the matrix and often ignore the spatial information of hyperspectral images (HSIs). In recent years, tensor-based methods have been gradually used in hyperspectral unmixing, owing to their ability to completely preserve the real spatial structure of HSIs. A blind unmixing method for HSIs based on an L₁ regular term and tucker tensor decomposition (BUTTDL1) is proposed, which describes the low rank of abundance by tucker tensor decomposition in the form of a third-order tensor and increases the sparse characterization of abundance by the L₁ norm. A comparative experiment is performed on two simulation datasets. Compared with the latest method unmixing with low-rank tensor regularization algorithm accounting for endmembers variability (ULTRA-V), in the simulation dataset Data Cube 1 (DC1), the endmember mean square error (MSE) of BUTTDL1 is decreased by 1.1, and the abundance MSE is decreased by 8.6. In the simulation dataset DC2, the endmember MSE is decreased by 2.4, and the abundance MSE is decreased by 6.63.

Simultaneous Non-Convex Low Rank Regularization for Fast Magnetic Resonance Spectroscopy Reconstruction

Simultaneous Non-Convex Low Rank Regularization for Fast Magnetic Resonance Spectroscopy Reconstruction

Superpixel-Based Collaborative and Low-Rank Regularization for Sparse Hyperspectral Unmixing

Sparse unmixing (SU) has been widely applied to remotely sensed hyperspectral images interpretation. Compared with traditional unmixing algorithms, SU does not need to extract pure signatures (endmembers) from the image. The endmember matrix is constructed by directly selecting spectra from a known library which is used to estimate the fractional abundances associated with endmembers. This avoids the problem of extracting virtual endmembers without physical meaning. However, SU does not generally include spatial information, which may limit its performance. In order to address this limitation and include local spatial information, low-rank and sparse features in local regions can be exploited. In this paper, we include spatial information in the traditional SU algorithm by extracting low rank and spatial information based on superpixels, and further propose an algorithm named superpixel-based collaborative sparse and low-rank regularization for sparse unmixing (SCLRSU) to improve the performance of the traditional spatial regularization-based SU methods. In our proposed method, we combine superpixel segmentation and structural sparsity. Experiments are carried out on two simulated datasets and two real hyperspectral image datasets, and our results are compared with those obtained by traditional SU methods. Our results indicate that our newly proposed method provides very competitive performance.

Improved Robust Low-Rank Regularization Tensor Completion

Improved Robust Low-Rank Regularization Tensor Completion

Exploring Nonlocal Group Sparsity Under Transform Learning for Hyperspectral Image Denoising

Hyperspectral image (HSI) denoising has been regarded as an effective and economical preprocessing step in data subsequent applications. Recent nonlocal low-rank approximation on each full band patch group has demonstrated their superiority for HSI denoising. These methods, however, directly design the low-rank regularization to the grouped patch image itself (i.e., original domain), which ignores the spatial information of the grouped patch image and cannot explores the potential structure. To address these issues, this paper proposes a nonlocal group sparsifying transform learning method (dubbed TLNLGS) for HSI denoising. Motivated by the global spectral correlation in the HSI, we firstly impose a certain low-dimensional subspace hypothesis over the HSI to prevent the heavy computation burden with the spectral band increases, and then explore a discriminatively intrinsic nonlocal group sparse prior of the reduced image by transform model. The learned group sparse prior can not only excavate the nonlocal self-similarity as recent nonlocal low-rank approximation methods but also preserve the local spatial smooth structure of the image. Moreover, compared with the fixed transform domain (e.g., gradient and discrete cosine transformation domains), the transform learning scheme can improve the sparse representation ability. An efficient block coordinate descent (BCD) algorithm is developed to solve the proposed model. Extensive experiments, including simulated and real HSI datasets, indicate the superiority of the proposed TLNLGS method over the state-of-the-art HSI denoising approaches.

CFAR Detection Based on the Nonlocal Low-Rank and Sparsity-Driven Laplacian Regularization for HFSWR

Target detection in high-frequency surface-wave radar (HFSWR) systems is a challenging task due to various sources of interference. In this article, we propose a robust constant false alarm rate (CFAR) detector to solve the problem of target detection for the HFSWR by exploiting the prior information on the detection environment. Clutter amplitudes in the HFSWR are commonly assumed to follow the Weibull distribution. Additionally, distribution parameters of the clutter have nonlocal self-similarity, and targets have sparsity as well as a certain geometric structure. We design an objective function based on the Bayesian framework, which consists of a data fidelity term and some regularization terms. Specifically, we use the nonlocal low-rank (NLR) regularization to represent the prior information regarding distribution parameters of clutter, and we adopt the combination of the sparsity and Laplacian regularization to represent the prior information regarding targets. The proposed objective function inherits advantages of the NLR, sparsity, and Laplacian regularization, thereby improving the performance of the proposed detector in the nonhomogeneous clutter. By optimizing the proposed objective function, we obtain estimates of clutter statistics for every cell under test and perform the CFAR detection. Experiments on both simulated data and experimental HFSWR data demonstrate that the proposed detector has a good false alarm regulation property and detection performance in the nonhomogeneous clutter.

Weighted truncated nuclear norm regularization for low-rank quaternion matrix completion

In recent years, quaternion matrix completion (QMC) based on low-rank regularization has been gradually used in image processing. Unlike low-rank matrix completion (LRMC) which handles RGB images by recovering each color channel separately, QMC models retain the connection of three channels and process them as a whole. Most of the existing quaternion-based methods formulate low-rank QMC (LRQMC) as a quaternion nuclear norm (a convex relaxation of the rank) minimization problem. The main limitation of these approaches is that they minimize the singular values simultaneously such that cannot approximate low-rank attributes efficiently. To achieve a more accurate low-rank approximation, we introduce a quaternion truncated nuclear norm (QTNN) for LRQMC and utilize the alternating direction method of multipliers (ADMM) to get the optimization in this paper. Further, we propose weights to the residual error quaternion matrix during the update process for accelerating the convergence of the QTNN method with admissible performance. The weighted method utilizes a concise gradient descent strategy which has a theoretical guarantee in optimization. The effectiveness of our method is illustrated by experiments on real visual data sets.

CT Image Reconstruction via Nonlocal Low-Rank Regularization and Data-Driven Tight Frame

X-ray computed tomography (CT) is widely used in medical applications, where many efforts have been made for decades to eliminate artifacts caused by incomplete projection. In this paper, we propose a new CT image reconstruction model based on nonlocal low-rank regularity and data-driven tight frame (NLR-DDTF). Unlike the Spatial-Radon domain data-driven tight frame regularization, the proposed NLR-DDTF model uses an asymmetric treatment for image reconstruction and Radon domain inpainting, which combines the nonlocal low-rank approximation method for spatial domain CT image reconstruction and data-driven tight frame-based regularization for Radon domain image inpainting. An alternative direction minimization algorithm is designed to solve the proposed model. Several numerical experiments and comparisons are provided to illustrate the superior performance of the NLR-DDTF method.

Multi-sensor data fusion of remotely-sensed images with sparse and logarithmic low-rank regularization for shadow removal and denoising

ABSTRACT Hyperspectral images have high spectral resolution but low spatial resolution, which results in a large number of mixed pixels. As an economical and effective means to improve image quality, the fusion of hyperspectral and multispectral data from different sensors can achieve the reconstruction of super-resolution images. As a representative of fusion method, coupled non-negative matrix factorization is an ill-posed problem, in which the number of endmembers was set to be no less than the groundtruth without requiring an accurate value. However, this often results in spectral shadows and spatial information redundancy, especially when the observed images are contaminated by noise. To address these problems above, this article incorporates sparse and low-rank regularization to reformulate a bi-convex fusion problem for the removal of shadows and noise, in which the logarithmic sum function is employed to suppress the small singular components of endmember and abundance matrices. Then, an efficient solver is designed to obtain the closed-form solutions via matrix-vector operators, in which the alternating direction method of multipliers is utilized to split the variables using equality constraints. The experimental results of real datasets demonstrate that the proposed fusion method can effectively enhance the quality of reconstructed super-resolution images especially in high-noise environments, which also verifies the validation of incorporated regularization.

Deep low-Rank plus sparse network for dynamic MR imaging

In dynamic magnetic resonance (MR) imaging, low-rank plus sparse (L+S) decomposition, or robust principal component analysis (PCA), has achieved stunning performance. However, the selection of the parameters of L+S is empirical, and the acceleration rate is limited, which are common failings of iterative compressed sensing MR imaging (CS-MRI) reconstruction methods. Many deep learning approaches have been proposed to address these issues, but few of them use a low-rank prior. In this paper, a model-based low-rank plus sparse network, dubbed L+S-Net, is proposed for dynamic MR reconstruction. In particular, we use an alternating linearized minimization method to solve the optimization problem with low-rank and sparse regularization. Learned soft singular value thresholding is introduced to ensure the clear separation of the L component and S component. Then, the iterative steps are unrolled into a network in which the regularization parameters are learnable. We prove that the proposed L+S-Net achieves global convergence under two standard assumptions. Experiments on retrospective and prospective cardiac cine datasets show that the proposed model outperforms state-of-the-art CS and existing deep learning methods and has great potential for extremely high acceleration factors (up to 24×).

Motion compensation combining with local low rank regularization for low dose dynamic CT myocardial perfusion reconstruction

Dynamic CT myocardial perfusion imaging (DCT-MPI) is a reliable examination tool for the assessment of myocardium and vascular, while its special scan protocol may result in excessive radiation exposure to patients and inevitable inter-frame motion. Lowering the tube current is a simple way to reduce radiation exposure. However, low mAs will certainly cause severe image noise, thus may further impact the accuracy of functional hemodynamic parameters, which are used for the assessment of blood supply. In this work, we present a novel scheme applying motion compensation and local low rank regularization (MC-LLR) for obtaining high quality motion compensated DCT-MPI images. Specifically, motion compensation by using robust data decomposition registration (RDDR) was introduced. Robust principal component analysis coupled with optical flow-based registration algorithm were used in RDDR. Then, the local low rank constraint on the motion compensated time series images was applied for the DCT-MPI reconstruction. One healthy mini pig and two patient datasets were used to evaluate the proposed MC-LLR algorithm. Results show that the present method achieved satisfactory image quality with higher CNRs, smaller rRMSEs, and more accurate hemodynamic parameter maps.

Nonlocal low-rank regularized two-phase approach for mixed noise removal

Removing mixed noise in images is a difficult problem which has been discussed in many recent papers. In this paper, we tackle the problem of having mixed additive Gaussian white noise and impulse noise. We propose to remove this mixed noise through a nonlocal low-rank regularized two-phase approach. In the first phase, we identify and label the pixels that are likely to be corrupted by the impulse noise. In the second phase, the image is restored through the unlabeled observed data. The restored image is achieved via solving an optimization problem whose objective function has an ℓ 1/ℓ 2 combined content-dependent fidelity term and a nonconvex nonlocal low-rank regularization term. Both terms are built on patch matrices formed from similar patches. Each patch matrix is considered to be low-rank according to the prior knowledge of images. We solve this nonconvex optimization through an iterative adaptive nuclear norm minimization algorithm and provide its convergence analysis. Our experiments show the proposed method outperforms the existing state-of-the-art algorithms in terms of three quantitative metrics, namely, the peak signal-to-noise ratio, the structural similarity and the feature similarity, and visual quality of the restored images.

Learnable low-rank latent dictionary for subspace clustering

Recently, Self-Expressive-based Subspace Clustering (SESC) has been widely applied in pattern clustering and machine learning as it aims to learn a representation that can faithfully reflect the correlation between data points. However, most existing SESC methods directly use the original data as the dictionary, which miss the intrinsic structure (e.g., low-rank and nonlinear) of the real-word data. To address this problem, we propose a novel Projection Low-Rank Subspace Clustering (PLRSC) method by integrating feature extraction and subspace clustering into a unified framework. In particular, PLRSC learns a projection transformation to extract the low-dimensional features and utilizes a low-rank regularizer to ensure the informative and important structures of the extracted features. The extracted low-rank features effectively enhance the self-expressive property of the dictionary. Furthermore, we extend PLRSC to a nonlinear version (i.e., NPLRSC) by integrating a nonlinear activator into the projection transformation. NPLRSC cannot only effectively extract features but also guarantee the data structure of the extracted features. The corresponding optimization problem is solved by the Alternating Direction Method (ADM), and we also prove that the algorithm converges to a stationary point. Experimental results on the real-world datasets validate the superior of our model over the existing subspace clustering methods.

Effective Refinement of Distinctive Analysis of the Facial Matrices for Automatic Face Annotation

We deal with real world images which contains numerous faces captioned with equivalent names, it may be wrongly annotated. The face naming technique that we propose, exploits the weakly labeled image dataset, and aims at labeling a face in the image accurately. We propose this efficient face naming technique which is self regulated and aims at correctly labeling a face in an image. This is a challenging task because of the very large appearance variation in the images, as well as the potential mismatch between images and their captions. This paper introduces a method called Refined Low-Rank Regularization (RLRR) which productively employs the weakly named image information to determine a low-rank matrix which is obtained by examining many subspace structures of the recreated data. From the recreation method used a discriminatory matrix is deduced. Also, Large Margin Nearest Neighbor (LMNN) method is used to label an image, which further leads to another kernel matrix, based on the Mahalanobis distances of the data and the two consistent facial matrices can be fused to enhance the quality of each other and it is used as a new reiterative method to infer the names of each facial image. Experimental results on synthetic and real world data sets validate the effectiveness of the proposed method.

A parallel spatial and Bloch manifold regularized iterative reconstruction method for MR Fingerprinting

Magnetic Resonance Fingerprinting (MRF) reconstructs tissue maps based on a sequence of very highly undersampled images. In order to be able to perform MRF reconstruction, state-of-the-art MRF methods rely on priors such as the MR physics (Bloch equations) and might also use some additional low-rank or spatial regularization. However to our knowledge these three regularizations are not applied together in a joint reconstruction. The reason is that it is indeed challenging to incorporate effectively multiple regularizations in a single MRF optimization algorithm. As a result most of these methods are not robust to noise especially when the sequence length is short. In this paper, we propose a family of new methods where spatial and low-rank regularizations, in addition to the Bloch manifold regularization, are applied on the images. We show on digital phantom and NIST phantom scans, as well as volunteer scans that the proposed methods bring significant improvement in the quality of the estimated tissue maps.

Hyperspectral image restoration via superpixel segmentation of smooth band

Hyperspectral images (HSIs) are inevitably degraded in the acquisition process by mixed noise including Gaussian noise, impulse noise, stripes, and so on. Recently, many low-rank regularization based HSI restoration methods have been proposed to powerfully remove the mixed noise. However, most of them use the square patch based denoising strategy, which destroyed the boundary information of the objects in the HSI. In this paper, we adopt superpixel segmentation to group the pixels of HSI with adjacent position, similar color, texture and luminance into a homogeneous region, whose shape is adaptive. Several homogeneous regions cover the full HSI. This is better than simply dividing the HSI into square patches. By taking advantage of both the low-rank property and the spectral smooth of the HSI, this approach can efficiently remove the mixed noise with few time. Several experiments verify the performance of the proposed approach for HSI restoration.

Single-Image Super-Resolution Using Panchromatic Gradient Prior and Variational Model

Single-image super-resolution (SISR) is a resolution enhancement technique and is known as an ill-posed problem. Motivated by the idea of pan-sharping, we propose a novel variational model for SISR. The structure tensor of the input low-resolution image is exploited to obtain the gradient of an imaginary panchromatic image. Then, by constraining the gradient consistency, the image edges and details can be better recovered during the procedure of restoration of high-resolution images. Besides, we resort to the nonlocal sparse and low-rank regularization of image patches to further improve the super-resolution performance. The proposed variational model is efficiently solved by ADMM-based algorithm. We do extensive experiments in natural images and remote sensing images with different magnifying factors and compare our method with three classical super-resolution methods. The subjective visual impression and quantitative evaluation indexes both show that our method can obtain higher-quality results.

Intra-class low-rank regularization for supervised and semi-supervised cross-modal retrieval

Cross-modal retrieval aims to retrieve related items across different modalities, for example, using an image query to retrieve related text. The existing deep methods ignore both the intra-modal and inter-modal intra-class low-rank structures when fusing various modalities, which decreases the retrieval performance. In this paper, two deep models (denoted as ILCMR and Semi-ILCMR) based on intra-class low-rank regularization are proposed for supervised and semi-supervised cross-modal retrieval, respectively. Specifically, ILCMR integrates the image network and text network into a unified framework to learn a common feature space by imposing three regularization terms to fuse the cross-modal data. First, to align them in the label space, we utilize semantic consistency regularization to convert the data representations to probability distributions over the classes. Second, we introduce an intra-modal low-rank regularization, which encourages the intra-class samples that originate from the same space to be more relevant in the common feature space. Third, an inter-modal low-rank regularization is applied to reduce the cross-modal discrepancy. To enable the low-rank regularization to be optimized using automatic gradients during network back-propagation, we propose the rank-r approximation and specify the explicit gradients for theoretical completeness. In addition to the three regularization terms that rely on label information incorporated by ILCMR, we propose Semi-ILCMR in the semi-supervised regime, which introduces a low-rank constraint before projecting the general representations into the common feature space. Extensive experiments on four public cross-modal datasets demonstrate the superiority of ILCMR and Semi-ILCMR over other state-of-the-art methods.

Sparse Constrained Low Tensor Rank Representation Framework for Hyperspectral Unmixing

Recently, non-negative tensor factorization (NTF) as a very powerful tool has attracted the attention of researchers. It is used in the unmixing of hyperspectral images (HSI) due to its excellent expression ability without any information loss when describing data. However, most of the existing unmixing methods based on NTF fail to fully explore the unique properties of data, for example, low rank, that exists in both the spectral and spatial domains. To explore this low-rank structure, in this paper we learn the different low-rank representations of HSI in the spectral, spatial and non-local similarity modes. Firstly, HSI is divided into many patches, and these patches are clustered multiple groups according to the similarity. Each similarity group can constitute a 4-D tensor, including two spatial modes, a spectral mode and a non-local similarity mode, which has strong low-rank properties. Secondly, a low-rank regularization with logarithmic function is designed and embedded in the NTF framework, which simulates the spatial, spectral and non-local similarity modes of these 4-D tensors. In addition, the sparsity of the abundance tensor is also integrated into the unmixing framework to improve the unmixing performance through the L2,1 norm. Experiments on three real data sets illustrate the stability and effectiveness of our algorithm compared with five state-of-the-art methods.

Whole Brain Myelin Water Mapping in One Minute Using Tensor Dictionary Learning With Low-Rank Plus Sparse Regularization.

The quantification of myelin water content in the brain can be obtained by the multi-echo [Formula: see text] weighted images ( [Formula: see text]WIs). To accelerate the long acquisition, a novel tensor dictionary learning algorithm with low-rank and sparse regularization (TDLLS) is proposed to reconstruct the [Formula: see text]WIs from the undersampled data. The proposed algorithm explores the local and nonlocal similarity and the global temporal redundancy in the real and imaginary parts of the complex relaxation signals. The joint application of the low-rank constraints on the dictionaries and the sparse constraints on the core coefficient tensors improves the performance of the tensor-based recovery. Parallel imaging is incorporated into the TDLLS algorithm (pTDLLS) for further acceleration. A pulse sequence is proposed to prospectively undersample the Ky-t space to obtain the whole brain high-quality myelin water fraction (MWF) maps within 1 minute at an undersampling rate (R) of 6.