Non-local Spatial Information Research Articles

Graph Information Aggregation Cross-Domain Few-Shot Learning for Hyperspectral Image Classification.

Most domain adaptation (DA) methods in cross-scene hyperspectral image classification focus on cases where source data (SD) and target data (TD) with the same classes are obtained by the same sensor. However, the classification performance is significantly reduced when there are new classes in TD. In addition, domain alignment, as one of the main approaches in DA, is carried out based on local spatial information, rarely taking into account nonlocal spatial information (nonlocal relationships) with strong correspondence. A graph information aggregation cross-domain few-shot learning (Gia-CFSL) framework is proposed, intending to make up for the above-mentioned shortcomings by combining FSL with domain alignment based on graph information aggregation. SD with all label samples and TD with a few label samples are implemented for FSL episodic training. Meanwhile, intradomain distribution extraction block (IDE-block) and cross-domain similarity aware block (CSA-block) are designed. The IDE-block is used to characterize and aggregate the intradomain nonlocal relationships and the interdomain feature and distribution similarities are captured in the CSA-block. Furthermore, feature-level and distribution-level cross-domain graph alignments are used to mitigate the impact of domain shift on FSL. Experimental results on three public HSI datasets demonstrate the superiority of the proposed method. The codes will be available from the website: https://github.com/YuxiangZhang-BIT/IEEE_TNNLS_Gia-CFSL.

Full-parameter adaptive fuzzy clustering for noise image segmentation based on non-local and local spatial information

The fuzzy clustering C-means (FCM) algorithm is a compelling image segmentation method in image segmentation. However, the algorithm is less robust for images containing noise. This paper proposes a non-local full-parameter adaptive spatial information combined with a local fuzzy factor for noisy image segmentation. Firstly, we realize the adaptive calculation of the smoothing parameter, search term window, and neighborhood window in the non-local spatial information by defining the smoothness and designing the adaptive matching function. Secondly, the image’s non-local and local spatial information is considered comprehensively to reduce noise interference and segmentation ambiguity. Finally, the weighted average membership linking is used as the denominator of the objective function to reduce the number of iterations. The results in synthetic noise image and color image segmentation experiments show that the proposed algorithm has outstanding performance in various evaluation metrics and visual effects, outperforming most other variants of fuzzy clustering-based algorithms.

Densely sampled light field reconstruction with transformers

Densely sampled light fields (LFs) are critical for their further applications, such as digital refocus and depth estimation. However, it is costly and time-consuming to capture them. LF reconstruction, which aims at reconstructing a densely sampled LF from a sparsely sampled one, has attracted extensive attention of researchers. Although existing methods have achieved significant progress, these methods synthesize novel views, either through depth estimation and image warping, which depend heavily on the accuracy of the depth maps and are prone to cause artifacts at occluded regions, or by stacking multi-layer convolutions to learn the inherent structure of the LF, which will result in blurring results due to limited receptive fields when processing scenes with large disparities. We propose a transformer-based neural network for LF reconstruction (termed as LFRTR). Specifically, two novel transformers are introduced, namely angular transformer and spatial transformer. The former can fully explore angular information and correlations among different views, whereas the latter can capture local and non-local spatial texture information within each view. Moreover, dense skip connections are employed to enhance information flow between different layers. Thanks to the inherent global modeling ability of self-attention, the proposed LFRTR can reconstruct high-quality densely sampled LF in complex scenarios, such as large disparity, occlusion, and reflection. Experimental results on both synthetic and real-world LF datasets show that the proposed LFRTR outperforms other state-of-the-art methods in terms of both visual and numerical evaluations.

Multifrequency PolSAR Image Fusion Classification Based on Semantic Interactive Information and Topological Structure

Compared with the rapid development of single-frequency polarimetric SAR (PolSAR) image classification technology, there is less research on the land cover classification of multi-frequency PolSAR (MF-PolSAR) images. And the deep learning methods among them are mainly based on convolutional neural networks (CNNs), only local spatiality is considered but the nonlocal relationship is ignored. Therefore, this paper proposes the MF semantics and topology fusion (MF-STF) model based on semantic interaction and nonlocal topological structure to improve MF-PolSAR classification performance. During MF-STF optimization, the semantic information-based classification (SIC) and topological property-based classification (TPC) work collaboratively, not only fully leveraging the complementarity of bands, but also combining local and nonlocal spatial information to improve the discrimination of different categories. For SIC, the designed cross-band interactive feature extraction (CIFE) module is embedded to explicitly model the deep semantic correlation among bands, thereby leveraging the complementarity of bands to make ground objects more separable. In TPC, the graph sample and aggregate network (GraphSAGE) is employed to dynamically capture the representation of nonlocal topological relations between land cover categories. In this way, the robustness of classification can be further improved by combining nonlocal spatial information. Finally, a MF weighted fusion (MFWF) strategy is proposed to merge inference from different bands, so as to make the MF joint classification decisions of SIC and TPC. Notably, its weights are adjusted based on the total model loss. The effectiveness of the proposed modules is proved by ablation experiments on three measured MF-PolSAR datasets. In addition, the comparative experiments show that MF-STF can achieve more competitive classification performance than some state-of-the-art methods.

Particle Competitive Mechanism Based Multiobjective Rough Clustering Algorithm for Image Segmentation

Rough clustering has attracted increasing attention due to well dealing with the fuzziness and uncertainty of data. It is well known that it needs to manually set the threshold to determine the upper and lower approximations of rough clusters, which may bring a great effect on the clustering performance. When applied to image segmentation, rough clustering is always sensitive to the initialized cluster centers and image noise. Furthermore, only one clustering criterion is considered in rough clustering, which cannot satisfy diverse practical requirements. To handle these issues, a particle competitive mechanism based multiobjective rough clustering algorithm (PCM-MORCA) for image segmentation is proposed. First, a rough intraclass compactness function considering the nonlocal spatial information derived from an image is constructed to overcome the sensitivity to image noise. Next, the constructed rough intraclass compactness function and an interclass separation function are optimized simultaneously to make cluster centers meet diverse segmentation requirements. Then, an adaptive threshold determination mechanism by which the threshold adaptively varies with the clustered data is presented to well determine the upper and lower approximations of rough clusters. After that, to effectively search appropriate cluster centers, a novel pair competition-based particle weight updating strategy is designed for multiobjective particle swarm optimization by improving the elite particle selection and particle update. Finally, a rough clustering index with the nonlocal spatial information is constructed for selecting the optimal solution for PCM-MORCA. Segmentation experiments on Berkeley and magnetic resonance images reveal that PCM-MORCA behaves well on the segmentation accuracy and noise robustness.

Video super-resolution with fused local and nonlocal feature

Video super-resolution (VSR) is a typical ill-posed problem aiming to increase spatial resolution of a video. To explore interframe information between adjacent frames, optical flow and deformable convolution are commonly used for temporal dependency modeling. However, they lack the ability to use nonlocal spatial information. We propose the dual-branch feature extraction network for VSR. Our method consists of feature extraction, feature fusion and refinement, and frame reconstruction. Specifically, the feature extraction is designed to explore both local and nonlocal information of input frames. For the local feature extraction branch, we use deformable convolutions to eliminate motion between frames. For the nonlocal feature extraction branch, we employ a nonlocal module to gather relevant features. We concatenate these extracted features on channel dimension for progressive feature fusion and refinement. The frame reconstruction part is responsible for increasing spatial resolution and reconstruction. Experimental results demonstrate the effectiveness of the proposed method by extensive quantitative and qualitative evaluations in public benchmark datasets.

SAR Image Segmentation by Efficient Fuzzy C-Means Framework with Adaptive Generalized Likelihood Ratio Nonlocal Spatial Information Embedded

The existence of multiplicative noise in synthetic aperture radar (SAR) images makes SAR segmentation by fuzzy c-means (FCM) a challenging task. To cope with speckle noise, we first propose an unsupervised FCM with embedding log-transformed Bayesian non-local spatial information (LBNL_FCM). This non-local information is measured by a modified Bayesian similarity metric which is derived by applying the log-transformed SAR distribution to Bayesian theory. After, we construct the similarity metric of patches as the continued product of corresponding pixel similarity measured by generalized likelihood ratio (GLR) to avoid the undesirable characteristics of log-transformed Bayesian similarity metric. An alternative unsupervised FCM framework named GLR_FCM is then proposed. In both frameworks, an adaptive factor based on the local intensity entropy is employed to balance the original and non-local spatial information. Additionally, the membership degree smoothing and the majority voting idea are integrated as supplementary local information to optimize segmentation. Concerning experiments on simulated SAR images, both frameworks can achieve segmentation accuracy of over 97%. On real SAR images, both unsupervised FCM segmentation frameworks work well on SAR homogeneous segmentation in terms of region consistency and edge preservation.

Fuzzy subspace clustering noisy image segmentation algorithm with adaptive local variance & non-local information and mean membership linking

The Fuzzy C-means (FCM) clustering algorithm is an effective method for image segmentation. Non-local spatial information considers more redundant information of the image thus is more robust to noise. However, under-segmentation of non-local spatial information may exist with higher noise density. The number of iteration steps is also significant in FCM, and employing membership linking can effectively reduce the number of iteration steps. Nonetheless, when there are outliers in the membership degree, the membership linking can make the algorithm converge prematurely before reaching the optimum, affecting segmentation performance. This paper presents a fuzzy subspace clustering noisy image segmentation algorithm with adaptive local variance & non-local information and mean membership linking (FSC_LNML). Firstly, local variance templates are utilized to eliminate the under-segmentation of non-local information, and local variance & non-local information are integrated into the FCM objective function to improve robustness. Secondly, the mean membership linking is employed as the denominator of the objective function to reduce the number of iterations and solve the problem that the algorithm converges early before reaching the optimum when the membership has an outlier. Thirdly, the absolute intensity difference between the original image and the local variance & non-local information and its inverse are used to adaptively constrain the original image and the local variance & non-local information. Finally, the concept of the subspace is introduced to adaptively assign appropriate weights to each dimension of the image to improve the segmentation performance of color images. The simulation results on noisy grayscale images and noisy color images show that the efficiency of the proposed method FSC_LNML is better than other fuzzy-based clustering algorithms. The convergence proof of the algorithm is also presented.

Semisupervised Hyperspectral Image Classification via Superpixel-Based Graph Regularization With Local and Nonlocal Features

Although a graph-based Semi-supervised learning (SSL) approach can utilize limited numbers of labeled samples for hyperspectral image (HSI) classification, it is difficult to use the large amount of pixels in an HSI to construct a large-scale graph. In this paper, we therefore propose a superpixel-based graph model for HSI classification, using anchor graph regularization to improve the extraction of local and non-local spatial information. To avoid the large-scale graph problem, the superpixel-based graph model constructs a scalable anchor graph using a small number of anchor points for graph-based SSL. The proposed approach consists of three key components: (1) First, local and non-local features are extracted from the principal components of an HSI using the locally grouped order pattern (LGOP) and the non-local binary pattern (NLBP) approaches. (2) Second, the extracted features are combined with the original spectral information and used as input to an improved simple line iteration clustering method (ISLIC) to obtain superpixels, the centers of which are used as anchors in the anchor graph approach. (3) Third, we use superpixel-based graph regularization (SGR) to model the global information among the superpixels for better predictions of the label of each unlabeled pixel sample. Experimental results demonstrate that the proposed approach outperforms other state-of-the-art Semi-supervised HSI classification approaches for cases with a limited quantity of labeled samples.

Ship Detection of Polarimetric SAR Images Using a Nonlocal Spatial Information-Guided Method

Ship detection of polarimetric synthetic aperture radar (PolSAR) plays an important role in marine monitoring and ocean protection. Over the past years, local spatial information around pixels has been successfully applied to this task. However, few works have been done on PolSAR ship detection using the nonlocal spatial information (NSI). Within this context, we here propose one NSI-guided ship detection method PMR. Briefly speaking, the feature power difference (PD) is first constructed by computing the total power difference between the center pixel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> and its most similar nonlocal pixel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> within a 7×7 window. Then, the polarimetric feature reflection symmetry (RS) is introduced into PD to construct the method PMR (i.e., PD Multiply RS) for further enhancing the target-to-clutter ratio (TCR) and improving the ship detection accuracy. Experiments carried out on three real PolSAR datasets show that, in comparison with some other methods, especially the recently proposed local neighborhood information-based ship detector PWF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>N</i></sub> , PMR is more apt for ship detection. On average, its figure of merit (FoM) and TCR values respectively surpass PWF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>N</i></sub> 0.24 and 12.83 dB.

Improved Fusion of Spatial Information into Hyperspectral Classification through the Aggregation of Constrained Segment Trees: Segment Forest

Compared with traditional optical and multispectral remote sensing images, hyperspectral images have hundreds of bands that can provide the possibility of fine classification of the earth’s surface. At the same time, a hyperspectral image is an image that coexists with the spatial and spectral. It has become a hot research topic to combine the spatial spectrum information of the image to classify hyperspectral features. Based on the idea of spatial–spectral classification, this paper proposes a novel hyperspectral image classification method based on a segment forest (SF). Firstly, the first principal component of the image was extracted by the process of principal component analysis (PCA) data dimension reduction, and the data constructed the segment forest after dimension reduction to extract the non-local prior spatial information of the image. Secondly, the images’ initial classification results and probability distribution were obtained using support vector machine (SVM), and the spectral information of the images was extracted. Finally, the segment forest constructed above is used to optimize the initial classification results and obtain the final classification results. In this paper, three domestic and foreign public data sets were selected to verify the segment forest classification. SF effectively improved the classification accuracy of SVM, and the overall accuracy of Salinas was enhanced by 11.16%, WHU-Hi-HongHu by 15.89%, and XiongAn by 19.56%. Then, it was compared with six decision-level improved space spectrum classification methods, including guided filtering (GF), Markov random field (MRF), random walk (RW), minimum spanning tree (MST), MST+, and segment tree (ST). The results show that the segment forest-based hyperspectral image classification improves accuracy and efficiency compared with other algorithms, proving the algorithm’s effectiveness.

A novel self-learning weighted fuzzy local information clustering algorithm integrating local and non-local spatial information for noise image segmentation

Fuzzy clustering algorithm (FCM) can be directly used to segment images, it takes no account of the neighborhood information of the current pixel and does not have a robust segmentation noise suppression. Fuzzy Local Information C-means Clustering (FLICM) is a widely used robust segmentation algorithm, which combines spatial information with the membership degree of adjacent pixels. In order to further improve the robustness of FLICM algorithm, non-local information is embedded into FLICM algorithm and a fuzzy C-means clustering algorithm has local and non-local information (FLICMLNLI) is obtained. When calculating distance from pixel to cluster center, FLICMLNLI algorithm considers two distances from current pixel and its neighborhood pixels to cluster center. However, the algorithm gives the same weight to two different distances, which incorrectly magnifies the importance of neighborhood information in calculating the distance, resulting in unsatisfactory image segmentation effects and loss of image details. In order to solve this problem, we raise an improved self-learning weighted fuzzy algorithm, which directly obtains different weights in distance calculation through continuous iterative self-learning, then the distance metric with the weights obtained from self-learning is embedded in the objective function of the fuzzy clustering algorithm in order to improve the segmentation performance and robustness of the algorithm. A large number of experiments on different types of images show that the algorithm can not only suppress the noise but also retain the details in the image, the effect of segmenting complex noise images is better, and it provides better image segmentation results than the existing latest fuzzy clustering algorithms.

Coarse–fine surrogate model driven multiobjective evolutionary fuzzy clustering algorithm with dual memberships for noisy image segmentation

To improve the segmentation performance and boost evolutionary efficiency of multiobjective evolutionary clustering algorithms on noisy images, this paper proposes a coarse–fine surrogate model driven multiobjective evolutionary fuzzy clustering algorithm with dual membership functions (CFS-MOEFC). The existing fuzzy clustering validity-based fitness functions generally consider one single fuzzy membership function, which cannot fully process the uncertainty in an image. Moreover, most existing fitness functions used in image segmentation only utilize one type of spatial information derived from the image, which cannot behave robust on images with uncertain types of noise. To deal with the above problems, dual fuzzy membership functions are first designed by utilizing the local and non-local image spatial information. Then, the designed dual membership functions and the two kinds of spatial information are both used to construct spatial information-motivated fitness functions for image segmentation. To promote the time efficiency, the coarse–fine surrogate model is employed to assist the evolutionary optimization process, such as approximating the fitness functions instead of real function evaluations and evolving satisfactory cluster centers for CFS-MOEFC. Besides, a dual memberships-driven cluster validity index combining the local and non-local spatial information is designed for selecting an optimal solution from the final non-dominated solution set of CFS-MOEFC. Experiments on Berkeley and Magnetic Resonance (MR) images indicate that CFS-MOEFC greatly improves the segmentation accuracy on images with multiple types of noise, preserves more significant detailed image information, and behaves well on time cost comparing with state-of-the-art methods.

Hybrid-Hypergraph Regularized Multiview Subspace Clustering for Hyperspectral Images

Clustering algorithms play an essential and unique role in classification tasks, especially when annotated data are unavailable or very scarce. Current clustering approaches in remote sensing are mostly designed for a single data source, such as hyperspectral image (HSI), while, nowadays, multisensor data are being routinely acquired. In this article, we propose a multiview subspace clustering model that exploits effectively the rich information from multiple features extracted either from a single data source (HSI) or from multiple sources that we call generically multiviews of the same scene. An important novelty of our approach is that it integrates local and nonlocal spatial information from each view in a unified framework. Our model learns a common intrinsic cluster structure from view-specific subspace representations by a new decomposition-based scheme. In addition, we develop innovative manifold-based spatial regularization as a hybrid hypergraph, which merges local and nonlocal spatial context and improves, thereby, the learning of view-specific structures. We develop an efficient algorithm to solve the resulting optimization problem. Extensive experiments on real data sets demonstrate the superior clustering performance over the state of the art.

Automatic segmentation of plantar thermograms using adaptive C means technique

Diabetic foot ulcer (DFU) is one of the major concern of diabetes and it is rapidly increasing, in worst case scenario this may lead to amputation. The DFU can be avoided by the early detection and proper diagnosis. Many of the studies carried out highlights that, thermography is the most useful technique to measure the changes in the temperature of plantar surface and alerts to indicate the risk associated with DFU. The distribution of temperature does not have a fixed pattern across the patients, hence it makes the difficulty in measuring the appropriate changes. This gap will provide a scope to improve the analysis technique so as to measure the plantar surface temperature effectively and identify any abnormal changes. In this paper, the segmentation algorithm namely adaptive C means (ACM) for the image segmentation is discussed. ACM is based on the spatial information and this method includes the two stage implementation. In the first stage, nonlocal spatial information is added and in the second stage, spatial shape information is used in order to refine the constraint of local spatial. Outcome of the proposed method shows that ACM is very much effective and it outperforms the other existing methods.

Knowledge-based transfer fuzzy clustering with non-local spatial information for surface roughness measurement

Surface roughness is an important quality indicator of machining processing, which is often needed as a feedback. To meet the requirement of on-line roughness measurement, the machine vision-based measurement method has emerged in recent decades. In this paper, a new surface roughness measurement method is developed. It measures the surface roughness by evaluating the aliasing degree of the images reflected by the machined surface. To analyze the aliasing degree more accurately, a knowledge-based transfer fuzzy clustering with non-local spatial information (KTFCM_NLS) is proposed. The KTFCM_NLS incorporates both the auxiliary knowledge and non-local spatial information into the fuzzy c-means (FCM) algorithm. It can preserve more image details, and is robust to image noise. The results show that the proposed algorithm performs better than other FCM algorithms, and the corresponding index is strongly related to surface roughness. The average relative error of the proposed method is only 5.95% on the testing samples.

Multiobjective fuzzy clustering with multiple spatial information for Noisy color image segmentation

Clustering method is a widely used and effective technique in color image segmentation. In general, traditional clustering-based image segmentation algorithms consider only one objective function and the segmentation performance is easily influenced by the noise in the image. Therefore, utilizing many clustering criteria and the neighborhood statistic information of pixels are more effective to improve the segmentation performance. In this paper, we put forward a multiobjective fuzzy clustering algorithm with multiple spatial information (MFCMSI) for noisy color image segmentation. Firstly, two conflicting fitness functions including the local spatial information with a circular neighborhood window and the non-local spatial information with circular search and similarity windows are designed to improve the noise-resistant ability and segmentation performance. In order to optimize these two fitness functions, the variable-length and cluster-center-based encoding strategy and some efficient evolutionary operations are utilized in each generation. Finally, a cluster validity index with multiple spatial information is constructed to select the best solution from the final non-dominated solution set of the last generation. Aiming to improve the robustness of MFCMSI, the ensemble strategy is introduced into MFCMSI and an ensemble version of MFCMSI is presented. Experimental results show that MFCMSI and its ensemble version behave well in evolving the number of segments and obtaining the satisfactory segmentation performance.

Infrared Small Maritime Target Detection Based on Integrated Target Saliency Measure

Robust and effective detection of a small target in an infrared maritime image is a key technology of maritime target search and tracking applications. Infrared small target detection is a challenging task due to the factors such as dim small targets and various complex backgrounds caused by sun glitters and strong waves. In this article, the integrated target saliency measure (ITSM) based on local and nonlocal spatial information is proposed to improve target detection performance. We combine local heterogeneity property of targets and nonlocal self-correlation property of background with targets' sparsity to separate real targets from background clutters. First, local heterogeneity calculation based on cross-window standard deviation (CSD) is proposed to extract candidate targets preliminarily, which enhances the local intensity difference between small targets and neighboring background. Meanwhile, low-rank representation (LRR) is applied to background prediction and removal, which is followed by adaptive threshold segmentation to enhance target saliency. Finally, we integrate the results obtained from the two steps mentioned above to further enhance targets and suppress background clutters. Then, real targets are extracted by an iterative threshold on the integrated map so as to generate the seed map, in addition, the target expansion strategy is exploited to keep full target areas. Experimental results on six datasets show that the proposed method outperforms the state-of-the-art methods in terms of robustness and target detection accuracy.

Self-Supervised Ultrasound to MRI Fetal Brain Image Synthesis.

Fetal brain magnetic resonance imaging (MRI) offers exquisite images of the developing brain but is not suitable for second-trimester anomaly screening, for which ultrasound (US) is employed. Although expert sonographers are adept at reading US images, MR images which closely resemble anatomical images are much easier for non-experts to interpret. Thus in this article we propose to generate MR-like images directly from clinical US images. In medical image analysis such a capability is potentially useful as well, for instance for automatic US-MRI registration and fusion. The proposed model is end-to-end trainable and self-supervised without any external annotations. Specifically, based on an assumption that the US and MRI data share a similar anatomical latent space, we first utilise a network to extract the shared latent features, which are then used for MRI synthesis. Since paired data is unavailable for our study (and rare in practice), pixel-level constraints are infeasible to apply. We instead propose to enforce the distributions to be statistically indistinguishable, by adversarial learning in both the image domain and feature space. To regularise the anatomical structures between US and MRI during synthesis, we further propose an adversarial structural constraint. A new cross-modal attention technique is proposed to utilise non-local spatial information, by encouraging multi-modal knowledge fusion and propagation. We extend the approach to consider the case where 3D auxiliary information (e.g., 3D neighbours and a 3D location index) from volumetric data is also available, and show that this improves image synthesis. The proposed approach is evaluated quantitatively and qualitatively with comparison to real fetal MR images and other approaches to synthesis, demonstrating its feasibility of synthesising realistic MR images.

Nonlocal Tensor-Based Sparse Hyperspectral Unmixing

Sparse unmixing is an important technique for analyzing and processing hyperspectral images (HSIs). Simultaneously exploiting spatial correlation and sparsity improves substantially abundance estimation accuracy. In this article, we propose to exploit nonlocal spatial information in the HSI for the sparse unmixing problem. Specifically, we first group similar patches in the HSI, and then unmix each group by imposing simultaneous a low-rank constraint and joint sparsity in the corresponding third-order abundance tensor. To this end, we build an unmixing model with a mixed regularization term consisting of the sum of the weighted tensor trace norm and the weighted tensor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2,1}$ </tex-math></inline-formula> -norm of the abundance tensor. The proposed model is solved under the alternating direction method of multipliers framework. We term the developed algorithm as the nonlocal tensor-based sparse unmixing algorithm. The effectiveness of the proposed algorithm is illustrated in experiments with both simulated and real hyperspectral data sets.