Nonnegative Matrix Factorization With Data-Guided Constraints For Hyperspectral Unmixing

Hyperspectral unmixing is an important technique for analyzing remote sensing images which aims to obtain a collection of endmembers and their corresponding abundances. In recent years, non-negative matrix factorization (NMF) has received extensive attention due to its good adaptability for mixed data with different degrees. The majority of existing NMF-based unmixing methods are developed by incorporating additional constraints into the standard NMF based on the spectral and spatial information of hyperspectral images. However, they neglect to exploit the nature of imbalanced pixels included in the data, which may cause the pixels mixed with imbalanced endmembers to be ignored, and thus the imbalanced endmembers generally cannot be accurately estimated due to the statistical property of NMF. To exploit the information of imbalanced samples in hyperspectral data during the unmixing procedure, in this paper, a cluster-wise weighted NMF (CW-NMF) method for the unmixing of hyperspectral images with imbalanced data is proposed. Specifically, based on the result of clustering conducted on the hyperspectral image, we construct a weight matrix and introduce it into the model of standard NMF. The proposed weight matrix can provide an appropriate weight value to the reconstruction error between each original pixel and the reconstructed pixel in the unmixing procedure. In this way, the adverse effect of imbalanced samples on the statistical accuracy of NMF is expected to be reduced by assigning larger weight values to the pixels concerning imbalanced endmembers and giving smaller weight values to the pixels mixed by majority endmembers. Besides, we extend the proposed CW-NMF by introducing the sparsity constraints of abundance and graph-based regularization, respectively. The experimental results on both synthetic and real hyperspectral data have been reported, and the effectiveness of our proposed methods has been demonstrated by comparing them with several state-of-the-art methods.

Hyperspectral unmixing (HU) is a research hotspot of hyperspectral remote sensing technology. As a classical HU method, the nonnegative matrix factorization (NMF) unmixing method can decompose an observed hyperspectral data matrix into the product of two nonnegative matrices, i.e., endmember and abundance matrices. Because the objective function of NMF is the traditional least-squares function, NMF is sensitive to noise. In order to improve the robustness of NMF, this paper proposes a maximum likelihood estimation (MLE) based NMF model (MLENMF) for unmixing of hyperspectral images (HSIs), which substitutes the least-squares objective function in traditional NMF by a robust MLE-based loss function. Experimental results on a simulated and two widely used real hyperspectral data sets demonstrate the superiority of our MLENMF over existing NMF methods.

Hyperspectral unmixing (HU) has become an important technique in exploiting hyperspectral data since it decomposes a mixed pixel into a collection of endmembers weighted by fractional abundances. The endmembers of a hyperspectral image (HSI) are more likely to be generated by independent sources and be mixed in a macroscopic degree before arriving at the sensor element of the imaging spectrometer as mixed spectra. Over the past few decades, many attempts have focused on imposing auxiliary regularizes on the conventional nonnegative matrix factorization (NMF) framework in order to effectively unmix these mixed spectra. As a promising step toward finding an optimum regularizer to extract endmembers, this paper presents a novel blind HU algorithm, referred to as Kurtosis-based Smooth Nonnegative Matrix Factorization (KbSNMF) which incorporates a novel regularizer based on the statistical independence of the probability density functions of endmember spectra. Imposing this regularizer on the conventional NMF framework promotes the extraction of independent endmembers while further enhancing the parts-based representation of data. Experiments conducted on diverse synthetic HSI datasets (with numerous numbers of endmembers, spectral bands, pixels, and noise levels) and three standard real HSI datasets demonstrate the validity of the proposed KbSNMF algorithm compared to several stateof- the-art NMF-based HU baselines. The proposed algorithm exhibits superior performance especially in terms of extracting endmember spectra from hyperspectral data; therefore, it could uplift the performance of recent deep learning HU methods which utilize the endmember spectra as supervisory input data for abundance extraction.

Nonnegative matrix factorization(NMF) has been applied to hyperspectral unmixing in recent years. Different constraints based on geometrical or statistical properties of endmember and abundance are incorporated into NMF model to improve unmixing result. In this paper, a new regularizer based on spectral cluster information is proposed to strengthen the constrained relationship between original image and abundance maps. The new algorithm makes abundances of similar pixels close and abundances of dissimilar pixels be separated completely. Additionally, L 1/2 sparsity constraint is adopted to make the solutions sparse. Comparative results on real and synthetic hyperspectral datasets prove our proposed method could improve the hyperspectral unmixing accuracy.

Hyperspectral unmixing refers to a source separation problem of decomposing a hyperspectral imagery (HSI) to estimate endmembers, and their corresponding abundances. Recently, matrix-vector nonnegative tensor factorization (MV-NTF) was proposed for unmixing to avoid structure information loss, which is caused by the HSI cube unfolding in nonnegative matrix factorization (NMF)-based methods. However, MV-NTF ignores local spatial information due to directly dealing with data as a whole, meanwhile, the forceful rank constraint in low-rank tensor decomposition loses some detailed structures. Unlike MV-NTF works at the original data, the pixel-based NMF is more adaptive to learn local spatial variations. Hence, from the perspective of multi-view, itis significant to utilize the complementary advantages of MV-NTF and NMF to fully preserve the intrinsic structure information, and exploit more detailed spatial information. In this article, we propose a sparsity-constrained coupled nonnegative matrix-tensor factorization (SCNMTF) model for unmixing, wherein MV-NTF and NMF are subtly coupled by sharing endmembers and abundances. Since the representations for abundances in MV-NTF and NMF are distinct, abundance sharing is achieved indirectly by introducing an auxiliary constraint. Furthermore, the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularizer is adopted to promote the sparsity of abundances. A series of experiments on synthetic and real hyperspectral data demonstrate the effectiveness of the proposed SCNMTF method.

Hyperspectral unmixing (HU) is a crucial technique for exploiting remotely sensed hyperspectral data, which aims to estimate a set of spectral signatures, called endmembers and their corresponding proportions, called abundances. Nonnegative matrix factorization (NMF) and its various robust extensions have been widely applied to HU. Most existing robust NMF methods consider that noises only exist in one kind of formulation. However, the hyperspectral images (HSIs) are unavoidably corrupted by noisy bands and noisy pixels simultaneously in the real applications. This paper proposes a novel spectral-spatial robust NMF model by incorporating 12,1 norm and 11,2 norm, which achieves robustness to band noise and pixel noise simultaneously. The Huber's M-estimator is integrated into the proposed model to achieve better assignations of weights for each pixel and band with various noise intensities, which avoids the singularity problem and effectively improves the unmixing performance. The elegant updating rules of the proposed spectral-spatial robust model are also efficiently learned and provided. Experiments are conducted on both synthetic and real hyperspectral data sets. The experimental results demonstrate the effectiveness of the proposed methods in unmixing performance.

Hyperspectral imaging captures detailed spectral data for remote sensing. However, due to the limited spatial resolution of hyperspectral sensors, each pixel of a hyperspectral image (HSI) may contain information from multiple materials. Although the hyperspectral unmixing (HU) process involves estimating endmembers, identifying pure spectral components, and estimating pixel abundances, existing algorithms mostly focus on just one or two tasks. Blind source separation (BSS) based on nonnegative matrix factorization (NMF) algorithms identify endmembers and their abundances at each pixel of HSI simultaneously. Although they perform well, the factorization results are unstable, require high computational costs, and are difficult to interpret from the original HSI. CUR matrix decomposition selects specific columns and rows from a dataset to represent it as a product of three small submatrices, resulting in interpretable low-rank factorization. In this paper, we propose a new blind HU framework based on CUR factorization called CUR-HU that performs the entire HU process by exploiting the low-rank structure of given HSIs. CUR-HU incorporates several techniques to perform the HU process with a performance comparable to state-of-the-art methods but with higher computational efficiency. We adopt a deterministic sampling method to select the most informative pixels and spectrum components in HSIs. We use an incremental QR decomposition method to reduce computation complexity and estimate the number of endmembers. Various experiments on synthetic and real HSIs are conducted to evaluate the performance of CUR-HU. CUR-HU performs comparably to state-of-the-art methods for estimating the number of endmembers and abundance maps, but it outperforms other methods for estimating the endmembers and the computational efficiency. It has a 9.4 to 249.5 times speedup over different methods for different real HSIs.

Nonnegative matrix factorization (NMF) based hyperspectral unmixing aims at estimating pure spectral signatures and their fractional abundances at each pixel. During the past several years, manifold structures have been introduced as regularization constraints into NMF. However, most methods only consider the constraints on abundance matrix while ignoring the geometric relationship of endmembers. Although such relationship can be described by traditional graph construction approaches based on k-nearest neighbors, its accuracy is questionable. In this paper, we propose a novel hyperspectral unmixing method, namely NMF with endmember sparse graph learning, to tackle the above drawbacks. This method first integrates endmember sparse graph structure into NMF, then simultaneously performs unmixing and graph learning. It is further extended by incorporating abundance smoothness constraint to improve the unmixing performance. Experimental results on both synthetic and real datasets have validated the effectiveness of the proposed method.

Hyperspectral unmixing (HU) is a crucial technique for exploiting remotely sensed hyperspectral data, which aims at estimating a set of spectral signatures, called endmembers and their corresponding proportions, called abundances. The performance of HU is often seriously degraded by various kinds of noise existing in hyperspectral images (HSIs). Most of existing robust HU methods are based on the assumption that noise or outlier only exists in one kind of formulation, e.g., band noise or pixel noise. However, in real-world applications, HSIs are unavoidably corrupted by noisy bands and noisy pixels simultaneously, which require robust HU in both the spatial dimension and spectral dimension. Meanwhile, the sparsity of abundances is an inherent property of HSIs and different regions in an HSI may possess various sparsity levels across locations. This article proposes a correntropy-based spatial-spectral robust sparsity-regularized unmixing model to achieve 2-D robustness and adaptive weighted sparsity constraint for abundances simultaneously. The updated rules of the proposed model are efficient to be implemented and carried out by a half-quadratic technique. The experimental results obtained by both synthetic and real hyperspectral data demonstrate the superiority of the proposed method compared to the state-of-the-art methods.

Hyperspectral unmixing (HU) is a technique for estimating a set of pure source signals (end members) and their proportions (abundances) from each pixel of the hyperspectral image. Non-negative matrix factorization (NMF) can decompose the observation matrix into the product of two non-negative matrices simultaneously and can be used in HU. Unfortunately, a limitation of many traditional NMF-based methods, i.e., the non-convexity of the objective function, may lead to a sub-optimal solution. Thus, we put forward a new unmixing method based on NMF under smoothing and sparse constraints to obtain a better solution. First, considering the sparseness of the abundance matrix, a weight sparse regularization is introduced into the NMF model to ensure the sparseness of the abundance matrix. Second, according to the similarity prior of the same feature in the adjacent pixels, a Total Variation regularization is further added to the NMF model to improve the smoothness of the abundance map. Finally, the signatures of each end member are modified smoothly in spectral space. Moreover, it is noticed that discontinuities may emerge due to the removal of noisy bands. Therefore, the spectral data are piecewise smooth in spectral space. Then, in this paper, a piecewise smoothness constraint is further applied to each column of the end-member matrix. Experiments are conducted to evaluate the effectiveness of the proposed method based on two different datasets, including a synthetic dataset and the real-life Cuprite dataset, respectively. Experimental results show that the proposed method outperforms several state-of-the-art HU methods. In the Cuprite hyperspectral dataset, the proposed method’s Spectral Angle Distance is 0.1694, compared to the TV-RSNMF method’s 0.1703, L1/2NMF method’s 0.1925, and VCA-FCLS method’s 0.1872.

Feature extraction is an important research topic in hyper-spectral image (HSI) classification. However, most of feature extraction methods only extract low-level features, which makes them not perform well in the applications of HSI. In this paper, we have proposed a non-negative matrix factorization (NMF) based deep feature extraction algorithm, namely deep NMF. Deep NMF tries to construct a deep feature representation by cascading multiple NMFs. Reconstruction residual of NMF is passed layer by layer to reduce information loss. Meanwhile, passing residuals between layers can construct a feature hierarchy from coarse to fine. Furthermore, activation functions are applied between adjacent layers to enhance the ability of non-linear feature extraction. Experimental results have also shown that our algorithm is computationally efficient and effective for HSI classification.

Hyperspectral unmixing is an important image interpre-tation technique that aims to estimate the pure constituent materials (endmembers) and their corresponding fractional abundances in each mixed pixel. Nonnegative matrix factorization (NMF) has attracted a lot of attention because of its ability to solve mixed pixel scenarios. The sparse NMF method achieves better unmixing results thanks to its full use of the sparse characteristics of the data. However, most existing sparse NMF unmixing techniques lack the consid-eration of spatial information. In fact, hyperspectral images contain intrinsic geometric information as well as rich spatial information. In this paper, a spatial graph regularized nonneg-ative matrix factorization unmixing framework (SGNMF) is established. For the proposed SGNMF, on the one hand, the graph regularization is introduced to characterize the latent manifold structure of the data, and on the other hand, the spatial weighting factor is used to mine the spatial correlation between pixels. The optimization problem of the SGNMF model can be solved by a multiplicative iterative rule. Exper-imental results on synthetic data sets indicate that the newly proposed SGNMF method is able to produce better results than other advanced spectral unmixing algorithms.

ABSTRACTHyperspectral unmixing (HU) has drawn remarkable attention because it can decompose mixed pixels into a set of endmembers and abundance fractions. And the nonnegative matrix factorization (NMF) algorithm has been widely used for solving hyperspectral spectral unmixing problem. This letter proposes a Hessian graph regularized NMF (HGNMF) algorithm, which relies upon the construction of a Hessian graph representation, to solve the hyperspectral unmixing problem. According to the HGNMF algorithm, the smoothness in the estimated abundance maps is well promoted. Moreover, the optimized problem of HGNMF algorithm is also solved by employing the multiplicative updating rules. Compared to other algorithms, the proposed HGNMF algorithm demonstrates lots of advantages based on the simulated results of the synthetic data and real data sets.

The hyperspectral and multispectral image fusion process, commonly known as the multisharpening process (also known as the hypersharpening process), results in the creation of unobservable high spectral and spatial resolution hyperspectral remote sensing data, by combining (or fusing) a hyperspectral image, characterized by a high spectral resolution, with, especially, a high spatial resolution multispectral one. There are several approaches in the literature performing such a process, in particular, those based on the spectral unmixing concept. The latter fusion methods aim to extract pure-material spectra from the high spectral resolution hyperspectral image and combine them with their corresponding proportions extracted from the high spatial resolution multispectral image. However, most of these methods, which mainly consider the standard linear mixing model, do not address the spectral variability phenomenon, precisely due to the considered data mixing model, which does not handle variations in illumination and atmospheric conditions or material composition inducing spectral variability. Neglecting such a phenomenon may lead to errors in the fusion process, thus decreasing the spatial and spectral qualities of sharpened products. In this paper, a new hyperspectral and multispectral data fusion method, based on the linear spectral unmixing concept that deals with spectral variability, is proposed. This approach, based on a multiplicative Nonnegative Matrix Factorization (NMF) technique, substantially extends the multiplicative Coupled NMF (CNMF) algorithm by considering a linear data mixing model that addresses the considered phenomenon. Indeed, the introduced approach minimizes new cost functions involving matrices with specific structures in order to deal with the spectral variability phenomenon. The proposed method is applied to synthetic and real data, and its effectiveness is evaluated with established performance criteria. Experimental results show that the designed algorithm globally yields sharpened hyperspectral data with fine spectral and spatial fidelities. Also, these results demonstrate that the proposed technique globally outperforms the tested literature approaches.

In this paper, we address the problem of spectral unmixing in urban hyperspectral images using a Maximum A Posteriori (MAP)-based Non-negative Matrix Factorization (NMF) approach. Considering a Linear-Quadratic (LQ) mixing model, we seek to decompose the spectrum observed in each pixel of the image into a set of pure material spectra, as well as their abundance fractions and the mixing coefficients associated with products of these pure material spectra. The main idea of the proposed method is to take into account the available prior information about the unknown parameters for a better estimation of them. To this end, we first derive a MAP-based cost function, then minimize it using a projected gradient algorithm by modifying a recently proposed NMF method adapted to LQ mixtures. Simulation results confirm the relevance of our approach.