Constrained Nonnegative Matrix Factorization for Blind Hyperspectral Unmixing Incorporating Endmember Independence

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 aims to estimate a set of endmembers and corresponding abundances in pixels. Nonnegative matrix factorization (NMF) and its extensions with various constraints have been widely applied to hyperspectral unmixing. L 1 / 2 and L 2 regularizers can be added to NMF to enforce sparseness and evenness, respectively. In practice, a region in a hyperspectral image may possess different sparsity levels across locations. The problem remains as to how to impose constraints accordingly when the level of sparsity varies. We propose a novel nonnegative matrix factorization with data-guided constraints (DGC-NMF). The DGC-NMF imposes on the unknown abundance vector of each pixel with either an L 1 / 2 constraint or an L 2 constraint according to its estimated mixture level. Experiments on the synthetic data and real hyperspectral data validate the proposed algorithm.

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.

Text embedding plays a crucial role in natural language processing (NLP). Among various approaches, nonnegative matrix factorization (NMF) is an effective method for this purpose. However, the standard NMF approach, fundamentally based on the bag-of-words model, fails to utilize the contextual information of documents and may result in a significant loss of semantics. To address this issue, we propose a new NMF scheme incorporating a regularization term based on the Wasserstein metric, which quantifies an approximation of a word-context matrix to leverage semantic information. Since the word-context matrix is typically symmetric and positive definite (SPD), the Wasserstein metric used in the regularization is related to a natural SPD manifold structure. We then build upon this manifold structure to integrate an explicit expression of the Wasserstein metric into the NMF framework. Additionally, we introduce modifications to the gradient computation to adapt three fundamental classes of numerical algorithms from ordinary NMF to tackle the Wasserstein-regularized NMF (NMF-WR) problem. To demonstrate the effectiveness of the NMF-WR model and the proposed algorithms, we conduct experiments on popular datasets, focusing on topic modeling and document clustering. The results indicate that the NMF-WR model exhibits superior performance compared to other conventional NMF models in processing text embeddings. These findings suggest that this novel NMF-WR framework not only enhances semantic representation but also underscores a commitment to the model’s reliability and interpretability.

Hyperspectral image unmixing is an important part of hyperspectral data analysis. The mixed pixel decomposition consists of two steps, endmember (the unique signatures of pure ground components) extraction and abundance (the proportion of each endmember in each pixel) estimation. Recently, a Discrete Particle Swarm Optimization algorithm (DPSO) was proposed for accurately extract endmembers with high optimal performance. However, the DPSO algorithm shows very high computational complexity, which makes the endmember extraction procedure very time consuming for hyperspectral image unmixing. Thus, in this paper, the DPSO endmember extraction algorithm was parallelized, implemented on the CUDA (GPU K20) platform, and evaluated by real hyperspectral remote sensing data. The experimental results show that with increasing the number of particles the parallelized version obtained much higher computing efficiency while maintain the same endmember exaction accuracy.

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.

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.

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.

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 imaging is concerned with the measurement, analysis, and interpretation of spectra acquired from a given scene (or specific object) at a short, medium or long distance by an airbone or satellite sensor. Over the last few years, hyperspectral image data sets have been collected for a great amount of locations over the world, using a variety of instruments for Earth observation. Despite the increasing importance of hyperspectral images in remote sensing applications, there is no common repository of hyperspectral data intended to distribute and share hyperspectral data sets in the community. Quite opposite, the hyperspectral data sets which are available for public use are spread among different storage locations and present significant heterogeneity regarding the storage format, associated meta-data (if any), or ground-truth availability. As a result, the development of a standardized hyperspectral data repository is a highly desired goal in the remote sensing community. In this paper, we take a necessary first step towards the development of a digital repository for remotely sensed hyperspectral data. The proposed system allows uploading new hyperspectral data sets along with meta-data, ground-truth and analysis results, with the ultimate goal of sharing publicly available hyperspectral images within the remote sensing community. The database has been designed in order to allow storing relevant information for the hyperspectral data available through the system, including basic image characteristics (width, height, number of bands, format) and more advanced meta-data (ground-truth information, publications in which the data has been used). The current implementation consists of a front-end to ease the management of images through a web interface, thus containing both synthetic and real hyperspectral images from two highly representative instruments, such as NASAs Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) over the Cuprite Mining District in Nevada. Most importantly, the developed system includes a spectral unmixing-based content based image retrieval (CBIR) functionality which allows searching for images on the spectral unmixing information (spectrally pure components or endmembers and their associated abundances in the scene). This information is stored as meta-data associated to each hyperspectral image instance, and then used to search and retrieve images based on information content. This paper presents the design of the system and a preliminary validation of the unmixing-based retrieval functionality using both synthetic and real hyperspectral images stored in the database.

Hyperspectral Unmixing (HU), also known as spectral mixture analysis, is a challenging problem that decomposes a mixed spectrum into a collection of endmembers and their abundance fractions. In this paper, we extended the autoencoder network (Palsson et al. in IEEE Access 6:25646–25656, [1]) for blind hyperspectral nonlinear unmixing. The proposed autoencoder architecture consists of two networks encoder and decoder. The encoder network has the same as the original architecture. The architecture of the decoder network was altered to handle nonlinear unmixing. Our proposed encoder network has four fully connected layers, each with the number of neurons equal to the dimension of the end members. Experiments were conducted using nonlinear synthetic dataset sampled from the USGS library, and the performance was evaluated using both SAD and SID metrics. Four categories of experiments were carried out; accuracy assessment, weight initialization techniques, learning rate, and robustness to the noise. Experimental results show that the proposed autoencoder outperforms traditional endmember extraction algorithms in nonlinear cases. Finally, we introduced the application of hyperspectral image unmixing algorithm in the Internet of things (IoT) environment.

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.

Hyperspectral image (HSI) change detection (CD) is an important technology for identifying surface changes using multi-temporal HSIs. Nevertheless, the high dimensionality of HSIs presents significant challenges for CD tasks, including issues such as lack of robustness and high computational costs in existing methods. To address those issues, this paper proposes an unsupervised simple and effective HSI CD model termed L2,1-norm regularized double non-negative matrix factorization (L2,1-DNMF). Specifically, the proposed model employs a symmetric double non-negative matrix factorization (NMF) framework to jointly analyze multitemporal HSIs, capturing their common and invariant structural information to construct a shared feature basis. Meanwhile, two non-negative feature weight matrices are learned to generate a differential image matrix that directly reflects the change regions. To enhance robustness against noise, an L2,1-norm constraint is imposed on the difference image matrix, ensuring that unchanged areas exhibit near-zero values while changed areas present nonzero values. Finally, comprehensive experiments performed on three benchmark hyperspectral datasets validated the efficacy of the proposed method, which is superior to some state-of-the-art ones regarding detection performance and computational cost.