Maximum Likelihood Estimation Based Nonnegative Matrix Factorization 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) 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.

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.

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.

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.

Semi-supervised graph regularized nonnegative matrix factorization with local coordinate for image representation

Non-negative matrix factorization (NMF) has been one of the most popular methods for feature learning in the field of machine learning and computer vision. Most existing works directly apply NMF on high-dimensional image datasets for computing the effective representation of the raw images. However, in fact, the common essential information of a given class of images is hidden in their low rank parts. For obtaining an effective low-rank data representation, we in this paper propose a non-negative low-rank matrix factorization (NLMF) method for image clustering. For the purpose of improving its robustness for the data in a manifold structure, we further propose a graph regularized NLMF by incorporating the manifold structure information into our proposed objective function. Finally, we develop an efficient alternating iterative algorithm to learn the low-dimensional representation of low-rank parts of images for clustering. Alternatively, we also incorporate robust principal component analysis into our proposed scheme. Experimental results on four image datasets reveal that our proposed methods outperform four representative methods.

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.

Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as the product of two smaller nonnegative matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, and with penalty terms in the objective function. By disjoint, we mean that each variable appears in at most one equality constraint. Our MU satisfy the set of constraints after each update of the variables during the optimization process, while guaranteeing that the objective function decreases monotonically. We showcase this framework on three NMF models, and show that it competes favorably the state of the art: (1)~$\beta$-NMF with sum-to-one constraints on the columns of $H$, (2) minimum-volume $\beta$-NMF with sum-to-one constraints on the columns of $W$, and (3) sparse $\beta$-NMF with $\ell_2$-norm constraints on the columns of $W$.

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 (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.

In order to improve the speech enhancement effect and overcome the weakness of falling into the local optimum of the traditional nonnegative matrix factorization, this paper presents speech enhancement methodology based on nonnegative matrix factorization combined with the improved genetic algorithm. The methodology consists of training stage and enhancement stage. In the training stage, instead of gradient descent, iterative learning is achieved by selecting several directions in super flat close to deflector cone as the mutation. By adding the mutation operator and simulated annealing operator to NMF, the proposed algorithm is used to acquire the accurate noise dictionary as prior information. In the de-noising stage, NMF-IGA is used to evaluate the dictionary matrix and the activation matrix of the de-noised speech. Then using the dictionary of de-noised speech to get the enhanced speech spectrogram and reconstruct the enhanced speech. It is verified that the proposed algorithm yields less residual noise and better speech quality than traditional nonnegative matrix factorization and Bayes nonnegative matrix factorization algorithm.

Profile-Decomposing Output of Multi-Channel Odor Sensor Array

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.

In this work, we will consider the dimension reduction of the set of time series, such as economic data, to find the meaningful basis vector for the set of data, and indicate which data use which basis vector. Usually each of the time series is analyzed independently in economics but here we will analyze the set of time series simultaneously. Since some of the economic data are measured as positive values and we want to decompose them as a mixture of the parts, we will apply non-negative matrix factorization to the economic data. Non-negative matrix factorization can compress dimensions by approximating a non-negative matrix with the product of two non-negative matrices. The two non-negative matrices are called the coefficient matrix and the basis matrix, and the basis matrix can be considered as a dimensionally compressed matrix. If the standard non-negative matrix factorization is used for economic data, the basis matrix may not be smooth. We think that the basis vectors should be smooth except a few special economical incidents. In the proposed method, a Graph-based non-negative matrix factorization is introduced to regularize the basis matrix of the time series. A path graph for representing the time series of economic data is incorporated into the non-negative matrix factorization as regularization. As a result, basis vectors that maintains the time series of economic data are decomposed. Furthermore, we propose to introduce a sparsity in the non-negative matrix factorization. Traditionally, the sparsity incorporated into non-negative matrix factorization has been used for basis vectors. However, the proposed method introduces the sparsity for coefficient vectors. Thus the proposed method, which simultaneously incorporates the sparsity for the coefficient vectors and the smoothness for the basis vectors, can extract the smooth basis vectors and the original economical data are approximated as the weighted sum of the few bases vectors. This allows us to discover economic trends and the best-fit trends for each data at the same time.