Hyperspectral Unmixing via $L_{1/2}$ Sparsity-Constrained Nonnegative Matrix Factorization
Hyperspectral unmixing is a crucial preprocessing step for material classification and recognition. In the last decade, nonnegative matrix factorization (NMF) and its extensions have been intensively studied to unmix hyperspectral imagery and recover the material end-members. As an important constraint for NMF, sparsity has been modeled making use of the <i>L</i><sub>1</sub> regularizer. Unfortunately, the <i>L</i><sub>1</sub> regularizer cannot enforce further sparsity when the full additivity constraint of material abundances is used, hence limiting the practical efficacy of NMF methods in hyperspectral unmixing. In this paper, we extend the NMF method by incorporating the <i>L</i><sub>1/2</sub> sparsity constraint, which we name <i>L</i><sub>1/2</sub> -NMF. The <i>L</i><sub>1/2</sub> regularizer not only induces sparsity but is also a better choice among <i>Lq</i>(0 <; <i>q</i> <; 1) regularizers. We propose an iterative estimation algorithm for <i>L</i><sub>1/2</sub>-NMF, which provides sparser and more accurate results than those delivered using the <i>L</i><sub>1</sub> norm. We illustrate the utility of our method on synthetic and real hyperspectral data and compare our results to those yielded by other state-of-the-art methods.
Constrained Nonnegative Matrix Factorization for Hyperspectral Unmixing
Hyperspectral unmixing is a process to identify the constituent materials and estimate the corresponding fractions from the mixture. During the last few years, nonnegative matrix factorization (NMF), as a suitable candidate for the linear spectral mixture model, has been applied to unmix hyperspectral data. Unfortunately, the local minima caused by the nonconvexity of the objective function makes the solution nonunique, thus only the nonnegativity constraint is not sufficient enough to lead to a well-defined problem. Therefore, in this paper, two inherent characteristics of hyperspectral data, piecewise smoothness (both temporal and spatial) of spectral data and sparseness of abundance fraction of every material, are introduced to NMF. The adaptive potential function from discontinuity adaptive Markov random field model is used to describe the smoothness constraint while preserving discontinuities in spectral data. At the same time, two NMF algorithms, nonsmooth NMF and NMF with sparseness constraint, are used to quantify the degree of sparseness of material abundances. A gradient-based optimization algorithm is presented, and the monotonic convergence of the algorithm is proved. Three important facts are exploited in our method: First, both the spectra and abundances are nonnegative; second, the variation of the material spectra and abundance images is piecewise smooth in wavelength and spatial spaces, respectively; third, the abundance distribution of each material is almost sparse in the scene. Experiments using synthetic and real data demonstrate that the proposed algorithm provides an effective unsupervised technique for hyperspectral unmixing.
Structured Sparse Method for Hyperspectral Unmixing
Hyperspectral Unmixing (HU) has received increasing attention in the past decades due to its ability of unveiling information latent in hyperspectral data. Unfortunately, most existing methods fail to take advantage of the spatial information in data. To overcome this limitation, we propose a Structured Sparse regularized Nonnegative Matrix Factorization (SS-NMF) method based on the following two aspects. First, we incorporate a graph Laplacian to encode the manifold structures embedded in the hyperspectral data space. In this way, the highly similar neighboring pixels can be grouped together. Second, the lasso penalty is employed in SS-NMF for the fact that pixels in the same manifold structure are sparsely mixed by a common set of relevant bases. These two factors act as a new structured sparse constraint. With this constraint, our method can learn a compact space, where highly similar pixels are grouped to share correlated sparse representations. Experiments on real hyperspectral data sets with different noise levels demonstrate that our method outperforms the state-of-the-art methods significantly. (C) 2013 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS) Published by Elsevier B.V. All rights reserved.
Nonlinear Hyperspectral Unmixing With Robust Nonnegative Matrix Factorization
We introduce a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures. The new model extends the commonly used linear mixing model by introducing an additional term accounting for possible nonlinear effects, that are treated as sparsely distributed additive outliers. With the standard nonnegativity and sum-to-one constraints inherent to spectral unmixing, our model leads to a new form of robust nonnegative matrix factorization with a group-sparse outlier term. The factorization is posed as an optimization problem, which is addressed with a block-coordinate descent algorithm involving majorization-minimization updates. Simulation results obtained on synthetic and real data show that the proposed strategy competes with the state-of-the-art linear and nonlinear unmixing methods.
Total Variation Regularized Reweighted Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing
Blind hyperspectral unmixing (HU), which includes the estimation of endmembers and their corresponding fractional abundances, is an important task for hyperspectral analysis. Recently, nonnegative matrix factorization (NMF) and its extensions have been widely used in HU. Unfortunately, most of the NMF-based methods can easily lead to an unsuitable solution, due to the nonconvexity of the NMF model and the influence of noise. To overcome this limitation, we make the best use of the structure of the abundance maps, and propose a new blind HU method named total variation regularized reweighted sparse NMF (TV-RSNMF). First, the abundance matrix is assumed to be sparse, and a weighted sparse regularizer is incorporated into the NMF model. The weights of the weighted sparse regularizer are adaptively updated related to the abundance matrix. Second, the abundance map corresponding to a single fixed endmember should be piecewise smooth. Therefore, the TV regularizer is adopted to capture the piecewise smooth structure of each abundance map. In our multiplicative iterative solution to the proposed TV-RSNMF model, the TV regularizer can be regarded as an abundance map denoising procedure, which improves the robustness of TV-RSNMF to noise. A number of experiments were conducted in both simulated and real-data conditions to illustrate the advantage of the proposed TV-RSNMF method for blind HU.
Robust Collaborative Nonnegative Matrix Factorization for Hyperspectral Unmixing
Spectral unmixing is an important technique for remotely sensed hyperspectral data exploitation. It amounts to identifying a set of pure spectral signatures, which are called endmembers, and their corresponding fractional, draftrulesabun-dances in each pixel of the hyperspectral image. Over the last years, different algorithms have been developed for each of the three main steps of the spectral unmixing chain: 1) estimation of the number of endmembers in a scene; 2) identification of the spectral signatures of the endmembers; and 3) estimation of the fractional abundance of each endmember in each pixel of the scene. However, few algorithms can perform all the stages involved in the hyperspectral unmixing process. Such algorithms are highly desirable to avoid the propagation of errors within the chain. In this paper, we develop a new algorithm, which is termed robust collaborative nonnegative matrix factorization (R-CoNMF), that can perform the three steps of the hyperspectral unmixing chain. In comparison with other conventional methods, R-CoNMF starts with an overestimated number of endmembers and removes the redundant endmembers by means of collaborative regularization. Our experimental results indicate that the proposed method provides better or competitive performance when compared with other widely used methods.
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