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Nonlinear Unmixing of Hyperspectral Images: Models and Algorithms
"... When considering the problem of unmixing hyperspectral images, most of the literature in the geoscience and image processing areas relies on the widely used linear mixing model (LMM). However, the LMM may be not valid and other nonlinear models need to be considered, for instance, when there are mul ..."
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When considering the problem of unmixing hyperspectral images, most of the literature in the geoscience and image processing areas relies on the widely used linear mixing model (LMM). However, the LMM may be not valid and other nonlinear models need to be considered, for instance, when there are multiscattering effects or intimate interactions. Consequently, over the last few years, several significant contributions have been proposed to overcome the limitations inherent in the LMM. In this paper, we present an overview of recent advances in nonlinear unmixing modeling.
Hierarchical Clustering of Hyperspectral Images Using RankTwo Nonnegative Matrix Factorization
 IEEE, Transactions on Geoscience and Remote Sensing
, 2015
"... In this paper, we design a hierarchical clustering algorithm for highresolution hyperspectral images. At the core of the algorithm, a new ranktwo nonnegative matrix factorizations (NMF) algorithm is used to split the clusters, which is motivated by convex geometry concepts. The method starts with ..."
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In this paper, we design a hierarchical clustering algorithm for highresolution hyperspectral images. At the core of the algorithm, a new ranktwo nonnegative matrix factorizations (NMF) algorithm is used to split the clusters, which is motivated by convex geometry concepts. The method starts with a single cluster containing all pixels, and, at each step, (i) selects a cluster in such a way that the error at the next step is minimized, and (ii) splits the selected cluster into two disjoint clusters using ranktwo NMF in such a way that the clusters are well balanced and stable. The proposed method can also be used as an endmember extraction algorithm in the presence of pure pixels. The effectiveness of this approach is illustrated on several synthetic and realworld hyperspectral images, and shown to outperform standard clustering techniques such as kmeans, spherical kmeans and standard NMF.
A Vavasis, “Semidefinite programming based preconditioning for more robust nearseparable nonnegative matrix factorization,” arXiv preprint arXiv:1310.2273
, 2013
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Blind Separation of QuasiStationary Sources: Exploiting Convex Geometry in Covariance Domain
, 2015
"... This paper revisits blind source separation of instantaneously mixed quasistationary sources (BSSQSS), motivated by the observation that in certain applications (e.g., speech) there exist time frames during which only one source is active, or locally dominant. Combined with nonnegativity of sourc ..."
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This paper revisits blind source separation of instantaneously mixed quasistationary sources (BSSQSS), motivated by the observation that in certain applications (e.g., speech) there exist time frames during which only one source is active, or locally dominant. Combined with nonnegativity of source powers, this endows the problem with a nice convex geometry that enables elegant and efficient BSS solutions. Local dominance is tantamount to the socalled pure pixel/separability assumption in hyperspectral unmixing/nonnegative matrix factorization, respectively. Building on this link, a very simple algorithm called successive projection algorithm (SPA) is considered for estimating the mixing system in closed form. To complement SPA in the specific BSSQSS context, an algebraic preprocessing procedure is proposed to suppress shortterm source crosscorrelation interference. The proposed procedure is simple, effective, and supported by theoretical analysis. Solutions based on volume minimization (VolMin) are also considered. By theoretical analysis, it is shown that VolMin guarantees perfect mixing system identifiability under an assumption more relaxed than (exact) local dominance—which means wider applicability in practice. Exploiting the specific structure of BSSQSS, a fast VolMin algorithm is proposed for the overdetermined case. Careful simulations using real speech sources showcase the simplicity, efficiency, and accuracy of the proposed algorithms.
Identifiability of the simplex volume minimization criterion for blind hyperspectral unmixing: The no purepixel case,” arXiv Preprint arXiv:1406.5273
, 2014
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SelfDictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search Are Related
"... Abstract—This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a selfdictionary multiple measurement vector (SDMMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SD ..."
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Abstract—This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a selfdictionary multiple measurement vector (SDMMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SDMMV formalism is special in that it allows simultaneous identification of the endmember spectral signatures and the number of endmembers. Previous SDMMV studies mainly focus on convex relaxations. In this study, we explore the alternative of greedy pursuit, which generally provides efficient and simple algorithms. In particular, we design a greedy SDMMV algorithm using simultaneous orthogonal matching pursuit. Intriguingly, the proposed greedy algorithm is shown to be closely related to some existing pure pixel search algorithms, especially, the successive projection algorithm (SPA). Thus, a link between SDMMV and pure pixel search is revealed. We then perform exact recovery analyses, and prove that the proposed greedy algorithm is robust to noiseincluding its identification of the (unknown) number of endmembersunder a sufficiently low noise level. The identification performance of the proposed greedy algorithm is demonstrated through both synthetic and realdata experiments. Index Terms—Greedy pursuit, hyperspectral unmixing, number of endmembers estimation, selfdictionary sparse regression.
A nonlocal structure tensor based approach for multicomponent image recovery problems
 IEEE Trans. Image Process
"... NonLocal Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTVbased regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. Th ..."
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NonLocal Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTVbased regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the nonlocal variations, jointly for the different components, through various `1,p matrix norms with p ≥ 1. To facilitate the choice of the hyperparameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the STNLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented thanks to the flexibility offered by recent primaldual proximal algorithms. Experiments are carried out for multispectral and hyperspectral images. The results demonstrate the interest of introducing a nonlocal structure tensor regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current stateoftheart methods. 1
Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation
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HYPERSPECTRAL SUPERRESOLUTION OF LOCALLY LOW RANK IMAGES FROM COMPLEMENTARY MULTISOURCE DATA
, 2014
"... Remote sensing hyperspectral images (HSI) are quite often locally low rank, in the sense that the spectral vectors acquired from a given spatial neighborhood belong to a low dimensional subspace/manifold. This has been recently exploited for the fusion of low spatial resolution HSI with high spatial ..."
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Remote sensing hyperspectral images (HSI) are quite often locally low rank, in the sense that the spectral vectors acquired from a given spatial neighborhood belong to a low dimensional subspace/manifold. This has been recently exploited for the fusion of low spatial resolution HSI with high spatial resolution multispectral images (MSI) in order to obtain superresolution HSI. Most approaches adopt an unmixing or a matrix factorization perspective. The derived methods have led to stateoftheart results when the spectral information lies in a low dimensional subspace/manifold. However, if the subspace/manifold dimensionality spanned by the complete data set is large, the performance of these methods decrease mainly because the underlying sparse regression is severely illposed. In this paper, we propose a local approach to cope with this difficulty. Fundamentally, we exploit the fact that real world HSI are locally low rank, to partition the image into patches and solve the data fusion problem independently for each patch. This way, in each patch the subspace/manifold dimensionality is low enough to obtain useful superresolution. We explore two alternatives to define the local regions, using sliding windows and binary partition trees. The effectiveness of the proposed approach is illustrated with synthetic and semireal data. 1
A PRECONDITIONED FORWARDBACKWARD APPROACH WITH APPLICATION TO LARGESCALE NONCONVEX SPECTRAL UNMIXING PROBLEMS
"... ABSTRACT Many inverse problems require to minimize a criterion being the sum of a non necessarily smooth function and a Lipschitz differentiable function. Such an optimization problem can be solved with the ForwardBackward algorithm which can be accelerated thanks to the use of variable metrics de ..."
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ABSTRACT Many inverse problems require to minimize a criterion being the sum of a non necessarily smooth function and a Lipschitz differentiable function. Such an optimization problem can be solved with the ForwardBackward algorithm which can be accelerated thanks to the use of variable metrics derived from the MajorizeMinimize principle. The convergence of this approach is guaranteed provided that the criterion satisfies some additional technical conditions. Combining this method with an alternating minimization strategy will be shown to allow us to address a broad class of optimization problems involving largesize signals. An application example to a nonconvex spectral unmixing problem will be presented.