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...tion. Therefore, given a hyperspectral image, NMF is able to compute the spectral signatures of the endmembers and simultaneously the abundance of each endmember in each pixel. We refer the reader to =-=[8, 90]-=- for recent surveys on blind HU techniques. 4 X(:, j)︸ ︷︷ ︸ spectral signature of jth pixel ≈ r∑ k=1 W (:, k)︸ ︷︷ ︸ spectral signature of kth endmember H(k, j)︸ ︷︷ ︸ abundance of kth endmember in jth ...

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... mixing model, a measured spectral vector can be represented by a convex combination of the so-called pure spectral vectors — spectral vectors that are purely constituted by a single surface material =-=[1]-=-. Pure pixels identification aims at determining the pure spectral vectors, or simply pure pixels, from the measured hyperspectral data, assuming that pure pixels exist in the measured data. It repres...

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...r unmixing techniques, as nonlinear models may provide more accurate estimations of mixtures, in particular when there are significant physical interactions of photons scattered by multiple materials =-=[1]-=-. Several algorithms have been proposed to address the nonlinear unmixing problem. For instance, bilinear models were considered to handle complex scenarios such as multilayered scenes [2, 3], by intr...

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... . , yP ] T into a collection of K individual spectra M = [m1, . . . , mK] T , called endmembers, and estimating their relative proportions (or abundances) A = [a1, . . . , aP ] T in each observation =-=[1]-=-. Most of the hyperspectral unmixing algorithms proposed in the signal & image processing and geoscience literatures rely on the commonly admitted linear mixing model (LMM), Y ≈ MA. Indeed, LMM provid...

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...is [16], for which we know the reference spectra but are uncertain about how to align them with the data because of wavelength misalignment. Another example we will consider is hyperspectral unmixing =-=[17, 18, 19]-=-. Multiple reference spectral signatures, or endmembers, may have been measured for the same material, and they may all be slightly different if they were measured under different conditions. We may n...

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...imaging, which consists of acquiring a same scene in several hundreds of contiguous spectral bands, has opened a new range of relevant applications, such as target detection [4] and spectral unmixing =-=[5]-=-. Naturally, to take advantage of the newest benefits offered by HS images, the problem of fusing HS and PAN images has been explored [6]. Capitalizing on decades of experience in MS pansharpening, mo...

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...training dataset of image patches to learn a dictionary D which yields sparse (albeit high-dimensional) representations. However, in contrast to the transformations discussed earlier, the dictionaries learned here do not have orthogonal elements and require the application of custom algorithms for sparse approximation, described in the sidebar Sparse recovery: Methods and guarantees. In recent years, sparsity has also been studied in contexts where the types of spectra (called endmembers) are known a priori and that each particular pixel is a linear combination of only a few of the endmembers [7, 8]. The sparse representation of the spectrum effectively identifies the component endmembers and their concentrations at each pixel, a process referred to as hyperspectral unmixing [8]. Various global sparsifying transforms, to be applied to the entire image rather than its patches, have also been proposed [9–11]. Unfortunately, the corresponding increase in dimensionality also increases the computational complexity of the transformation and approximation; furthermore, the improvements in approximation error are often not found to be significant enough to warrant the additional computational lo...