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Hyperspectral unmixing via sparse regression. Optimization problems and algorithms Hyperspectral unmixing is the decomposition of the pixel spectra from a hyperspectral image into a collection of constituent spectra, or spectral signatures, called endmembers, and the fractional abundances, one set per pixel. Hyperspectral unmixing has recently been approached in a semi-supervised fashion, by assuming that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectro-radiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In practice, this is a combinatorial problem which calls for efficient linear sparse regression techniques based on sparsity-inducing regularizers. In this talk, I will address a set of recently introduced hyperspectral unmixing techniques based on sparse regression. In this approach, the unmixing is obtained by solving a convex optimization problem, where the objective function is a sum of convex terms with possibly convex constraints. Usually, one of terms in the objective function measures the data fidelity, while the others, jointly with the constraints, enforce some type of sparsity on the solution. Several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this talk, I will present a new class of algorithms to handle these convex optimization problems. The proposed class of algorithms is an instance of the so-called alternating direction method of multipliers (ADMM), for which convergence sufficient conditions are known. We show that these conditions are satisfied by the proposed algorithms. The effectiveness of the proposed approach is illustrated in a series of hyperspectral unmixing problems. |