In a plethora of applications dealing with inverse problems, e.g. in imageprocessing, social networks, compressive sensing, biological data processingetc., the signal of interest is known to be structured in several ways at thesame time. This premise has recently guided the research to the innovative andmeaningful idea of imposing multiple constraints on the parameters involved inthe problem under study. For instance, when dealing with problems whoseparameters form sparse and low-rank matrices, the adoption of suitably combinedconstraints imposing sparsity and low-rankness, is expected to yieldsubstantially enhanced estimation results. In this paper, we address thespectral unmixing problem in hyperspectral images. Specifically, two novelunmixing algorithms are introduced, in an attempt to exploit both spatialcorrelation and sparse representation of pixels lying in homogeneous regions ofhyperspectral images. To this end, a novel convex mixed penalty term is firstdefined consisting of the sum of the weighted $\ell_1$ and the weighted nuclearnorm of the abundance matrix corresponding to a small area of the imagedetermined by a sliding square window. This penalty term is then used toregularize a conventional quadratic cost function and impose simultaneouslysparsity and row-rankness on the abundance matrix. The resulting regularizedcost function is minimized by a) an incremental proximal sparse and low-rankunmixing algorithm and b) an algorithm based on the alternating minimizationmethod of multipliers (ADMM). The effectiveness of the proposed algorithms isillustrated in experiments conducted both on simulated and real data.
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