  In a plethora of applications dealing with inverse problems, e.g. in image
processing, social networks, compressive sensing, biological data processing
etc., the signal of interest is known to be structured in several ways at the
same time. This premise has recently guided the research to the innovative and
meaningful idea of imposing multiple constraints on the parameters involved in
the problem under study. For instance, when dealing with problems whose
parameters form sparse and low-rank matrices, the adoption of suitably combined
constraints imposing sparsity and low-rankness, is expected to yield
substantially enhanced estimation results. In this paper, we address the
spectral unmixing problem in hyperspectral images. Specifically, two novel
unmixing algorithms are introduced, in an attempt to exploit both spatial
correlation and sparse representation of pixels lying in homogeneous regions of
hyperspectral images. To this end, a novel convex mixed penalty term is first
defined consisting of the sum of the weighted $\ell_1$ and the weighted nuclear
norm of the abundance matrix corresponding to a small area of the image
determined by a sliding square window. This penalty term is then used to
regularize a conventional quadratic cost function and impose simultaneously
sparsity and row-rankness on the abundance matrix. The resulting regularized
cost function is minimized by a) an incremental proximal sparse and low-rank
unmixing algorithm and b) an algorithm based on the alternating minimization
method of multipliers (ADMM). The effectiveness of the proposed algorithms is
illustrated in experiments conducted both on simulated and real data.
