Estimation of the number of endmembers existing in a scene constitutes acritical task in the hyperspectral unmixing process. The accuracy of thisestimate plays a crucial role in subsequent unsupervised unmixing steps i.e.,the derivation of the spectral signatures of the endmembers (endmembers'extraction) and the estimation of the abundance fractions of the pixels. Acommon practice amply followed in literature is to treat endmembers' numberestimation and unmixing, independently as two separate tasks, providing theoutcome of the former as input to the latter. In this paper, we go beyond thiscomputationally demanding strategy. More precisely, we set forth a multipleconstrained optimization framework, which encapsulates endmembers' numberestimation and unsupervised unmixing in a single task. This is attained bysuitably formulating the problem via a low-rank and sparse nonnegative matrixfactorization rationale, where low-rankness is promoted with the use of asophisticated $\ell_2/\ell_1$ norm penalty term. An alternating proximalalgorithm is then proposed for minimizing the emerging cost function. Theresults obtained by simulated and real data experiments verify theeffectiveness of the proposed approach.
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