Title: Low-rank and Sparse NMF for Joint Endmembers' Number Estimation and Blind Unmixing of Hyperspectral Images,
Abstract: Estimation of the number of endmembers existing in a scene constitutes a
critical task in the hyperspectral unmixing process. The accuracy of this
estimate 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. A
common practice amply followed in literature is to treat endmembers' number
estimation and unmixing, independently as two separate tasks, providing the
outcome of the former as input to the latter. In this paper, we go beyond this
computationally demanding strategy. More precisely, we set forth a multiple
constrained optimization framework, which encapsulates endmembers' number
estimation and unsupervised unmixing in a single task. This is attained by
suitably formulating the problem via a low-rank and sparse nonnegative matrix
factorization rationale, where low-rankness is promoted with the use of a
sophisticated $\ell_2/\ell_1$ norm penalty term. An alternating proximal
algorithm is then proposed for minimizing the emerging cost function. The
results obtained by simulated and real data experiments verify the
effectiveness of the proposed approach.