Multi and hyperspectral image unmixing with spatial coherence by extended blind end-member and abundance extraction

Blind linear unmixing (BLU) methods decompose multi and hyperspectral datasets into end-members and abundance maps with an unsupervised perspective. However, due to measurement noise and model uncertainty, the estimated abundance maps could exhibit granularity, which causes a loss of detail that could be crucial in certain applications. To address this problem, in this paper, we present a BLU proposal that considers spatial coherence (SC) in the abundance estimates. The proposed BLU formulation is based on the extended blind end-member and abundance extraction (EBEAE) methodology, and is denoted as EBEAE-SC. In this proposed method, the energy functional of EBEAE-SC includes new variables, which are denoted as internal abundances, to induce SC in the BLU approach. The new formulation of the optimization problem is solved by a coordinate descent algorithm, constrained quadratic optimization, and the split Bregman formulation. We present a comprehensive validation process that considers synthetic and experimental datasets at different noise types and levels, and a comparison with five state-of-the-art BLU methods. In our results, EBEAE-SC can significantly decrease the granularity in the estimated abundances, without losing detail of the structures present in the multi and hyperspectral images. In addition, the resulting complexity of EBEAE-SC is analyzed and compared it to the original formulation of EBEAE, and also the numerical convergence of the resulting iterative process is evaluated. Hence, by our analysis, EBEAE-SC allows blind estimates of end-members and abundances in the studied datasets of diverse applications, producing linearly independent and non-negative end-members, as well as non-negative abundances, with lower estimation errors and computational times compared to five methodologies in the state-of-the-art.