Improved Local Spectral Unmixing of hyperspectral data using an algorithmic regularization path for collaborative sparse regression

Local Spectral Unmixing (LSU) methods perform the unmixing of hyperspectral data locally in regions of the image. The endmembers and their abundances in each pixel are extracted region-wise, instead of globally to mitigate spectral variability effects, which are less severe locally. However, it requires the local estimation of the number of endmembers to use. Algorithms for intrinsic dimensionality (ID) estimation tend to overestimate the local ID, especially in small regions. The ID only provides an upper bound of the application and scale dependent number of endmembers, which leads to extract irrelevant signatures as local endmembers, associated with meaningless local abundances. We propose a method to select in each region the best subset of the locally extracted endmembers. Collaborative sparsity is used to detect spurious endmembers in each region and only keep the most influent ones. We compute an algorithmic regularization path for this problem, giving access to the sequence of successive active sets of endmembers when the regularization parameter is increased. Finally, we select the optimal set in the sense of the Bayesian Information Criterion (BIC), favoring models with a high likelihood, while penalizing those with too many endmembers. Results on real data show the interest of the proposed approach.