Robust and Unified Semi-Supervised Unmixing of Hyperspectral Imaging for Linear and Multilinear Models

The spectral unmixing paradigm is an important analysis tool for hyperspectral (HS) images which allows one to decompose the 2D spatial information from the basic spectral signatures or end-members. In this work, we introduce a semi-supervised perspective for spectral unmixing, where some end-members are known a priori, while the rest are estimated from the HS image. The proposal is relevant in unmixing scenarios where there is only available partial information of end-members, or when the known end-members are not fully representative of the scene. Our formulation simultaneously addresses linear and multilinear mixing models in a unified fashion. The proposed algorithms are referred as ESSEAE (Extended Semi-Supervised End-members and Abundance Extraction) for the linear model, and NESSEAE (Non-linear Extended Semi-Supervised End-members and Abundance Extraction) for the multilinear one. The estimation process is presented as a weighted optimal approximation problem with regularization terms for abundances, end-members and sparse noise components, which is solved by a cyclic coordinate descent optimization (CCDO) scheme. In this work, we derive closed-solutions at each step of the CCDO scheme, and just for the multilinear model, the end-members estimation involves a gradient descent scheme with optimal linear search. We validate first our contributions with synthetic HS images that include Gaussian and sparse noise components to evaluate their robustness, and compare them with supervised and unsupervised perspectives. In addition, we validated the linear scheme with a breast histological sample, and the multilinear approach with the Urban dataset. The use of two datasets from different fields guarantees the generalizability of the proposed formulation. In general, our semi-supervised spectral unmixing schemes provide accurate and robust results with a fast computational time, and as expected, present an overall performance in between the supervised and unsupervised approaches. All scripts for the proposed algorithms are freely available in https://github.com/Nicothe4th/ESSEAE-NESSEAE.