Combinatorial Nonnegative Matrix-Tensor Factorization for Hyperspectral Unmixing Using a General $\ell _{q}$ Norm Regularization

Hyperspectral unmixing (HU), an essential procedure for various environmental applications, has garnered significant attention within remote sensing communities. Among different groups of HU methods, nonnegative matrix factorization (NMF)-based ones have gained widespread popularity due to their high capability of simultaneously extracting endmembers and their corresponding abundances. However, converting a 3-D hyperspectral data cube into a 2-D matrix format leads to the loss of spatial and potential correlation information. Consequently, in recent years, nonnegative tensor factorization (NTF) methods, which preserve the 3-D nature of hyperspectral data cube, have been extensively embraced by numerous researchers. Nevertheless, incorporating prior information into NTF-based problems faces limitations owing to the inconsistency of such information, particularly concerning $\ell _{1}$ norm sparsity and the abundance sum-to-one constraint (ASC). To address this limitation, our study introduces a novel general regularization term. This term leverages sparsity and ASC simultaneously, integrating it into a matrix-tensor factorization framework. Our proposed method, named a matrix-tensor-based HU method with general $\ell _{q}$ norm regularization (MTUHL$_{q}$), is established on the block term decomposition (BTD) paradigm, which ensures physical interpretability and simple implementation. To investigate the performance of the proposed MTUHL$_{q}$, a series of experiments on both synthetic and real hyperspectral datasets were conducted. The results of the implemented experiments indicated that the proposed method outperformed other state-of-the-art HU methods.