Tensor singular value decomposition and low-rank representation for hyperspectral image unmixing

Hyperspectral unmixing (HU) finds pure spectra (endmembers) and their proportions (abundances) in hyperspectral images (HSIs). The matrix–vector non-negative tensor factorization (MV-NTF) describes the HSI as the sum of the outer products of the endmembers and their corresponding abundance maps. Concatenating these abundance maps in the third dimension is precisely the abundance tensor. Many subsequent studies have focused on exploiting different priors to improve the accuracy of MV-NTF. Most of them, however, explore the properties of abundance matrices or abundance maps, which is hard to fully utilize the structural similarity in abundance tensors corresponding to HSIs containing mixed materials. In this paper, we use the tensor singular value decomposition (T-SVD) to directly exploit the structural information in the abundance tensor. For this purpose, we propose a new low-rank representation by dividing the abundance tensor into a main feature tensor and a disturbance term. We characterize the low-rank property of the feature tensor after performing T-SVD and characterize the sparsity of the disturbance term. In this vein, we establish a model named abundance low-rank structure based on T-SVD (ALRSTD) and propose the solution algorithm. Experiments show that ALRSTD has better unmixing effect compared with several state-of-the-art methods, especially in the abundance estimation and the computation speed.