Subspace matching pursuit for sparse unmixing of hyperspectral data

Sparse unmixing assumes that each mixed pixel in the hyperspectral image can be expressed as a linear combination of only a few spectra (endmembers) in a spectral library, known a priori. It then aims at estimating the fractional abundances of these endmembers in the scene. Unfortunately, because of the usually high correlation of the spectral library, the sparse unmixing problem still remains a great challenge. Moreover, most related work focuses on the l₁ convex relaxation methods, and little attention has been paid to the use of simultaneous sparse representation via greedy algorithms (GAs) (SGA) for sparse unmixing. SGA has advantages such as that it can get an approximate solution for the l₀ problem directly without smoothing the penalty term in a low computational complexity as well as exploit the spatial information of the hyperspectral data. Thus, it is necessary to explore the potential of using such algorithms for sparse unmixing. Inspired by the existing SGA methods, this paper presents a novel GA termed subspace matching pursuit (SMP) for sparse unmixing of hyperspectral data. SMP makes use of the low-degree mixed pixels in the hyperspectral image to iteratively find a subspace to reconstruct the hyperspectral data. It is proved that, under certain conditions, SMP can recover the optimal endmembers from the spectral library. Moreover, SMP can serve as a dictionary pruning algorithm. Thus, it can boost other sparse unmixing algorithms, making them more accurate and time efficient. Experimental results on both synthetic and real data demonstrate the efficacy of the proposed algorithm.