Spectral Unmixing via Compressive Sensing

The recently developed theory of compressive sensing (CS) exhibits enormous potentials in signal recovery. In this paper, we investigate its application on spectral unmixing, which appears in hyperspectral data analysis and is usually based on a linear mixture model (LMM) that assumes that a mixed pixel is a linear combination of a set of pure spectral signatures (called endmembers) weighted by their corresponding abundances. Unlike the classical LMM that is a compact representation, we first extend it to a sparse representation (SR) by using a redundant and known endmember set instead of the complete one. Then, the SR model is multiplied by a random Gaussian measurement matrix, so spectral unmixing is casted in the framework of CS. Finally, the ℓ
₁
-minimization algorithms are used to recover the nonnegative abundances by solving the SR model and our proposed model named CS+SR, respectively. Experimental results on both simulated and real hyperspectral data demonstrate that the CS+SR model, formed by multiplying a random Gaussian matrix on the SR model, can improve, at least in the sense of probability, the ability of the ℓ
₁
-minimization algorithms for recovering the nonnegative sparse abundances.