Simultaneous sparse recovery for unsupervised hyperspectral unmixing

Spectral pixels in a hyperspectral image are known to lie in a low-dimensional subspace. The Linear Mixture Model states that every spectral vector is closely represented by a linear combination of some signatures. When no prior knowledge of the representing signatures available, they must be extracted from the image data, then the abundances of each vector can be determined. The whole process is often referred to as unsupervised endmember extraction and unmixing. The Linear Mixture Model can be extended to Sparse Mixture Model R=MS + N, where not only single pixels but the whole hyperspectral image has a sparse representation using a dictionary M made of the data itself, and the abundance vectors (columns of S) are sparse at the same locations. The endmember extraction and unmixing tasks then can be done concurrently by solving for a row-sparse abundance matrix S. In this paper, we pose a convex optimization problem, then using simultaneous sparse recovery techniques to find S. This approach promise a global optimum solution for the process, rather than suboptimal solutions of iterative methods which extract endmembers one at a time. We use l1l2 norm of S to promote row-sparsity in simultaneous sparse recovery, then impose additional hyperspectral constraints to abundance vectors (such as non-negativity and sum-to-one).