Hyperspectral Data Reconstruction Combining Spatial and Spectral Sparsity

This report introduces a novel sparse decomposition model for hyperspectral image reconstruction. The model integrates two well-known sparse structures of hyperspectral images: a small set of signature spectral vectors span all spectral vectors (one at each pixel), and like a standard image, a hyperspectral image is spatially redundant. In our model, a threedimensional hyperspectral cube X is first decomposed into a small number of endmembers by X = Hβ + n, where H is the endmember dictionary, β contains the coefficients, and n are errors and noise. Then β, which is a 3D cube with the same spatial dimensions as X , is further decomposed into overlapping cubelets {βi}, which are sparsely represented by a common dictionary D, i.e., βi = Dαi where αi is a set of sparse coefficients. This model not only exploits spectral sparsity to the original hyperspectral cube X to the smaller cube β but also applies latest image sparse representation techniques to β. Given a corrupted hyperspectral cube with noise and missing voxels, our method reconstructs the cube by learning H , D, and αi’s from the data itself. These parameters are statistically modeled such that αi’s are sparse and Bayesian inferences have closed-form formulas and are thus easy to compute. Numerical simulations were performed on AVIRIS images. We show that merely 5% randomly selected voxels are enough for the proposed method to returned state-of-theart reconstruction results.