Collaborative Sparse Hyperspectral Unmixing Using $l_{0}$ Norm

Abstract

Sparse unmixing has been applied on hyperspectral imagery popularly in recent years. It assumes that every observed signature is a linear combination of just a few spectra (end-members) from a known spectral library. However, solving the sparse unmixing problem directly (using l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm to control the sparsity of solution at a low level) is NP-hard. Most related works focus on convex relaxation methods, but the sparsity and accuracy of results cannot be well guaranteed. Under these circumstances, this paper proposes a novel algorithm termed collaborative sparse hyperspectral unmixing using l0 norm (CSUnL0), which aims at solving l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> problem directly. First, it introduces a row-hard-threshold function. The row-hardthreshold function makes it possible to combine l0 norm, instead of its approximate norms, with alternating direction method of multipliers. Compared with the convex relaxation methods, the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm constraint guarantees sparser and more accurate results. Moreover, the antinoise ability of CSUnL0 also gets improved. Second, CSUnL0 uses l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm of each end-members' abundance across the whole map as a collaborative constraint, which can take advantage of the hyperspectral data's subspace property. The experimental results indicate that l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm contributes to acquiring a more sparser solution and helps CSUnL0 to enhance calculation accuracy.