A Nonconvex Framework for Sparse Unmixing Incorporating the Group Structure of the Spectral Library

Abstract

Sparse unmixing (SU) has been widely investigated for hyperspectral analysis with the aim to find the optimal subset of spectral signatures in a spectral library (known in advance) that can optimally model each pixel of the given hyperspectral image. Usually, the available spectral library organizes spectral signatures in groups. However, most existing strategies do not take full advantage of the inherent properties in the library. In this article, we design a convex framework for SU that incorporates the group structure of the spectral library. The convex framework includes two kinds of algorithms derived from either the primal or the dual form of the alternating direction method of multipliers (ADMM). Then, the convergence properties of the convex framework are established. Based on the convex framework, a novel nonconvex framework is developed for unmixing, which provides a new manner to enhance the sparsity of solution. The core of the nonconvex framework is to design a nonconvex penalty function for efficient minimization utilizing the generalized shrinkage mapping. The penalty function can be regarded as a closer approximation of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{0}$ </tex-math></inline-formula> norm. Experiments conducted on simulated and real hyperspectral data demonstrate the superiority and effectiveness of the proposed nonconvex framework in improving the unmixing performance and enhancing the sparsity of solution with respect to state-of-the-art techniques.