Adaptive ${L}_{\bf 1/2}$ Sparsity-Constrained NMF With Half-Thresholding Algorithm for Hyperspectral Unmixing

Abstract

Hyperspectral unmixing is an important step for material classification and recognition. Recently, nonnegative matrix factorization (NMF) has been utilized to unmix the hyperspectal imagery due to the advantage that it needs no assumption for the presence of pure pixels and can determine the endmembers and abundances simultaneously. In order to improve its unmixing performance, sparsity-constrained NMF has been demonstrated to be an efficient approach. The very recent study on L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularization theory in compressive sensing (CS) and sparsity-constrained NMF show that the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularizer can yield stronger sparsity-promoting solutions than L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> or L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> regularizer. However, the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularization can result in a complex nonconvex optimization problem that is hard to solve efficiently. In this paper, we propose a fast and efficient adaptive half-thresholding algorithm for hyperspectral unmixing based on L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> sparsity-constrained NMF. In the proposed algorithm, iterative half-thresholding procedure that has been proved to be an efficient method for solving L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularization problem in CS is hybridized with the multiplicative update rule of standard NMF to deal with the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> sparsity-constrained NMF, which can give sparser and better unmixing results than the alternative algorithms. Furthermore, the data sparsity information can be incorporated into the algorithm to adaptively adjust the regularization parameter of the model to improve algorithm performance and usability. The effectiveness of proposed algorithm was demonstrated by comparing with the representative algorithms on synthetic and real hyperspectral data.