Hyperspectral Unmixing via $L_{1/2}$ Sparsity-Constrained Nonnegative Matrix Factorization

Hyperspectral unmixing is a crucial preprocessing step for material classification and recognition. In the last decade, nonnegative matrix factorization (NMF) and its extensions have been intensively studied to unmix hyperspectral imagery and recover the material end-members. As an important constraint for NMF, sparsity has been modeled making use of the
$L_{1}$
regularizer. Unfortunately, the
$L_{1}$
regularizer cannot enforce further sparsity when the full additivity constraint of material abundances is used, hence limiting the practical efficacy of NMF methods in hyperspectral unmixing. In this paper, we extend the NMF method by incorporating the
$L_{1/2}$
sparsity constraint, which we name
$L_{1/2}$
-NMF. The
$L_{1/2}$
regularizer not only induces sparsity but is also a better choice among
$L_{q}(0 < q < 1)$
regularizers. We propose an iterative estimation algorithm for
$L_{1/2}$
-NMF, which provides sparser and more accurate results than those delivered using the
$L_{1}$
norm. We illustrate the utility of our method on synthetic and real hyperspectral data and compare our results to those yielded by other state-of-the-art methods.