Minimum Dispersion Constrained Nonnegative Matrix Factorization to Unmix Hyperspectral Data

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper considers the problem of unsupervised spectral unmixing for hyperspectral image analysis. Each observed pixel is assumed to be a noisy linear mixture of pure material spectra, namely, <emphasis emphasistype="boldital">endmembers</emphasis>. The mixing coefficients, usually called <emphasis emphasistype="boldital">abundances</emphasis>, are constrained to positive and summed to unity. The proposed unmixing approach is based on the nonnegative matrix factorization (NMF) framework, which considers the physical constraints of the problem, including the positivity of the endmember spectra and abundances. However, the basic NMF formulation has degenerated solutions and suffers from nonconvexity limitations. We consider here a regularization function, called <emphasis emphasistype="boldital">dispersion</emphasis>, which favors the solution such that the endmember spectra have minimum variances. Such a solution encourages the recovered spectra to be flat, preserving the possible spectral singularities (peaks and sharp variations). The regularized criterion is minimized with a projected gradient (PG) scheme, and we propose a new step-size estimation technique to fasten the PG convergence. The derived algorithm is called MiniDisCo, for minimum dispersion constrained NMF. We experimentally compare MiniDisCo with the recently proposed algorithm. It is shown to be particularly robust to the presence of flat spectra, to a possible <emphasis emphasistype="boldital">a priori</emphasis> overestimation of the number of endmembers, or if the amount of observed spectral pixels is low. In addition, experiments show that the considered regularization correctly overcomes the degeneracy and nonconvexity problems, leading to satisfactory unmixing accuracy. We include a comparative analysis of a real-world scene. </para>