Constrained Nonnegative Matrix Factorization for Hyperspectral Unmixing

Hyperspectral unmixing is a process to identify the constituent materials and estimate the corresponding fractions from the mixture. During the last few years, nonnegative matrix factorization (NMF), as a suitable candidate for the linear spectral mixture model, has been applied to unmix hyperspectral data. Unfortunately, the local minima caused by the nonconvexity of the objective function makes the solution nonunique, thus only the nonnegativity constraint is not sufficient enough to lead to a well-defined problem. Therefore, in this paper, two inherent characteristics of hyperspectral data, piecewise smoothness (both temporal and spatial) of spectral data and sparseness of abundance fraction of every material, are introduced to NMF. The adaptive potential function from discontinuity adaptive Markov random field model is used to describe the smoothness constraint while preserving discontinuities in spectral data. At the same time, two NMF algorithms, nonsmooth NMF and NMF with sparseness constraint, are used to quantify the degree of sparseness of material abundances. A gradient-based optimization algorithm is presented, and the monotonic convergence of the algorithm is proved. Three important facts are exploited in our method: First, both the spectra and abundances are nonnegative; second, the variation of the material spectra and abundance images is piecewise smooth in wavelength and spatial spaces, respectively; third, the abundance distribution of each material is almost sparse in the scene. Experiments using synthetic and real data demonstrate that the proposed algorithm provides an effective unsupervised technique for hyperspectral unmixing.