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.
Blind Spectral Unmixing Based on Sparse Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is a widely used method for blind spectral unmixing (SU), which aims at obtaining the endmembers and corresponding fractional abundances, knowing only the collected mixing spectral data. It is noted that the abundance may be sparse (i.e., the endmembers may be with sparse distributions) and sparse NMF tends to lead to a unique result, so it is intuitive and meaningful to constrain NMF with sparseness for solving SU. However, due to the abundance sum-to-one constraint in SU, the traditional sparseness measured by L0/L1-norm is not an effective constraint any more. A novel measure (termed as S-measure) of sparseness using higher order norms of the signal vector is proposed in this paper. It features the physical significance. By using the S-measure constraint (SMC), a gradient-based sparse NMF algorithm (termed as NMF-SMC) is proposed for solving the SU problem, where the learning rate is adaptively selected, and the endmembers and abundances are simultaneously estimated. In the proposed NMF-SMC, there is no pure index assumption and no need to know the exact sparseness degree of the abundance in prior. Yet, it does not require the preprocessing of dimension reduction in which some useful information may be lost. Experiments based on synthetic mixtures and real-world images collected by AVIRIS and HYDICE sensors are performed to evaluate the validity of the proposed method.
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