Abstract
Hyperspectral unmixing is a powerful method of the remote sensing image mining that identifies the constituent materials and estimates the corresponding fractions from the mixture. We consider the application of nonnegative matrix factorization (NMF) for the mining and analysis of spectral data. In this paper, we develop two effective active set type NMF algorithms for hyperspectral unmixing. Because the factor matrices used in unmixing have sparse features, the active set strategy helps reduce the computational cost. These active set type algorithms for NMF is based on an alternating nonnegative constrained least squares (ANLS) and achieve a quadratic convergence rate under the reasonable assumptions. Finally, numerical tests demonstrate that these algorithms work well and that the function values decrease faster than those obtained with other algorithms.
Highlights
Hyperspectral data are acquired by high-resolution imaging sensors that record hundreds of continuous narrow spectral band images. e goals of hyperspectral imaging are to obtain the spectrum for each pixel in the image of the scene and identify materials or process objects
We present two active set identification techniques that help estimate the active set of the optimal solution of the alternating nonnegative constrained least squares (ANLS) problem. e active set strategy will improve the accuracy of the ANLS solution, and the computational cost is reduced
When dealing with hyperspectral data, we found that projected gradient (PG) and projected Newton (PN) are sensitive to the initial matrices U0 and V0 and the values of the parameters in the algorithm
Summary
Hyperspectral data are acquired by high-resolution imaging sensors that record hundreds of continuous narrow spectral band images. e goals of hyperspectral imaging are to obtain the spectrum for each pixel in the image of the scene and identify materials or process objects. We propose an active set type Newton method to solve nonnegative least squares problems. Because the combination coefficients in the abundance matrix are generally sparse, active set type algorithms are a good choice for solving NMF problems [37,38,39,40]. 3. Active Set Type Newton Algorithm for ANLS e columns of V are used to generate a vector vec(V), and the ANLS can be written as follows: UTU f(U, V). It follows that ANLS is a strictly convex optimization problem and after the Newton method is applied, a second order local convergent rate is obtained.
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