Gradient Type Methods for Linear Hyperspectral Unmixing (CSIAM Transactions on Applied Mathematics, 2022)
Hyperspectral unmixing (HU) plays an important role in terrain classification, agricultural monitoring, mineral recognition and quantification, and militarysurveillance. The existing model of the linear HU requires the observed vector to be alinear combination of the vertices. Due to the presence of noise, or any other perturbation source, we relax this linear constraint and penalize it to the objective function.The obtained model is solved by a sequence of gradient type steps which contain aprojection onto the simplex constraint. We propose two gradient type algorithms forthe linear HU, which can find vertices of the minimum volume simplex containingthe observed hyper-spectral vectors. When the number of given pixels is huge, thecomputational time and complexity are so large that solving HU efficiently is usuallychallenging. A key observation is that our objective function is a summation of manysimilar simple functions. Then the computational time and complexity can be reducedby selecting a small portion of data points randomly. Furthermore, a stochastic variance reduction strategy is used. Preliminary numerical results showed that our newalgorithms outperformed state-of-the-art algorithms on both synthetic and real data.
Keywords: Hyperspectral unmixing; minimum volume simplex; linear mixture model; alternating minimization; proximal gradient method; adaptive moments method; stochastic variance reduction strategy.