A fast geometric algorithm for solving the inversion problem in spectral unmixing

A well-known problem in hyperspectral unmixing is the estimation of the abundances once the endmembers are known, while respecting the constraints on these abundances. Recently, we have presented the simplex projection unmixing algorithm for solving this inversion problem, based on the equivalence of the constrained unmixing problem and geometric projection onto a simplex. This algorithm however does not yield the correct solution in all cases, and counter-examples can be easily found. In this paper, we integrate the simplex projection algorithm with an efficient algorithm for validating candidate solutions. When a solution is rejected, the algorithm can be restarted from a better starting point, until a correct solution is found. The results of this validated simplex projection algorithm are shown to be identical to those obtained via other methods, over a wide variety of configurations. Furthermore, we show that this algorithm outperforms the fully constrained least-squares algorithm, except when the number of endmembers is high.