A Fully Constrained Linear Spectral Unmixing Algorithm Based on Distance Geometry

Under the linear spectral mixture model, hyperspectral unmixing can be considered as a convex geometry problem, in which the endmembers are located in the vertices of simplex enclosing the hyperspectral data set and the barycentric coordinates of observation pixels with respect to the simplex correspond to the abundances of endmembers. Based on distance geometry theory, in this paper we propose a new approach for abundance estimation of mixed pixels in hyperspectral images. With the endmember signatures, which is known a priori or can be obtained from the endmember extraction algorithms, the proposed method automatically estimates the abundances of endmembers at each pixel using convex geometry concepts and distance geometry constraints. In the algorithm, denoting the pairwise distances with Cayley-Menger matrix makes it easy to calculate the barycentric coordinates of the observation pixels. Another characteristic of this algorithm is that the optimal estimated points of observation pixels as well as the least distortion in geometric structure of original data set can be obtained with the distance geometry constraint. Simultaneously, the use of barycenter of simplex builds an accurate and efficient method to estimate endmembers with zero abundance and, as a result, the subsimplex containing the estimated points is obtained. A comparative study and analysis based on Monte Carlo simulations and real data experiments is conducted among the proposed algorithm and three state-of-the-art algorithms: fully constrained least squares (FCLS), FCLS computed using constrained sparse unmixing by variable splitting and augmented Lagrangian, and simplex-projection unmixing (SPU). The experimental results show that the proposed algorithm always provides the best unmixing accuracy and when the number of endmembers is not very large the algorithm has a lower computational complexity.