Spectral Unmixing Cluster Validity Index for Multiple Sets of Endmembers

A hyperspectral pixel is generally composed of a relatively small number of endmembers. Several unmixing methods have been developed to enforce this concept through sparsity promotion or piece-wise convex mixing models. Piece-wise convex unmixing methods often require as parameters the number of endmembers and the number of sets of endmembers needed. However, these values are often unknown in advance and difficult to estimate. In this article, a new cluster validity index for multiple sets of endmembers is developed. The proposed index is used to evaluate spectral unmixing results and identify optimal parameter sets for piece-wise convex unmixing methods. No other conventional cluster validity index is directly applicable or theoretically well-suited for the piece-wise convex model. Specifically, we focus on addressing cases in which endmembers may or may not be located in a dense region of the data. Additionally, we focus on cases in which hyperspectral data is well distributed within a convex cluster (not exhibiting significant holes or gaps). The proposed validity index is applied to both simulated and real hyperspectral data. Results show that the proposed method consistently selects the best parameter set.