A method to compute the n-dimensional solid spectral angle between vectors and its use for band selection in hyperspectral data

This study presents the calculation of spectral angle beyond two endmember vectors to the n-dimensional solid spectral angle (NSSA). The calculation of the NSSA is used to characterize the local spectral shape difference among a set of endmembers, leading to a methodology for band selection based on spectral shape variations of more than two spectra. Equidistributed sequences used in the quasi-Monte Carlo method (ESMC) for numerical simulations are shown to expedite the calculation of the NSSA. We develop a band selection method using the computation of NSSA(?_n) in the context of a sliding window. By sliding the window over all bands available for varying band intervals, the calculated solid spectral angle values can capture the similarity of the endmembers over all spectral regions available and for spectral features of varying widths. By selecting a subset of spectral bands with largest solid spectral angles, a methodology can be developed to capture the most important spectral information for the separation or mapping of endmembers. We provide an example of the merits of the NSSA-ESMC method for band selection as applied to linear spectral unmixing. Specifically, we examine the endmember abundance errors resulting from the NSSA band selection method as opposed to using the full spectral dimensionality available.