An analysis of the probability distribution of spectral angle and Euclidean distance in hyperspectral remote sensing using microspectroscopy

Determining the probability distribution of hyperspectral imagery (HSI) data and of the results of algorithms applied to those data, is critical to understanding algorithm performance and for establishing performance metrics such as probability of detection, false alarm rate, and minimum detectable and identifiable quantities. The results of analyses of visible/near-infrared (VNIR; 400 nm to 900 nm) HSI microscopy data of small fragments (~1.25 cm in size) of minerals are presented. HSI microscopy, also known as microspectroscopy, is the acquisition of HSI data cubes of fields of view ranging from centimeters to millimeters in size. It is imaging spectrometry but at a small spatial scale. With HSI microspectroscopy, several thousand spectral signatures may be easily acquired of individual target materials—samples of which may be quite small. With such data, probability distributions may be very precisely determined. For faceted/irregularly shaped samples and mixtures (checkerboard, intimate, or microscopic), HSI microscopy data readily facilitate a detailed assessment of the contribution of the materials, their morphology, spectral mixing interactions, radiative transfer processes, view/illumination geometry contributions, etc., to the observed probability distribution(s) of the HSI data and of algorithm output. Here, spectral angle, the individual components of spectral angle (e.g., the inner product or numerator of the spectral angle equation), Euclidean distance, and L1 norm values are calculated. Regions of interest (ROI) on the fragments are easily defined that contain thousands of spectra far from the fragments' edges though translucency sometimes remains a factor impacting spectral signatures. The aforementioned metrics are derived for the spectra in an ROI of individual mineral fragments; across ROIs of different minerals; and with an ROI of an inert background. The resulting probability distributions of the various populations of the metrics are decidedly non-Gaussian though the precise probability distribution is difficult to determine. Spectral angle values appear to be most closely related to beta distributed. The HSI microscopy method is described as are the results of the analyses applied to the data of the mineral fragments. The interpretation of the microspectroscopy data is considered within the ongoing investigation into determining how the spectral variability on the ~10 micrometer spatial scale relates to the spectral variability on larger scales such as those acquired by airborne remote sensing systems.