The complex spectral analysis method (CSAM): a new approach to the analysis of spectral information

Abstract

Algorithms for the analysis of spectral information operate almost exclusively with real-number arithmetic on real-number entities; i.e. spectra. This is certainly so in the exploitation of imaging spectrometer data. Motivated by a desire to take advantage of the power of complex mathematics for spectral analysis and an attempt to incorporate additional spectral information into traditional exploitation techniques, a method to transform real-number spectra into complex numbers is presented. The esssence of the complex spectral analysis method (CSAM)is the population of the imaginary part of a complex number representation of a spectrum; the original, untransformed spectrum forms the real part of the complex number. Here, the imaginary part is assigned the magnitude of the separation between successive points in a spectrum; its sign is equal to the sign of the slope of the line segment connecting the two points. This parameterization is chosen to pack more information on spectral shape into signatures operated on by algorithms modified to process complex data. A spectral library of mineral signatures was converted to complex spectra. A confusion matrix of complex spectral angles was constructed for the transformed library. A confusion matrix of real spectral angles between the original, untransformed library signatures was generated for comparison. The CSAM signatures provide greater spectral separability with a lower number of hits (i.e. number of confusion matrix cells with angle values less than or equal to the threshold) per threshold angle compared to the real spectral angles. CSAM separability also exceeds that of appended spectra where an extended real spectrum is created by appending the transformed information onto the end of the original, real spectrum rather than creating a complex spectrum. Numerous spectral parameterizations (and information sources) for building complex spectra are suggested as is the utility of CSAM for hyperspectral information (HSI) analysis and exploitation.