A Theory of High-Order Statistics-Based Virtual Dimensionality for Hyperspectral Imagery

Virtual dimensionality (VD) has received considerable interest in its use of specifying the number of spectrally distinct signatures present in hyperspectral data. Unfortunately, it never defines what such a signature is. For example, various targets of interest, such as anomalies and endmembers, should be considered as different types of spectrally distinct signatures and have their own different values of VD. Specifically, these targets are insignificant in terms of signal energies due to their relatively small populations. Accordingly, their contributions to second-order statistics (2OS) are rather limited. In this case, 2OS-based methods such as eigen-approaches to determine VD may not be effective in determining how many such type of signal sources as spectrally distinct signatures are. This paper develops a new theory that expands 2OS-VD theory to a high-order statistics (HOS)-based VD, called HOS-VD theory. Since there is no counterpart of the characteristic polynomial equation used to find eigenvalues in 2OS available for HOS, a direct extension is inapplicable. This paper re-invents a wheel by finding actual targets directly from the data rather than eigenvectors/singular vectors used in 2OS-VD theory which do not represent any real targets in the data. Consequently, comparing to 2OS-VD theory which can only be used to estimate the value of VD without finding real targets, the developed HOS-VD theory can accomplish both of tasks at the same time, i.e., determining the value of VD as well as finding actual targets directly from the data.