Geometric estimation of the inherent dimensionality of single and multi-material clusters in hyperspectral imagery

The inherent dimensionality of a spectral image can be estimated in a number of ways, primarily based on statistical measures of the data cloud in the hyperspace. Methods using the eigenvalues from Principal Components Analysis, a Minimum Noise Fraction transformation, or the Virtual Dimensionality algorithm are widely applied to entire hyperspectral images typically with the goal of reducing the overall dimensionality of an image. Additionally, in complex scenes containing non-natural materials, the lack of multivariate normality of the data set implies that a statistically-based estimation is less than optimal. However, it is desirable to understand the dimensionality of individual components and small multi-material clusters within a hyperspectral scene, as there is no a priori reason to expect all distinct material classes in the scene to have the same inherent dimensionality. For this reason, the inherent dimensionality may be useful as an indicator of the presense of manmade or natural materials within small subsets of the image. Here, a geometric approach is developed based on the spatially local estimation of dimensionality in the native data hyperspace. It will be shown that the dimensionality of a collection of data points in the full n dimensions ( where n is the number of spectral channels measured) can be estimated by calculating the change in point density as a function of distance in the full n dimensional hyperspace. Simple simulated examples to demonstrate the concept will be shown, as well as applications to real hyperspectral imagery collected with the HyMAP sensor.