Hyperspectral interactions: Invariance and scaling

We present new invariance and scaling results for scale space analysis of hyperspectral data. First, we note that a hyperspectral curve can be segmented into independent regions selected by features of scale space fingerprints. These fingerprint features are persistent inflection points that precisely locate major atmospheric features that define the regions. The strength and location of hyperspectral features in one atmospheric region are independent of features in other regions; as a result, hyperspectral analysis can be simplified to a region-by-region analysis. We then generate simple scaling and invariance rules for features within such a spectral region. We show that the scale of individual features is independent of the details of feature shape and depends only on the area of the feature. Interacting features in turn exhibit a fascinating bifurcation behavior: at large separations features behave independently; at smaller separations features interact and their scales are damped; below a critical separation distance (the bifurcation point) the features nest. The scales of features above the bifurcation point, the scales of the nested features, and the location of the bifurcation point depend only on the feature areas and not on shape-associated parameters of the individual features.