Non-Euclidean structures in PCA-based descriptions of multispectral databases

• Traditional color science is based on human color vision. For many applications (for example remote sensing, medical imaging, industrial inspection) where spectral properties of the radiation are of interest, human color vision based systems are too restrictive. Therefore multispectral approaches are becoming more and more popular.In such applications, the investigation of spectral data sets requires different color systems. These systems are often built with a small number of color channels that are not necessarily related to human color perception. Of those, systems based on Principal Component Analysis (PCA) are widely used.Considering spectra as non-negative signals, we show that PCA based systems lead to conical structures in the coordinate space. Several geometrical properties of this coordinate space such as convexity, boundaries, and different partitions are also discussed. Spaces with conical structures arc usually seen as models of non-Euclidean geometries. The special case of three dimensional conical structures is often investigated with the help of the SU(1,1) or SO(2,1) groups.Motivated by these observations, we introduce a new framework to investigate illumination spectra. In this framework we start by using a PCA based system to describe spectra by their coordinates in a conical vector space. In this conical space, we apply group theoretical tools to investigate sequences of coordinate vectors. We describe these sequences of spectra coordinate vectors with special continuous subgroups of SU(1,1) acting on this non-Euclidean space. Then we show how to compute the one-parameter subgroup of SU(1,1) from a given set of illumination spectra.In our experiments we investigate the following spectra sequences: black-body radiation spectra, daylight/twilight spectra sequences measured in Norrköping, Sweden and in Granada, Spain, and spectra generated by the SMARTS2 simulation program. The results show that important properties of sequences of illumination spectra can be modelled in this framework.Finally, we illustrate the usefulness of the framework by deriving illumination invariants and an efficient visualization implementation.

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