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It is known that for every selection of illumination spectra there is a coordinate system such that all coordinate vectors of these illumination spectra are located in a cone. A natural set of transformations of this cone are the Lorentz transformations. In this paper we investigate if sequences of illumination spectra can be described by one-parameter subgroups of Lorentz-transformations. We present two methods to estimate the parameters of such a curve from a set of coordinate points. We also use an optimization technique to approximate a given set of points by a one-parameter curve with a minimum approximation error. In the experimental part of the paper we investigate series of blackbody radiators and sequences of measured daylight spectra and show that one-parameter curves provide good approximations for large sequences of illumination spectra.

In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows from the Perron-Frobenius (and Krein-Rutman theory). Experiments show also that for stochastic processes with nonnegative signals the mean vector is often very similar to the first eigenvector. This is not true in general, but we first give a heuristical explanation why we can expect such a similarity. We then derive a connection between the dominance of the first eigenvalue and the similarity between the mean and the first eigenvector and show how to check the relative size of the first eigenvalue without actually computing it. In the last part of the paper we discuss the implication of theoretical results for multispectral color processing.