Spaces of Spectral Distributions and Their Natural Geometry

It is well-known that the knowledge of the natural geometry of a problem is often crucial in finding solutions. Problems involving functions on a circle are, for example, often solved using the theory of Fourier series. This is the mathematical explanation of the enormous success of the DFT, FFT and DCT-based methods. Another example is the relation between scaling properties and wavelet theory. In this paper we show that spaces of spectral distributions, like color stimuli, have a natural cone-like structure. We use the framework of the Karhunen-Loéve transform in a Hilbert space context to describe this cone-like structure and demonstrate how to compute natural coordinate systems from empirical data, like multi-spectral measurements and images. We will illustrate the theoretical findings with databases consisting of collections of multispectral measurements of color chips from color systems like Munsell, NCS and Pantone, multi-channel images of natural scenes, satellite data and daylight spectra. We will also comment on the possible application of group theoretical methods in color science based on those findings. 1. PCA for spectral distributions Color spectra are by definition non-negative functions where is the wavelength variable with values in the visible range of the spectrum. Typically takes values in the interval: . The space of such spectra is an infinite-dimensional Hilbert space and therefore one can describe these spectra by the coefficients in a series expansion: