Derivation of efficient color-space conversion formulae for n-dimensional interpolation

Abstract

Conversions from one color-space to another are frequently used in image processing. Many conversion methods have been described mathematically, a few of which have been elevated to be international standards. However, for some conversions, purely mathematically based methods have been found wanting, and the method of measurement and interpolation has been used to produced more accurate results. Nowhere is this method more prevalently used than in dealing with colors resulting from combinations of inks or toners applied to paper. Although ink or toner colors of Cyan, Magenta, Yellow and Black are most commonly used, the use of a larger number of inks or toners that expand the color gamut is becoming more important for high fidelity color printing. The subject of this paper is the derivation of true n-dimensional linear interpolation formulae that are much more efficient than "tri-linear" or "quad-linear" and that can be used to convert from one n-dimensional color-space to another of equal, fewer or greater dimensions. The mathematical principle to be used for deriving the formulae is called axiom based induction. An interesting application of these formulae might be the conversion of a seven-dimensional color-space to a four dimensional color-space that would allow a seven-color master image to be "re-purposed" for printing by a less costly four-color method. Another application might be the use of a seven-dimensional interpolation, applied iteratively to produce a "corner turn," that could allow the direct mapping of three-dimensional color into seven-dimensional ink densities. An example of interpolation errors resulting from round-trip conversion of three to four and back to three dimensions will be given.