Spectral Gamut Mapping and Gamut Concavity

A spectral gamut-mapping algorithm is introduced that works well for printers with a large number of inks. It finds the best mapping onto the convex hull of the printer spectral gamut while preserving color defined in CIE XYZ as much as possible. The technique employs a non-negative least-square fit. Since the gamut-mapping algorithm depends on the common assumption that the gamut is convex, an experimental study of the degree of gamut concavity is conducted. It finds that there is a significant amount of concavity, and that that the degree does not appear to change much as the number of inks is increased. Finally, the performance of the gamut-mapping algorithm and gamut coverage in spectral space is compared for 3-, 4-, 5and 6-ink printers using both synthetic ink models and real ink data. Introduction In comparison to standard color printing, spectral printing aims to reproduce a given reflectance spectrum rather than produce a metameric reflectance spectrum that simply matches a given color. Spectral printing aims to reduce a problem that can arise in metameric color printing which is that the reproduced color may match under one illuminant, but not match well under some other illuminant. Clearly, if the printed output reflectance matches the input reflectance, the printed color will match the input color under all illuminants. Spectral printing requires a significantly larger number of inks than the standard CMYK ones, but this increases the computational complexity of printing algorithms in terms of both time and space. In particular, standard gamut-mapping algorithms map colors within a 3-dimensional space. Generally, their computational complexity increases rapidly with dimension, so that they become intractable for the gamut-mapping of spectra represented in, say, 11 dimensions. For example, a gamutmapping algorithm that relies on the computation of the convex hull of the measured gamut will not work since computing a ddimensional convex hull of n points requires order O(n**floor(d/2)+1) operations. Bakke et al. [12] address this problem by reducing the dimensionality via principal components analysis and then computing up the convex hull in up to 8 dimensions. The first part of this paper introduces a spectral gamutmapping algorithm that projects an out-of-gamut spectrum onto the printer gamut's convex hull without having to calculate the hull explicitly. It is based on a non-negative least-squares solution to a set of constraints that Finlayson et. al. [1] originally proposed as part of a color constancy method. The computational requirements of the proposed gamut-mapping algorithm are reasonable for the higher number of dimensions required. A modification to the algorithm is then presented that maps out-ofgamut spectra to the closest spectra on the gamut's convex hull subject to the constraint that it preserve XYZ tristimulus values. The second part of the paper evaluates the validity of the assumption that the printer gamut is convex. To what extent is the printer gamut concave, and does the degree of concavity vary as a function of the number of inks? Experimental results on the concavity of the gamuts of 3 through 6 inks are presented. Spectral Gamut Mapping Using Non-Negative Least Squares For a point • inside a convex gamut, it can be represented as a convex combination of other points, qi, within the gamut: • = • •iqi, •i •0, ••i = 1 (1) The •i’s are weights, and the restrictions on the weights ensure that • does not lie outside the convex hull of the qi. For a point • outside a convex gamut, we can find the closest point to • lying on the convex hull of the gamut by finding •i minimizing the distance e: e = | • • •iqi |2, •i •0, ••i =1 (2) Finlayson et. al. [1] showed that (2) can be rewritten to include a weight W as an extra dimension in the input data, and that the revised equations can then be solved by standard nonnegative least squares. Their derivation is as follows.