Colour Theory for Painters

Abstract: Using an inversion of the Munsell renotation, this paper calculates that a colour's shadow series is approximately a straight line in the Munsell system. The line starts at the colour's Munsell specification and ends about one value step below N0, on the neutral axis. The colour's hue in shadow shifts slightly towards the yellow part of the spectrum. The calculations suggest that ideal black belongs at about N(-1) in the Munsell system, rather than at N0, if equality of perceptual steps is to be maintained. Similarly, ideal white should be slightly lighter than N10.

Here is the set of 182 shadow series, along with approximating least squares lines, plotted in terms of their values and chromas, but disregarding hue, in the Munsell system (Fig. 4 of the article):



Note that the bulk of the extrapolated shadow series cross the neutral axis between 0 and 2 value steps below N0. The data used to generate this figure is contained in the file ShadowSeriesInMunsellSystem.txt, which others are free to use for further research.

Abstract: The 1943 Munsell renotation includes a table that converts 2,734 Munsell specifications into xyY coordinates, along with a graphical interpolation method, and a graphical inversion method, that converts xyY coordinates back into Munsell specifications. The current paper presents open-source computer code, running in Matlab or Octave, that both interpolates and inverts the Munsell renotation automatically. The steps in both algorithms are described in detail. Like previous inversion algorithms, it relies on interpolations between entries in the 1943 table. For colours near the MacAdam limits, the inversion also requires extrapolations beyond the 1943 entries. The outputs of the current implementation do not differ significantly from the outputs of other inversion algorithms. The main distinguishing feature of the current algorithm is that both the algorithm and code implementation are publicly available.

This second article is of more interest to colour scientists than to painters. An Inverse Table for the Munsell Renotation gives a table of inverse renotation values. The main routine for the code implementation is xyYtoMunsell.m, which is written in Matlab/Octave. This routine runs in the ColorLab project; supporting routines and more information are available below.

Abstract: This paper demonstrates that the CIE XYZ colour solid is a zonoid. An approximating zonohedral colour solid is constructed explicitly from a set of generating vectors, which are integrals of colour-matching functions over narrow intervals of the visible spectrum. The zonohedral approach yields an intuitive, constructive proof of the Optimal Colour Theorem: the reflectance function of an optimal colour takes on only the values 0 or 1, with at most two transition wavelengths. In addition, zonohedral techniques can simplify computations: for example, optimal colours can be found without calculating transition wavelengths. Finally, zonohedra provide a simple, unified approach to colour space, and eliminate much of the confusion arising from chromaticity diagrams.

Abstract: In a display with more than three primaries (called a multi-primary display), a color can be expressed as multiple combinations (called control sequences) of primaries. This paper presents an algorithm for assigning control sequences, that preserves current assignments when further primaries are added. We call these control sequences extensible. It is shown that the gamut of any number of primaries is a zonohedron, which can be dissected into parallelepipeds. Control sequences are assigned within each parallelepiped. The current parallelepipeds remain when more primaries are added, so the current assignments are preserved. Multi-primary displays can also cause unwanted metamerism, and make continuous color scales appear discontinuous. The algorithm avoids these problems. When viewed through natural filters, such as yellowed ocular lenses, multi-primary displays can sometimes make two different colors appear identical. If the primaries satisfy the Binet-Cauchy criterion, which is always the case when all primaries are monochromatic, then these spurious matches are avoided.

Abstract: A control sequence gives the intensities of the primaries for a pixel of a display device. The display gamut, i.e. the set of all the colours that a display can produce, is a zonohedral subset of CIE XYZ space, and contains both boundary and interior colours. Displays with four primaries or more exhibit metamerism, in which different control sequences produce colours that appear identical to an observer. This paper shows mathematically that, provided no three primaries are linearly dependent, metamerism can only occur for interior colours. When there are four or more primaries, metamers can always be found for interior colours. A colour on the gamut boundary, by contrast, is only produced by a unique control sequence. The proof used for displays can be extended to object-colour solids, to show that optimal colours, which are on the boundary of an object-colour solid, have unique reflectance functions.

Abstract: An object colour's CIE $XYZ$ coordinates can change when it is viewed under different illuminants. The set of $XYZ$ coordinates for all object colours, which is called the object-colour solid, likewise varies under different illuminants. This paper shows that, despite these changes, some properties are invariant under illuminant transformations. In particular, as long as the illuminant is nowhere zero in the visible spectrum, optimal colours take the same Schr\"odinger form, and no two optimal colours are metameric. Furthermore, all object-colour solids have the same shape at the origin: they all fit perfectly into the convex cone (which we will call the spectrum cone) generated by the spectrum locus. The spectrum cone, itself, does not vary when the illuminant changes. The object-colour solid for one illuminant can be transformed into the solid for another illuminant, by an easily visualized sequence of expansions and contractions along irregular rings, called zones.

web counter