Non uniform colour space

Although the CIE system is based upon human visual responses, it is of little use if unmodified, to specify colour tolerances. Equal visual steps in colour, in any of the three dimensions or in any combination of them do not correspond with equal numerical steps in chromaticity coordinates and CIE lightness values. To be of any use the CIE system must be transformed so that equal numerical steps in the system are visually equal.

It is not possible to do this exactly as CIE colour space is Euclidean in nature whereas visual colour space is of a much more complex Riemannian form. What we are attempting is to embed a 3-dimensional Riemannian space into a 3-dimensional Euclidean space. A geometric theorem states that if 'n' is the number of dimensions of the Riemannian space the required number of dimensions of the Euclidean space 'm' embedding the Riemannian space may be as high as:-

A famous study by David MacAdam with Nutting as his observer concerning visual perception limits was carried out in two dimensions of colour space only - hue and chroma with the lightness dimension (value) being kept at about 5. The 2-dimensional visual Riemannian space required 3 dimensions of Euclidean or colorimetric space for its full description. As 3 dimensions of visual colour space would require 6 dimensions of Euclidean space and 3-dimensional system transposing from one colour space to the other can only be approximate.