Page 98 - Computational Colour Science Using MATLAB

CATS 85
approximation (Wandell, 1995). However, Finlayson has shown that a diagonal
3
mapping is always possible between two real three-dimensional spaces (< ) if the
spaces are first subject to a specific linear transformation. He argues that if the
tristimulus values or cone responses are first transformed by a linear transform
into a suitable RGB space, then a diagonal transform can effectively discount the
illumination (Finlayson and Su ¨ sstrunk, 2000). The first linear transform is
sometimes called a sharp transform since it can be shown to convert the cone
responses into a set of channels whose spectral sensitivities are sharper than those
that have been measured for humans. We can therefore consider a generalized
CAT based upon Equation (6.2) where c and c refer to the tristimulus values of
1 2
the sample under the two illuminants,

c 2 ¼ M 1 DM CAT c 1 ,
CAT ð6:2Þ
and the diagonal matrix D is now composed from the white points of the two
illuminants in the RGB space. In Equation (6.2) the tristimulus values are first
subject to a linear transform (M CAT ) which converts them into RGB space and
then to a diagonal transform (D) to apply the illuminant correction, and finally a
linear transform (M 1 ) to convert back to tristimulus space. Finlayson has
CAT
derived the RGB or sharp transform as given by M ¼ M ,
CAT SHARP
2 3
1.2694 0.0988 0.1706
0.8364 1.8006 0.0357 .
4 5
M SHARP ¼
0.0297 0.0315 1.0018
The most popular CATs are consistent with Finlayson’s idea, and the
procedure of subjecting the tristimulus values of a stimulus under one illuminant
by a 363 linear transform M , followed by a diagonal transform D, and
CAT
finally followed by the inverse linear transform M 1 to return to tristimulus
CAT
space is ubiquitous in CAT research. Often researchers refer to the RGB space in
which the diagonal transform takes place as cone space, although the term is
being used loosely in this sense.
A number of CATs are currently in use and most transform the tristimulus
values into an RGB space before applying the diagonal transform. The RGB
space differs slightly between the different transforms; that is, the 363 linear
transform M CAT is different for each CAT. However, more significant differences
between the transforms are found in the way in which the elements of the
diagonal transform are computed and in which properties of the observing field
are used to compute these elements.
Hunt (1998) classified the observing field into five areas: the colour element,
the proximal field, the background, the surround and the adapting field, and
these areas are shown schematically in Figure 6.2. The colour element is the
central area of the observing field and this typically is a uniform patch of
approximately 28 of visual angle. The proximal field is the immediate