Page 178 - Computational Colour Science Using MATLAB

COMPUTATIONAL COLOUR CONSTANCY AND LINEAR MODELS 165
Suppose we have a device with three colour sensors, whose spectral
responsivities are R (l), k ¼ 1,2,3. The three sensor responses for a colour signal
k
S(l) will be
X
R 1 ðlÞSðlÞ,
r 1 ¼
l
X
R 2 ðlÞSðlÞ,
r 2 ¼
ð10.1Þ
l
X
R 3 ðlÞSðlÞ,
r 3 ¼
l
which we can group into a single matrix equation:
r ¼ Ms, ð10:2Þ
where r is a 361 matrix containing the sensor responses, the rows of the 3631
matrix M are the sensor spectral responsivities and the 3161 matrix s is the
colour signal (this assumes that the spectral sensitivities of the channels and
the spectral power of the colour signal are represented at 31 wavelengths in the
visible spectrum). The key question is: Is it possible to compute s, given both r
and M? Mathematically, we can rearrange Equation (10.2) by multiplying each
side of the equation by the pseudoinverse of the matrix M so that
s ¼ M r. ð10.3Þ
þ
Unfortunately, since M is not a square matrix it is not trivial to compute the
inverse and computational procedures must be used to estimate the inverse
+
matrix M . Estimates of s from Equation (10.3) are likely to be widely
inaccurate. In simple terms, Equations (10.2) and (10.3) represent a set of three
simultaneous equations with 31 unknowns; in mathematical terms this is an
under-determined system.
To simplify the problem, imagine that the surface is viewed under an equal
energy light source so that E(l) ¼ 1 for all l. We can now write
r ¼ Mp, ð10.4Þ

where the 3161 matrix p is the spectral reflectance. Although it may still appear
that three sensor responses are insufficient to estimate p we know there are strong
constraints on the variability of the spectral reflectances of surfaces (Maloney,
1986). [There are also known constraints (Judd et al., 1964) on the variability of
natural daylight.]
The natural constraints on surfaces and lights are usefully illustrated through
their representation by linear models. Typically, a set of basis functions B (where
j
j is 1, . . ., n) are defined such that, for example, each reflectance spectrum P is
i
defined by a linear sum of the basis functions,
P i ¼ B j a i,j , ð10.5Þ