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We need one more step to turn the three SL functions into a convenient
basis for a color space. Since I consider the and
functions to be color opponent functions, I would like each function's two
phases to cancel each other out in response to ``white'' (flat spectrum)
light, i.e., to give a zero response. This corresponds to perceptual
experience: white light does not seem to contain any color, either red,
green, blue, yellow (the four primary colors), or any other one. I want
the color model to reflect this directly. From
Figure
(p.
), we can see
that the phases of the opponent functions are not equal in
size.
I therefore apply a normalization step as follows:
where is any of the six SOL response functions as discussed in the
previous section,
is its normalized version,
is relative
radiance,
is wavelength as before,
is the sigmoidification
function as in equations
ff (p.
ff),
is an
energy equalization constant, and
is the integral over
positive values of
only.
The constant
is discussed
in more detail in Section
(p.
). This
normalization makes sure that the color-space dimensions are ``square'',
since the maximum response of each function is obtained in response to a
stimulus like
where is the zero-crossing wavelength of the function, i.e., the
wavelength at which the response changes from inhibition to
excitation.
The normalization step makes sure that these maximum responses
are the same for all functions. When the normalized functions are added
pairwise, we get the normalized versions of the SL basis functions that I
will refer to as the SLN functions :
with symbols as above, and and
representing the
maximum negative and positive response, respectively (see
Section
p.
). The normalization
applied to the L and D functions is slightly different because the --L (or
D) function is an inhibitor only, and has a much more limited range than
the other five. The overall scaling factors
,
, and
are
set such that the maximum response of the
function is 2, and the
maximum for each opponent function is 1. The response to an equal-energy
``white'' spectrum (defined by
) must therefore be close
to
. It is not exactly equal to that because of the
effect of the inhibitory phases of the opponent functions. The gray axis of
the color space (the path defined by the coordinates of equal-energy
spectra of increasing radiance) must therefore be close to the line through
(black) and
(white). Later I will examine the shape of the gray axis in more
detail. Figure
shows 3D plots of the resulting
functions.