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To visualize the general ``shape'' of the NPP color space, I have computed
the shape of the Optimal Color Stimuli (OCS) Surface in NPP space. We can
represent all physically possible surface-spectral reflectance
functions in a solid known as the Object Color Solid. The surface
of this solid represents the limit of physically realizable surface colors,
known as Optimal Color Stimuli, and can be generated by computing the
response of a given set of sensors to a continuum of two kinds of spectra:
The spectral reflectanceI have used the following differentiable approximations to these two types of reflectance functions:is either zero or unity, and in moving through the visible spectrum, there are generally not more than two transitions between these values. Optimal color stimuli are imaginary stimuli in the sense that no actual object surfaces have reflectance curves with abrupt transitions of this kind. However, they are of considerable interest because they represent limiting cases of all (non-fluorescent) object-color stimuli. [
] Two types of curves must be distinguished; the first has zero reflectance (or transmittance) at wavelengths
and
, the second at wavelengths
. [Wyszecki \& Stiles 1982][p.181 ff.]
where is wavelength in nm as usual,
is the width of the
``gap'' in nm, and
is the start of the ``gap'' in nm. Some typical
examples of reflectance spectra generated by these functions are shown in
Figure
.
If we assume a flat-spectrum (white) light source, defined by
, the light reaching the sensors has a spectrum
identical to the reflectance function,
and
we can compute Optimal Color Stimulus coordinates for linearly
responding sensors as follows:
where is a list of expressions with index variable
ranging from 1 to N, N is the number of dimensions (basis functions) of the
color space,
is the spectral sensitivity of each of the
basis functions, and
and
represent the lower and
upper limit of sensitivity for the sensors used, typically in the
neighborhood of 300 and 800 for the human visual system, respectively. By
varying
and
over the visible wavelength range, and
plotting the resulting points
and
, we can compute the shape of
the OCS surface. It is made up of two ``halves'' that fit together like
clam shells, corresponding to the set of points
and
.
Now we need to choose sets of basis functions . If we
choose the standard CIE XYZ functions (Section
p.
), we get the result shown in
Figure
. This is the typical ``torpedo-like shape'' that
[Wyszecki \& Stiles 1982] refer to. For the actual computations involved in
creating figures
ff, I used a computationally more
efficient technique than suggested by equations
ff, making
use of the special properties of the functions
and
and
using a list of partial integrals as a kind of cache.
The surface color in
Figure
is (necessarily) only an approximation, derived as
follows:
where are
normalized RGB coordinates,
is a
limiting function serving to limit RGB coordinates to the gamut of the
display device,
is a linear transform from XYZ to ``typical
computer monitor'' RGB coordinates such as the ones given in
[Rogers 1985] or [Hill 1990],
are the CIE XYZ
coordinates computed with equations
ff (p.
),
and
is the set of RGB coordinates corresponding to a maximum
radiance flat spectrum (white). The latter is used as a normalization
factor for display purposes, which is basically Von Kries
adaptation [Wyszecki \& Stiles 1982]. It is clear from these equations that
the displayed color has to be approximate, because of the limitations of
the gamut of the display device, the ``typical'' transform used, and the
inability to control for such things as gamma correction. Nevertheless, the
rendered color is a reasonable indication of the ``real'' color
corresponding to that particular point on the OCS surface.
It is interesting to note that only a relatively small volume of the XYZ
cube actually corresponds to physically realizable colors. The same is true
of course if we compute the OCS surface in RGB coordinates, but the
intersection of the volume enclosed by the OCS surface with the
all-positive (i.e., displayable) RGB subspace is even smaller than in the
XYZ case (Figure ).
Choosing different basis functions results in different shapes of the OCS surface. The computation for the SL functions is somewhat more involved then for the CIE XYZ functions, because of the nonlinearities. The equations for computing points on the surface are
where symbols are as in equations ff (p.
),
being the sigmoid component, and
being the linear component
of the basis functions. Since the value of the sigmoid function is a
non-linear function of its input, there is an issue here with respect to
scaling the linear responses that was not relevant for the linear CIE XYZ
functions. We have to determine values for the constants
. The method
I have used is essentially scaling with respect to the equal-energy (EE) or
``white'' response. The argument goes as follows. We want to scale the
brightness (Br) dimension to range from perceived black to white, without
affecting the adaptive state of the visual system. Once I have determined a
scaling factor for the Br dimension I will use it to scale the other
dimensions as well. It is justifiable to use the same scaling factor for
all color-space dimensions, since the functions I derived in
sections
ff (p.
ff) preserve the
ratios of responses among the cell types involved. Assuming (as usual)
that the perception of ``white'' arises from the viewing of a stimulus with
a flat equal-energy spectrum
, we can normalize
responses with respect to this type of stimulus.
Since ``white'' is at the top of the Br dimension (always within
the adaptive range), a flat spectrum ``white'' (maximum relative radiance)
stimulus must result in the maximum response for the Br function, which is
the function value at the wavelength of maximum response and at maximum
relative radiance:
The response of the linear Br function to a flat-spectrum ``white''
stimulus is given by
And the desired scaling factor is then just the quotient of the two:
The normalized response of a linear function to a stimulus
is then simply
The final SL normalized function value is
as discussed above. Applying this normalization to the SL functions and
computing the OCS surface results in the shape shown in
Figure .
For the SLN functions the normalization process is as described for the SL
functions, except the scaling factor is computed using the function,
and each of the six response functions is normalized individually before
being combined into 3 dimensions. The resulting OCS surface is shown in
Figure
.
Since the white point is slightly off-center (i.e., not on the
axis) in the SLN space as shown above, I will
apply a final linear transform to the color-space coordinates to compensate
for this. Since the SLN coordinates are in the range
(Section
p.
), the transform we want is given by
where and
are the
and
coordinates of
the white point, respectively. I will refer to the transformed coordinates
as the
coordinates, and the corresponding color space as the NPP
color space. The OCS surface in
coordinates is shown in
Figure
.
The gray axis is now perfectly vertical, the coordinates of black being
, and white being
. The
extrema of the OCS surface coordinates have now changed from
to
, which makes the complete color stimuli solid fit
inside a unit magnitude cube.
For comparative purposes, the OCS surface is represented in CIE L*a*b*
coordinates (Figure ), which represents an attempt to
create a perceptually equidistant color space for reflected light
(Section
p.
). The L*a*b* model is
based on psychophysical principles only, however, not on neurophysiological
data as the NPP space is.
Although the order of hues around the NPP and L*a*b* spaces is the same,
there are marked differences in the overall shapes (e.g. the sharp
protrusion of the L*a*b* space in the blue region is absent in the NPP
space), and the relative positions and areas that certain colors occupy on
the respective surfaces. These differences and their implications for theories of color
perception remain to be investigated in detail, but that is outside of the
scope of this dissertation. In the next section we will look at
similarities between the NPP and other spaces to the Munsell system,
another often-used psychological color-order system.
In [Wyszecki \& Stiles 1982], only black and white hand-drawn approximations of the OCS shape (in different color spaces, but not NPP of course) are shown, and I am not aware of any attempts to define the complete shape and its surface color analytically as I have done. Later, I will use the OCS surface as a frame of reference to investigate the distribution of basic color categories. It is well suited for that purpose, since it represents the limit of all physically possible surface reflectances (giving rise to color perception when viewed under an appropriate light source), and I can represent it in the neurophysiologically-based NPP space.