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7.3. Colorimetry 211
Figure 7.7. The spectrum for a ripe brown banana under white light [409].
and color matching experiments were performed using them. In color
matching, three colored lights are projected on a white screen so that
their colors add together and form a patch. A test color to match is
projected next to this patch. The test color patch is of a single wave-
length from the spectrum. The observer can then change the three col-
ored lights using knobs calibrated to a range weighted [−1, 1] until the
test color is matched. A negative weight is needed to match some test
colors, and such a weight means that the corresponding light is added in-
stead to the wavelength’s test color patch. One set of test results for three
Figure 7.8. The r, g,andb 2-degree color matching curves, from Stiles & Burch (1955).
Values are shown in terms of energy, not the [−1, 1] knob range. These color matching
curves are not to be confused with the spectral distributions of the light sources used in
the color matching experiment.
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212 7. Advanced Shading
lights, called r, g,andb, is shown in Figure 7.8. The lights were almost
monochromatic, with the energy distribution of each narrowly clustered
around one of the following wavelengths: r = 645 nm, g = 526 nm, and
b = 444 nm.
What these curves give is a way to convert a spectrum to three values.
Given a single wavelength of light, the three colored light settings can be
read off the graph, the knobs set, and lighting conditions created that will
give an identical sensation from both patches of light on the screen. Given
an arbitrary spectrum, these curves can be multiplied by the spectrum and
the area under each resulting curve (i.e., the integral) gives the relative
amounts to set the colored lights to match the perceived color produced by
the spectrum. Very different spectra can resolve to the same three weights;
in other words, they look the same to an observer. Matching spectra are
called metamers.
The three weighted r, g,andb values cannot directly represent all visible
colors, as their color matching curves have negative weights for various
wavelengths. The CIE proposed three different hypothetical light sources
that did not use monochromatic light. These color matching curves are
denoted
x(λ), y(λ), and z(λ), and their spectra are shown in Figure 7.9.
The color matching function
y(λ) is, in fact, one and the same as the
photometric curve (see page 209), as radiance is converted to luminance
with this curve. Given a surface reflectance and light source, the product
of these two defines a color function C(λ), i.e., a spectrum. By again
multiplying this spectrum by a color matching curve and integrating, a
Figure 7.9. The Judd-Vos-modified CIE (1978) 2-degree color matching functions. Note
that the two
x’s are part of the same curve. These color matching curves are not to be
confused with the spectral distributions of the light sources used in the color matching
experiment.
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7.3. Colorimetry 213
single value is computed for each curve:
X =
780
380
C(λ)x(λ)dλ,
Y =
780
380
C(λ)y(λ)dλ,
Z =
780
380
C(λ)z(λ)dλ.
(7.7)
These X, Y ,andZ tristimulus values are weights that define a color in
CIE XYZ space. This three-dimensional space is awkward to work in, so
the plane where X + Y + Z = 1 is used. See Figure 7.10. Coordinates in
this space are called x and y, and are computed as follows:
x =
X
X + Y + Z
,
y =
Y
X + Y + Z
,
z =
Z
X + Y + Z
=1− x −y.
(7.8)
Figure 7.10. On the left, the RGB color cube is shown in XYZ space, along with its
gamut projection (in violet) and chromaticity diagram (in blue) onto the X + Y + Z =1
plane. Lines radiating from the origin have a constant chromaticity value, varying
only in luminance. On the right, seven points on the plane X + Y + Z =1,andso
in xyz coordinates, are projected onto the XY plane (by dropping the z value). The
chromaticity diagram outline is shown for both planes, with the resulting diagram shown
in gray.
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214 7. Advanced Shading
Figure 7.11. The chromaticity diagram. The curve is labeled with the corresponding pure
wavelength colors, and a typical gamut is shown with the triangle. (Image generated by a
Java applet from the Computer Science Department, Rochester Institute of Technology.)
The z value gives no additional information, so it is normally omitted.
The plot of the chromaticity coordinates x and y values is known as the
CIE chromaticity diagram. See Figure 7.11. The curved line in the diagram
shows where the colors of the spectrum lie, and the straight line connecting
the ends of the spectrum is called the purple line. Illuminant E, the equal
energy spectrum, is often used to define white, at the point x = y = z =
1
3
.
For a computer monitor, the white point is the combination of the three
color phosphors at full intensity, i.e., the brightest white.
Given a color point (x,y), draw a line from the white point through this
point to the spectral line. The relative distance of the color point compared
to the distance to the edge of the region is the saturation of the color. The
point on the region edge defines the hue of the color.
5
The chromaticity diagram describes a plane. The third dimension
needed to fully describe a color is the Y value, luminance. These then
define what is called the xyY coordinate system.
5
Saturation and hue are the commonly used terms for these definitions. The proper
colorimetric terms are excitation purity and dominant wavelength, but these are rarely
used in graphics.
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7.3. Colorimetry 215
The chromaticity diagram is important in understanding how color is
used in rendering, and the limits of the rendering system. A computer
monitor presents colors by using some settings of R, G,andB color values.
Each color channel controls some physical phenomenon, e.g., exciting a
phosphor, which in turn emits light in a particular spectrum. By exciting
three phosphors, three spectra are emitted, which are added together and
create a single spectrum that the viewer perceives. The eye’s three types
of cones also each have their own color matching functions, i.e., each cone
responds to the incoming spectrum and sends a signal to the brain.
The display system is limited in its ability to present different colors by
the spectra of its three channels. An obvious limitation is in luminance, as a
computer monitor is only so bright, but there are also limits to saturation.
As seen earlier, the r, g,andb color matching system needed to have
negative weights for some of the spectrum colors to be matched. These
colors with negative weights are colors that could not be displayed by a
combination of the three lights. Video displays and printers are limited in
a similar fashion.
The triangle in the chromaticity diagram represents the gamut of a
typical computer monitor. The three corners of the triangle are the most
saturated red, green, and blue colors the monitor can display. An important
property of the chromaticity diagram is that these limiting colors can be
joined by straight lines to show the limits of the monitor as a whole. The
straight lines represent the limits of colors that can be displayed by mixing
these three primaries. See Stone’s book [1224] for more information.
Conversion from XY Z to RGB space is linear and can be done with a
matrix [287, 349]:
⎛
⎝
R
G
B
⎞
⎠
=
⎛
⎝
3.240479 −1.537150 −0.498535
−0.969256 1.875992 0.041556
0.055648 −0.204043 1.057311
⎞
⎠
⎛
⎝
X
Y
Z
⎞
⎠
. (7.9)
This conversion matrix is for a monitor with a D65 white point, a particular
definition of the “full on” white color produced by the monitor. Some XYZ
values can transform to RGB values that are negative or greater than one.
These are colors that are out of gamut, not reproducible on the monitor.
The inverse RGB to XYZ conversion is
⎛
⎝
X
Y
Z
⎞
⎠
=
⎛
⎝
0.412453
0.357580
0.180423
0.212671 0.715160 0.072169
0.019334 0.119193 0.950227
⎞
⎠
⎛
⎝
R
G
B
⎞
⎠
. (7.10)
A common conversion is to transform an RGB color to a grayscale lumi-
nance value, which is simply the middle row of the previous equation:
Y =0.212671R +0.715160G +0.072169B. (7.11)
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