45
3

PHOTOMETRY AND COLORIMETRY
Chapter 2 dealt with human vision from a qualitative viewpoint in an attempt to
establish models for monochrome and color vision. These models may be made
quantitative by specifying measures of human light perception. Luminance mea-
sures are the subject of the science of photometry, while color measures are treated
by the science of colorimetry.
3.1. PHOTOMETRY
A source of radiative energy may be characterized by its spectral energy distribution
, which specifies the time rate of energy the source emits per unit wavelength
interval. The total power emitted by a radiant source, given by the integral of the
spectral energy distribution,
(3.1-1)
is called the radiant flux of the source and is normally expressed in watts (W).
A body that exists at an elevated temperature radiates electromagnetic energy
proportional in amount to its temperature. A blackbody is an idealized type of heat
radiator whose radiant flux is the maximum obtainable at any wavelength for a body
at a fixed temperature. The spectral energy distribution of a blackbody is given by
Planck's law (1):
(3.1-2)
C λ()
PCλ()λd
0
∞
∫
=
C λ()
C
1

5
C
2
λT⁄{}exp
----------------------------------------=
PHOTOMETRY
47
this figure, approximating the measured data, is a 6000 kelvin (K) blackbody curve.
Incandescent lamps are often approximated as blackbody radiators of a given tem-
perature in the range 1500 to 3500 K (3).
The Commission Internationale de l'Eclairage (CIE), which is an international
body concerned with standards for light and color, has established several standard
sources of light, as illustrated in Figure 3.1-2 (4). Source S
A
is a tungsten filament
lamp. Over the wavelength band 400 to 700 nm, source S
B
approximates direct sun-
light, and source S
C
approximates light from an overcast sky. A hypothetical source,
called Illuminant E, is often employed in colorimetric calculations. Illuminant E is
assumed to emit constant radiant energy at all wavelengths.
Cathode ray tube (CRT) phosphors are often utilized as light sources in image
processing systems. Figure 3.1-3 describes the spectral energy distributions of
common phosphors (5). Monochrome television receivers generally use a P4 phos-
phor, which provides a relatively bright blue-white display. Color television displays
utilize cathode ray tubes with red, green, and blue emitting phosphors arranged in
triad dots or strips. The P22 phosphor is typical of the spectral energy distribution of
commercial phosphor mixtures. Liquid crystal displays (LCDs) typically project a

0
∞
∫
=
V λ() K
m
K
m
COLOR MATCHING
49
3.2. COLOR MATCHING
The basis of the trichromatic theory of color vision is that it is possible to match
an arbitrary color by superimposing appropriate amounts of three primary colors
(10–14). In an additive color reproduction system such as color television, the
three primaries are individual red, green, and blue light sources that are projected
onto a common region of space to reproduce a colored light. In a subtractive color
system, which is the basis of most color photography and color printing, a white
light sequentially passes through cyan, magenta, and yellow filters to reproduce a
colored light.
3.2.1. Additive Color Matching
An additive color-matching experiment is illustrated in Figure 3.2-1. In
Figure 3.2-1a, a patch of light (C) of arbitrary spectral energy distribution , as
shown in Figure 3.2-2a, is assumed to be imaged onto the surface of an ideal
diffuse reflector (a surface that reflects uniformly over all directions and all
wavelengths). A reference white light (W) with an energy distribution, as in
Figure 3.2-2b, is imaged onto the surface along with three primary lights (P
1
),
(P
2

C()A
3
C()
T
1
C()T
2
C()T
3
C()
T
1
C()
A
1
C()
A
1
W()
----------------= T
2
C()
A
2
C()
A
2
W()
----------------= T
3

) and (P
2
) plus (C). If
a match is achieved by this configuration, tristimulus value will be negative.
If this configuration fails, a match is attempted between (P
2
) plus (P
3
) and (P
1
) plus
(C). A correct match is denoted with a negative value for .
FIGURE 3.2-2. Spectral energy distributions.
T
1
C()
A
1
C()
A
1
W()
----------------= T
2
C()
A
2
C()
A
2

The process described above is a direct method for specifying a color quantita-
tively. It has two drawbacks: The method is cumbersome and it depends on the per-
ceptual variations of a single observer. In Section 3.3 we consider standardized
quantitative color measurement in detail.
3.2.2. Subtractive Color Matching
A subtractive color-matching experiment is shown in Figure 3.2-3. An illumination
source with spectral energy distribution passes sequentially through three dye
filters that are nominally cyan, magenta, and yellow. The spectral absorption of the
dye filters is a function of the dye concentration. It should be noted that the spectral
transmissivities of practical dyes change shape in a nonlinear manner with dye con-
centration.
In the first step of the subtractive color-matching process, the dye concentrations
of the three spectral filters are varied until a perceptual match is obtained with a refer-
ence white (W). The dye concentrations are the matching values of the color match
, , . Next, the three dye concentrations are varied until a match is
obtained with a desired color (C). These matching values , are
then used to compute the tristimulus values , , , as in Eq. 3.2-1.
FIGURE 3.2-3. Subtractive color matching.
T
1
C()
A
1
C()
A
1
W()
----------------= T
2
C()

C(),,
T
1
C()T
2
C()T
3
C()
COLOR MATCHING
53
It should be apparent that there is no fundamental theoretical difference between
color matching by an additive or a subtractive system. In a subtractive system, the
yellow dye acts as a variable absorber of blue light, and with ideal dyes, the yellow
dye effectively forms a blue primary light. In a similar manner, the magenta filter
ideally forms the green primary, and the cyan filter ideally forms the red primary.
Subtractive color systems ordinarily utilize cyan, magenta, and yellow dye spectral
filters rather than red, green, and blue dye filters because the cyan, magenta, and
yellow filters are notch filters which permit a greater transmission of light energy
than do narrowband red, green, and blue bandpass filters. In color printing, a fourth
filter layer of variable gray level density is often introduced to achieve a higher con-
trast in reproduction because common dyes do not possess a wide density range.
3.2.3. Axioms of Color Matching
The color-matching experiments described for additive and subtractive color match-
ing have been performed quite accurately by a number of researchers. It has been
found that perfect color matches sometimes cannot be obtained at either very high or
very low levels of illumination. Also, the color matching results do depend to some
extent on the spectral composition of the surrounding light. Nevertheless, the simple
color matching experiments have been found to hold over a wide range of condi-
tions.
Grassman (15) has developed a set of eight axioms that define trichromatic color

units of P:
(3.2-9)
With Grassman's laws now specified, consideration is given to the development of a
quantitative theory for color matching.
3.3. COLORIMETRY CONCEPTS
Colorimetry is the science of quantitatively measuring color. In the trichromatic
color system, color measurements are in terms of the tristimulus values of a color or
a mathematical function of the tristimulus values.
Referring to Section 3.2.3, the axioms of color matching state that a color C can
be matched by three primary colors P
1
, P
2
, P
3
. The qualitative match is expressed as
(3.3-1)
where , , are the matching values of the color (C). Because the
intensities of incoherent light sources add linearly, the spectral energy distribution of
a color mixture is equal to the sum of the spectral energy distributions of its compo-
nents. As a consequence of this fact and Eq. 3.3-1, the spectral energy distribution
can be replaced by its color-matching equivalent according to the relation
(3.3-2)
M() N()◊[]N() P()◊[]∩ M() P()◊⇒
cC• mM()•[]nN()•[]pP()•[]⊕⊕◊
cC()•[]mM()•[]nN()•[]pP()•[]⊕◊⊕
cC()•[]mM()•[]nN()•[]⊕⊕ pP()•[]◊
C() A
1
C() P

λ()++ A
j
C()P
j
λ()
j 1
=
3
∑
=◊
COLORIMETRY CONCEPTS
55
Equation 3.3-2 simply means that the spectral energy distributions on both sides of
the equivalence operator evoke the same color sensation. Color matching is usu-
ally specified in terms of tristimulus values, which are normalized matching values,
as defined by
(3.3-3)
where represents the matching value of the reference white. By this substitu-
tion, Eq. 3.3-2 assumes the form
(3.3-4)
From Grassman's fourth law, the luminance of a color mixture Y(C) is equal to
the luminance of its primary components. Hence
(3.3-5a)
or
(3.3-5b)
where is the relative luminous efficiency and represents the spectral
energy distribution of a primary. Equations 3.3-4 and 3.3-5 represent the quantita-
tive foundation for colorimetry.
3.3.1. Color Vision Model Verification
Before proceeding further with quantitative descriptions of the color-matching pro-

3
∑
◊
YC() C λ()V λ()λd
∫
A
j
C()P
j
λ()V λ()λd
∫
j 1
=
3
∑
==
YC() T
j
C()A
j
W()P
j
λ()V λ()λd
∫
j 1
=
3
∑
=
V λ() P

=
56
PHOTOMETRY AND COLORIMETRY
If a viewer observes the primary mixture instead of C, then from Eq. 3.3-4, substitu-
tion for should result in the same cone signals . Thus
(3.3-7a)
(3.3-7b)
(3.3-7c)
Equation 3.3-7 can be written more compactly in matrix form by defining
(3.3-8)
Then
(3.3-9)
or in yet more abbreviated form,
(3.3-10)
where the vectors and matrices of Eq. 3.3-10 are defined in correspondence with
Eqs. 3.3-7 to 3.3-9. The vector space notation used in this section is consistent with
the notation formally introduced in Appendix 1. Matrices are denoted as boldface
uppercase symbols, and vectors are denoted as boldface lowercase symbols. It
should be noted that for a given set of primaries, the matrix K is constant valued,
and for a given reference white, the white matching values of the matrix A are con-
stant. Hence, if a set of cone signals were known for a color (C), the corre-
sponding tristimulus values could in theory be obtained from
(3.3-11)
C λ() e
i
C()
e
1
C() T
j

e
3
C() T
j
C()A
j
W()P
j
λ()s
3
λ() λd
∫
j 1
=
3
∑
=
k
ij
P
j
λ()s
i
λ()λd
∫
=
e
1
C()
e

3
W()
T
1
C()
T
2
C()
T
3
C()
=
e C() KAt C()=
e
i
C()
T
j
C()
t C() KA[]
1
–
e C()=
COLORIMETRY CONCEPTS
57
provided that the matrix inverse of [KA] exists. Thus, it has been shown that with
proper selection of the tristimulus signals

, any color can be matched in the
sense that the cone signals will be the same for the primary mixture as for the actual

C()
T
s
1
λ() T
s
2
λ() T
s
3
λ() λ
C
ψ
ψ
δλ ψ–()
e
i
C
ψ
() δλψ–()s
i
λ()λd
∫
A
j
W()P
j
λ()T
s
j

s
j
ψ()s
i
λ() λd
∫
j 1
=
3
∑
=
ψ
C ψ()δλ ψ–()s
i
λ()λψdd
∫∫
e
i
C() A
j
W()P
j
λ()C ψ()T
s
j
ψ()s
i
λ()ψd λd
∫∫
j 1

can be considered to form the three axes of
a color space as illustrated in Figure 3.3-2. A particular color may be described as a
a vector in the color space, but it must be remembered that it is the coordinates of
the vectors (tristimulus values), rather than the vector length, that specify the color.
In Figure 3.3-2, a triangle, called a Maxwell triangle, has been drawn between the
three primaries. The intersection point of a color vector with the triangle gives an
indication of the hue and saturation of the color in terms of the distances of the point
from the vertices of the triangle.
FIGURE 3.3-1. Tristimulus values of typical red, green, and blue primaries required to
match unit energy throughout the spectrum.
FIGURE 3.3-2 Color space for typical red, green, and blue primaries.
COLORIMETRY CONCEPTS
59
Often the luminance of a color is not of interest in a color match. In such situa-
tions, the hue and saturation of color (C) can be described in terms of chromaticity
coordinates, which are normalized tristimulus values, as defined by
(3.3-16a)
(3.3-16b)
(3.3-16c)
Clearly, , and hence only two coordinates are necessary to describe a
color match. Figure 3.3-3 is a plot of the chromaticity coordinates of the spectral
colors for typical primaries. Only those colors within the triangle defined by the
three primaries are realizable by physical primary light sources.
3.3.3. Luminance Calculation
The tristimulus values of a color specify the amounts of the three primaries required
to match a color where the units are measured relative to a match of a reference
white. Often, it is necessary to determine the absolute rather than the relative
amount of light from each primary needed to reproduce a color match. This informa-
tion is found from luminance measurements of calculations of a color match.
FIGURE 3.3-3. Chromaticity diagram for typical red, green, and blue primaries.

T
2
T
3
++
------------------------------≡
t
3
1 t
1
t
2
––=
60
PHOTOMETRY AND COLORIMETRY
From Eq. 3.3-5 it is noted that the luminance of a matched color Y(C) is equal to
the sum of the luminances of its primary components according to the relation
(3.3-17)
The integrals of Eq. 3.3-17,
(3.3-18)
are called luminosity coefficients of the primaries. These coefficients represent the
luminances of unit amounts of the three primaries for a match to a specific reference
white. Hence the luminance of a matched color can be written as
(3.3-19)
Multiplying the right and left sides of Eq. 3.3-19 by the right and left sides, respec-
tively, of the definition of the chromaticity coordinate
(3.3-20)
and rearranging gives
(3.3-21a)
Similarly,

()T
2
C()YP
2
()T
3
C()YP
3
()++=
t
1
C()
T
1
C()
T
1
C() T
2
C() T
3
C()++
----------------------------------------------------------=
T
1
C()
t
1
C()YC()
t

3
()++
--------------------------------------------------------------------------------------------------=
T
3
C()
t
3
C()YC()
t
1
C()YP
1
()t
2
C()YP
2
()t
3
C()YP
3
()++
--------------------------------------------------------------------------------------------------=
TRISTIMULUS VALUE TRANSFORMATION
61
3.4. TRISTIMULUS VALUE TRANSFORMATION
From Eq. 3.3-7 it is clear that there is no unique set of primaries for matching colors.
If the tristimulus values of a color are known for one set of primaries, a simple coor-
dinate conversion can be performed to determine the tristimulus values for another
set of primaries (16). Let (P

λ()P
2
λ()P
3
λ()
A
1
W()A
2
W()A
3
W()
P
˜
1
()P
˜
2
()P
˜
3
() P
˜
1
λ()
P
˜
2
λ()P
˜

T
˜
3
C()
e C() KA W()t C() K
˜
A
˜
W
˜
()t
˜
C()==
W
˜
e W
˜
() KA W()t W
˜
() K
˜
A
˜
W
˜
()t
˜
W
˜
()==

() KA W()t P
1
˜
() K
˜
A
˜
W
˜
()t
˜
P
1
˜
()==
e P
2
˜
() KA W()t P
2
˜
() K
˜
A
˜
W
˜
()t
˜
P

W
˜
()
----------------
0
0
=
t
˜
P
˜
2
()
0
1
A
2
W
˜
()
----------------
0
=
t
˜
P
˜
2
()
0