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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

3

PHOTOMETRY AND COLORIMETRY

Chapter 2 dealt with human vision from a qualitative viewpoint in an attempt toestablish models for monochrome and color vision These models may be madequantitative 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 wavelengthinterval The total power emitted by a radiant source, given by the integral of thespectral 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

where is the radiation wavelength, T is the temperature of the body, and and

are constants Figure 3.1-1a is a plot of the spectral energy of a blackbody as a

function of temperature and wavelength In the visible region of the electromagneticspectrum, the blackbody spectral energy distribution function of Eq 3.1-2 can be

approximated by Wien's radiation law (1):

(3.1-3)

Wien's radiation function is plotted in Figure 3.1-1b over the visible spectrum The most basic physical light source, of course, is the sun Figure 2.1-1a shows a

plot of the measured spectral energy distribution of sunlight (2) The dashed line in

FIGURE 3.1-1 Blackbody radiation functions.

FIGURE 3.1-2 CIE standard illumination sources.

C2

C( )λ C1

λ5exp{C2⁄λT} -

=

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 internationalbody 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 imageprocessing systems Figure 3.1-3 describes the spectral energy distributions ofcommon phosphors (5) Monochrome television receivers generally use a P4 phos-phor, which provides a relatively bright blue-white display Color television displaysutilize cathode ray tubes with red, green, and blue emitting phosphors arranged intriad dots or strips The P22 phosphor is typical of the spectral energy distribution ofcommercial phosphor mixtures Liquid crystal displays (LCDs) typically project awhite light through red, green and blue vertical strip pixels Figure 3.1-4 is a plot oftypical color filter transmissivities (6)

Photometric measurements seek to describe quantitatively the perceptual ness of visible electromagnetic energy (7,8) The link between photometric mea-surements and radiometric measurements (physical intensity measurements) is the

bright-photopic luminosity function, as shown in Figure 3.1-5a (9) This curve, which is a

CIE standard, specifies the spectral sensitivity of the human visual system to optical

radiation as a function of wavelength for a typical person referred to as the standard

FIGURE 3.1-3 Spectral energy distribution of CRT phosphors.

observer In essence, the curve is a standardized version of the measurement of cone

sensitivity given in Figure 2.2-2 for photopic vision at relatively high levels of mination The standard luminosity function for scotopic vision at relatively low

illu-levels of illumination is illustrated in Figure 3.1-5b Most imaging system designs are based on the photopic luminosity function, commonly called the relative lumi- nous efficiency.

The perceptual brightness sensation evoked by a light source with spectral energydistribution is specified by its luminous flux, as defined by

(3.1-4)

where represents the relative luminous efficiency and is a scaling stant The modern unit of luminous flux is the lumen (lm), and the correspondingvalue for the scaling constant is = 685 lm/W An infinitesimally narrowbandsource of 1 W of light at the peak wavelength of 555 nm of the relative luminousefficiency curve therefore results in a luminous flux of 685 lm

con-FIGURE 3.1-4 Transmissivities of LCD color filters.

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 acolored 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 (P1),

(P2), (P3) whose spectral energy distributions are sketched in Figure 3.2-2c to e.

The three primary lights are first overlapped and their intensities are adjusted untilthe overlapping region of the three primary lights perceptually matches thereference white in terms of brightness, hue, and saturation The amounts of thethree primaries , , are then recorded in some physical units,such as watts These are the matching values of the reference white Next, theintensities of the primaries are adjusted until a match is achieved with

the colored light (C), if a match is possible The procedure to be followed

if a match cannot be achieved is considered later The intensities of the primaries

FIGURE 3.1-5 Relative luminous efficiency functions.

C( )λ

A1( ) A W 2( ) A W 3( )W

, , when a match is obtained are recorded, and normalized ing values , , , called tristimulus values, are computed as

If a match cannot be achieved by the procedure illustrated in Figure 3.2-1a, it is often possible to perform the color matching outlined in Figure 3.2-1b One of the primaries, say (P3), is superimposed with the light (C), and the intensities of all

three primaries are adjusted until a match is achieved between the overlapping

region of primaries (P1) and (P2) with the overlapping region of (P3) and (C) If

such a match is obtained, the tristimulus values are

(3.2-2)

In this case, the tristimulus value is negative If a match cannot be achieved

with this geometry, a match is attempted between (P1) plus (P3) and (P2) plus (C) If

a match is achieved by this configuration, tristimulus value will be negative

If this configuration fails, a match is attempted between (P2) plus (P3) and (P1) plus

(C) A correct match is denoted with a negative value for

FIGURE 3.2-2 Spectral energy distributions.

Finally, in the rare instance in which a match cannot be achieved by either of the

configurations of Figure 3.2-1a or b, two of the primaries are superimposed with (C)

and an attempt is made to match the overlapped region with the remaining primary

In the case illustrated in Figure 3.2-1c, if a match is achieved, the tristimulus values

quantita-3.2.2 Subtractive Color Matching

A subtractive color-matching experiment is shown in Figure 3.2-3 An illuminationsource with spectral energy distribution passes sequentially through three dyefilters that are nominally cyan, magenta, and yellow The spectral absorption of thedye filters is a function of the dye concentration It should be noted that the spectraltransmissivities 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 , arethen used to compute the tristimulus values , , , as in Eq 3.2-1

FIGURE 3.2-3 Subtractive color matching.

It should be apparent that there is no fundamental theoretical difference betweencolor matching by an additive or a subtractive system In a subtractive system, theyellow dye acts as a variable absorber of blue light, and with ideal dyes, the yellowdye effectively forms a blue primary light In a similar manner, the magenta filterideally forms the green primary, and the cyan filter ideally forms the red primary.Subtractive color systems ordinarily utilize cyan, magenta, and yellow dye spectralfilters rather than red, green, and blue dye filters because the cyan, magenta, andyellow filters are notch filters which permit a greater transmission of light energythan do narrowband red, green, and blue bandpass filters In color printing, a fourthfilter 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 ing have been performed quite accurately by a number of researchers It has beenfound that perfect color matches sometimes cannot be obtained at either very high orvery low levels of illumination Also, the color matching results do depend to someextent on the spectral composition of the surrounding light Nevertheless, the simplecolor matching experiments have been found to hold over a wide range of condi-tions

match-Grassman (15) has developed a set of eight axioms that define trichromatic colormatching and that serve as a basis for quantitative color measurements In thefollowing presentation of these axioms, the symbol indicates a color match; thesymbol indicates an additive color mixture; the symbol indicates units of acolor These axioms are:

1 Any color can be matched by a mixture of no more than three colored lights

2 A color match at one radiance level holds over a wide range of levels

3 Components of a mixture of colored lights cannot be resolved by the human eye

4 The luminance of a color mixture is equal to the sum of the luminance of itscomponents

5 Law of addition If color (M) matches color (N) and color (P) matches color (Q), then color (M) mixed with color (P) matches color (N) mixed with color (Q):

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 P1, P2, P3 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 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

Equation 3.3-2 simply means that the spectral energy distributions on both sides ofthe equivalence operator evoke the same color sensation Color matching is usu-ally specified in terms of tristimulus values, which are normalized matching values,

From Grassman's fourth law, the luminance of a color mixture Y(C) is equal to

the luminance of its primary components Hence

3.3.1 Color Vision Model Verification

Before proceeding further with quantitative descriptions of the color-matching cess, it is instructive to determine whether the matching experiments and the axioms

pro-of color matching are satisfied by the color vision model presented in Section 2.5 Inthat model, the responses of the three types of receptors with sensitivities ,, are modeled as

(3.3-6a)(3.3-6b)(3.3-6c)

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

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

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 thesense that the cone signals will be the same for the primary mixture as for the actual

color C Unfortunately, the cone signals are not easily measured physicalquantities, and therefore, Eq 3.3-11 cannot be used directly to compute the tristimu-lus values of a color However, this has not been the intention of the derivation.Rather, Eq 3.3-11 has been developed to show the consistency of the color-match-ing experiment with the color vision model

3.3.2 Tristimulus Value Calculation

It is possible indirectly to compute the tristimulus values of an arbitrary color for aparticular set of primaries if the tristimulus values of the spectral colors (narrow-band light) are known for that set of primaries Figure 3.3-1 is a typical sketch of thetristimulus values required to match a unit energy spectral color with three arbitraryprimaries These tristimulus values, which are fundamental to the definition of a pri-mary system, are denoted as , , , where is a particular wave-length in the visible region A unit energy spectral light ( ) at wavelength withenergy distribution is matched according to the equation

Integrating each side of Eq 3.3-13 over and invoking the sifting integral gives the

cone signal for the color (C) Thus

(3.3-14)

By correspondence with Eq 3.3-7, the tristimulus values of (C) must be equivalent

to the second integral on the right of Eq 3.3-14 Hence

From Figure 3.3-1 it is seen that the tristimulus values obtained from solution of

Eq 3.3-11 may be negative at some wavelengths Because the tristimulus valuesrepresent units of energy, the physical interpretation of this mathematical result isthat a color match can be obtained by adding the primary with negative tristimulusvalue to the original color and then matching this resultant color with the remainingprimary In this sense, any color can be matched by any set of primaries However,from a practical viewpoint, negative tristimulus values are not physically realizable,and hence there are certain colors that cannot be matched in a practical color display(e.g., a color television receiver) with fixed primaries Fortunately, it is possible tochoose primaries so that most commonly occurring natural colors can be matched

The three tristimulus values T1, T2, T'3 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 ofthe 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 anindication of the hue and saturation of the color in terms of the distances of the pointfrom 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.

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

3.3.3 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 referencewhite Often, it is necessary to determine the absolute rather than the relativeamount 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.

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 referencewhite Hence the luminance of a matched color can be written as

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 dinate conversion can be performed to determine the tristimulus values for another

coor-set of primaries (16) Let (P1), (P2), (P3) be the original set of primaries with tral energy distributions , , , with the units of a match determined

spec-by a white reference (W) with matching values , , Now, consider

a new set of primaries , , with spectral energy distributions ,, Matches are made to a reference white , which may be differentthan the reference white of the original set of primaries, by matching values ,, Referring to Eq 3.3-10, an arbitrary color (C) can be matched by the

tristimulus values , , with the original set of primaries or by thetristimulus values , , with the new set of primaries, according tothe matching matrix relations

(3.4-1)

The tristimulus value units of the new set of primaries, with respect to the originalset of primaries, must now be found This can be accomplished by determining thecolor signals of the reference white for the second set of primaries in terms of bothsets of primaries The color signal equations for the reference white become

(3.4-2)

where Finally, it is necessary to relate the two sets ofprimaries by determining the color signals of each of the new primary colors ,, in terms of both primary systems These color signal equations are

(3.4-3a)(3.4-3b)(3.4-3c)where

01

A2( )W˜ -0

001

A3( )W˜ -

=

Matrix equations 3.4-1 to 3.4-3 may be solved jointly to obtain a relationshipbetween the tristimulus values of the original and new primary system:

(3.4-4a)

(3.4-4b)

(3.4-4c)

where denotes the determinant of matrix T Equations 3.4-4 then may be written

in terms of the chromaticity coordinates , , of the new set of maries referenced to the original primary coordinate system

pri-With this revision,

by its chromaticity values , and its luminance Y(C) Appendix 2 presents

formulas for color coordinate conversion between tristimulus values and ity coordinates for various representational combinations A third approach in speci-fying a color is to represent the color by a linear or nonlinear invertible function ofits tristimulus or chromaticity values

chromatic-In this section we describe several standard and nonstandard color spaces for therepresentation of color images They are categorized as colorimetric, subtractive,video, or nonstandard Figure 3.5-1 illustrates the relationship between these colorspaces The figure also lists several example color spaces

m ij ∆ij

∆i -

Natural color images, as opposed to computer-generated images, usually nate from a color scanner or a color video camera These devices incorporate threesensors that are spectrally sensitive to the red, green, and blue portions of the lightspectrum The color sensors typically generate red, green, and blue color signals thatare linearly proportional to the amount of red, green, and blue light detected by eachsensor These signals are linearly proportional to the tristimulus values of a color at

origi-each pixel As indicated in Figure 3.5-1, linear RGB images are the basis for the

gen-eration of the various color space image representations

3.5.1 Colorimetric Color Spaces

The class of colorimetric color spaces includes all linear RGB images and the

stan-dard colorimetric images derived from them by linear and nonlinear intercomponenttransformations

FIGURE 3.5-1 Relationship of color spaces.

nonstandard

colorimetric

linear

subtractive CMY/CMYK

colorimetric

nonlinear

video gamma RGB

colorimetric linear RGB

video gamma luma/chroma YCC linear intercomponent transformation

linear point transformation nonlinear intercomponent transformation

nonlinear point transformation

R C G C B C Spectral Primary Color Coordinate System In 1931, the CIE developed a

standard primary reference system with three monochromatic primaries at lengths: red = 700 nm; green = 546.1 nm; blue = 435.8 nm (11) The units of the

wave-tristimulus values are such that the wave-tristimulus values R C , G C , B C are equal when

matching an equal-energy white, called Illuminant E, throughout the visible spectrum.

The primary system is defined by tristimulus curves of the spectral colors, as shown inFigure 3.5-2 These curves have been obtained indirectly by experimental color-match-ing experiments performed by a number of observers The collective color-matching

response of these observers has been called the CIE Standard Observer Figure 3.5-3 is

a chromaticity diagram for the CIE spectral coordinate system

FIGURE 3.5-2 Tristimulus values of CIE spectral primaries required to match unit energy

throughout the spectrum Red = 700 nm, green = 546.1 nm, and blue = 435.8 nm

FIGURE 3.5-3 Chromaticity diagram for CIE spectral primary system.

R N G N B N NTSC Receiver Primary Color Coordinate System Commercial

televi-sion receivers employ a cathode ray tube with three phosphors that glow in the red,green, and blue regions of the visible spectrum Although the phosphors ofcommercial television receivers differ from manufacturer to manufacturer, it iscommon practice to reference them to the National Television Systems Committee(NTSC) receiver phosphor standard (14) The standard observer data for the CIEspectral primary system is related to the NTSC primary system by a pair of linearcoordinate conversions

Figure 3.5-4 is a chromaticity diagram for the NTSC primary system In thissystem, the units of the tristimulus values are normalized so that the tristimulus

values are equal when matching the Illuminant C white reference The NTSC

phosphors are not pure monochromatic sources of radiation, and hence the gamut ofcolors producible by the NTSC phosphors is smaller than that available from thespectral primaries This fact is clearly illustrated by Figure 3.5-3, in which the gamut

of NTSC reproducible colors is plotted in the spectral primary chromaticity diagram

(11) In modern practice, the NTSC chromaticities are combined with Illuminant D65.

Broad-cast Union (EBU) has established a receiver primary system whose chromaticitiesare close in value to the CIE chromaticity coordinates, and the reference white isIlluminant C (17) The EBU chromaticities are also combined with the D65 illumi-nant

Interna-tional Telecommunications Union (ITU) issued its Recommendation 601, which

FIGURE 3.5-4 Chromaticity diagram for NTSC receiver phosphor primary system.