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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 measures are the subject of the science of photometry, while color measures are treated by the science of colorimetry.

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Digital Image Processing: PIKS Inside, Third Edition. William K. Pratt Copyright © 2001 John Wiley & Sons, Inc. ISBNs: 0-471-37407-5 (Hardback); 0-471-22132-5 (Electronic) 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 C ( λ ) , 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, ∞ P = ∫0 C(λ ) d λ (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): C1 C ( λ ) = ----------------------------------------------------- (3.1-2) 5 λ [ exp { C 2 ⁄ λT } – 1 ] 45
46 PHOTOMETRY AND COLORIMETRY FIGURE 3.1-1. Blackbody radiation functions. where λ is the radiation wavelength, T is the temperature of the body, and C 1 and C 2 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 electromagnetic spectrum, the blackbody spectral energy distribution function of Eq. 3.1-2 can be approximated by Wien's radiation law (1): C1 C ( λ ) = --------------------------------------- - (3.1-3) 5 λ exp { C 2 ⁄ λT } 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 3.1-2. CIE standard illumination sources.
PHOTOMETRY 47 FIGURE 3.1-3. Spectral energy distribution of CRT phosphors. 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 SA is a tungsten filament lamp. Over the wavelength band 400 to 700 nm, source SB approximates direct sun- light, and source SC 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 white light through red, green and blue vertical strip pixels. Figure 3.1-4 is a plot of typical color filter transmissivities (6). Photometric measurements seek to describe quantitatively the perceptual bright- ness of visible electromagnetic energy (7,8). The link between photometric mea- surements and radiometric measurements (physical intensity measurements) is the 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
48 PHOTOMETRY AND COLORIMETRY FIGURE 3.1-4. Transmissivities of LCD color filters. 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 illu- mination. The standard luminosity function for scotopic vision at relatively low 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 energy distribution C ( λ ) is specified by its luminous flux, as defined by ∞ F = Km ∫ C ( λ )V ( λ ) d λ (3.1-4) 0 where V ( λ ) represents the relative luminous efficiency and K m is a scaling con- stant. The modern unit of luminous flux is the lumen (lm), and the corresponding value for the scaling constant is K m = 685 lm/W. An infinitesimally narrowband source of 1 W of light at the peak wavelength of 555 nm of the relative luminous efficiency curve therefore results in a luminous flux of 685 lm.
COLOR MATCHING 49 FIGURE 3.1-5. Relative luminous efficiency functions. 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 C ( λ ) , 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 until the overlapping region of the three primary lights perceptually matches the reference white in terms of brightness, hue, and saturation. The amounts of the three primaries A 1 ( W ) , A 2 ( W ) , A3 ( W ) are then recorded in some physical units, such as watts. These are the matching values of the reference white. Next, the intensities 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
50 PHOTOMETRY AND COLORIMETRY FIGURE 3.2-1. Color matching. A1 ( C ), A 2 ( C ), A 3 ( C ) when a match is obtained are recorded, and normalized match- ing values T1 ( C ) , T 2 ( C ) , T3 ( C ) , called tristimulus values, are computed as A1 ( C ) A2 ( C ) A3 ( C ) T 1 ( C ) = --------------- - T 2 ( C ) = --------------- - T 3 ( C ) = --------------- - A1 ( W ) A2 ( W ) A3( W ) (3.2-1)
COLOR MATCHING 51 FIGURE 3.2-2. Spectral energy distributions. 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 A1 ( C ) A2 ( C ) – A3 ( C ) T 1 ( C ) = --------------- - T 2 ( C ) = --------------- - T 3 ( C ) = ----------------- - (3.2-2) A1 ( W ) A2 ( W ) A3( W ) In this case, the tristimulus value T3 ( C ) 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 T2 ( C ) 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 T1 ( C ) .
52 PHOTOMETRY AND COLORIMETRY 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 become A1 ( C ) –A2 ( C ) – A3 ( C ) T 1 ( C ) = --------------- - T 2 ( C ) = ----------------- - T 3 ( C ) = ----------------- - (3.2-3) A1 ( W ) A2 ( W ) A3( W ) If a match is not obtained by this configuration, one of the other two possibilities will yield a match. 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 E ( λ ) 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 A1 ( W ) , A 2 ( W ) , A 3 ( W ) . Next, the three dye concentrations are varied until a match is obtained with a desired color (C). These matching values A1 ( C ), A2 ( C ), A3 ( C ) , are then used to compute the tristimulus values T1 ( C ) , T2 ( C ), T 3 ( C ), as in Eq. 3.2-1. FIGURE 3.2-3. Subtractive color matching.
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 matching and that serve as a basis for quantitative color measurements. In the following presentation of these axioms, the symbol ◊ indicates a color match; the symbol ⊕ indicates an additive color mixture; the symbol • indicates units of a color. 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 its components. 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): ( M ) ◊ ( N ) ∩ ( P ) ◊ ( Q) ⇒ [ ( M) ⊕ ( P) ] ◊ [ ( N ) ⊕ ( Q )] (3.2-4) 6. Law of subtraction. If the mixture of (M) plus (P) matches the mixture of (N) plus (Q) and if (P) matches (Q), then (M) matches (N): [ (M ) ⊕ (P )] ◊ [(N ) ⊕ ( Q) ] ∩ [( P) ◊ (Q) ] ⇒ ( M) ◊ (N ) (3.2-5) 7. Transitive law. If (M) matches (N) and if (N) matches (P), then (M) matches (P):
54 PHOTOMETRY AND COLORIMETRY [ (M) ◊ (N)] ∩ [(N) ◊ (P) ] ⇒ (M) ◊ (P) (3.2-6) 8. Color matching. (a) c units of (C) matches the mixture of m units of (M) plus n units of (N) plus p units of (P): c • C ◊ [m • (M )] ⊕ [n • ( N) ] ⊕ [p • (P ) ] (3.2-7) or (b) a mixture of c units of C plus m units of M matches the mixture of n units of N plus p units of P: [c • (C )] ⊕ [m • ( M) ] ◊ [n • (N)] ⊕ [ p • (P) ] (3.2-8) or (c) a mixture of c units of (C) plus m units of (M) plus n units of (N) matches p units of P: [c • (C )] ⊕ [m • ( M) ] ⊕ [n • (N )] ◊ [ p • (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 P1, P2, P3. The qualitative match is expressed as ( C ) ◊ [ A 1 ( C ) • ( P 1 ) ] ⊕ [ A 2 ( C ) • ( P 2 ) ] ⊕ [ A3 ( C ) • ( P 3 ) ] (3.3-1) where A 1 ( C ) , A2 ( C ) , A 3 ( C ) 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 C ( λ ) can be replaced by its color-matching equivalent according to the relation 3 C ( λ ) ◊ A 1 ( C )P 1 ( λ ) + A2 ( C )P2 ( λ ) + A 3 ( C )P 3 ( λ ) = ∑ A j ( C )Pj ( λ ) (3.3-2) j =1
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 Aj ( C ) T j ( C ) = -------------- - (3.3-3) Aj ( W ) where A j ( W ) represents the matching value of the reference white. By this substitu- tion, Eq. 3.3-2 assumes the form 3 C(λ) ◊ ∑ Tj ( C )Aj ( W )Pj ( λ ) (3.3-4) j =1 From Grassman's fourth law, the luminance of a color mixture Y(C) is equal to the luminance of its primary components. Hence 3 Y(C ) = ∫ C ( λ )V ( λ ) dλ = ∑ ∫ Aj ( C )Pj ( λ )V ( λ ) dλ (3.3-5a) j =1 or 3 Y(C ) = ∑ ∫ Tj ( C )Aj ( W )Pj ( λ )V ( λ ) dλ (3.3-5b) j =1 where V ( λ ) is the relative luminous efficiency and P j ( λ ) 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- cess, it is instructive to determine whether the matching experiments and the axioms of color matching are satisfied by the color vision model presented in Section 2.5. In that model, the responses of the three types of receptors with sensitivities s 1 ( λ ) , s 2 ( λ ) , s 3 ( λ ) are modeled as e 1 ( C ) = ∫ C ( λ )s 1 ( λ ) d λ (3.3-6a) e 2 ( C ) = ∫ C ( λ )s 2 ( λ ) d λ (3.3-6b) e 3 ( C ) = ∫ C ( λ )s 3 ( λ ) d λ (3.3-6c)
56 PHOTOMETRY AND COLORIMETRY If a viewer observes the primary mixture instead of C, then from Eq. 3.3-4, substitu- tion for C ( λ ) should result in the same cone signals e i ( C ) . Thus 3 e1 ( C ) = ∑ Tj ( C )Aj ( W ) ∫ Pj ( λ )s1 ( λ ) d λ (3.3-7a) j =1 3 e2( C ) = ∑ Tj ( C )Aj ( W ) ∫ Pj ( λ )s2 ( λ ) d λ (3.3-7b) j =1 3 e3 ( C ) = ∑ Tj ( C )Aj ( W ) ∫ Pj ( λ )s3 ( λ ) d λ (3.3-7c) j =1 Equation 3.3-7 can be written more compactly in matrix form by defining k ij = ∫ Pj ( λ )si ( λ ) dλ (3.3-8) Then e1 ( C ) k 11 k 12 k 13 A1( W ) 0 0 T1 ( C ) e2 ( C ) = k 21 k 22 k 23 0 A2( W ) 0 T2 ( C ) (3.3-9) e3 ( C ) k 31 k 32 k 33 0 0 A3( W ) T3 ( C ) or in yet more abbreviated form, e ( C ) = KAt ( C ) (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 e i ( C ) were known for a color (C), the corre- sponding tristimulus values Tj ( C ) could in theory be obtained from –1 t ( C ) = [ KA ] e ( C ) (3.3-11)
COLORIMETRY CONCEPTS 57 provided that the matrix inverse of [KA] exists. Thus, it has been shown that with proper selection of the tristimulus signals Tj ( C ) , any color can be matched in the sense that the cone signals will be the same for the primary mixture as for the actual color C. Unfortunately, the cone signals e i ( C ) are not easily measured physical quantities, 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 a particular 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 the tristimulus values required to match a unit energy spectral color with three arbitrary primaries. These tristimulus values, which are fundamental to the definition of a pri- mary system, are denoted as Ts1 ( λ ) , T s2 ( λ ) , T s3 ( λ ) , where λ is a particular wave- length in the visible region. A unit energy spectral light ( C ψ ) at wavelength ψ with energy distribution δ ( λ – ψ ) is matched according to the equation 3 e i ( Cψ ) = ∫ δ ( λ – ψ )si ( λ ) d λ = ∑ ∫ Aj ( W )Pj ( λ )Ts ( ψ )si ( λ ) d λ j (3.3-12) j=1 Now, consider an arbitrary color [C] with spectral energy distribution C ( λ ) . At wavelength ψ , C ( ψ ) units of the color are matched by C ( ψ )Ts1 ( ψ ) , C ( ψ )Ts2 ( ψ ) , C ( ψ )T s ( ψ ) tristimulus units of the primaries as governed by 3 3 ∫ C ( ψ )δ ( λ – ψ )si ( λ ) d λ = ∑ ∫ Aj ( W )Pj ( λ )C ( ψ )Ts ( ψ )si ( λ ) d λ j (3.3-13) j =1 Integrating each side of Eq. 3.3-13 over ψ and invoking the sifting integral gives the cone signal for the color (C). Thus 3 ∫ ∫ C ( ψ )δ ( λ – ψ )si ( λ ) d λ dψ = ei ( C ) = ∑ ∫ ∫ Aj ( W )Pj ( λ )C ( ψ )Ts ( ψ )si ( λ ) dψ d λ j j =1 (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 Tj ( C ) = ∫ C ( ψ )Ts ( ψ ) dψ j (3.3-15)
58 PHOTOMETRY AND COLORIMETRY FIGURE 3.3-1. Tristimulus values of typical red, green, and blue primaries required to match unit energy throughout the spectrum. 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 values represent units of energy, the physical interpretation of this mathematical result is that a color match can be obtained by adding the primary with negative tristimulus value to the original color and then matching this resultant color with the remaining primary. 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 to choose 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 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 3.3-2 Color space for typical red, green, and blue primaries.
COLORIMETRY CONCEPTS 59 FIGURE 3.3-3. Chromaticity diagram 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 T1 t 1 ≡ ----------------------------- - (3.3-16a) T 1 + T 2 + T3 T2 t 2 ≡ ----------------------------- - (3.3-16b) T 1 + T 2 + T3 T3 t 3 ≡ ----------------------------- - (3.3-16c) T 1 + T 2 + T3 Clearly, t3 = 1 – t 1 – t2 , 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.
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 Y(C ) = ∑ Tj ( C ) ∫ Aj ( C )Pj ( λ )V ( λ ) d λ (3.3-17) j =1 The integrals of Eq. 3.3-17, Y ( Pj ) = ∫ Aj ( C )Pj ( λ )V ( λ ) d λ (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 Y ( C ) = T 1 ( C )Y ( P1 ) + T 2 ( C )Y ( P 2 ) + T 3 ( C )Y ( P 3 ) (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 T1 ( C ) t 1 ( C ) = --------------------------------------------------------- - (3.3-20) T 1 ( C ) + T 2 ( C ) + T3 ( C ) and rearranging gives t 1 ( C )Y ( C ) T 1 ( C ) = ------------------------------------------------------------------------------------------------- - (3.3-21a) t 1 ( C )Y ( P1 ) + t 2 ( C )Y ( P2 ) + t 3 ( C )Y ( P 3 ) Similarly, t 2 ( C )Y ( C ) T 2 ( C ) = ------------------------------------------------------------------------------------------------- - (3.3-21b) t 1 ( C )Y ( P1 ) + t 2 ( C )Y ( P2 ) + t 3 ( C )Y ( P 3 ) t 3 ( C )Y ( C ) T 3 ( C ) = ------------------------------------------------------------------------------------------------- - (3.3-21c) t 1 ( C )Y ( P1 ) + t 2 ( C )Y ( P2 ) + t 3 ( C )Y ( P 3 ) Thus the tristimulus values of a color can be expressed in terms of the luminance and chromaticity coordinates of the color.
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 (P1), (P2), (P3) be the original set of primaries with spec- tral energy distributions P1 ( λ ), P2 ( λ ), P3 ( λ ), with the units of a match determined by a white reference (W) with matching values A 1 ( W ), A 2 ( W ), A 3 ( W ). Now, consider ˜ ˜ ˜ ˜ a new set of primaries ( P 1 ) , ( P2 ) , ( P3 ) with spectral energy distributions P1 ( λ ) , ˜ ˜ ˜ P2 ( λ ), P 3 ( λ ). Matches are made to a reference white ( W ) , which may be different than the reference white of the original set of primaries, by matching values A1 ( W ), ˜ ˜ ˜ A2 ( W ), A3 ( W ). Referring to Eq. 3.3-10, an arbitrary color (C) can be matched by the tristimulus values T 1 ( C ) , T2 ( C ) , T 3 ( C ) with the original set of primaries or by the ˜ ˜ ˜ tristimulus values T1 ( C ) , T 2 ( C ) , T 3 ( C ) with the new set of primaries, according to the matching matrix relations ˜ ˜ ˜ ˜ e ( C ) = KA ( W )t ( C ) = K A ( W )t ( C ) (3.4-1) The tristimulus value units of the new set of primaries, with respect to the original set of primaries, must now be found. This can be accomplished by determining the color signals of the reference white for the second set of primaries in terms of both ˜ sets of primaries. The color signal equations for the reference white W become ˜ ˜ ˜ ˜ ˜ ˜ ˜ e ( W ) = KA ( W )t ( W ) = K A ( W )t ( W ) (3.4-2) ˜ ˜ ˜ ˜ ˜ ˜ where T 1 ( W ) = T 2 ( W ) = T 3 ( W ) = 1. Finally, it is necessary to relate the two sets of primaries by determining the color signals of each of the new primary colors ( P1 ) , ˜ ˜ ˜ ( P 2 ) , ( P3 ) in terms of both primary systems. These color signal equations are ˜ ˜ ˜ ˜ ˜ ˜ ˜ e ( P 1 ) = KA ( W )t ( P 1 ) = K A ( W )t ( P1 ) (3.4-3a) ˜ ˜ ˜ ˜ ˜ ˜ ˜ e ( P 2 ) = KA ( W )t ( P 2 ) = K A ( W )t ( P2 ) (3.4-3b) ˜ ˜ ˜ ˜ ˜ ˜ ˜ e ( P 3 ) = KA ( W )t ( P 3 ) = K A ( W )t ( P3 ) (3.4-3c) where 1 0 0 --------------- - ˜ 0 ˜( P1 ) = A1( W ) t ˜ ˜ ˜ 1 - t ( P2 ) = --------------- t ˜ ˜( P2 ) = 0 A2 ( W ) ˜ 1 --------------- - A3( W ) ˜ 0 0
62 PHOTOMETRY AND COLORIMETRY Matrix equations 3.4-1 to 3.4-3 may be solved jointly to obtain a relationship between the tristimulus values of the original and new primary system: T1( C ) ˜ ˜ T1 ( P 2 ) T1 ( P 3 ) T2 ( C ) ˜ ˜ T2 ( P 2 ) T2 ( P 3 ) T3 ( C ) T3 ( P 2 ) T3 ( P 3 ) ˜ ˜ ˜ T 1 ( C ) = ------------------------------------------------------------------------ - (3.4-4a) T (W ˜ ) T (P ) T (P ) ˜ ˜ 1 1 2 1 3 ˜ ˜ ˜ T2 ( W ) T2 ( P 2 ) T2 ( P 3 ) ˜ ˜ ˜ T 3 ( W ) T3 ( P 2 ) T3 ( P 3 ) ˜ T1 ( P 1 ) T1 ( C ) ˜ T1 ( P 3 ) ˜ T2 ( P 1 ) T2 ( C ) ˜ T2 ( P 3 ) ˜ T3 ( P 1 ) T3 ( C ) T3 ( P 3 ) ˜ ˜ T 2 ( C ) = ------------------------------------------------------------------------ - (3.4-4b) ˜ T ( P ) T ( W) T (P ) ˜ ˜ 1 1 1 1 3 ˜ ˜ ˜ T 2 ( P1 ) T 2 ( W ) T2 ( P 3 ) ˜ T 3 ( P1 ) ˜ ˜ T 3 ( W ) T3 ( P 3 ) ˜ T1 ( P1 ) ˜ T 1 ( P2 ) T1 ( C ) ˜ T2 ( P1 ) ˜ T 2 ( P2 ) T2( C ) ˜ T3 ( P1 ) T 3 ( P2 ) T 3 ( C ) ˜ ˜ T 3 ( C ) = ------------------------------------------------------------------------ - (3.4-4c) ˜ T (P ) T (P ) T (W) ˜ ˜ 1 1 1 2 1 ˜ T2 ( P 1 ) ˜ T2 ( P 2 ) ˜ T2 ( W ) ˜ T3 ( P 1 ) ˜ T3 ( P 2 ) ˜ T3 ( W ) where T denotes the determinant of matrix T. Equations 3.4-4 then may be written ˜ ˜ ˜ in terms of the chromaticity coordinates ti ( P 1 ), ti ( P 2 ), ti ( P 3 ) of the new set of pri- maries referenced to the original primary coordinate system. With this revision, ˜ T1 ( C ) m 11 m 12 m 13 T1( C ) ˜ T (C) = m 21 m 22 m 31 T2( C ) (3.4-5) 2 ˜ T3 ( C ) m 31 m 32 m 33 T3( C )
COLOR SPACES 63 where ∆ ij m ij = ------ ∆i and ˜ ˜ ˜ ∆1 = T 1 ( W )∆ 11 + T 2 ( W )∆ 12 + T 3 ( W )∆ 13 ˜ ˜ ˜ ∆2 = T 1 ( W )∆ 21 + T 2 ( W )∆ 22 + T 3 ( W )∆ 23 ˜ ˜ ˜ ∆3 = T 1 ( W )∆ 31 + T 2 ( W )∆ 32 + T 3 ( W )∆ 33 ˜ ˜ ˜ ˜ ∆ 11 = t 2 ( P2 )t 3 ( P3 ) – t3 ( P 2 )t 2 ( P 3 ) ˜ ˜ ˜ ˜ ∆ 12 = t 3 ( P2 )t 1 ( P3 ) – t1 ( P 2 )t 3 ( P 3 ) ˜ ˜ ˜ ˜ ∆ 13 = t 1 ( P2 )t 2 ( P3 ) – t2 ( P 2 )t 1 ( P 3 ) ˜ ˜ ˜ ˜ ∆ 21 = t 3 ( P1 )t 2 ( P3 ) – t2 ( P 1 )t 3 ( P 3 ) ˜ ˜ ˜ ˜ ∆ 22 = t 1 ( P1 )t 3 ( P3 ) – t3 ( P 1 )t 1 ( P 3 ) ˜ ˜ ˜ ˜ ∆ 23 = t 2 ( P1 )t 1 ( P3 ) – t1 ( P 1 )t 2 ( P 3 ) ˜ ˜ ˜ ˜ ∆ 31 = t 2 ( P 1 )t 3 ( P2 ) – t3 ( P 1 )t2 ( P 2 ) ˜ ˜ ˜ ˜ ∆ 32 = t 3 ( P 1 )t 1 ( P2 ) – t1 ( P 1 )t3 ( P 2 ) ˜ ˜ ˜ ˜ ∆ 33 = t 1 ( P 1 )t 2 ( P2 ) – t2 ( P 1 )t1 ( P 2 ) Thus, if the tristimulus values are known for a given set of primaries, conversion to another set of primaries merely entails a simple linear transformation of coordinates. 3.5. COLOR SPACES It has been shown that a color (C) can be matched by its tristimulus values T1 ( C ) , T 2 ( C ) , T 3 ( C ) for a given set of primaries. Alternatively, the color may be specified by its chromaticity values t 1 ( C ) , t 2 ( C ) and its luminance Y(C). Appendix 2 presents formulas for color coordinate conversion between tristimulus values and chromatic- 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 of its tristimulus or chromaticity values. In this section we describe several standard and nonstandard color spaces for the representation of color images. They are categorized as colorimetric, subtractive, video, or nonstandard. Figure 3.5-1 illustrates the relationship between these color spaces. The figure also lists several example color spaces.
64 PHOTOMETRY AND COLORIMETRY nonstandard linear and nonlinear intercomponent colorimetric transformation linear linear intercomponent transformation colorimetric subtractive linear CMY/CMYK nonlinear RGB intercomponent transformation linear point nonlinear transformation point nonlinear colorimetric transformation intercomponent nonlinear transformation video video gamma gamma luma/chroma RGB YCC linear intercomponent transformation FIGURE 3.5-1. Relationship of color spaces. Natural color images, as opposed to computer-generated images, usually origi- nate from a color scanner or a color video camera. These devices incorporate three sensors that are spectrally sensitive to the red, green, and blue portions of the light spectrum. The color sensors typically generate red, green, and blue color signals that are linearly proportional to the amount of red, green, and blue light detected by each sensor. These signals are linearly proportional to the tristimulus values of a color at 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 intercomponent transformations.