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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 λ 5 C 2 λT⁄{}exp 1–[] -----------------------------------------------------= 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) 46 PHOTOMETRY AND COLORIMETRY 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 electromagnetic spectrum, 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. λ C 1 C 2 C λ() C 1 λ 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 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 FIGURE 3.1-3. Spectral energy distribution of CRT phosphors. 48 PHOTOMETRY AND COLORIMETRY 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 is specified by its luminous flux, as defined by (3.1-4) where represents the relative luminous efficiency and is a scaling con- stant. The modern unit of luminous flux is the lumen (lm), and the corresponding value for the scaling constant is = 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. FIGURE 3.1-4. Transmissivities of LCD color filters. C λ() FK m C λ()V λ()λd 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 ), (P 3 ) 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 , , 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 FIGURE 3.1-5. Relative luminous efficiency functions. C λ() A 1 W()A 2 W()A 3 W() 50 PHOTOMETRY AND COLORIMETRY , , when a match is obtained are recorded, and normalized match- ing values , , , called tristimulus values, are computed as (3.2-1) FIGURE 3.2-1. Color matching. A 1 C()A 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 C() A 3 C() A 3 W() ----------------= COLOR MATCHING 51 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 (P 3 ), 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 (P 1 ) and (P 2 ) with the overlapping region of (P 3 ) 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 (P 1 ) plus (P 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 W() ----------------= T 3 C() A– 3 C() A 3 W() ------------------= T 3 C() T 2 C() T 1 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 (3.2-3) 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 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() A– 2 C() A 2 W() ------------------= T 3 C() A– 3 C() A 3 W() ------------------= E λ() A 1 W()A 2 W()A 3 W() A 1 C()A 2 C()A 3 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 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): (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): (3.2-5) 7. Transitive law. If (M) matches (N) and if (N) matches (P), then (M) matches (P): ◊ ⊕• M() N()◊ P() Q()◊ M() P()⊕[]N() Q()⊕[]◊⇒∩ M() P()⊕[]N() Q()⊕[]◊ P() Q()◊[]∩ M() N()◊⇒ 54 PHOTOMETRY AND COLORIMETRY (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): (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: (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: (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 1 ()•[]A 2 C() P 2 ()•[]A 3 C() P 3 ()•[]⊕⊕◊ A 1 C() A 2 C() A 3 C() C λ() C λ() A 1 C()P 1 λ() A 2 C()P 2 λ() A 3 C()P 3 λ()++ A j C()P j λ() j 1 = 3 ∑ =◊ [...]... 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... tristimulus values for another set of primaries (16) Let (P1), (P2), (P3) be the original set of primaries with spectral 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... 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 ˜ ˜ ˜( P1 ) = A1( W ) t 0 0 0 1 ˜