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Effect of Luminance of Samples on Color Discrimination Ellipses: Analysis and Prediction of Data Ralph W. Pridmore,1* Manuel Melgosa2 1

Central Houses P/L, 8c Rothwell Rd., Turramurra, New South Wales 2074, Australia

2

´ ptica, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Departamento de O

Received 31 January 2004; revised 6 August 2004; accepted 24 September 2004

Abstract: Four data sets are analyzed to quantify three effects of luminance of samples on chromaticity discrimination: on ellipse area, axis dimensions (a and b), and a/b ratio. Ellipses for aperture, surface, and simulated surface colors in CIE 1931 and 1964 x, y, Y color spaces are shown to reduce axis dimensions with higher luminance by different functions for the major and minor axes. Reduction is greater for major than minor axes, thus improving ellipse circularity. The functions plot straight lines in log-log scale as power law equations, except luminances below 3 cd/m2. We give formulae to predict a and b axes, a/b ratio, and ellipse area for almost any luminance in x, y, Y spaces. Effect of luminance is remarkable on ellipse area, which on average halves with every 3.5 times higher luminance. To illustrate the substantial effects of luminance, RIT-DuPont ellipses are predicted for three levels of equal luminance at 42, 212, and 2120 cd/m2. In the latter, ellipses are much smaller and are nearer circular than in the former. Higher luminance is known to improve color discrimination, so reduced ellipse area is to be expected but does not occur in CIELAB and DIN99 spaces because of lack of luminancelevel dependency. We discuss our results’ implications on uniform color space. Weber fraction ⌬Y/Y indicates brightness discrimination decreases with increasing luminance and is thus independent of chromaticity discrimination. © 2005 Wiley Periodicals, Inc. Col Res Appl, 30, 186 –197, 2005; Published online in Wiley InterScience (www.interscience.wiley. com). DOI 10.1002/col.20107

Key words: color appearance; color matching; color difference ellipses

*Correspondence to: Ralph W. Pridmore (e-mail: rpridmo@bigpond. net.au) © 2005 Wiley Periodicals, Inc.

186

INTRODUCTION

Luminance is a major inﬂuence on chromaticity discrimination. The most common example, experienced by ordinary people in their everyday work as well as by scientists in the laboratory, is that discrimination improves under stronger illumination. It follows that chromaticity discrimination ellipses become smaller with higher luminance, up to a limit. Commencing with Brown’s seminal data on the effect of luminance on chromaticity discrimination ellipses,1 the effect has been reported severally but few conclusions have been drawn. Brown was the ﬁrst to demonstrate that lower luminance increases ellipse dimensions in CIE x, y, Y space, especially at luminance levels below 3 cd/m2. Luo and Rigg made perhaps the ﬁrst attempt to formulate this effect,2 assuming equal functions for the major and minor axes (a and b), to predict ellipses at an equal luminance. Since then, the effects of luminance have been reported explicitly or implicitly in several papers3– 8 and in several color spaces, including CIE 1931, CIE 1964, and CIELAB. The reports demonstrate that discrimination ellipses in CIE x, y, Y spaces reduce size with higher luminance3,6,7 but remain about the same in CIELAB space.3,6 However, the effect was not formulated and the effect on a and b axes (and thus a/b ratio) was not differentiated. The a/b ratio is signiﬁcant because if a/b decreases with say higher luminance, ellipses become more circular. These effects may bear on current efforts to develop new color difference formulae and spaces. The CIE and many workers1–9 over the past 50 years have called for more rigorous study of the effects of luminance on chromaticity discrimination. There are now sufﬁcient data in the literature to enable meaningful analysis and general conclusions on the effect of luminance in the parameters of ellipse area, axis dimensions a and b, and a/b ratio. The crucial parameter is the axis dimensions, on COLOR research and application

TABLE I. Characteristics of the four data sets analyzed.

Data set

Mode of appearance

Brown et al.1

Method to deﬁne ellipse

Aperture (dark surround)

Distribution of matched results (including L direction) Simulated object on CRT Discrimination probability of (brighter surround) samples from the reference (including L direction) Object (neutral surround Probability that difference in with L* ⫽ 38; sample pair is judged larger simulated daylight of than the difference of * ⫽ 1.02) 6100 K; 2000 1x) anchor pair (⌬Eab (including L direction) Aperture (darker Discrimination probability of surround) samples from the reference (not including L)

Melgosa et al.3 RIT-DuPont5,6

Yebra et al.7,8

which the other parameters depend. The present article analyzes and formulates (for CIE 1931 and 1964 x, y, color spaces) the effect in four data sets covering aperture, surface, and simulated surface colors,1,3,5– 8 referred to here as Brown, Melgosa, RIT-DuPont, and Yebra data sets. (RITDuPont data5 were converted to ellipsoids at a later date,6 thus allowing comparison.) We give equations for each of the four data sets and average equations applicable to most data sets to enable prediction of a and b axis dimensions as a function of luminance from 3.0 to 10,000 cd/m2. Equations for area and a/b ratio are also given. The equations represent a ﬁrst approach to formulating these three effects of luminance, and may need adjustment to accommodate subsequent data or analyses. The RIT-DuPont set of ellipses are predicted in x, y, Y space at three equal luminances (about 40, 200, and 2000 cd/m2) to illustrate the substantial effects of luminance. ANALYSIS OF DATA SETS

The major characteristics of the four data sets to be analyzed below, including methodology and mode of appearance, are summarized in Table I. The numbers of centers are those analyzed here, rather than those in the original data. Brown Data Set Brown’s color stimuli subtended a 20 angle and were observed with a dark surround.1 His data for two observers (himself and the renowned scientist D.L. MacAdam) are well known to demonstrate the decrease in ellipse areas and axis dimensions with higher luminance. His data cover the range from 0.1 to 24 cd/m2. He noted that below 1 or 2 cd/m2 the size of ellipses increased rapidly and ascribed this to a normal eye’s tritanopic errors in dimly lighted ﬁelds. For this reason, his very low luminance data are omitted from analysis here. His graphs are in the CIE 1931 x, y, chromaticity diagram, and distance in this space is measured as x, y, distances equivalent in any direction. Although ellipse sizes may vary by chromaticity area of CIE diagram, the present study concerns the relative variation of size with Volume 30, Number 3, June 2005

Target

Number Number L levels of of per observers centers center

L range (cd/m2)

2° circular

2

4

3 to 6

0.1 to 24.0

3° by 6° rectangular

2

5

5

2.4 to 55.3

10° by 3.8° rectangular (test pair)

50

4

2 or 3

11.1 to 401.5

2

8

2 or 3

2.4 to 19.8

2° circular

luminance more than absolute size. The focus is on axis dimensions (note we use the axis rather than semiaxis dimensions in the ﬁgures below). Figure 1(A) shows Brown’s ellipse axis dimensions plotted by luminance and size, that is, CIE x, y, distances. Luminance is to log scale. The data points show the mean of two observers, with a and b axis dimensions multiplied by 100 (as in all ﬁgures below) to give more manageable numbers. Data for some intermediate luminances are omitted to simplify the curves. The major axis a and the minor axis b are shown separately, for each of Brown’s principal color centers labeled #4, #19, #31, and #34. Over most of his luminance range 0.1–24 cd/m2, a given luminance applies to two or three of the four colors. The indicated mean curves a and b are obtained by arithmetic means and curve smoothing and for the most part represent means of three colors. For each of the four colors, major axis a reduces size more rapidly than minor axis b, indicating that ellipses becomes closer to circular with higher luminance. Color #4 (572 c, purple in this light source) is too low in luminance to be of much use in averaging. Its axis a reduces size more rapidly than the other three colors; #34 is blue 462 nm, #31 is green 501 nm, and #19 is white (the source is about 3400 K correlated color temperature), so the coverage of chromaticity diagram is rather limited. The mean a and b curves for the latter three colors, from 2 to 24 cd/m2, are formulated as power law functions and extrapolated to 100 cd/m2. Figure 1(B) shows the data in log-log scale. Below about 2 cd/m2, the mean a and b curves remain curved, and above 2 cd/m2 they become straight lines, representing power law functions (labeled). Evidently, as Brown noted, color discrimination is a quite different function for luminances ⬍2 cd/m2. The mean a and b curves from 2 to 24 cd/m2 are straight lines that exactly align with their power law functions in this log-log scale. So it is reasonable to extrapolate the lines according to their functions to higher luminances to indicate approximately the respective ellipse dimensions. The a and b lines will intersect at some point representing unit ratio in a/b dimensions. It is well known that higher luminance improves color discrimination, that is, reduces ellipse area,10 up to a very high luminance (say, about 187

of both observers. The general trend of the curves shows axis dimensions decrease with higher luminance, and that axis a decreases more rapidly than axis b, similarly to Brown’s data. Figure 2(B), in log-log scale, shows power law trend lines for each color’s a and b axes. As in Brown’s data, the mean line for each axis is found by arithmetic means, giving equal weight to each color center, the power law function is determined (as labelled), and the two lines are extrapolated to 10,000 cd/m2. They intersect at 8200 cd/m2, very similar to Brown’s data, but the curve slopes are considerably steeper. RIT-DuPont Data Set RIT-DuPont data do not explicitly report on luminance effects, but several of their color centers are of sufﬁciently similar chromaticity to allow general deductions. These are valuable in extending in principle the accurate deductions from other data sets to the much higher luminances of RIT-DuPont data. The similar chromaticities, using the number order in both Refs.5,6 are as follows: #3, #10, and #18, Gray; #11 and #13, both Cyan 487– 487.5 nm; #1 and

FIG. 1. (A) Brown’s data1 for the means of two observers, showing axis (not semiaxis) dimensions a (dashed lines) and b (solid lines) at 100 times actual size, for four color centers #4 (purple), #19 (white), #31 (green), and #34 (blue), plotted to luminance in log scale. The indicated “smoothed mean curves” generally represent the latter three colors and are extrapolated by their power law eqns (see B) to 100 cd/m2. (B) As in A but showing axes a and b to log-log scale for three colors (excluding #4). The curves become straight lines in log-log for luminance ⬎ 2 cd/m2; their power law functions are labeled and extrapolated to their intersection at 8,100 cd/m2.

25,000 cd/m2) where glare starts to degrade color discrimination. The straight lines intersect at 8100 cd/m2, implying that the mean ellipse of Brown’s data would be a circle at this luminance. However, surface colors are rarely such high luminances, implying they will not normally give circular ellipses. Melgosa Data Set Melgosa et al.3 give detailed data on the luminance effect on ellipsoids for ﬁve color centers on a CRT device for two observers. The authors employed ﬁve levels of luminance for each of the ﬁve CIE-recommended color centers:9 Gray (varying from 2.4 to 55 cd/m2), Green (13– 43 cd/m2), Red (6 –26 cd/m2), Blue (6 –22 cd/m2), and Yellow (10 –53 cd/m2). They noted the decrease in area of ellipses in the CIE x, y diagram when luminance was increased toward that of the surround. Figure 2(A) shows the ellipse dimensions a (dashed line) and b (solid line) in semilog scale for the mean 188

FIG. 2. The Melgosa data set,3 showing ellipse dimensions (100 times actual size) for the means of two observers. (A) Showing the a (dashed line) and b (solid line) ellipse dimensions of ﬁve colors. (B) As in A but in log-log scale and with power law trend lines (solid) ﬁtted to the data (all dashed). The mean a and mean b of the trend lines are extended by their power law functions (labeled) to their intersection at 8,200 cd/m2.

COLOR research and application

#14, both Blue 472– 476 nm; and #15 and #16, both redPurple 489 – 491.5 c. The latter pair are rather different purities but close enough in wavelength to be suitable, and the large luminance difference (15–166 cd/m2) assures any change in ellipse dimensions will be in the correct direction. Other pairs of similar chromaticities are too similar in luminance to allow safe deductions. The above wavelengths pertain to the CIE 1964 x10, y10 space indicated in Ref. 5 and would be some 6 nm longer in wavelength in CIE 1931 space. Figure 3(A) shows the four color areas (as distinct from color centers) and their two or three data points each to semilog scale. Figure 3(B) shows the data in log-log scale with power law trend lines ﬁtted to the Gray color’s axes [as also in Fig. 3(A)]. The b dimension of the Blue line is eccentric in rising slightly instead of falling, possibly due to the wavelength difference (4 nm) of colors #1 and #14. Like #15 and #16, this pair of colors can only allow broad deductions as to luminance effect. The mean a and b lines (determined as above) to each set of four lines are extrapolated to 10,000 cd/m2. Their point of intersection, calculated from their functions, is at 12,800 cd/m2. The slopes of the two mean lines show, like the other data sets, a distinctly steeper slope for a than for b dimensions. The slopes are similar to Brown’s data. FIG. 4. Ellipse dimensions (100 times actual size) for eight color centers from the Yebra data7 for means of two observers. (A) As in Fig. 2(A), but with mean trend lines for a and b axis dimensions. (B) As Fig. 2(B), in log-log scale and with mean trend lines for a and b axis dimensions. The dash-dot lines indicate extrapolated mean curves for the asterisked data in Table I.

Yebra Data Set

FIG. 3. Ellipse dimensions (100 times actual size) for four color areas, labeled, from the RIT-DuPont data.5,6 (A) As in Fig. 2(A). (B) As in Fig. 2(B), in log-log scale. The point of intersection is 12,800 cd/m2.

Volume 30, Number 3, June 2005

Yebra et al.7 give detailed data on 18 color centers and their color discrimination ellipses in relation to luminance at various levels from 2 to 20 cd/m2. They concluded that ellipse size increases with decreased luminance, particularly below 3 or 4 cd/m2 in agreement with Brown. Table II lists ellipse dimensions and a/b ratios8 as not available in the general literature. The authors found no clear a/b relationship as a function of luminance. As Table II shows, for the selected ellipses there are eight instances of decreasing a/b ratio (becoming more circular) and ﬁve instances of increasing ratio, with higher luminance. However, when graphed in Fig. 4(A) the mean curve (found as above) for axis a is clearly steeper than for axis b; that is, the mean a/b ratio clearly decreases with higher luminance. Color center #1 is omitted from Fig. 4 because of its very large interobserver variability: observer JA indicates that a/b ratio increases with each higher luminance (by an unusually large factor, four times overall), whereas YA indicates a decrease in ratio. Figure 4(B) shows the same functions and mean curves (with functions labeled) in log-log scale. Extrapolated by their functions, they intersect at 153,000 cd/m2, off the graph. This is a much higher luminance than in Figs. 1–3. It 189

TABLE II. Semiaxis dimensions a and b, and a/b ratios, of ellipses in the Yebra data (100 ⫻ actual sizes)7,8 for two observers. CIE 1931 Color 1 1 1 2 2 3 3 3 4 4 4 5 5 6 6 6 7 7 8 8 9 9

observer JA y

Y cd/m2

a1

b1

a/b

a2

b2

a/b

Mean a/b

0.175 green

0.567

0.175 green 0.378 yellow

0.413

0.356 yellow

0.38

0.333 gray 0.45 orange

0.333

0.395 red 0.175 green 0.423 red

0.302

2.41 13.02 19.83 2.41 13.02 2.41 13.02 19.83 2.41 13.02 19.83 2.41 13.02 2.41 13.02 19.83 2.41 13.02 3.87 19.83 3.87 19.83

3.26 3.36 4.07 3.04 1.7 2.85 1.58 1.7 3.43 1.67 1.09 3.5 2.14 2.5 2.32 1.7 2.22 1.08 2.51 2.34 3.31 0.99

1.99 0.91 0.65 2.49 1.25 0.94 0.65 0.7 1.24 0.97 0.94 0.8 0.75 0.95 0.86 0.54 0.82 0.48 2.01 1.38 0.89 0.46

1.64 3.69 6.26 1.22 1.36 3.03 2.43 2.43 2.77 1.72 1.16 4.37 2.85 2.63 2.7 3.15 2.71 2.25 1.25 1.7 3.72 2.15

4.29 2.39 1.87 3.12 1.82 3.27 1.29 2.39 3.36 2.84 3.41 5.49 2.48 6.57 2.51 2.37 4.67 1.07 2.54 2.83 2.51 0.89

2.14 1.31 1.1 2.78 1.24 1.05 0.74 0.43 1.24 1.05 0.92 0.88 0.72 0.92 0.83 0.69 0.84 0.48 2.18 0.93 0.65 0.44

2 1.82 1.7 1.12 1.47 3.11 1.74 5.56 2.71 2.7 3.71 6.24 3.44 7.14 3.02 3.43 5.56 2.23 1.17 3.04 3.86 2.02

1.82 2.75 3.98 1.17 1.41 3.07 2.08 3.99 2.74 2.21 2.43 5.3 3.14 4.88 2.86 3.29 4.13 2.24 1.21 2.37 3.79 2.08

x

0.427

0.378

0.487 0.341

Ratio diff. from min Y

observer AY

a

Larger Larger Larger* Smaller* Larger Smaller* Smaller Smaller* Smaller* Smaller Smaller* Larger* Smaller* 2

Their data for intermediate luminances L that are not at least 5 times greater than the minimum available L (2.41 or 3.87 cd/m ) are omitted from this table. The last column notes whether the mean a/b ratio is larger or smaller than in the minimum luminance; where the note is true (or not signiﬁcantly contradictory) of both observers and thus more reliable, it is asterisked (see Fig. 4).

may be seen from Table II that there is considerable interobserver variation in some a/b ratios, sometimes disagreeing on whether the ratio for a much higher luminance is larger or smaller ratio than in the minimum luminance. If these disagreed ratios are omitted, and the remainder (asterisked in Table I) are plotted to Fig. 4(B) (dash-dot lines),

their mean lines for the a and b axes intersect, when extrapolated, about 3,000 cd/m2. (In determining an average equation for all data sets, below, the full set of data in Table I excluding color #1 will be used.) Summary The four data sets are compared in Fig. 5, which shows the mean a and b axis dimensions for each data set to the same scale (100 times actual size). The mean dimension is represented by its power law function (labeled, from Figs 1 to 4) and extrapolated to 10,000 cd/m2 and to 3 cd/m2 if necessary. All data sets agree that axis dimensions decrease as a function of higher luminance and that the a function gives a steeper gradient than the b function. DISCUSSION OF DATA

FIG. 5. Mean a and b curves for Brown, Melgosa, RITDuPont, and Yebra data sets compared at the same scale, that is, the given axis dimensions (or semiaxis ⫻ 2) times 100. All curves are extrapolated (by the labeled functions, as straight lines in log-log scale) to 10,000 cd/m2. Steepest slopes, in decreasing order, are for Melgosa, Yebra, RITDuPont, and Brown data. Dashed black lines denote the average or General Eqn.

190

Four data sets are analyzed above to determine how axis dimensions a and b vary as a function of luminance. The results agree in demonstrating that (1) ellipse area decreases with higher luminance, and (2) a/b ratio decreases with higher luminance, thus improving circularity. Deduction (1) has been made several times in the literature,2,3,7 usually in respect of the far larger areas for luminances ⬍ 3 cd/m2. Yebra et al.7 focused on the range 2–20 cd/m2, over a large variety of hues, to conﬁrm that a similar effect exists in this intermediate range. The present study analyzes data over a wide range of luminance from 2 to 400 cd/m2 (i.e., up to 55 cd/m2 in the Melgosa data and up to 400 cd/m2 in the RIT-DuPont data) and conﬁrms the consistent decrease of COLOR research and application

ellipse area with higher luminance. Deduction (2) is novel in the literature, and its implications on uniform color space are discussed under Conclusions and Implications. The variation in a/b ratio as a function of chromaticity has been variously hypothesized, for example, along lines of deuteranopic and tritanopic confusion lines in CIE x, y space.7,11,12 Ellipses in the Brown data set (gained from color matching) are somewhat smaller than the other data sets, which are all gained from color discrimination. The variation in ellipse size between data sets and methodologies (let alone between chromaticities) is often discussed in the literature but remains unresolved in quantitative terms. Indow et al.11 noted their results for aperture color ellipses obtained by direct matching (e.g., using a CRT or a colorimeter as in MacAdam’s12 and in Brown’s1 experiments) are not signiﬁcantly different in size from simulated surface color ellipses obtained by the same method. Indow noted that his aperture color ellipses (about 30 cd/m2) were slightly larger than MacAdam ellipses,12 but noted the latters’ higher luminance (50 cd/m2); the slight difference in size can be understood from the present results on luminance effect. The MacAdam ellipses (for which MacAdam was not the observer) are of similar size to Brown’s ellipses for similar chromaticities, although MacAdam’s luminance is about two times higher than Brown’s highest luminances. Brown’s data may be taken as generally representative of data from direct matching.11,12 Indow noted that ellipses obtained from matching were considerably smaller than those obtained by paired comparison (for real surface colors)6,2 but that the different methodologies made direct comparison of sizes problematical. The present results (Fig. 5) indicate that Brown’s ellipses (from color matching) are rather smaller than other data, taken at an equal luminance. As to ellipse dimensions as a function of chromaticity, Luo and Rigg2 note that ellipse size varies systematically with chromaticity, as does a/b ratio. However, the system varies between sets: for example, in their data2 the smallest ellipses are for gray or gray-blue with larger ellipses for saturated blues, whereas the smallest ellipses in MacAdam data12 are for saturated blues. This issue is discussed later, relative to RITDuPont ellipses predicted at equal luminances. Present results on ellipse dimensions with varying luminance (Figs 1– 4) give no consistent indication of whether any hues give steeper slopes than others. For example, Blue ellipses display steep a and b slopes in Fig. 1 but ﬂat slopes in Figs 2 and 3. In Fig. 4, Green ellipses display both steep and ﬂat slopes; the Gray ellipse shows steep slopes here and in Fig. 2 but ﬂattish slopes in Figs 1 and 3. Though more data are required it seems that all colors, of all purities including gray, reduce axis dimensions as similar functions of luminance. The present results demonstrate not only that ellipses become nearer to circles with higher luminance but also that larger ellipses (in a given data set) reduce size more rapidly than smaller ellipses. This means that extreme a/b ratios automatically adjust toward unit ratio with higher Volume 30, Number 3, June 2005

luminance. This is further discussed below relative to predicted ellipses (Fig. 6). Though ellipses may vary area and shape as functions of chromaticity in x, y space the available data, as already mentioned, do not allow a convincing and quantitative hypothesis. In any case, such differences reduce substantially with higher luminance. In the interests of simplicity, a set of general equations for all chromaticities seems appropriate at this stage of knowledge. This appears to be the ﬁrst formulation of the separate luminance effects on a and b ellipse dimensions from analysis of several data sets and represents therefore a ﬁrst approach to the issue. It may eventuate that the equations represent only a ﬁrst order of accuracy and require modiﬁcation as more data become available. DERIVATION AND APPLICATION OF FORMULAE

This section formulates the effect of luminance on ellipse axis dimensions for each data set and as an average of the four data sets. The data sets’ individual equations are given as Eq. (1)–(4). The average is termed the general equation, as Eq. (5). These equations apply to luminances from 3.0 to 10,000 cd/m2, or possibly greater, although the slope must level off to horizontal at some luminance, say about 30,000 cd/m2, before reversing slope. Equations (1)–(5) all apply to ellipse axis dimensions times 100, as in Figs 1–5. If used for semiaxes, the factors (but not the exponents) should be divided by 2; if used for actual dimensions the factors should be divided by 100. Equation (1) derives from the Brown data (Fig. 1), Eq. (2) from the Melgosa data (Fig. 2), Eq. (3) from the RITDuPont data (Fig. 3), and Eq. (4) from the Yebra data (Fig. 4). Equations (a) and (b) refer to axes a and b in all cases. The symbol x refers to luminance (cd/m2), and a or b refers to the axis (not semiaxis) dimension in CIE x, y, space. a ⫽ 1.426 x ⫺0.27

(1a)

b ⫽ 0.4667 x ⫺0.146

(1b)

a ⫽ 10.261 x ⫺0.494

(2a)

b ⫽ 2.181 x ⫺0.322

(2b)

a ⫽ 8.42 x ⫺0.33

(3a)

b ⫽ 1.65 x ⫺0.17

(3b)

a ⫽ 9.359 x ⫺0.339

(4a)

b ⫽ 2.939 x ⫺0.242

(4b)

The general equation [Eq. (5)] represents the average function of the above individual equations and thus represents most situations. It may be used to predict ellipse axes whose functions with luminance fall between Eqs. (1)–(4) a ⫽ 7.366 x ⫺0.358

(5a)

b ⫽ 1.809 x ⫺0.220

(5b) 191

TABLE III. Results from Eqs. (1) to (4) showing the percentage relative error and the percentage interobserver variability for the four data sets and the luminance where axes a ⫽ b (i.e., a perfect circle). % Interobserver variability

% Relative error

Brown Melgosa Yebra RIT-Dupont AVERAGE

a

b

a

b

23.6 29.9 24.0 23.1 25.2

23.2 30.7 30.9 42.5 31.8

23.0 20.8 31.3 — 25.0

13.0 24.6 14.5 — 17.4

Luminance cd/m2 for a ⫽ b 8,100 8,200 153,000 12,800

Table III shows for each data set and the corresponding Eqs. (1)–(4) the percentage relative error, deﬁned as 100 times the absolute value of the difference between experimental and predicted values divided by the experimental values. The percentage of interobserver variability, deﬁned as 100 times the absolute value of the difference between two observers divided by its average, is also shown (except for the RIT-DuPont data set, where only average data are provided), as well as the luminances where axes a ⫽ b. Table IV shows analogous results to those in Table III but using the general equation [Eq. (5)], after adjustment of predicted values by appropriate scale factors, deﬁned as the average of the ratio of experimental and predicted values. From Tables III and IV it can be concluded that predictions made by Eqs. (1) to (5) have a similar relative error to that attributable to interobserver variability, predictions of axis b being in general worse than those of axis a. Equations (6a) and (6b) give factors A and B to be applied to a given ellipse and luminance to calculate that particular ellipse’s dimensions in a target luminance. Their use is exempliﬁed in predicting RIT-DuPont ellipses at equal luminances in Fig. 6 as follows: A ⫽ a 2/a 1

(6a)

B ⫽ b 2/b 1

(6b)

where a1 and a2 denote a values (axis size) at the original and the target luminances respectively as calculated by Eq. (5a); b1 and b2 similarly denote the b values calculated from Eq. (5b); and A and B denote the factors by which to multiply the original axis dimensions to predict them at the target luminance. Figure 6 shows examples of the equations’ application, converting/predicting all RIT-DuPont ellipses to three levels of equal luminance, using Eqs. (5a) and (5b). A worked example follows. The reference white for RIT-DuPont data is 2000 lux at the adapting surround, and taking it as a Lambertian surface its luminance L is 636.6 cd/m2 per Eq. (7).10 E/ ␲ ⫽ L

(7)

where E is illuminance lux (or lumens/m2) and L is luminance cd/m2. A normalized Yo⫽100 has been employed.6 192

Consider RIT-DuPont color #1, of Y factor 8.67. This Y divided by 100 and times 636.6 (or 2000 lux /␲) gives L 55.2 cd/m2. The target luminance is chosen to be 212 cd/m2 (as in Fig. 6C). The a axis dimension is calculated by Eq. (5a), for the original and target luminances of color #1, and the ratio of dimensions becomes factor A in Eqn (6a). From Eq. (5a), the dimension of a at 55.2 cd/m2 is 1.7523, and the dimension of a at 212 cd/m2 is 1.0824. From Eq. (6a), A ⫽ 1.0824/1.7523, that is, 0.618. The factor A times the original ellipse axis a (Ref. 6’s Table III gives semiaxes in x, y space) gives its new size at 212 cd/m2. This is plotted in Fig. 6 (times 3 for clarity). Similarly, axis b is found from Eqs. (5b) and (6b), and an ellipse is ﬁtted to both axes by computer graphics. Figure 6(A) shows the RIT-DuPont ellipses (times 3) in CIE 1964 space at the semiaxis dimensions and Y factors given in Ref. 6. For clarity, the three overlapping Gray ellipses are shown in the insets to Figs. 6(A)–(D), multiplied a further 2 or 3 times as indicated. The mean Y of the 19 color centers in Ref. 6 is 24, that is, about 1⁄4 of Y0 ⫽ 100. The adapting illuminance in Figs. 6(B)–(D) is taken as similarly 4 times the equal illuminance level for colors in those ﬁgs. Figure 6(B) shows the 19 color centers predicted by Eqs. (5a), (5b), (6a), and (6b) at equal Y factor 6.6, that is, 42 cd/m2. [Fig. 6(B)’s Y factor of 6.6 was chosen as 1⁄4 of the mean Y (i.e., 24) and of color #3 (Y ⫽27.4), the Gray recommended by CIE.9] Because this Y is lower than most of the various Ys (from Ref. 6) in Fig. 6(A), most ellipses are larger than in Fig. 6(A) and a few are smaller. The original ellipse orientations6 are employed throughout Fig. 6, because prediction of reorientation with luminance remains uncertain,3 and is not within this article’s scope. Figure 6(C) shows the same color centers at equal 33.3 Y, that is, 212 cd/m2 (5 times higher than that in Fig. 6(B); the difference from exact times 5 is due to rounding of Y and cd/m2). Note most ellipses are now smaller (e.g., #13 and #15) than their original size, a few are bigger (#8, #18, and #19), and some have hardly altered (e.g., #9 and #5, whose original Y factors were near 33.3 Y). Red-purple #15, the largest ellipse in Fig. 6A, has reduced to near the average size, demonstrating that its original size is because of its very low luminance. Figure 6(D) shows the situation at 333 Y, that is, 2120 cd/m2, where all ellipses are smaller than their original sizes. Half the ellipses are near circular (say, a/b ⬍ 1.5:1), 58% are ⬍ 1.6:1, and only 16% (#1, #17, and #18) are ⬎ 2:1 ratio (#1 and #17 are worst, at 2.9:1). Gray ellipses remain distinctly elliptical, though #10 is near circular at 1.48:1. Yellow #19, Magenta #15, and Cyan #13, maintain a relatively large size at all equal luminances, suggesting they are innately large ellipses at all luminances. Note that ellipses #15 and #16, of similar chromaticities but 1:11 ratio in luminance and 3:1 ratio in area in Fig. 6(A), are of similar size at any level of equal luminance (Figs. 6(B)–(D), as one would expect. Together with similar indications elsewhere, a principle is demonstrated: that a dimensional relationship COLOR research and application

FIG. 6. The 19 color discrimination ellipses (3 times actual size) of the RIT-DuPont data set5,6 at their various luminances. Insets show Gray color centers, further multiplied by 2 or 3 as indicated. (A) The original ellipses plotted to CIE 1964 color space from data.6 The central cross is the color center for grays #3 and #10; lower bar to the cross is center of gray #18. (B) Dimensions of the 19 ellipses predicted by Eqs. (5a), (5b), and (6) for equal luminance factor 6.6 Y (42 cd/m2). (C) As for (B) but predicted for equal luminance factor 33.3 Y (212 cd/m2). (D) As for (B) but predicted for equal luminance factor 333 Y (2120 cd/m2).

between ellipses at equal luminances does not hold at different luminances; or a relationship between ellipses at a

TABLE IV. Results from the general equation Eq. (5), showing % relative error in predicting actual data of the indicated data set, the scale factors used to adjust predicted data (to allow for different relative sizes of ellipse dimensions between data sets), and the % interobserver variability. Scale factors

Brown Melgosa Yebra RIT-Dupont AVERAGE

% Relative error

% Interobserver variability

a

b

a

b

a

b

0.23 0.97 1.44 1.30 0.99

0.32 0.94 1.76 1.11 1.03

29.0 28.7 26.0 24.8 27.1

25.4 33.9 36.0 39.9 33.8

23.0 20.8 31.3 — 25.0

13.0 24.6 14.5 — 17.4

Volume 30, Number 3, June 2005

certain ratio of luminances does not hold at any different ratio of luminances. The implications to uniform color space are discussed later. Concerning extreme a/b ratios, note the very narrow ellipses #1 and #17 in Figs. 6(A) and 6(B) (where the a/b ratio is 5:1) and their more uniform ratios in Fig. 6(D) (where the ratio is 2.9:1). With no special provision in equations, extreme ratios rapidly adjust toward uniform ratios with increasing luminance. In plotting the ellipses to Fig. 6(D) for high luminances, a dilemma arose when the a axis of an ellipse (#4) was predicted by the equations to become smaller than the b axis. (To lesser degree, the same dilemma arose for #15 and #16; other such cases arise for yet higher luminances.) Would this be the case in real vision, or would the a slope [e.g., in Fig. 1(B)], on intersecting the b axis, adopt the same slope as b, thus maintaining circularity into higher luminances? If the latter were the case, one by one all ellipses 193

would become circular, giving a high luminance uniform color space (UCS). The alternative is that ellipses attain circularity at various luminances (depending on how near to circular the original ellipses are) and at that point swap axes a for b and then degrade in circularity. The former is the more desirable as it offers a UCS. On this assumption, ellipse #4 was held to circularity with its a axis adopting the b axis’ more gradual reduction in size [per Eq. (5)], both giving the same dimension at 2120 cd/m2. Sadly, this ideal situation can be shown to be false [e.g., by Eqs. (9) and (10)] in that it causes major eccentricities in areas and a/b ratios. It is certain that ellipses become circles only momentarily (one recalls the adage that the only constant in nature is change.) For the Eqs. (5a) and (5b), where ratio a/b is 3.49905 at 3 cd/m2 and 1.0 at 26,000 cd/m2, the average dimension ratio a/b (say, D) may be predicted by Eq. (8) as follows: D ⫽ 4.0725 x ⫺0.1381

(8)

where x is luminance cd/m . Some guidelines are D ⫽ 3.0 at 10 cd, 2.0 at 170 cd, 1.7 at 500 cd, and 1.57 at 1,000 cd/m2. D reduces by 1.374 times whenever luminance increases by 10 times. Dimension ratios a/b for RIT-DuPont ellipses in Fig. 6 were analyzed for the three levels of equal luminance. The average relationship to luminance, for a given ellipse, is as follows: 2

共L 2/L 1兲

0.135

⫽ D 1/D 2

(9)

with L2 ⬎ L1, and where D1 is a/b ratio at luminance L1 and D2 is a/b ratio at L2. Equation (9) permits prediction of D for a given ellipse at any luminance, for example, as: D2 ⫽ D1/ (L2 / L1)0.135. Ratio D predicted by Eq. (9) is very similar (within 3%) to D predicted by Eq. (8) but can be applied to a particular chromaticity or ellipse of given dimensions, whereas Eq. (8) cannot. The average relationship between ellipse area (A) and luminance, again derived from the RIT-DuPont ellipses in Fig. 6, is calculated as follows: 共L 2/L 1兲 0.58 ⫽ A 1/A 2

(10)

with L2 ⬎ L1 and where A1 is ellipse area at luminance L1 and A2 is area at luminance L2. From Eqs. (5a) and (5b), where the average ellipse area (remember that ellipse dimensions in all equations are 100 times actual x, y distances) is 5.546 at 3 cd/m2 and 0.0294 at 26,000 cd/m2, ellipse area A may be predicted as follows, where x is luminance cd/m2: A ⫽ 10.464 x ⫺0.578

(11)

Hence area A reduces by 3.78 times when luminance increases by 10 times, for any luminance range; or by 2.53 times when luminance increases by 5 times. This is a surprisingly large effect. Ellipse area is affected by luminance much more (nearly 3 times more) than is a/b ratio. Ratio A predicted by Eq. (11) is very similar (within 4%) to A predicted by Eq. (10). Of the four data sets analyzed, the 194

lowest rate of change of A with luminance is for the Brown data, for which the exponent in Eq. (10) becomes 0.418 (rather than 0.58); even here, A reduces by 2.06 times (i.e., it effectively halves) whenever luminance increases by 3.5 times. The ratio between the areas of two given ellipses in a given luminance is slightly larger than in a 10 times higher luminance, so although ellipses become distinctly more circular they become only slightly more similar in size in higher luminances. Equations (8) and (11) derive directly from Eqs. (5a) and (5b). Because they derive from ellipses predicted from Eq. (5), Eqs. (9) and (10) derive indirectly from Eqs. (5a) and (5b) but with the limitation that the original ellipse dimensions are those of the RIT-DuPont data. As the data sets show (e.g., Table III), there is considerable interellipse variability. For example RIT-DuPont ellipses #10, #3, #18, are very similar (gray) color centers at three very different luminances (11, 174, 401 cd/m2). But though ellipse #10 and #3’s (original) a/b ratios and areas relate to luminance by functions similar to Eqs. (9) and (10) they do not so relate with ellipse #18, whose (original) a/b ratio and area are both much larger than would be predicted by Eqs. (9) and (10) from either ellipse #10 or #3. Hence #18 is significantly eccentric to the other two gray ellipses. CONCLUSIONS AND IMPLICATIONS

A total of 21 color centers from four data sets are graphed in Figs 1– 4. Each data set demonstrates a decrease of axis dimensions, a/b ratio, and ellipse area with higher luminance. The present results extend previous reports by formulating these three luminance effects from several data sets and by demonstrating the luminance effect is greater for the major than the minor axis. The mean functions for the a and b axes (shown in Figs. 1– 4) plot straight lines in log-log space, in the form of power law functions, for luminances greater than 2 or 3 cd/m2 and up to the luminance limits of the data (e.g., 400 cd/m2 for RIT-DuPont data, Fig. 3). Formulae are given enabling prediction of a and b axes with varying luminance. The relative error in predicting actual data is calculated in Table III and is similar to interobserver error. The relationship between ellipse area and luminance, and between a/b ratio and luminance, is also formulated. The results agree with conclusions by Melgosa et al.3, Yebra et al.7, and with other reports,1,2,13–15 on the reduction of ellipse area with higher luminance over a broad range of chromaticity. This effect is in general agreement with the smaller Munsell Chroma contours at higher Munsell Value levels in CIE x, y, space.10 The results for a/b ratio are similar to previous results16; average a/b ratio for ellipses at 100 cd/m2 is 2.16 per Eq. (8) and is an average 2.2 per Kuehni’s analysis16 of ﬁve data sets of various lightnesses. It is of interest whether this reduction in axis size with increasing luminance applies also to the third dimension (Y) of ellipsoids in x, y, Y space. From three of the data sets used here1,3,6 we ascertained that luminance discrimination decreases with increasing luminance, and in the same manner COLOR research and application

as previous published data obtained from measurements of the Weber fraction, ⌬Y/Y, with monochromatic or white ﬁelds. Hence in general brightness discrimination decreases, whereas chromaticity discrimination increases, with higher sample luminance. The present results are independent of chroma or hue, because the equations predict ellipses (for various luminances) from any given ellipse at a speciﬁed chromaticity and luminance. The effect of chroma and of hue on ellipses is, however, of interest, but is complicated by the color space used. Kuehni found that in CIELAB, ellipse area increases proportionally to chroma, expressing a chroma crispening effect,16 but as mentioned above this effect is not so consistent for MacAdam ellipses in CIE x, y, and u, v, spaces. Kuehni found no correlation between ellipse area and hue in CIELAB, but this does not necessarily apply to other CIE spaces due to differences in wavelength distribution. It is concluded, from each data set, that ellipses become nearer circular with higher luminance. Brown1 and recently Carren˜o and Zoido4 have noted that symmetry increases with higher luminance, without formulating the effect. By predicting ellipses at higher luminances [Fig. 6D)] than currently available from data, it is concluded that circularity occurs at some luminance limit dependent on the data set and the particular color center. Circularity commences to degrade beyond that limit [i.e., about 2,000 cd/m2 for RITDuPont ellipses #4, #15, and #16, or 26,000 cd/m2 for the average ellipse, according to Eq. (5)]. According to MacAdam limits10, the lighter a color is, the more restricted is its range of chromaticity. Thus, dependent on the data set, some of these luminance limits indicate that ellipse circularity will occur at such high luminances as to appear achromatic, whereas other ellipses in the original data are already near circular (particularly in CIELAB space) at moderate luminances. This luminance limit may possibly depend on the chromaticity area, because RIT-DuPont data (and to some degree the Luo and Rigg data2) suggest ellipses in cyan, yellow, and magenta areas (in x, y, Y space) are already nearer circular at low luminances than other ellipses and thus attain circularity at a lower luminance limit than others: see ellipses #4 and #13 (cyan), #15 and #16 (magenta), and #19 (yellow), in Figs. 6(A) and 6(B), and in high luminance in Fig. 6(D). In contrast, saturated reds (ellipses #7 and #12) and blues (ellipses #1, #14, #17, from 468 to 476 nm, i.e., 474 – 482 nm in CIE 1931 space) are narrower ellipses, as also in Luo and Rigg data. However, the evidence is somewhat mixed; for example, ellipse #11 contrasts with other cyans and ellipse #18 contrasts with other grays. Further, a/b ratio depends on the color space. CIELUV may be more suitable for such comparisons than CIE 1931 or 1964 spaces. Robertson17 has plotted MacAdam ellipses in CIELUV. These, for equal luminance 50 cd/m2, are nearer circular for yellow and red-magenta and narrowest for blue areas, which is rather similar to RIT-DuPont ellipses in Fig. 6. However, the varying data on ellipse dependence on chromaticity,2,7,12,15,16 together with the dependence on the Volume 30, Number 3, June 2005

color space, seem to disallow safe and consistent deductions at this stage. The possible dependence on chromaticity is allowed for in the present study by using Eq. (6) to adjust each ellipse, in its particular chromaticity, to its predicted dimensions at target luminances, as shown in Fig. 6. The relatively large size of cyan, magenta, and yellow ellipses (say, the CMY effect) in CIE x, y, and u, v, spaces coincides with the various data sets on uniform hue difference (e.g., Munsell and OSA-UCS) and wavelength discrimination,10 which all demonstrate that wavelengths about 485 and 580 nm are the most accurately discriminated and those in B, G, R hues are less discriminable. The explanation of large CMY ellipses (in the hue angle dimension) is the nonuniform distribution of wavelength in these CIE spaces, with the greatest spacing (in terms of hue angle) about 490 and 575 nm, and the closest spacing near 530 nm and the spectrum ends. If the wavelength distribution were uniform (impossible because of complementary wavelength pairing), cyan and yellow ellipses would be narrower in the hue angle dimension and blue ellipses (e.g., #1 and #17) broader. Hence ellipse size is relative to the color space’s wavelength distribution. Of course, a uniform color space such as CIELAB is not necessarily constrained to straight lines between complementary wavelength pairs and can reduce this CMY effect. CIE colorimetry does not provide for all luminance effects on chromaticity. Color discrimination data, and the predictions in Fig. 6, apply to the given x, y chromaticity centers for all luminances. But a center’s perceived hue will change slightly in most cases (excepting four invariant hues) according to data on hue shift of color stimuli observed at different luminances at separate times (as distinct from simultaneously as in the Bezold–Brucke contrast effect).18 –20 This implies that ellipses for a speciﬁed color center will refer to slightly different perceived chromaticities in different luminances. Strictly speaking the data, together with the present results, apply to constant CIE colorimetric chromaticities at different luminances rather than to constant perceived chromaticities. The slight shift of chromaticity may well cause some of the effect on ellipse orientation, which changes with chromaticity and with luminance.1,4,7 Our graphs of axis dimensions with luminance, which (for luminance ⬎2 cd/m2) plot straight lines in log-log space and are power functions, support the Stevens power law.10,16,21 It is clear that the relative size and shape of ellipses cannot be reliably judged unless at some level of equal luminance. For example, Fig. 6(C) for 212 cd/m2 gives smaller and more uniform ellipses than the considerably larger and narrower ellipses shown in Fig. 6(B) for 42 cd/m2. It is common practice to employ ellipses of widely varying luminances to judge (1) color discrimination as a function of chromaticity, (2) uniform color spaces, or (3) color difference formulae.22,23 The present results indicate this practice may allow unreliable deductions. Though based on the two CIE x, y, Y, spaces, the results hold true in 195

general since color discrimination indubitably improves with higher luminance.1,4,12 In contrast, it is well known6,16,22,23 that higher luminance does not consistently reduce axis dimensions in CIELAB or any color-difference space that similarly lacks luminancelevel dependency. For example, Kuehni’s analysis16 of two data sets2,6 found that ellipse area in CIELAB space remains constant or slightly larger with higher metric lightness, except (interestingly) for metric chroma values 25– 45, where ellipse area reduces just as it does in x, y, Y space. Hence CIELAB seems unsuitable for consistently representing luminance effects in color appearance; this failing is not a desirable characteristic in a color appearance or uniform color space (UCS) but is perhaps unavoidable (see below). The same is true of color difference ellipses in UCSs based on DIN99 and similarly derived formulae.22 For example, RIT-Dupont ellipses #15 and #16, of similar chromaticity but 1:11 luminance ratio, plot almost the same size in such spaces.22 Fairchild24 ascribes this and similar failings of CIELAB to the fact that CIELAB incorporates no luminance-level dependency and no surround dependency. He notes that the CIE (Publication 15.2) speciﬁcally states that CIELAB is designed to represent color differences viewed in identical white to middle-gray surroundings. This lack of luminance dependency is a vexed question. It is perhaps a necessary characteristic of a UCS (such as CIELAB) designed to represent color to an eye adapted to the surround; arguably, the fully adapted eye discriminates color similarly in any adapting luminance level within the normal photopic range. CIELUV space gives a consistent variation of ellipse size with luminance similar to CIE x, y, space, and ellipses appear more circular than in CIE x, y, as Robertson has shown.10,17 But CIELUV has its own failings in respect of representing color appearance in artiﬁcial illuminants.24 It is desirable that a UCS be luminance-sensitive but if this requirement mitigates against uniformity of color space (as indicated by CIELAB) then perhaps the solution is purposespeciﬁc UCSs, such as CIELAB (or equivalent) for the adapted eye and CIELUV (or a space similar to color appearance model CIECAM02) for luminance-dependent effects. The present results show that the inﬂuence of luminance is substantial and too large to be ignored as it has in the past. If the effect is excluded by a UCS, the UCS will give misleading impressions of ellipse sizes and shapes. An ellipse at a given luminance which appears near-circular (say, a/b⫽1.5) and similar in size to other ellipses in such a UCS, may actually (in a luminance-sensitive space) have an a/b ratio of 2.1 and nearly four times the area, at 1/10 that luminance [per Eqs. (8)–(11) above]. The Appendix describes some examples of CIELAB’s misrepresentation of luminance-dependent chromaticity. With higher luminance, ellipses reduce in area so severely [by half with a 3.5 times higher luminance, per Eqs. (10) or (11)] that the effect cannot reasonably be ignored in future color appearance spaces. The effect may be reduced for the purposes of a UCS so long as the relative effect is represented. Given the 196

several reports over 50 years on the effect of luminance on ellipse dimensions,1– 8,13–15 and the CIE’s interest in such luminance effects,9 it is hoped the CIE (e.g., TC 1–55) will ﬁnd it possible to incorporate luminance-level dependency in future UCSs or in one UCS of a set of purpose-speciﬁc UCSs. After all, the practical use of a UCS that cannot predict the substantial effect of luminance on color difference is severely limited. That discrimination ellipses are nearer circular in high luminances has interesting implications. It means a UCS, where ellipses ideally approximate equal circles, is unlikely except for colors of very high luminance. With increasing luminance, near-circular ellipses at low luminances (e.g., #4, #15, and #16 in Fig. 6) become circles earlier and then degrade, and narrow ellipses do the same but later. If a luminance-sensitive UCS is designed for low-to-moderate luminances it will not necessarily represent a uniform space in higher luminances. So compromises may be required in pursuing the ideal of a UCS for all luminances (L). Ideally a UCS should represent a broad variance of L, say 10 –1000 cd/m2. Possibly two UCSs may be appropriate: one designed for a higher L level (say 500 cd/m2), as the optimally uniform space, and one for a moderate L level (say 100 cd/m2) for most applications. This is a similar concept to the two CIE standard observers, for small and for large visual ﬁelds. The above discussion infers that ellipses vary consistently as a function of luminance only in CIE color-mixture spaces or linear transforms such as CIELUV. Is this necessarily so? If so, it would constrain the design of luminance-sensitive UCSs to linear transforms of x, y, spaces. Testing various experimental color spaces indicates that allowing curvature to the lines through the white point between complementary wavelengths reduces the consistent effect of luminance on ellipse dimensions. Why this is so is an open question. Empirically, a crucial parameter in maintaining a proper luminance effect appears to be linearity between pairs of complementary wavelengths rather than a color space’s totally linear relationship with CIE x, y space. However, as mentioned above, linearity between complementary wavelength pairs causes a nonuniform wavelength distribution, giving greater hue angle to wavelengths in the CMY hues and thus larger ellipses, with narrower ellipses in the B, G, and R hues. Hence, to display all these as similar sized ellipses, compromises appear necessary. Naturally, the degree of compromise to linearity carries a cost in misrepresenting not only the luminance effect but such aspects as metameric pairs. UCS design will need such compromises but there seems considerable room to maneuver in designing a luminance-sensitive UCS. It has been a common though not universal view that luminance has rather little effect on chromaticity in general. That view is reﬂected in the fact that, despite some efforts made,25 CIE colorimetry does not fully provide for the inﬂuence of luminance on chromaticity.24 Similarly some of the literature still accepts the notion that varying luminance has little or no signiﬁcant effect on the perceived chromaticity of object colors.26 The CIE and others1–10,13–16 have COLOR research and application

called for rigorous studies of the effects of luminance on chromaticity discrimination. We believe the present study, given its reasonably adequate data bank, is a useful ﬁrst approach to quantifying those effects.

4.

APPENDIX: CIELAB MISREPRESENTATION OF CHROMATICITY

6.

The present results and equations allow quantitative analysis of the type and degree of misrepresentation occurring in UCSs that lack luminance-level dependency (such as CIELAB and DIN99). The CIE is moving toward ﬁnding a better UCS than CIELAB, but the latter will sufﬁce as an example. CIELAB represents a given color center, such as Gray (RIT-DuPont ellipses #3 and #10, whose luminances are a 16:1 ratio), as ellipses of about the same shape and size (see Table II of Ref. 6) whereas CIE x, y space shows ellipse #10 occupies 5 times the area of ellipse #3 [see Fig. 6(A)]. By showing it as only 1/5 that area, 80% of ellipse #10’s chromaticity variance has been omitted or compressed in CIELAB (or alternatively CIELAB gives ellipse #3 ﬁve times its actual chromaticity variance). Where has this chromaticity area, extending well into the blue and yellow areas, gone? Is it so insigniﬁcant that it may be omitted or compressed? Apparently, it is compressed into ellipse #3’s area, where it certainly does not belong. This does not seem a satisfactory method of representing chromaticity. Another example: RIT-DuPont color centers vary from 11 cd/m2 (ellipse #10, gray) to 400 cd/m2 (ellipse #18, gray). The gray ellipse at 11 cd/m2 is 14 times bigger in CIE x, y space (per Eqs 10 or 11) than it would be at 400 cd/m2, yet CIELAB represents both ellipses as similar sizes. It is completely unable (by its design) to represent such luminance-dependent effects on chromaticity and hence contradicts the major effect of luminance: that is, higher luminance improves color discrimination thus reducing ellipses. Ellipses reduce in area so severely [see Eqs. (10) or (11)] that the effect cannot be reasonably ignored in future UCSs. ACKNOWLEDGMENTS

This work was partly supported by research project FIS2004 – 05537, Ministerio de Educacion y Ciencia (Spain). 1. Brown WRJ. The inﬂuence of luminance level on visual sensitivity to color differences. J Opt Soc Am 1951;41:684 – 688. 2. Luo MR, Rigg B, Chromaticity-discrimination ellipses for surface colours. Color Res Appl 1986;11:25– 42. 3. Melgosa M, Pe´rez MM, El Moraghi A, Hita E. Color discrimination

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