Testing the Robustness of CIECAM02

C. J. Li,*1,2 M. R. Luo2 1Shenyang Institute of Automation, Chinese Academy of Science, Shenyang, China

2Department of Colour and Polymer Chemistry, University of Leeds, Leeds LS2 9JT, UK

Received 14 July 2003; revised 22 March 2004; accepted 28 June 2004

Abstract: CIE TC8-01 has adopted a new color appearance tional complexity, whereby industrial applications require model: CIECAM021 replaces the CIECAM97s.2 The new the color appearance model to be not only accurate in model consists of a number of refinements and simplifica- predicting the perceptual attributes but also simple in com- tions of the CIECAM97s color appearance model. This putation complexity. To this end, its nonlinear chromatic article describes further tests to the robustness of the for- adaptation transform must be simplified. Li, Luo, Rigg, and ward and reverse modes. © 2005 Wiley Periodicals, Inc. Col Res Hunt5 first considered the linearization of the CMCCAT974 Appl, 30, 99–106, 2005; Published online in Wiley InterScience (www. using the available experimental data sets. The newly de- interscience.wiley.com). DOI 10.1002/col.20087 rived linear chromatic adaptation transform (CMC- Key words: CIECAM02; CIECAM97s; forward mode; re- CAT2000) is not only simpler but also more accurate than 4 6,7 8 verse mode; sRGB space; object color solid CMCCAT97 and CIECAT94. Fairchild and Hunt, Li, Juan, and Luo9 made further revisions to the CIECAM97s using different linear chromatic adaptation transforms. The INTRODUCTION former used a modified linear chromatic transformation, and The CIE in 1997 recommended a simple version of the color the latter used CMCCAT2000. The Fairchild linear chro- appearance model.2 It is named CIECAM97s. It has a for- matic transformation has the same structure as CMC- ward mode and a reverse mode. After the CIECAM97s was CAT2000 but optimizes the matrix coefficients to agree the published, some shortcomings of the CIECAM97s have final predictions as close as to those predicted by CMC- been found. For example, when the tristimulus value Y CAT97, that is, the results are compatible with reaches zero, the model (forward mode) predicted lightness CIECAM97s. In addition, the saturation scale(s) was J approaches a constant that is different from zero and changed,9 resulting in much better performance in predict- dependent on viewing conditions and luminance level. This ing visual saturation results. One of the most important may lead to negative tristimulus values given by the reverse improvements over CIECAM97s was given by Hunt, Li, mode, which is unacceptable. For example, if Y goes to and Luo.10 The models were named Power Model (PM) and zero, the predicted J approaches 5, and then the reverse modified PM. It was found that for a fixed set of chroma- model may lead to negative tristimulus values when starting ticity coordinates x, y, the perceptual attributes, saturation with J between 0 and 5. It was also found that the color (s) and hue (h), vary with the change of tristimulus value Y. gamut volume of the CIECAM97s under dim surround is Ideally these percepts should not be affected by changing of larger than the one under the average surround. In fact it Y. This problem is overcome by using the new models.10 should be the opposite case. Li, Luo, and Hunt3 made the CIE TC 8-01, Color Appearance Model for Color Manage- above corrections. In addition, because of the nonlinear ment Applications, has recently recommended a new color chromatic adaptation transform: CMCCAT974 embedded in appearance model, CIECAM02,1 to replace the existing the CIECAM97s, it makes the reverse mode more compli- CIECAM97s. The new model corrects all the shortcomings of cated and inexact. An iterative method was also proposed the CIECAM97s. It is not only simpler, but also equally or for the reverse mode to achieve the exact reverse in the more accurate than the CIECAM97s.11 It is also found that the same article.3 However, this is at the expense of computa- new CIECAM02 can provide a more uniform color space.12 This article describes the robustness of the forward and *Correspondence to: C. J. Li (e-mail: [email protected]) reverse modes by testing them using comprehensive data © 2005 Wiley Periodicals, Inc. sets.

Volume 30, Number 2, April 2005 99 THE CONDITIONS FOR THE EXISTENCE OF THE Ј ϩ Ј ϩ ͑ ͒ Ј ϭ ͱ 2 ϩ 2 R a G a 21/20 B a p1 a b . (3.3) FORWARD MODE The lightness attribute correlate ( J) of the forward mode of The denominator in Eq. (3) (denoted by td) of the parameter the CIECAM021 is calculated using Eq. (1) as follows: t must be nonzero. In fact it must be positive, otherwise, the computation of the chroma C will fail. Therefore, for the cz J ϭ 100͑ A/Aw͒ (1) given input tristimulus values X, Y, and Z, the forward mode works (or exists) if where A ϭ ͓ p Ϫ 0.305͔N Ն 0 (4) 1 2 bb A ϭ ͫ2 RЈ ϩ GЈ ϩ BЈ Ϫ 0.305ͬN (1.1) a a 20 a bb ϭ Ј ϩ Ј ϩ Ј Ͼ td R a G a p3B a 0. (5) and Thus for a given set of tristimulus values X, Y, and Z under 1 a set of defined viewing conditions in terms of reference A ϭ ͫ2 RЈ ϩ GЈ ϩ BЈ Ϫ 0.305ͬN . (1.2) w wa wa 20 wa bb white, luminance level, background, and surround condition, as long as Eqs. (4) and (5) are satisfied, the forward If we set mode can successfully predict the perceptual attributes correlates: lightness ( J), chroma (C), hue (h), hue composi- 1 ϭ Ј ϩ Ј ϩ Ј tion (H), colorfulness (M), brightness (Q), and saturation p2 2 R wa G wa B wa (1.3) 20 (s). p ϭ it follows from Eq. (1.1) that Note that the parameter 3 21/ 20 rather than one was included in the CIECAM97s for concerning the reversibility A problem as pointed by Luo and Hunt.2 p ϭ ϩ 0.305. (1.4) 2 N Because of the nonlinear functions involved in computing bb Ј Ј Ј Ј Ј Ј R a, G a, and B a from R , G , and B [Eqs. (7.15)–(7.17)] in Ref. 1] respectively, Eqs. (4) and (5) cannot be simply Note that the symbols for the parameters used in this article verified without carrying real computations so that compre- are the same as those in Ref. 1. hensive data sets were generated to provide severe test for In Eq. (1), because the c multiplying z is always a the forward and reverse modes of the CIECAM02. positive real number, the forward mode would fail for a negative ratio of A and Aw. The Aw is calculated from the tristimulus values of the test illuminant and is always THE REVERSIBILITY OF THE FORWARD MODE greater than zero. For details see the appendix. Hence, the forward mode fails if A is negative. However, when A ϭ 0, The input values to the reverse mode of CIECAM02 are as described in Ref. 1, the lightness J will be zero. Hence, three perceptual attributes correlates, say J, C, and h. It will chroma, C, and colorfulness, M, and brightness, Q, will all output the tristimulus values X, Y, and Z. If the parameters ͌ be zero, which leads to saturation s ϭ 100 M/Q unde- of the viewing conditions such as reference white, lumi- fined. In this case (when A ϭ 0), the saturation should then nance level, background, and surround condition are the be set to zero. Another possible failure is the prediction of same for the forward and reverse modes, then the tristimulus the chroma C as given in Eq. (2). values calculated by the reverse mode should be the same as those inputted to the forward mode. In the CIECAM02 reverse mode, computations could J C ϭ t0.9 ͱ ͑1.64 Ϫ 0.29n͒0.73, (2) have a problem when computing a (redness-greenness) and 100 b (yellowness-blueness) given in the following formulae. where t is given by Eq. (3) ͑ ͒ ͑ 2 ϩ 2͒1/ 2 50000/13 NcNcbet a b Case 1: ͉sin(h )͉ Ն ͉cos(h )͉, t ϭ r r Ј ϩ Ј ϩ ͑ ͒ Ј . (3) ͑ ϩ ͒͑ ͒ R a G a 21/20 B a p2 2 p3 460/1403 b ϭ Setting db a ϭ b͓cos͑hr͒/sin͑hr͔͒ 21 p ϭ (3.1) where 3 20 ϭ ϩ͑ ϩ ͒͑ ͓͒ ͑ ͒ ͑ ͔͒ db p4 2 p3 220/1403 cos hr /sin hr and Ϫ͑ ͒ϩ ͑ ͒ 27/1403 p3 6300/1403 (6) ͑50000/13͒ N N e p ϭ c cb t ϭ 1 p1 (3.2) p4 t sin͑hr͒ ͉ ͉ Ͻ ͉ ͉ gives Case 2: sin(hr) cos(hr) ,

100 COLOR research and application ͑ ϩ ͒͑ ͒ p2 2 p3 460/1403 Because sin(hr) and cos(hr) cannot be zero at the same a ϭ time, therefore, for Case 1, the following condition can be da deduced from Eq. (10):

b ϭ a͓sin͑hr͒/cos͑hr͔͒ ͉sin͑hr͉͒ Ͼ 0 and b 0. where Thus, from Eq. (10), Eq. (3.3) becomes the following: d ϭ p ϩ ͑2 ϩ p ͒͑220/1403͒ Ϫ ͓͑27/1403͒ a 5 3 b Ϫ ͑ ͔͓͒ ͑ ͒ ͑ ͔͒ Ј ϩ Ј ϩ Ј ϭ p3 6300/1403 sin hr /cos hr (7) R a G a p3B a p1 (11.1) sin͑hr͒ p ϭ 1 and p5 cos͑hr͒ cos͑h ͒ ϭ r In both cases, h is given by the following: a b . (11.2) r sin͑hr͒ ␲ Substituting Eqs. (11.1) and (11.2) into Eq. (11) and rear- h ϭ h . (8) r 180 ranging Eq. (11) result in the following: Thus, if the denominator d of b in Case 1 or the denomi- p 220 cos͑h ͒ 27 6300 b ͫ 1 ϩ ͑ ϩ ͒ r Ϫ ͩ Ϫ ͪͬ 2 p3 p3 b nator da of a in Case 2 is zero, then the reverse mode will sin͑hr͒ 1403 sin͑hr͒ 1403 1403 fail. 460 Note that hr, a, and b satisfy the following: ϭ p ͑2 ϩ p ͒ . 2 3 1403 b b ͑ ͒ ϭ ͑ ͒ ϭ 1/cz sin hr , cos hr , Note that A ϭ A ( J/100) and A Ͼ 0, and J is ͱa2 ϩ b2 ͱa2 ϩ b2 w w nonnegative. Thus, p2 of Eq. (1.4) and p3 of Eq. (3.1) are b a always positive. Therefore, the right-hand side of the above ͱ 2 ϩ 2 ϭ ϭ or a b . (9) equation will never be zero. Hence the factor in the square sin͑hr͒ cos͑hr͒ bracket in the left-hand side of the above equation will not Now we need to show that if the three perceptual attribute be zero, which is the denominator db in Eq. (6) for calcu- correlates, say, for example, J, C, h, are the output of the lating b. Thus the denominator of the formula for comput- forward mode with a set of tristimulus values as input, then ing b is nonzero in Case 1. Similarly, the denominator da for there is no problem for the computations of a and b when computing a is nonzero as far as the inputs J, C, h to the starting from J, C, h for reverse mode as input. In fact, by reverse mode are the outputs from the forward mode. the forward mode, a and b satisfy the following: The above evidence shows that the reversibility of the forward mode does not depend on the choice of p . There- 1 3 2 RЈ ϩ GЈ ϩ BЈ ϭ p fore, the reversibility of the forward mode will not be a a 20 a 2 ϭ ϭ affected regardless of p3 21/20 or p3 1. Note that p ϭ 21/ 20 was introduced in CIECAM97s2 for facilitat- 12 1 3 RЈa Ϫ GЈa ϩ BЈa ϭ a ing the reverse of the model. It seems that it is not the case. 11 11 Second, the results also show that as long as the forward 1 mode does not fail, it is always reversible in the reverse ͑RЈ ϩ GЈ Ϫ 2BЈ͒ ϭ b. mode. In other words, if the three attribute correlates J, C, 9 a a a h inputted to the reverse mode are the same as the output Thus, from the forward mode then the reverse mode should not be failed. However, Eqs. (4) and (5) for the forward mode are 460 451 288 difficult to be verified analytically. The forward mode was RЈ ϭ p ϩ a ϩ b a 1403 2 1403 1403 tested by using the data in practical applications to ensure the successful application of the model. 460 891 261 Furthermore, it is also noted that if a neutral color J, C, GЈ ϭ p Ϫ a Ϫ b (10) a 1403 2 1403 1403 h with C ϭ 0 was entered the reverse mode, t value in Eq. (3.2) will be zero because it is calculated using the follow- 460 220 6300 BЈ ϭ p Ϫ a Ϫ b. ing formula: a 1403 2 1403 1403 C 1/0.9 Ј t ϭ . Multiplying p3 by the third equation (B a) and adding the J ΂ n 0.73΃ first and second equations yield, ͱ ͑1.64 Ϫ 0.29 ͒ 100 ͑ Ј ϩ Ј ϩ Ј͒ ϭ ͑ ϩ ͒ 1403 R a G a p3B a p2 2 p3 460 Thus, computing p1 will be failed. In this case, the chro- Ϫ ͑ ϩ ͒ ϩ ͑ Ϫ ͒ 220 2 p3 a 27 p36300 b. (11) matic components a and b should be set to zero.

Volume 30, Number 2, April 2005 101 cated in the gray triangle area. However, the above does demonstrate that it is necessary to test the forward mode. Two tests were conducted. Test 1 was based on sRGB13 space and Test 2 was based on the object color solid.14 The sRGB space is an international standard color space for transmitting color images in open computer systems. It is a simple and robust device independent space representing all available CRT colors and providing good quality and back- ward compatibility with minimum transmission and system overhead. Thus Test 1 is designed to test the color appearance model in imaging applications. Test 2 was designed to test the color appearance model with a comprehensive set of tristimulus values (much larger than that of sRGB) representing all realistic and theoretical surface colors. The color gamuts of the two are plotted in Fig. 2, where the gamuts of the sRGB and optimum color solid are plotted in the x, y chromaticity diagram. However, the tests do not include cases where ﬂuorescence occurs.

FIG. 1. The chromaticity locus of the CIE1931 standard. The CIECAM02 forward model fails with the samples in the Test 1: sRGB Space area of gray triangle. [Color ﬁgure can be viewed in the online issue, which is available at www.interscience.wiley. This test is based on sRGB space. Set R ϭ i, G ϭ j, B ϭ com.] k, in the space, and i, j, k ϭ 0, 1, 2, . . . , 255. Then the R, G, B values were transformed to the tristimulus values

Xijk, Yijk, Zijk under D65/2° conditions. The tristimulus TESTING THE FORWARD AND REVERSE MODES values Xijk, Yijk, Zijk were then inputted to the forward mode to calculate the perceptual attribute correlates J , In this section, both the forward and reverse modes of ijk Cijk, hijk. Finally, these correlates were inputted to the CIECAM02 are tested using practical data. The tests were reverse mode to obtain the predicted tristimulus values. In focused on the accuracy of the reversibility and the robust- total, 224 Ϫ 1 sets of tristimulus values with the exclusion of ness of forward mode. The accuracy of the reversibility ϭ ϭ ϭ X000 Y000 Z000 0 were generated. (The reason to were evaluated in terms of CIELAB (1976) color difference ϭ ϭ ϭ exclude the sample with X000 Y000 Z000 0is between the inputted tristimulus values to the forward mode because it is not a realistic color. Unlike the CIECAM97s, and those calculated by the reverse mode with the same set of viewing conditions: reference white, background, surround, and luminance level. In theory, the color difference should be zero for a perfect agreement between two sets of tristimulus values. However, this is not the case due to ﬁnite digits used in the computation. To test the forward mode requires testing samples in terms of tristimulus values. Hence there is a need to sample a comprehensive testing data. Will any set of tristimulus values X, Y, Z satisfying the following:

0 Ͻ X Յ XW,0Ͻ Y Յ YW,0Ͻ Z Յ ZW be sufﬁcient (where XW, YW, ZW are the tristimulus values of the test illuminant). The answer is negative. For example, for a sample having X ϭ 16.158, Y ϭ 2.0, Z ϭ ϭ 103.4389, under viewing conditions of XW 95.047, ϭ ϭ ϭ ϭ Yw 100.0, ZW 108.83, Yb 20, LA 80, and average surround, A equals Ϫ0.173. In this case the forward mode fails because Eq. (4) is not satisﬁed. However, the above color does not exist, because its chromaticity coordinates: ( x, y) ϭ (0.13288, 0.01645), denoted in Fig. 1 by a black triangle point and located outside the chromatic- FIG. 2. The color gamuts of the sRGB space (thin curve) ity locus of CIE1931 standard. In fact, the forward model and the optimum color solid (thick curve) with Y ϭ 50 in failed with all samples with chromaticity coordinates lo- chromaticity diagram under D65.

102 COLOR research and application TABLE I. Performance of the CIECAM02 forward and reverse modes based on the samples from sRGB space ϭ ϭ with LA 80, average surround, Yb 18 under D65/2° condition. ⌬ p3 Amin tdmin dbmin damin Emax

1.0 0.4099 0.8469 6.7880 12.4654 0.0005 21/20 0.4099 0.8558 6.9011 12.6731 0.0005

ϭ ϭ the CIECAM02 works for the sample with X000 Y000 stimuli arising by the reflectance (or transmission) of inci- ϭ ϭ Z000 0 as long as a test about Q 0 is used before the dent flux by objects. Thus the object color is defined by a calculation of saturation. In this case, all the attributes A, J, spectral reflectance function R(␭) across the visible spec- C, M, Q will be zero. As mentioned before, saturation s trum. For a given light source, and a CIE standard color- should be set to zero as well. Other attributes such as H and metric observer, the tristimulus values X, Y, Z of a partic- h have no meaning in this case. The tristimulus values ular object are given in Eq. (12) as follows: X ϭ Y ϭ Z ϭ 0 are generated if we put J ϭ C ϭ 000 000 000 ϭ ͸ 0 and any h, into the reverse mode.) Each set of the X WX,iri i tristimulus values has an A and a td value calculated in Eqs. (4) and (5) from the forward mode and d (the denominator b ϭ ͸ ͉ ͉ Ն ͉ ͉ Y WY,iri (12) of b formula for Case 1: sin(h) cos(h) ) and da (the denominator of a formula for Case 2: ͉cos(h)͉ Ͼ ͉sin(h)͉) i A t d calculated from the reverse mode. The min, dmin, bmin Z ϭ ͸ W r , ͉ ͉ Z,i i and damin, which are the smallest values of the A, td, db , i ͉ ͉ and da values for all the tested samples respectively are used to indicate whether the forward mode is successfully where WX,i, WY,i, WZ,i are weighting tables at 1-nm interval Ͻ Յ run, that is, if Amin 0ortdmin 0, the forward mode will depending on relative spectral power distribution of the Ј Ј Ј fail. Let X ijk, Y ijk, Z ijk be the output of the reverse mode light source and CIE standard colorimetric observer, and ri ⌬ and let Eijk be CIELAB color difference between Xijk, are the reflectance values between zero and one at wave- Ј Ј Ј ␭ ϭ ␭ Yijk, Zijk and X ijk, Y ijk, Z ijk. The maximum value of all length i, that is, ri R( i). The data points obtained by ⌬ ⌬ Eijk values, Emax, was also reported. The smaller value taking all suitable reflectance values to define a solid in (X, ⌬ 14 of Emax, the more accurate the reverse mode is. All the test Y, Z) space is called the object color solid. results are summarized in Tables I and II for average and A Y value between zero and Yw is first defined. Subse- dark surrounds respectively. The results calculated using the quently, the boundary of XZ plane of the object color solid ϭ ϭ p3 1 and p3 21/ 20 in the model are both reported. is found as shown in Fig. 3. If XL and XU are defined as the From the results of Tables I and II, the forward mode minimum and maximum X values from the boundary, then successfully predicted the perceptual attribute correlates for they satisfy: ϭ each of the inputted tristimulus values no matter p3 1or ϭ ϭ X ϭ ͑͸ W r ͒ X ϭ ͑͸ W r ͒ p3 21/20. The Amin 0.4099 for both p3 cases in Table L min X,i i , U max X,i i ϭ i i I and Amin 0.1150 for both cases in Table II proves that the A function is independent of p . The results also show 3 ϭ ͸ ⌬ Y WY,iri a satisfactory accuracy performance with a Emax of 0.005 for all cases. Comparing results from two tables differing in Subject to i 0 Յ r Յ 1 surround conditions, it can be seen that under the dark i surround conditions (Table II), A and td are smaller, min min Second, an X value between XL and XU is chosen. Again, and db and da are larger than those corresponding min min the corresponding maximum and minimum values ZL and results under the average conditions (Table I). ZU are found so that points (X, ZL) and (X, ZU) are located on the boundary of XZ plane as shown in Fig. 3 and they Test 2: Object Color Solid satisfy: ϭ ͑͸ ͒ ϭ ͑͸ ͒ The second test was conducted by selecting testing sam- ZL min WZ,iri , ZU max WZ,iri ples from the so-called object color solid,14 which are color i i

TABLE II. Performance of the CIECAM02 forward and reverse modes based on the samples from sRGB space ϭ ϭ with LA 0.001, dark surround, Yb 0.01 under D65/2° condition. ⌬ p3 Amin tdmin dbmin damin Emax

1.0 0.1150 0.3339 9.0236 14.8322 0.0005 21/20 0.1150 0.3392 9.1740 15.0794 0.0005

Volume 30, Number 2, April 2005 103 TABLE IV. Performance of the CIECAM02 forward and reverse modes based on the object color solid ϭ ϭ with LA 0.001, dark surround, Yb 0.01 under D65/2° condition.

⌬ p3 Amin tdmin dbmin damin Emax

1.0 0.1952 0.3338 4.99995 8.5457 0.001 21/20 0.1952 0.3388 5.0828 8.6882 0.001

then rk is a proper reﬂectance vector and satisfy:

T T T T W rk ϭ W ͑␣krL ϩ ͑1 Ϫ ␣krU͒ ϭ ␣kW rL ϩ ͑1 Ϫ ␣k͒W rU T T T ϭ ␣k͑Xi, Yj, ZL͒ ϩ ͑1 Ϫ ␣k͒͑Xi, Yj, ZL͒ ϭ ͑Xi, Yj, Zk͒

indicating that (Xi, Yj, Zk) is the tristimulus values of an object color. Thus, the tristimulus values (Xi, Yj, Zk) were used as input to the forward mode. All test results are listed in Tables III and IV. The measures reported are the same as those used in Test 1. Once again, the forward mode worked successfully and did not fail for all the sampled tristimulus FIG. 3. Boundary (thick line) of the XZ plane of the object values. As for the accuracy, the worst case had a value of color solid with Y ϭ 50 under the D65 and CIE1931 stan- 0.001 CIELAB color difference unit for both p3 cases. dard colorimetric observer. Similarly, the model was tested using the same principle but under CIE illuminant A and 1931 standard observer and results are listed in Tables V and VI. It can be seen from X ϭ ͸ W r X,i i Tables III–VI that the model has the same performance i under the two illuminants. Subject to Y ϭ ͸ W r Y,i i Furthermore, the CIECAM02 was also tested with adap- i tation factor D ϭ 1.0 additional to the D value calculated 0 Յ r Յ 1 i using the given formula. The results showed that both The samples were selected by first varying Y from 0.01 to 1 methods to calculate D gave very similar results. with step size of 0.01 and from 1 to Yw with a step size of 1. In total, there are 199 Y values, which are denoted by Yj, ϭ ϭ CONCLUSION j 1, 2, . . . , 199. For any Yj, choose Xi, i 1, 2, . . . , 100, uniformly distributed between XL and XU. Finally, for This article investigates the robustness of the forward mode ϭ any given Yj and Xi, choose Zk, k 1, 2, . . . , 100, of the CIECAM02 and tests its reversibility. The results uniformly distributed between ZL and ZU. It can be shown showed that the forward mode may theoretically fail in ϭ that (Xi, Yj, Zk), k 1, 2, . . . , 100, are object colors. In some color regions. However, these colors cannot be veri- fact, by construction, there are reflectance vectors rL and rU fied analytically without computation using some real test- so that ing samples. Hence, two methods were developed to sample test data based on more realistic samples, that is, sRGB ͑X Y Z ͒T ϭ WTr ͑X Y Z ͒T ϭ WTr i, j, L L, i, j, U U space for CRT applications and optimum color solid. The ␣ testing results were highly satisfactory for which the model and an k between zero and 1 so that passed all the tests using these testing samples. In addition, ϭ ␣ ϩ ͑ Ϫ ␣ ͒ Zk kZL 1 k ZU. the reverse mode is reversible as long as the forward mode Let does not fail. Furthermore, the reversibility does not depend ϭ ϭ on the choice of parameter p3 1orp3 21/ 20. This ϭ ␣ ϩ ͑ Ϫ ␣ ͒ rk krL 1 k rU, implies that p3 can be set to 1 for simplicity.

TABLE III. Performance of the CIECAM02 forward TABLE V. Performance of the CIECAM02 forward and reverse modes based on the object color solid and reverse modes based on the object color solid ϭ ϭ ϭ ϭ with LA 80, average surround, Yb 18 under with LA 80, average surround, Yb 18 under A/2° D65/2° condition. condition.

⌬ ⌬ p3 Amin tdmin dbmin damin Emax p3 Amin tdmin dbmin damin Emax

1.0 0.6847 0.8436 1.9797 4.4908 0.001 1.0 0.7949 0.8833 5.1408 7.0033 0.002 21/20 0.6847 0.8479 2.0127 4.5657 0.001 21/20 0.7949 0.8919 5.2264 7.1201 0.002

104 COLOR research and application ϭ Ϫ Ϫ TABLE VI. Performance of the CIECAM02 forward tristimulus values of the test illuminant and zw 1 xw and reverse modes based on the object color solid y Ն 0. ϭ ϭ w with LA 0.001, dark surround, Yb 0.001 under If we let ⍀ϭ{(x, y)} be the set of the chromaticity A/2° condition. coordinates ( x, y) satisfying: ⌬ p3 Amin tdmin dbmin damin Emax 0.46839x ϩ 0.76766y Ն 0.07868 (A2) 1.0 0.2213 0.3363 7.8693 9.8627 0.002 Ϫ ϩ Ն Ϫ 21/20 0.2213 0.3415 8.0005 10.0270 0.002 0.27622x 1.13699y 0.04641 and 0 Յ x Յ 1, 0 Յ y Յ 1, ACKNOWLEDGMENTS then, all points in ⍀ϭ{(x, y)} will locate on or above the The authors thank the two referees and Professor Robert two broken lines shown in Fig. A1. The two broken lines are W. G. Hunt for their valuable comments and suggestions for as follows: improving the quality of the paper. y ϭ 0.10249 Ϫ 0.61015x (A3) ϭ Ϫ ϩ APPENDIX: THE POSITIVITY OF AW y 0.04082 0.24294x Ͼ Ն ϭ Proposition: Aw 0 if the chromaticity coordinates ( xw, yw) derived from inequality (A2) by changing to and have of the light source is inside the area circled by the chroma- a crossing point at ( x, y) ϭ (0.16799, 0.00001). The ticity locus of the CIE standard. horseshoe shape curve with a straight line connecting the Proof: By Ref. 1, two ends of the curve in Fig. A1 is the chromaticity locus of CIE1931 standard at a 5-nm interval. Because the two ends Rwc 1 Rw 1 points are (0.17556, 0.005294) and (0.73469, 0.26531), ͩ Gwc ͪ ϭ Y Dͩ 1 ͪ ϩ ͑1 Ϫ D͒ ͩ Gw ͪ ϭ Y D ͩ 1 ͪ w w therefore, the straight line connecting the two ends of the Bwc 1 Bw 1 horseshoe shape has the equation: Xw y ϭ 0.005294 ϩ 0.465037x. (A4) ϩ ͑1 Ϫ D͒ M ͩ Yw ͪ CAT02 Zw Thus, it can be veriﬁed that all the points of the CIE1931 chromaticity coordinates are located above the two broken thus, lines as shown by Fig. A1. Figure A2 is the enlarged part of Fig. A1 around the corner of the V-shape of the two broken RЈw Rwc 1.00001 Ј Ϫ1 lines and clearly shows all the points of the CIE1931 chro- ͩ G w ͪ ϭ M M ͩ Gwc ͪ ϭ Y D ͩ 1 ͪ HEP CAT02 w maticity coordinates are located above the two broken lines. BЈw Bwc 1

Xw ϩ ͑1 Ϫ D͒ M ͩ Yw ͪ. (A1) HEP Zw Let u be a vector and u Ն 0 means each component of vector u is not less than zero. Next we want to show

Xw M ͩ Yw ͪ HEP Zw

0.38971Xw ϩ 0.68898Yw Ϫ 0.07868Zw Ϫ ϩ ϩ ϭ ͩ 0.22981Xw 1.18340Yw 0.04641Zw ͪ Ն 0. Zw The above inequality is equivalent to

0.38971xw ϩ 0.68898yw Ϫ 0.07868zw Ϫ ϩ ϩ ͩ 0.22981xw 1.18340yw 0.04641zw ͪ zw

0.46839xw ϩ 0.76766yw Ϫ 0.07868 Ϫ ϩ ϩ ϭ ͩ 0.27622xw 1.13699yw 0.04641 ͪ Ն 0. zw FIG. A1. The chromaticity locus of the CIE1931 standard. The square symbols represent the chromaticity coordinates Here, xw and yw are the chromaticity coordinates of the of the light sources of A, C, D50, D65, D75, F1, F2, to F11.

Volume 30, Number 2, April 2005 105 Ј Ј Ј is an increasing function of R w, and similarly, G aw, B aw are Ј Ј increasing functions of G w and B w respectively. Thus if we let

400͑F Y D/100͒0.42 ϭ L w ϩ g 0.42 0.1, 27.13 ϩ ͑FLYwD/100͒

then g Ͼ 0.1. Hence,

RЈaw g Ј ͩ G aw ͪ Ն ͩ g ͪ BЈaw g

and

Aw ϭ ͓2 RЈaw ϩ GЈaw ϩ BЈaw/20Ϫ 0.305͔Nbb

Ն ͓3.05g Ϫ 0.305͔Nbb Ͼ 0,

concluding the proof. FIG. A2. The enlarged part of Fig. A1 around the corner of the V-shape of the two broken lines. 1. CIE TC8–01 Technical Report (Draft 11): A color appearance model for color management systems: CIECAM02, 2003. 2. Luo MR, Hunt RWG. The structures of the CIE 1997 colour appear- The points with square symbols in Figs. A1 and A2 repre- ance model (CIECAM97s). Color Res Appl 1998;23:138–146. sent the chromaticity coordinates of the light sources A, C, 3. Li CJ, Luo MR, Hunt RWG. A revision of the CIECAM97s model. D50, D55, D65, D75, F1, F2, to F11 under the CIE1931 Color Res Appl 2000;25:260–266. standard observer. Similar results can also be found with the 4. Luo MR, Hunt RWG. A chromatic adaptation transform and a colour inconstancy index. Color Res Appl 1998;23:154–158. CIE1964 standard observer and chromaticity coordinates. 5. Li CJ, Luo MR, Rigg B, Hunt RWG. CMC 2000 Chromatic Adapta- So as long as the chromaticity coordinates of the light tion Transform: CMCCAT2000. Color Res Appl 2002;27:49–58. source is inside the area cycled by the chromaticity locus of 6. CIE. A method of predicting corresponding colors under different the CIE chromaticity coordinates, the corresponding chromatic and illuminant adaptations. CIE Tech. Rep. 109, Vienna. 7. Nayatani, Yano T, Hashimoto K, Sobagaki. Proposal of an abridged

Xw color-appearance model CIECAT94LAB and its field trials. Color Res Appl 1999;26:422–438. M ͩ Yw ͪ HEP Z 8. Fairchild MD. A revision of CIECAM97s for practical applications. w Color Res Appl 2001;26:418–427. 9. Hunt RWG, Li CJ, Juan LY, Luo MR. Further improvements to 0.38971Xw ϩ 0.68898Yw Ϫ 0.07868Zw Ϫ ϩ ϩ CIECAM97s. Color Res Appl 2002;27:164–170. ϭ ͩ 0.22981Xw 1.18340Yw 0.04641Zw ͪ Ն 0. 10. Hunt RWG, Li CJ, Luo MR. Dynamic cone response functions for Zw models of colour appearance. Color Res Appl 2003;28:82–88. Therefore, by inequality A1, we always have 11. Li CJ, Luo MR, Hunt RWG, Moroney N, Fairchild MD, Newman T. The performance of CIECAM02. Proc. IS&T/SID 10th Color Imaging RЈ R 1 Conference, IS&T, Springfield, VA; 2002; pp 28–32. w wc 12. Li CJ, Luo MR, Cui GH. Colour-difference evaluation using colour Ј Ϫ1 ͩ G w ͪ ϭ M M ͩ Gwc ͪ Ն Y D ͩ 1 ͪ. th HEP CAT02 w appearance models, Proc IS&T/SID 11 Color Imaging Conference, Ј B w Bwc 1 IS&T, Springfield, VA; 2003, p 127–131. 13. IEC/3WD 61966-2.1: Color measurement and management in multi- Conversely, media systems and equipment—Part 2.1: Draft RGB color space— sRGB, 1998. 400͑F RЈ/100͒0.42 Ј ϭ L w ϩ 14. Wyszecki G, Stiles WS. Color science, 2nd edition. New York: John R aw 0.42 0.1 27.13 ϩ ͑FLRЈw/100͒ Wiley and Sons; 1982.

106 COLOR research and application