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 condi- tion, as long as Eqs. (4) and (5) are satisfied, the forward If we set mode can successfully predict the perceptual attributes cor- relates: 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 .