Gaussian Illuminants and Reflectances for Colour Signal Prediction

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1 Gaussian Illuinants and Reflectances for Colour Signal Prediction Haidreza Mirzaei, Brian Funt; Sion Fraser University; Vancouver, BC, Canada Abstract An alternative to the von Kries scaling underlying the chroatic adaptation transfors found in colour appearance odels such as CIECAM0 is suggested for predicting what the colour signal (e.g., XYZ) reflected fro a surface under a first illuinant is likely to becoe when lit instead by a second illuinant. The proposed ethod, G M, eploys etaeric Gaussian-like functions to odel the illuinant and reflectance spectra. The ethod s prediction is based on relighting the Gaussian-like reflectance spectru with the second Gaussian-like illuinant. Tests show that the proposed G M ethod significantly outperfors CIECAT0. Introduction The colour signal (i.e., cone response triple or CIE XYZ) observed reflected fro an object s surface will generally change when the spectru of the incident light is changed. This paper investigates the proble of predicting the new colour signal given as input the original colour signal along with the colour signal of the incident light as reflected by an ideal reflector. Many ethods have been proposed for aking such prediction of which von Kries scaling [1] is perhaps the ost coon. Often the scaling is perfored in a sharpened [] basis, as is done, for exaple, in the chroatic adaptation transfors of Bradford [3] and the CIECAT0 of CIECAM0 [4]. In a different vein, Fairchild [5] proposed a spectral adaptation ethod based on the ratio of soothed spectra. As an alternative to von Kries scaling, the Gaussian Metaer (GM) ethod [6] instead predicts the new colour signal by finding a wraparound Gaussian reflectance [7] that is etaeric to the given colour signal, coputationally relighting that reflectance and then coputing the resulting colour signal. The ean prediction error for the GM ethod was reported to be roughly half that of either von Kries or Bradford. The Gaussian Metaer ethod, however, requires knowledge of the full spectru of the incident illuination, not just its colour signal. To eliinate the need for the full illuinant spectru, we propose replacing the true illuinant spectru with a etaeric Gaussian spectru. Although the prediction error increases when using a etaeric illuinant, it reains, nonetheless, less than two thirds of the CIECAT0 error. The details are presented below. Although predicting the colour signal under an illuinant change is potentially related to chroatic adaptation, it is not the sae as predicting the resulting colour appearance [8]. Neither is it the sae as predicting what huan subjects ay see as corresponding colours on displays since then the corresponding colours are the colours of lights, not object colours. One exception in this regard is Fairchild s [5] Spectral Adaptation ethod and data. The bulk of the Luo et al. LUTCHI dataset [9] is based on the atching of lights, with a relatively sall portion perfored using objects. The existence of etaer isatching also eans there is no unique colour signal to predict. Since there are any reflectances that can lead to the sae colour signal under the first light, the set of possible colour stiuli that results under the second light fors a volue in colour space, often referred to as the etaer isatch volue. Any colour signal in the etaer isatch volue is a possible correct answer. Our goal is siply to find the colour signal that fits the experiental data best on average, with the potential error in any particular case liited only by the size of the associated etaer isatch volue. Gaussian Reflectance and Illuinant Spectra Logvinenko [7] defines a set of Gaussian-like spectral reflectance functions defined in ters of their scaling, k, standard deviation, σ, and peak wavelength, μ. These Gaussian-like functions are not strictly Gaussians, but rather are defined on a finite wavelength interval and in soe cases wraparound at the ends of the interval, hence the nae wraparound Gaussians. Although the equations defining the are piecewise and a bit coplex, intuitively they siply describe a Gaussian centered at μ on the hue circle. Following Logvinenko, the reflectance functions are defined by Eqs. 1 to 4. If ( )/ : ax in For / in For / ax If ( )/ we have two cases: ax in For / in For / ax where ax and in are the ends of the visible spectru, ax in and 1/ (1). For 0 k 1, () (3) (4) Society for Iaging Science and Technology

2 and positive, we have a Gaussian-like in ax reflectance function. Now consider also the three-paraeter set of illuinant functions of the sae for: If ( )/ : l ax in For / in l (5) For / l ax l If ( )/ we have two cases: l ax in For / in l For / l ax Note that for the spectral power distributions the restriction is on the scaling is siply kl >= 0 since the intensity of the light is not liited. We will refer to triples (k, σ, μ) and (kl, σl, μl) as the KSM coordinates of the reflectance and light, respectively. Proposed Method Given the colour signal specified in CIE XYZ (or cone LMS) coordinates of light reflected fro a surface and the spectra of the first (F) and second (S) illuinants, the first step in the original GM ethod is to deterine the KSM coordinates of the wraparound Gaussian reflectance using a fast interpolation ethod [10] that is etaeric (i.e., of identical XYZ) to the given surface under F. This etaeric reflectance spectru is relit in other words, ultiplied by the full spectru of S and the colour signal under S is then calculated using the CIE 1931 x y z colour atching functions. The proposed new ethod odels the surface reflectance as well as both illuinants using wraparound Gaussian etaers and will be denoted G M. The first step in the G M ethod is to deterine the KSM coordinates (kf, σf, μf), again using the fast interpolation ethod, of the Gaussian illuinant spectru that is etaeric to F. The second step is to find the KSM coordinates (k, σ, μ) of the Gaussian reflectance that under Gaussian illuinant (kf, σf, μf) has the sae XYZ as the given surface under F. The third step is to find the Gaussian illuinant with coordinates (ks, σs, μs) that is etaeric to S. The fourth step is to relight the Gaussian reflectance (k, σ, μ) using the Gaussian illuinant (ks, σs, μs) and deterine its resulting XYZ colour signal. (6) (7) (8) Tests We copare the prediction results using GM and G M to those of CIECAT0, which is a chroatic adaptation transfor and the first step in the CIECAM0 colour appearance odel [4]. The three ethods predictions are copared to the coputed ground-truth values under the second illuinant (i.e., XYZ of the actual reflectance spectra ultiplied by the spectru of the second illuinant). Munsell Papers under CIE Illuinants In the first test we consider the set of 1600 Munsell papers [11] under CIE D50 as the first illuinant and CIE A and CIE D65 as two different second illuinants. The accuracy of each colour signal prediction is easured in ters of the CIEDE000 colour difference easure. Table 1 lists the results where it can be seen that the GM and G M predictions are better than those of CIECAT0 using coplete adaptation. Although the perforance of G M is, as expected, slightly worse than that of GM, the difference is surprisingly sall given that in G M the spectra of both illuinants are replaced with wraparound Gaussians. Table 1: CIEDE000 prediction errors of CIECAT0, GM and G M for the case of the 1600 Munsell papers with the illuinant changing fro CIE D50 to CIE A and to CIE D th To Method Median Mean Percentile GM A G M CAT GM D65 G M CAT Although GM and G M ake better predictions than CIECAT0, an additional advantage of these ethods is that their predictions are guaranteed to be within the etaer set so long as the KSM coordinates have k 1 since that condition ensures that the resulting wraparound Gaussian is a reflectance function (i.e., strictly within the range 0 to 1). For a very few colour signals the KSM coordinates have k > 1. For any such case, we use a rectangular etaer function fro Logvinenko s original object colour atlas [1] in place of a wraparound Gaussian. Of the 1600 colour signals of the Munsell papers under D50, GM found the KSM coordinates of only 44 of the had k > 1 and G M found only 43. For these few cases rectangular etaer functions were used in place of wraparound Gaussian functions. Since the rectangular reflectance atlas is coplete, we are guaranteed that there will always be a rectangular reflectance function that is etaeric to any given colour signal. As a result, all colour signal predictions are guaranteed to be feasible in that they lie within the etaer isatch volue of the given colour signal. nd Color and Iaging Conference Final Progra and Proceedings and nd Congress of the International Acadey of Digital Pathology 13

3 Chroatic Illuinants and Varying Chroa/Value As a further test of the proposed G M ethod of colour signal prediction, we consider the set of illuinants Logvinenko and Tokunaga used in their asyetric colour atching experient [13]. The spectra of these illuinants are shown in Fig. 1, where the colour of the curves indicates their corresponding spectra, except for the neutral, which is plotted in gray. Since the two reds, R1 and R, are siilar, we use only R1. For a change of illuinant fro G, B, Y, or R1 to N, Table gives the edian and average CIEDE000 error in the predictions taken across the 0 Munsell hues at a given value and chroa. It can be seen that both GM and G M ethods consistently outperfor CIECAT0. To assess the results visually, consider the exaple of G to N prediction. Fig. plots the GM, G M, and CIECAT0 predictions in chroaticity space for the 0 hues at value 7, chroa 8. The average error over the 0 papers in this case is 8.8 CIEDE000 for GM, 9.7 for G M, for CIECAT0. In ters of how the prediction error varies with chroa, Fig. 3 plots the edian error for the sae 0 hues at value 7, with the chroa varying over, 4, 6, and 8. The illuinant change is fro G to N. The error tends to increase with increasing chroa for all three ethods; however, the GM and G M errors are significantly less than those of CIECAT0 in all cases. Conclusion Fig. 1. Spectral power distributions of the green (G), blue (B), neutral (N), yellow (Y), first red (R1) and second red (R) illuinants used in Logvinenko and Tokunaga s experients [13]. The plotted colours identify the associated spectru along with grey indicating N. The solid red indicates R1 and the dashed red indicates R. We considered 0 different hues fro the Munsell Book of Color that saple the full hue circle. They are: 5 R, 10 R, 5 YR, 10 YR, 5 Y, 10 Y, 5 GY, 10 GY, 5 G, 10 G, 5 BG, 10 BG, 5 B, 10 B, 5 PB, 10 PB, 5 P, 10 P, 5 RP, and 10 RP. To evaluate the effect of Munsell chroa and value on the predictions, we test the three ethods at these hues while varying the chroa over, 4, 6, and 8, and value over 5 and 7. A new ethod of predicting the change in colour signal of the light reflected fro a surface was presented and shown to ake significantly ore accurate predictions than CIECAT0 with full adaptation. The proposed G M ethod eliinates the requireent of the previous Gaussian Metaer (GM) ethod [6] that full spectral power distributions of the two illuinants be known. Although the tests show that the accuracy of G M is soewhat less than that of GM, it is still significantly ore accurate than CIECAT0 as easured in ters of CIEDE000 colour differences. G M shares with GM the fact that, unlike von-kries-based prediction ethods, all of its predictions are guaranteed to represent a physically realizable change in colour signal. Fig.. Colour signal prediction for the 0 Munsell papers (of chroa 8 and value 7) when the illuinant is changed fro G (green) to N (neutral). Left GM, center G M and right CIECAT0. Plot is in CIE xy-chroaticity space. An arrow tail indicates the actual chroaticity of the paper under the neutral illuinant with the corresponding arrow head its predicted chroaticity. The red and green curves siply connect all the arrow tails and arrow heads for clarity. The red curves are the sae in all 3 panels Society for Iaging Science and Technology

4 Table : The edian and average prediction error in CIEDE000 for the change fro each of the 5 different chroatic illuinants to N ( white ). Each row is for papers of the 0 Munsell hues at the specified value and chroa. The last row reports the ean of the values in the corresponding colun. First Munsell Attribute Median CIEDE000 Mean CIEDE000 Illuinant Value Chroa GM G M CAT0 GM G M CAT0 G G G G G G G G B B B B B B B B Y Y Y Y Y Y Y Y R R R R R R R R Mean Fig. 3. Different ethods edian CIEDE000 prediction error as a function of Munsell chroa (, 4, 6, 8) at value 7 for a change of illuinant fro G to N. The GM, G M, and CIECAT0 results are plotted in solid green, dotted red, and dashed blue, respectively. References [1] von Kries, J. (1970) Chroatic adaptation. In D. L. MacAda (Ed.), Sources of colour vision. Cabridge, MA: MIT Press, pp [] Finlayson, G. D., Drew, M. S., & Funt, B. V. (1994). Spectral sharpening: sensor transforations for iproved colour constancy. J. Optical Society of Aerica A, vol. 11, no. 5, pp [3] La, K. M. (1985). Metaeris and colour constancy. Doctoral dissertation, University of Bradford. [4] Moroney, N., Fairchild, M., Hunt, R., Li, C., Luo, M., and Newan, T. (00) The CIECAM0 colour appearance odel. Proc. Tenth IS&T Color Iaging Conference. [5] Fairchild, M. D. (006). Spectral adaptation: A reason to use the wavenuber scale. Proc. Fourteenth IS&T Color Iaging Conf. pp [6] Mirzaei, H. and Funt. B. (011) Gaussian-etaer-based prediction of colour signal change under illuinant change, AIC Midter Meeting, International Colour Association. [7] Logvinenko, A. D. (013). Object-colour anifold. International journal of coputer vision, vol. 101, no. 1, pp nd Color and Iaging Conference Final Progra and Proceedings and nd Congress of the International Acadey of Digital Pathology 15

5 [8] Fairchild, M. D. (005). Color appearance odels. John Wiley & Sons. New York. [9] Luo, M. R., Clarke, A. A., Rhodes, P. A., Schappo, A., Scrivener, S. A., & Tait, C. J. (1991). Quantifying colour appearance. Part I. LUTCHI colour appearance data. Color Research & Application. vol. 16, no. 3, pp [10] Godau, C., and Funt. B, (010) XYZ to ADL: Calculating Logvinenko s Object Colour Coordinates. Eighteenth IS&T Color Iaging Conference. [11] University of Eastern Finland, Joensuu spectral database. (01) Available online: Accessed: Deceber 11, 01. [1] Logvinenko, A. D. (009). An object-colour space. J. of Vision, vol. 9, no. 11, pp [13] Logvinenko, A. D., & Tokunaga, R. (011). Colour constancy as easured by least dissiilar atching. Seeing and Perceiving, 4(5), Society for Iaging Science and Technology