Performance of a color-difference formula based on OSA-UCS space using small medium color differences

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1 Huertas et al. Vol. 23, o. 9/ September 2006/J. Opt. Soc. Am. A 2077 Performance of a color-difference formula based on OSA-UCS space using small medium color differences Rafael Huertas and Manuel Melgosa Departamento de Óptica, Facultad de Ciencias, Universidad de Granada, Granada, Spain Claudio Oleari Dipartimento di Fisica, Istituto azionale di Fisica per la Materia, Università degli Studi di Parma, I Parma, Italy Received June 10, 2005; revised December 18, 2005; accepted December 29, 2005; posted March 28, 2006 (Doc. ID 62734) An investigation of the color metrics and the complexity of the CIEDE2000 formula shows that CIELAB space is inadequate to represent small medium color differences. The OSA-UCS (Uniform Color Space) Committee has shown that no space with uniform scale for large color differences exists. Therefore the practical way for color-difference specification is a color-difference formula in a nonuniform space. First, the BFD (Bradford University) ellipses are considered in the OSA-UCS space, and their very high regularity suggests a new and very simple color-difference formula at constant luminance. Then the COM (combined) data set used for the development of the CIEDE2000 formula is considered in the OSA-UCS space, and the color-difference formula is extended to sample pairs with a different luminance factor. The value of the performance factor PF/ 3 for the proposed OSA-UCS-based formula shows that the formula performs like the more complex CIEDE2000 formula for small medium color differences Optical Society of America OCIS codes: , , , ITRODUCTIO The definition of a color-difference formula for small medium color differences is a highly debated problem in color science during the past few decades and is still open today. 1 The complexity of the CIEDE formula induces us to consider the CIELAB space, in which the formula is defined, inadequate to represent the small medium color differences and to remeditate the whole problem. If a uniform, Euclidean color space exists, then a color difference or threshold is represented by the equation for the surface of a sphere. The huge effort to realize an empirical uniform color space for large color differences has shown that the best space obtained, i.e., the OSA-UCS 3 5 (Uniform Color Space) one, has nonuniform scales. In this space, although nonuniform, the straight lines radiating from any color sample are geodesic lines, each with a uniform scale. This property induces us to expect that the small medium color differences are represented in this space by a highly regular formula 6,7 (the conversion formulas from tristimulus values to OSA-UCS coordinates are listed in Appendix A). With the aim of arriving at this formula, first the BFD (Bradford University) chromaticity-discrimination ellipses 8 are considered in this space. The BFD ellipses, defined at a constant luminance factor, show a very high regularity. Subsequently, the data sets used to develop the CIEDE2000 formula 9 are considered, i.e., BFD-P, 8 Leeds, 10 RIT DuPont, 11 and Witt 12 data sets. These latter data sets are combined in a set termed COM. 9 This work considers data sets of visually perceived color differences V i (the foot index i indicates each one of the color pairs considered) and the corresponding sets of color differences E i computed by a color-difference formula. The agreement between these two data sets indicates the goodness of the formula, which has to be evaluated by a statistical measure. The statistical measure here considered is the percentage performance factor PF/3 proposed by Guan and Luo 13 and shown in Appendix B. The parameters of the color-difference formulas here developed are obtained in two different ways: (1) by linear fitting of semiaxis values of the BFD ellipses and of analogous quantities related to the COM data set and (2) by minimizing the PF/3 13 evaluated on the experimental BFD ellipses and COM data sets. Our color-difference formula, evaluated on all the experimental data considered, produces a PF/ 3 index, which is generally competitive with that of the CIEDE2000 formula. Our formula is an advance in the understanding of the color discrimination phenomena because it has high regularity and simplicity in the OSA- UCS space, whose coordinates and the conversion formulas from tristimulus values are in strong relation with the psychophysics and physiology of the color vision. All this analysis holds true for the CIE 64 supplementary standard observer and the illuminant D COLOR-DIFFERECE FORMULA AD BFD ELLIPSES The whole BFD data set 8 has been obtained by putting together the BFD-P color-difference data set, representing /06/ /$ Optical Society of America

2 2078 J. Opt. Soc. Am. A/ Vol. 23, o. 9/ September 2006 Huertas et al. the perceptibility, and the BFD-A, representing the acceptability. The ellipses associated with these two data sets are considered separately and together. The evident result is that BFD-P and BFD-A ellipses, defined at a constant luminance factor, appear highly regular in OSA- UCS space (Fig. 1): (1) the long axes of the ellipses lie on lines radiating from the achromatic point; (2) these ellipses no longer contain the well-known tilt, evident in the blue region of the CIELAB a *, b * plane 7,14 ; and (3) both axes have lengths that appear hue independent and linearly chroma dependent. These properties are easily represented in the following formula designed with the foot index GP (from 2 Granada Parma): E GP 2 =10 L OSA + C OSA where 2 k L S L 2 k C S C + H OSA k H S H 2, 1. L OSA, C OSA, and H OSA are independent color differences in the OSA-UCS space related to differences of lightness, chroma, and hue, respectively (Appendix A); 2. The factor 10 2 is introduced because the OSA-UCS unit of distance is equal to 10 just-noticeable differences (jnds); Fig. 1. J,G plane of the OSA-UCS space with the BFD ellipses 8 : BFD-P (thick ellipses) and BFD-A (thin ellipses) S C, and S H are linear functions of the chroma C OSA,s of the color standard s; 4. S L cannot be defined because the BFD ellipses have constant luminance; 5. The parametric factors, k L, k C, and k H, introduced as correction terms for variation in experimental conditions, are all set equal to 1 in the current paper. A first technique for deriving the weighting functions S C and S H assumes that these functions are linear functions of the chroma C OSA,s of the color standard s and therefore are obtained by fitting the long and short semiaxes of the ellipses of the whole BFD set and the BFD-P and BFD-A subsets, considered separately (Table 1, left side). S C and S H are almost equal for all three sets of ellipses (BFD, BFD-P, and BFD-A), and this means that a very good general agreement and high compatibility exist between the BFD-P and BFD-A ellipses (Fig. 1). Figure 2 shows the S C and S H functions fitted to the whole BFD data set. The weights S C and S H are obtained by considering the semiaxis lengths, i.e., only two points per ellipse. Figure 3 compares the empirical BFD ellipses with those represented by Eq. (1), using the S C and S H functions given in Table 1, left side, for BFD. The second technique used for obtaining the line equation parameters of S C and S H, consists of minimizing the PF/3 index for 100 points per ellipse (not 2 as for the first technique) with angular uniform spacing and corresponding to a visual difference V i always assumed equal to 1. The value of the PF/3 index is almost constant for a number of color pairs greater than eight per ellipse. Because the PF/3 index is independent of a scale factor equal for all the unknown parameters, it is possible to define these parameters up to a scale factor. This scale factor is defined in a second step by minimizing the quantity i V i E GP,i, where the addition is made over all the color pairs considered. The minimization is obtained by applying the simplex algorithm, 15 which uses the weights evaluated by the first technique as starting values. The results related to the three BFD sets of ellipses are in Table 1, right side. The BFD ellipses are defined at a constant luminance factor, and thus they cannot be used to produce the S L function. This induces us to employ the different data sets Table 1. S C and S H Weights and Corresponding Linear Correlation Indices r Obtained by Linear Fits of the Long and Short Semiaxes of the Ellipses and Obtained by Minimizing the PF/3 Index as Functions of the Chroma C OSA,s of the Color Standard s (the Center of the Ellipse) a Linear Fitting PF/ 3 Optimization Set of Ellipses Weighting Functions and Linear Correlation Index PF/ 3 Weighting Functions PF/ 3 BFD S C = C OSA,s,r= S C = C OSA 21.4 S H = C OSA,s,r=0.81 S H = C OSA BFD-P S C = C OSA,s,r= S C = C OSA 22.7 S H = C OSA,s,r=0.82 S H = C OSA BFD-A S C = C OSA,s,r= S C = C OSA 21.7 S H = C OSA,s,r=0.78 S H = C OSA a Factor 10 is due to the unit of measure of C OSA,s equal to 10 jnd. The minimization of the PF/3 index and the PF/3 13 related to the three sets of ellipses BFD, BFD-P, and BFD-A are evaluated on 100 points per ellipse.

3 Huertas et al. Vol. 23, o. 9/ September 2006/J. Opt. Soc. Am. A 2079 Fig. 2. Long (solid squares) and short (open squares) semiaxes of the whole BFD 8 set of ellipses as functions of the chroma C OSA,s of the center of the ellipse. Fig. 3. J,G plane of the OSA-UCS space with the BFD ellipses (thick ellipses) and the ellipses representing the color-difference formula E GP (thin ellipses), assuming the S C and S H functions given in Table 1, left side, for BFD. used to define globally the CIEDE2000 formula 9 in order to achieve in this way the lightness-difference dependence in OSA-UCS space. 3. COLOR-DIFFERECE FORMULA BASED O BFD-P, LEEDS, RIT DUPOT, WITT, AD COM COLOR PAIRS The BFD-P, 8 Leeds, 10 RIT DuPont, 11 and Witt 12 data sets, used to define the CIEDE2000 formula, 2,9 are constituted by pairs of different colors representing small medium distinction perceptibility. In this case, the weighting functions S L, S C, and S H are defined below as functions of the average coordinates of the two color samples of any pair [L OSA= L OSA,s +L OSA,b /2, J = J s +J b /2, Ḡ= G s +G b /2], not from ellipses (foot index s means standard and b means batch). Also in this case the weighting functions S L, S C, and S H are evaluated by two different techniques: (1) linear fitting of quantities related to ellipsoid semiaxes and (2) minimizing the PF/ 3 index. The results obtained by the second technique are better than those obtained by the first one. The reasons for considering also the first technique are two: the first reason is to show that S C and S H have linear dependence on the chroma and S L is independent of the chroma, and the second reason is because the weights evaluated by the first technique are the input numbers to the second technique. With regard to the first technique, as in previous works, 9 S L, S C, and S H are computed separately by selecting color pairs by properly defined filters. ot all the color pairs from a data set are good for knowing the semiaxis lengths of the ellipsoids and for evaluating the weighting functions; strictly speaking, only those with two of the differences L OSA, C OSA,or H OSA equal to zero are suitable. For instance, if L OSA =0 and H OSA =0, the measured color difference V between the two colors, evaluated according to Eq. (1), results in V 2 =100 C OSA /S C 2 and then S C =10 C OSA / V. Analogously, S H =10 H OSA / V for L OSA =0 and C OSA =0, and S L =10 L OSA / V for C OSA =0 and H OSA =0. Practically, the requirement that two of the three differences L OSA, C OSA, and H OSA be equal to zero reduces the sets of useful color pairs to almost empty sets. Therefore, for the evaluation of S L, S C, and S H, color pairs are chosen with at least 90% of the total color difference attributable to only L OSA, C OSA,or H OSA, respectively. This is achieved by the filters L OSA / E 0 0.9, C OSA / E 0 0.9, and H OSA / E 0 0.9, where E 0 is the Euclidean distance in OSA-UCS space computed by Eq. (A6). Table 2 shows the number of color pairs selected by these filters in each data set used for the evaluation of S L, S C, and S H (let us call these combined data sets COM1). Table 2 also shows the different weights assigned to each one of the four data sets in the realization of COM1 data sets, following a procedure analogous to the one employed in the development of the CIEDE2000 formula. 9 Otherwise, if the four data sets were put together without any weight, the result would be biased toward the BFD-P data set, which always has the greatest number of color pairs. If the quantities L OSA / V, H OSA / V, and C OSA / V are considered functions of the averages L OSA, J, and Ḡ of the coordinates of the filtered color pairs, the main dependences are that H OSA / V and C OSA / V are linear functions of C OSA and L OSA / V is a linear function of L OSA. The S L, S C, and S H weighting functions, computed for each set of filtered color pairs, COM1 data sets included, are shown in Figs. 4 6 and in Table 3, left side. As for the BFD ellipses, the second technique used for obtaining the line equation parameters of S C and S H consists of minimizing the PF/3 index for all the color pairs Table 2. umber of Color Pairs Corresponding to the Filtered Data Sets Used for the Computation of the Weighting Functions S L, S C,andS H (Table 3, Left Side) a Data Set L OSA / E C OSA / E H OSA / E BFD-P (1) 556 (1) 406 (1) Leeds (2) 81 (7) 17 (24) RIT DuPont (11) 20 (28) 17 (24) Witt (7) 133 (4) 65 (6) a E 0 is the Euclidean distance in OSA-UCS space. The factors in parentheses must be applied in order to give approximately the same weight to the four data sets when they are put together to constitute the COM1 data sets, used for the computation of the weighting functions in COM data sets.

4 2080 J. Opt. Soc. Am. A/ Vol. 23, o. 9/ September 2006 Huertas et al. Fig. 4. Plot of L OSA / V as a function of the average lightness L OSA for the color pairs with predominant lightness differences (i.e., L OSA / E 0 0.9). of the data sets considered. As previously, the unknown parameters of the weights S C and S H are obtained up to a scale factor, which is defined in a second step by minimizing the quantity i V i E GP,i, where the addition is made over all the color pairs considered. The results are shown in Figs. 7 9 and in Table 3, right side. For any comment, only the results obtained by the second technique are considered because they are based on all the data of the COM data set and therefore have higher meaning. First, the mutual agreement between the results based on different data sets and related to the weight S L is lower than those related to the two weights S C and S H. Probably, this phenomenon depends on the luminance crispening. 16 The OSA-UCS space has been defined including the luminance crispening effect produced by comparing the color samples on a gray surround with 30% reflectance. The color pairs considered in the different data sets have been judged while not being exactly in the same viewing situation required by the OSA-UCS system. Probably, this is the main cause of the discrepancies existing among the different data sets. Certainly, with regard to S L, the luminance crispening effect should be considered. The lightness weighting function of the CIEDE2000 formula is a V-shaped curve with the lowest point at the CIELAB lightness of 50, describing the crispening effect. 16 This lightness corresponds to a luminance factor Y=18.4, which is far from the surround luminance factor of 30 of the OSA-UCS system. Moreover the surround luminance used for the different data sets is not the same. The results related to the two weights S C and S H are in general good agreement. The comparison of these weights with those obtained from the BFD ellipses shows that these weights are in the average of the others and with almost equal slope. Fig. 5. Plot of C OSA / V as a function of the average lightness C OSA for the color pairs with predominant chroma differences (i.e., C OSA / E 0 0.9). Fig. 6. Plot of H OSA / V as a function of the average lightness C OSA for the color pairs with predominant hue differences (i.e., H OSA / E 0 0.9). 4. PERFORMACE OF THE FORMULAS EVALUATED BY THE PF/3 IDEX Tables 1, 3, and 4 show the PF/3 (Ref. 13) values computed for different color-difference formulas and different data sets. The color-difference formulas CIELAB 1976 and CIEDE2000 are considered in the CIELAB space, while the Euclidean formula E 0 of Eq. (A6) and the formula E GP of Eq. (1), with the weights obtained by two different techniques, are considered in the OSA-UCS space. The results are the following: 1. The Euclidean formula E 0 evaluated in the OSA- UCS space has a lower PF/ 3 value than the CIELAB 1976 Euclidean formula in the CIELAB space, except for the Leeds data set, and this indicates that the OSA-UCS space is more appropriate than CIELAB 1976 to represent small medium color differences; 2. As expected, E GP strongly improves the results achieved by E 0 in all data sets with the PF/3 value equal to 35.6 for the COM 2,9 data set and weights obtained by the second technique, which is very close to the value 32.6 of the CIEDE2000; 3. The lowest PF/3 value, 19.2, is in regard to the RIT DuPont data set and E GP (Table 3);

5 Huertas et al. Vol. 23, o. 9/ September 2006/J. Opt. Soc. Am. A 2081 Table 3. Weighting Functions S L, S C,andS H Obtained by Two Different Techniques and Computed for Different Subsets as Functions of the Average Lightness L OSA and Average Chroma C OSA a Linear Fitting PF/ 3 Optimization Data Set Weighting Functions and Linear Correlation Index PF/ 3 Weighting Functions PF/ 3 BFD-P 8 S L = L OSA,r= S L = L OSA S C = C OSA,r=0.66 S C = C OSA S H = C OSA,r=0.53 S H = C OSA Leeds 10 S L = L OSA,r= S L = L OSA S C = C OSA,r=0.88 S C = C OSA S H = C OSA,r=0.30 S H = C OSA RIT DuPont 11 S L = L OSA,r= S L = L OSA S C = C OSA,r=0.51 S C = C OSA S H = C OSA,r=0.63 S H = C OSA Witt 12 S L = L OSA,r= S L = L OSA S C = C OSA,r=0.29 S C = C OSA S H = C OSA,r=0.27 S H = C OSA COM 2,9 S L = L OSA,r= S L = L OSA S C = C OSA,r=0.60 S C = C OSA S H = C OSA,r=0.48 S H = C OSA a For the first technique, linear correlation indices r are given. Values of the PF/3 13 computed for the corresponding different color-difference formulas related to different data sets BFD-P, 8 Leeds, 10 RIT DuPont, 11 and Witt 12 constituting the COM data set. 4. The PF/3 value is very low, lower than 23, based on the three sets of BFD ellipses, for both the CIEDE2000 and the E GP formulas; 5. The PF/3 values for the subsets and the E GP formula range from 21.7 to 41.7, while for the CIEDE2000 formula they range from 19.3 to Fig. 7. Plots of the S L weighting functions obtained from different data sets by minimizing the PF/3 index. 5. COCLUSIOS The analysis carried out in this work is based on the most comprehensive set of experimental data available on small medium color differences represented in the OSA- UCS space. The weighting functions S H, S C, and S L of Eq. (1), obtained from the different data sets here considered, show a general qualitative agreement among these data sets, although with different PF/ 3 values: 1. Particularly strong and evident is the result obtained by the use of the BFD ellipses, quantified by very high correlation-index values and very low PF/ 3-index values; 2. The dependence of S H and S C on the chroma is the most important one for all the data sets considered; 3. The dependence of S H and S C on the lightness is slight; 4. The dependence of S L on the lightness is weak; 5. The PF/3 values provided by Eq. (1) with appropriate weighting functions are close to or better than those provided by CIEDE2000. Fig. 8. Plots of the S C weighting functions obtained from different data sets by minimizing the PF/3 index. This analysis is promising, and, at the moment, the OSA-UCS space appears to be the best candidate to represent the small medium color differences in daylight adaptation by a very simple color-difference formula such as

6 2082 J. Opt. Soc. Am. A/ Vol. 23, o. 9/ September 2006 Huertas et al. Eq. (1). The CIEDE2000 formula is generally considered the most advanced formula but is now in competition with the new E GP formula: S C = C OSA, 3 1. The difference between the values of the PF/3 index evaluated for the E GP formula and for the CIEDE2000 one is lower than 0.8 percentage units for all the BFD data sets, is lower than 3 units for the RIT DuPont and Witt data sets, is equal to 8 units for the Leeds data set, and is equal to 3 for the COM data set (Table 4); 2. The E GP formula is much simpler than the CIEDE2000 one. All this induces us to consider the OSA-UCS space and the formula E GP, with the weighting functions based on the weighted COM experimental data sets, following the same methodology on which CIEDE2000 was based, as the most appropriate for further testing, i.e., Eq. (1) with S L = L OSA, 2 S H = C OSA, with L OSA and C OSA the average coordinates of the two color samples of any pair. APPEDIX A At least four different transformation formulas 4,7,17 19 have been proposed for the conversion from the tristimulus space to the OSA-UCS one. These formulas are in very good mutual agreement in the region of the space where the OSA-UCS color samples are given, and this region is almost the same as that considered for the definition of a color-difference formula. Here, the last proposed transformation formula 18 is used, in which the main practical properties are the analytical backconversion and the simplicity. Let Y 10,x 10,y 10 and L OSA,J,G be the color specification in the two spaces. The unit of distance in the OSA- UCS space is approximately 10 jnd, assuming that this space conforms to Euclidean metrics. The lightness is defined as in the original OSA-UCS formula: 4 L OSA = 5.9 Y 0 1/ Y / , Fig. 9. Plots of the S H weighting functions obtained from different data sets by minimizing the PF/3 index. with Table 4. Values of the PF/3 a Computed for Different Color-Difference Formulas and Related to the Three Sets of Ellipses BFD, BFD-P, and BFD-A (Evaluated on 100 Points per Ellipse) and Different Data Sets (BFD-P, b Leeds, c RIT DuPont, d and Witt e ) Constituting the COM Data Set f PF/3 a A1 Data Set umber of Color Pairs CIELAB 76 CIEDE E 0 E GP BFD-P ellipses BFD-A ellipses BFD ellipses 13, BFD-P b Leeds c RIT DuPont d Witt e COM 2,9 11,273 (1:9:18:7) a Ref. 13. b Ref. 8. c Leeds, Ref. 10. d RIT DuPont, Ref. 11. e Witt, Ref. 12. f In the column of the umber of Color Pairs a set of four numbers is reported in parentheses, representing the relative weights to be applied to the color pairs of the subsets BFD-P, Leeds, RIT DuPont, and Witt, to ensure that all of them have nearly the same weight in the PF/3 computation. The PF/3 index is evaluated with weights proposed by Eqs. 2 4.

7 Huertas et al. Vol. 23, o. 9/ September 2006/J. Opt. Soc. Am. A 2083 Y 0 = Y x y x 10 y x y The lightness L OSA has no simple analytical conversion from the OSA-UCS space to the tristimulus space. 19 The coordinates J and G, which correspond to the empirical j and g of the OSA-UCS system, are obtained by a sequence of linear transformations and a logarithmic compression: A B C = X10 Y Z 10, A2 G = J S J 0 sin cos 0 S G sin cos ln A/B A n /B n ln n B/C B n /C L OSA ln A/B ln B/C = L OSA A3 The analytical reversibility of this transformation is straightforward, excluding the conversion of L OSA. The chroma and hue angle are obtained in OSA-UCS space from J and G coordinates as in CIELAB from a * and b *, i.e., C OSA = J 2 + G 2, h OSA = arctan J A4 G deg. Consequently, the Euclidean distance between standard s and batch b samples in the Cartesian reference frame L OSA,J,G is E 0 2 = L OSA,s L OSA,b 2 + J s J b 2 + G s G b 2 A5 and in pseudocylindrical coordinates L OSA,C OSA,H OSA is where E 0 2 = L OSA 2 + C OSA 2 + H OSA 2, L OSA = L OSA,s L OSA,b, A6 A7 that different statistical measures could lead to different conclusions. To minimize such a divergence, Guan and Luo 13 combined three different measures by a suitably weighted value, termed performance factor PF/ 3 and represented by the following percentage equation: 1 + V AB CV PF/3 = 100, B1 3 where, V AB, and CV are functions of two data sets: V i is constituted by the visually perceived color differences and E i is constituted by the color differences calculated from a given color-difference formula (the foot index i indicates each one of the color pairs considered). In formula (B1) is a statistical measure proposed by Coates et al. 20 and represents the antilogarithm of the root-mean-square value of the decimal logarithm of the ratios E i / V i : with C OSA = C OSA,s C OSA,b, A8 C OSA,s = J s 2 + G s 2, C OSA,b = J b 2 + G b 2, A9 log 10 = 1 i=1 log 10 E i V i log 10 E i V i 2. B2 H OSA 2 = E 0 2 L OSA 2 C OSA 2. A10 V AB is the statistical measure proposed by Schultze, 21 APPEDIX B Different statistical measures have been defined for the evaluation of the agreement between two data sets, one constituted by measured data and the other constituted by data computed by a mathematical formula. It happens with V AB = 1 i=1 E i F V i 2 E i F V i,

8 2084 J. Opt. Soc. Am. A/ Vol. 23, o. 9/ September 2006 Huertas et al. = i=1 F i=1 E i / V i V i / E i B3 and CV is a percentage statistical measure proposed by Coates et al., 20 with CV = i=1 f = E i f V i 2 E i V i i=1 2 V i i=1 E 2,. B4 Perfect agreement between two data sets exists for =1 and V AB =CV=0, which corresponds to PF/3=0, while a higher PF/ 3 value indicates a worse agreement. ACKOWLEDGMETS The authors thank the following organizations for their support: the Ministero dell Istruzione, dell Università e della Ricerca, Italia (MIUR) for Azioni Integrate Italia- Spagna IT928, Cofinanziamento MIUR 2002, Cofinanziamento MIUR 2003, and Cofinanziamento MIUR 2005; Ministerio de Ciencia y Tecnología, Spain, for Acción Integrada España-Italia HI , and the Ministerio de Educación y Ciencia, Spain for Research project FIS Address correspondence to Claudio Oleari, Università degli Studi di Parma, Dipartimento di Fisica, Parco Area delle Scienze 7/A, I Parma, Italy. Anyone interested in having the BASIC language algorithms can request it by claudio.oleari@fis.unipr.it. REFERECES 1. R. Huertas, M. Melgosa, and C. Oleari, A new colourdifference formula defined in the OSA-UCS space, in Proceedings of the Tenth Congress of the International Colour Association (AIC Colour, 2005), pp Improvement to Industrial Color-Difference Evaluation, CIE Publication 142 (CIE Central Bureau, 2001). 3. D. L. MacAdam, Color Measurement (Springer-Verlag, 1985), pp D. L. MacAdam, Uniform color scales, J. Opt. Soc. Am. 64, (1974). 5. D. L. MacAdam, Colorimetric data for samples of OSA uniform color scales, J. Opt. Soc. Am. 68, (1978). 6. R. G. Kuehni, Towards an improved uniform color space, Color Res. Appl. 24, (1999). 7. C. Oleari, Comparison between color-space scales, uniform color-scales atlases and color-difference formulae, Color Res. Appl. 26, (2001). 8. M. R. Luo and B. Rigg, Chromaticity-discrimination ellipses for surface colors, Color Res. Appl. 11, (1986). 9. M. R. Luo, G. Cui, and B. Rigg, The development of CIE 2000 colour-difference formula: CIEDE2000, Color Res. Appl. 26, (2001). 10. D. H. Kim and J. H. obbs, ew weightings functions for the weighted CIELAB color difference formula, in Proceedings of the 8th Congress of the International Colour Association (AIC Colour 97) (Association Internationale de la Couleur, 1997), Vol. 1, pp R. S. Berns, D. H. Alman, L. Reniff, G. D. Snyder, and M. R. Balonon-Rosen, Visual determination of suprathreshold color tolerances using probit analysis, Color Res. Appl. 16, (1991). 12. K. Witt, Geometric relations between scales of small colour differences, Color Res. Appl. 24, (1999). 13. S. S. Guan and M. R. Luo, Investigation of parametric effects using small colour-differences, Color Res. Appl. 24, (1999). 14. A. Yebra, R. Huertas, M. M. Pérez, and M. Melgosa, On the relationship between tilt of a * b * tolerance ellipses in blue region and tritanopic confusion lines, Color Res. Appl. 27, (2002). 15. J. A. elder and R. Mead, A simplex method for function minimization, Comput. J. 7, (1965). 16. G. Cui, M. R. Luo, and B. Rigg, Crispening effect on lightness differences, in 9th Congress of the International Colour Association, R. Chung and A. Rodrigues, eds. Proc. SPIE 4421, (2002). 17. T. Seim and A. Valberg, Towards a uniform color space: a better formula to describe the Munsell and OSA color scales, Color Res. Appl. 11, (1986). 18. C. Oleari, Color opponencies in the system of the uniform color scales of the Optical Society of America, J. Opt. Soc. Am. A 21, (2004). 19. M. Kobayasi and K. Yosiki, Effective conversion algorithm from OSA-UCS to CIEXYZ, in 9th Congress of the International Color Association, R. Chung and A. Rodrigues, eds., Proc. SPIE 4421, (2002). 20. E. Coates, K. Y. Fong, and B. Rigg, Uniform lightness scales, J. Soc. Dyers Colour. 97, (1981). 21. W. Schultze, The usefulness of colour-difference formulas for fixing colour tolerances, in Colour Metrics, Proceedings of the Helmholtz Memorial Symposium, J.J.Vos.L.F.C. Friele, and P. L. Walraven, eds. (Association Internationale de la Couleur (AIC), 1971), pp