From a set of gonioapparent automotive samples from different manufacturers we selected 28 low-chroma color pairs with relatively small color differences predominantly in lightness. These color pairs were visually assessed with a gray scale at six different viewing angles by a panel of 10 observers. Using the Standardized Residual Sum of Squares (STRESS) index, the results of our visual experiment were tested against predictions made by 12 modern color-difference formulas. From a weighted STRESS index accounting for the uncertainty in visual assessments, the best prediction of our whole experiment was achieved using AUDI2000, CAM02-SCD, CAM02-UCS and OSA-GP-Euclidean color-difference formulas, which were no statistically significant different among them. A two-step optimization of the original AUDI2000 color-difference formula resulted in a modified AUDI2000 formula which performed both, significantly better than the original formula and below the experimental inter-observer variability. Nevertheless the proposal of a new revised AUDI2000 color-difference formula requires additional experimental data.
© 2014 Optical Society of America
Color-difference formulas try to predict visually-perceived color differences between two stimuli for an average observer under specified illumination and viewing conditions . A color-difference formula provides a number ΔE from instrumental color specifications of two given color stimuli having a visually-perceived color difference ΔV. Note that while ΔV is the subjective answer of our complex human visual system, ΔE is a simple objective instrumental measurement. In industrial applications related to quality control it is important to use appropriate color-difference formulas to accurately predict subjective assessments of color differences performed by colorists or final users. For example, in the automotive industry, the different parts of cars are produced by different manufactures using different materials, and reliable color-difference formulas are necessary for automatic pass/fail color decisions [2, 3].
During the last 40 years advances have been made in the development of successful color-difference formulas for industrial applications , and the last CIE-recommended color-difference formula CIEDE2000  has been recently adopted as a joint CIE/ISO standard . Nevertheless it must be recognized that CIEDE2000 is not a completely successful formula [7, 8], and other recent color-difference formulas like CMC , CIE94 , DIN99d , CAM02 , or OSA-GP-Euclidean  perform similarly to CIEDE2000 for different experimental data sets . In addition to the simple situation of samples with homogeneous colors (also called solid colors), research on color difference evaluation has also faced the most important problem of non-homogeneous samples with additional appearance attributes to color (e.g. gloss, texture, etc.), as well as color differences in complex images [15–17].
In the middle of the 90’s AUDI developed a tolerance formula for the approval of effect paint batches when colors with strong flop effects were important in the automotive sector . Later this formula was modified to predict color tolerances for solids as well as for effect colors, leading to the AUDI2000 color-difference formula , currently employed by different manufacturers in the automotive industry. Gonioapparent materials or materials with flop effects represent a big challenge for color-difference evaluation in the automotive industry, and nowadays AUDI2000 is the only available color-difference formula considering such effects. According to ASTM , “appearance” can be defined as “the aspect of visual experience by which things are recognized”, “goniochromatism” as the “change in any of all attributes of color of a specimen on change in angular illuminating-viewing conditions but without change in light source or observer”, and “flop” as “a difference in appearance of a material viewed over two widely different aspecular angles”. While preliminary tests of AUDI2000 with homogeneous (solid) color pairs provided satisfactory results [21, 22], the goal of the current paper is to perform a color-difference experiment using automotive samples with lightness flop effects, analyzing the performance of AUDI2000 and other advanced color-difference formulas with respect to our experimental results. As requested by CIE , it is desirable to have new reliable experimental color-difference data sets to test and improve current color-difference formulas, and color-difference data sets involving gonioapparent materials are very scarce in current literature .
2. Materials and method
2.1 Selection of color pairs
Color measurements of 277 samples from different automotive suppliers were carried out with a BYK-mac 23 mm multi-angle spectrophotometer, assuming D65 illuminant and CIE 1964 standard colorimetric observer. This instrument provides color measurements considering a light source placed 45° with respect to the perpendicular to the sample, and detection at six different angles : −15°, + 15°, + 25°, + 45°, + 75° and + 110°. The negative/positive signs of these six angles indicate clockwise/counterclockwise rotation angles with respect to the specular reflection of the incident light (see Fig. 1). These six illumination-detection geometries are also designed by CIE  as 45°x:-60°, 45°x:-30°, 45°x:-20°, 45°x:0°, 45°x:30° and 45°x:65°, respectively.
From measurements at + 45° (i.e. the direction perpendicular to the sample), a set of 28 color pairs were selected for visual assessment. These color pairs met the next requirements: 1) the two samples in the pair were in the achromatic region (C*ab<10); 2) their color differences were relatively small (ΔE*ab<5); 3) the pairs had predominant lightness differences (ΔL*<2[|ΔC*ab| + |ΔH*ab|]); 4) the two samples in the pair came from the same supplier and showed very similar texture (non-perceptible in most cases). These requirements were adopted according to our main goal: to test the performance of most recent color-difference formulas disregarding texture effects, and analyze the lightness flop term proposed by the AUDI2000 color-difference formula. The selected 28 color pairs (Fig. 2) were well distributed in CIELAB color space, particularly with respect to the presence of pairs with small, medium and high L* values.
2.2 Visual assessments
Using a gray scale method , each one of the selected 28 color pairs was visually assessed in a multi-angle byko-spectra effect light booth at the same six viewing angles previously mentioned: −15°, + 15°, + 25°, + 45°, + 75° and + 110° (see Fig. 1). Two issues of the SDC Grey Scale for Change in Colour  were used to provide the observer a set of 9 gray color pairs with increasing color differences. These 9 color pairs were placed on the same plane than the test color pair, the distance between the center of the test pair and the central gray color pair #3 being 7.5 cm. The illuminance at the center of the test pair was 1800 lx. Bearing in mind that our selected samples had different sizes, looking also for more accurate visual assessments, we used a black gray mask providing a rectangular 3.5 cm x 3.7 cm test pair, with the same size than the color pairs in the gray scale. The color and texture of this mask (L* = 18.6, a* = 0.2, b* = 0.1) were very similar to the ones of the background in the light booth (L* = 24.2, a* = 0.5, b* = −0.6).
The observer’s task was to assess the magnitude of the color difference in the test pairs in comparison with the color differences shown in the different gray pairs. Observers were encouraged to report intermediate assessments values between contiguous gray pairs. For a given test color pair, the six viewing angles were presented to the observer in random order, and the different test pairs were also presented in random order. Observers have a fixed viewing position in the light booth, in such a way that the distance from the observer to the test pair was 45 cm, and the viewing angle subtended by the test pair at + 45° was 4.5 degrees, allowing us to use the CIE 1964 standard colorimetric observer. Each observer performed a total of 504 visual assessments corresponding to 28 color pairs x 6 viewing angles x 3 replications, which were distributed across sessions of around 30 minutes, for a total of 10 sessions. Experiments were held in a dark room, and each observer adapted to the background of the light booth for 2 min before each experimental session. A panel of 10 observers (4 males and 6 females) with normal color vision (Ishihara test) participated in our experiment. They were staff and graduate students at the Department of Optics of the University of Granada, 7 of them with previous experience in color-difference assessments. All observers were subjected to a training round before their first observation session.
2.3 Spectroradiometric color measurements
It must be noted  that the light source in the byko-spectra effect light booth is close to TL84 while measurements with the BYK-mac spectrophotometer assumed D65, and also that the illumination in this light booth is not uniform, in particular at the extreme color pairs #1 and #5 of the gray scale. Considering that it is convenient to measure just what observers see, we decided to measure the color of all our samples in the byko-spectra effect light booth, using a Konica Minolta CS2000 spectroradiometer placed at the same position than the head of the observer, and use these measurements in all our next computations of performance of color-difference formulas. Therefore, while the selection of the color pairs used in our visual experiment was based on measurements made with the BYK-mac spectrophotometer at + 45°, as explained in section 2.1, the analyses of all current experimental results will be based only on spectroradiometric measurements performed under the same experimental conditions than visual assessments. The spectroradiometer was fixed on a robust tripod and it was slightly tilted to measure each one of the samples in both the test pairs and the gray scale pairs inside the light booth. A PTFE reference white placed at the same position than the samples was also measured with the spectroradiometer to allow transformations to CIELAB. On average, color differences measured with the Konica Minolta spectroradiometer and the BYK-mac spectrophotometer differed in 3.4 CIELAB units, the maximum differences being in L* values and for the −15° angle.
From our spectroradiometric measurements at the six viewing angles we found that color differences in our 28 color pairs were below 6.0 CIELAB units in more than 83% of the cases [Fig. 3(left)], which is an appropriate range considering that the maximum color differences in our gray scale were in the range 10-15 CIELAB units, depending on the viewing angle we consider. We also found from our spectroradiometric measurements that the average CIELAB lightness difference in our 28 color pairs was high (71% of the total color difference), as desired in our experiment. Finally, from spectroradiometric measurements in the light booth, we measured the lightness flop (defined as the difference between maximum and minimum L* values in the six measured angles) of the color samples involved in our selected 28 color pairs [Fig. 3(right)], obtaining an average value of 33.4 with a high standard deviation of 32.2. The samples with moderate lightness flop were predominant, and this flop range was considered appropriate to test the performance of the lightness flop term in AUDI2000.
The raw gray scale values reported by the observers were transformed into true ΔV values, as usually made in the literature , using a fourth polynomial function fitted from spectroradiometric CIELAB color differences measured in the gray scale color pairs at each one of the six observation angles (Fig. 4). The color pairs of the SDC gray scale  had different color differences at different viewing angles. This is not surprising bearing in mind that gonioapparent effects have been also reported for other reflectance standards .
2.4 Performance of color-difference formulas
The results of our visual experiment were tested against predictions made by 12 modern color-difference formulas, which have been also tested with other experimental data sets in recent color-difference research [14, 31]: CIELUV , CIELAB , CMC , BFD , CIE94 , AUDI2000 , CIEDE2000 , CAM02-UCS , CAM02-SCD , DIN99d , DIN99b  and OSA-GP-Euclidean . Parametric factors in all these formulas were kept as kL = kC = kH = 1.0 in our current analyses. We paid special attention to the performance of the AUDI2000 color-difference formula, defined as follows :Eqs. (3) and (4) are averaged and this average value replaces both previous SC,γi and SC,γi values. In previous equations, γi + 1 represents the next higher angle in the sequence + 15°, + 25°, + 45°, + 75° (e.g. γi + 1 = + 25° when γi = + 15°, or γi + 1 = + 110° when γi = + 75°), and L* and C*ab correspond to CIELAB lightness and chroma, respectively. Here the first term in Eq. (2) will be called lightness flop term, and the first terms in Eqs. (3) and (4) will be called chroma flop terms.
When γi = −15°, the SL,-15°, SC,-15° and SH,-15° functions are equal to SL, + 15°, SC, + 15° and SH, + 15° respectively, multiplying the slopes of all flop terms by a factor 1.2. That is, for −15° the slopes of each one of the flop terms are changed as follows: 1.0 x 1.2 = 1.2 = a5, 1.478 x 1.2 = 1.774 and 0.800 x 1.2 = 0.960. Analogously, when γi = + 110°, the SL,110°, SC,110° and SH,110° functions are equal to SL, + 75°, SC, + 75° and SH, + 75° respectively, multiplying the slopes of each one of the flop terms by a factor 0.5. That is, for + 110° the slopes of each one of the flop terms are changed as follows: 1.0 x 0.5 = 0.5 = a6, 1.478 x 0.5 = 0.739 and 0.800 x 0.5 = 0.400.
In summary, coefficients a1 and a2 are the slope and exponent in the lightness flop term, respectively, coefficients a3 and a4 are not in the lightness flop term but in the weighting function for lightness [Eq. (2)], and coefficients a5 and a6, are the slopes of the lightness flop terms for angles −15° and + 110°, respectively. The values of the six ai (i = 1,…,6) coefficients in the original AUDI2000 color-difference formula will be optimized later.
In usual automotive practice there is a master or reference sample, and Eqs. (2) to (4) are computed using color measurements from this master sample. However, in our current experiment we have two different samples and there is no reason to establish one of them as master. Therefore, here we computed Eqs. (2) to (4) for the two samples in each color pair and the corresponding arithmetical means were adopted as the final weighting functions in computations using Eq. (1). Note also that Eq. (1) provides a different value for each one of the six γi angles: the average and maximum of these six values are usually employed to establish pass/fail tolerances in the automotive industry.
The STRESS index  has been employed to measure the goodness of the predictions made by each tested color-difference formulas with respect to the whole set of results in our experiment (i.e. 28 color pairs x 6 geometries = 168 color pairs). The ΔVi (i = 1,…,168) values in our STRESS computations were the average of true ΔV values reported by our 10 observers. Low STRESS values (always in the range 0-100) indicate better performance of a color-difference formula. F-tests can be used with the STRESS index in order to know whether two color-difference formulas are or not statistically significant different : the square of the ratio of the STRESS values from two color-difference formulas is compared with a specific confidence interval, which in our case was [0.74 ; 1.36], assuming a 95% confidence interval and taking into account that we have 168 color pairs. We have also used the STRESS index to compute intra- and inter-observer variability in our experiment . For each observer, intra-observer variability was computed as the average of STRESS values of each one of the 3 replications made by this observer with respect to the average of the 3 replications, while inter-observer variability was computed as the STRESS value considering the average result of the 3 replications of this observer with respect to the average results of all 10 observers. Final intra- and inter-observer variability in our experiment were defined as the average intra- and inter-observer variability STRESS values from the 10 observers, respectively. A weighted STRESS, WSTRESS , has been also computed here bearing in mind that it is convenient to take into account the consistency of visual assessments made by the observers, giving a higher weight to color pairs where visual assessments have low standard deviations. Specifically, we used as appropriate weight the ratio ΔVi /4σi (i = 1,…,168), where ΔVi and σi are the average and standard deviation of true ΔV values from the 10 observers in each color pair, respectively. In our experiment the average weight was 0.90 with a standard deviation of 0.50.
3. Results and discussion
STRESS values corresponding to intra- and inter-observer variability in our whole experiment were 25.2 and 23.2, respectively. Similar values can be found in recent literature .
Table 1 shows STRESS and WSTRESS values for the whole results of our experiment and different tested color-difference formulas. F-tests from STRESS values shown in Table 1 indicate that the best color-difference formulas were AUDI2000 and CAM02-SCD (in this order), and they were not statistically significant differences among them. Analogously, F-tests from WSTRESS values shown in Table 1 indicate that the best color-difference formulas were AUDI2000, CAM02-SCD, CAM02-UCS and OSA-GP-Euclidean (in this order), and they were not statistically significant differences among them. While low performance (high STRESS or WSTRESS values) was expected for the CIELUV and CIELAB formulas, it was surprising to find also low performance for the CIE94 and CIEDE2000 formulas, which may be attributed to the fact that some conditions in our experiment (e.g. light source different than D65 illuminant, background behind the color pairs with L* very different to 50, etc.) were not those recommended for these two color-difference formulas [5, 10]. The best performance of AUDI2000 may be attributed to the fact that this is the only color-difference formula considering flop effects and different lightness weighting functions in each viewing angle.
Bearing in mind that results in Table 1 show STRESS and WSTRESS values for the AUDI2000 color-difference formula higher than our inter-observer variability (23.2 STRESS units), the six ai (i = 1,…,6) coefficients of the weighting function for lightness in AUDI2000 were optimized to achieve the lowest STRESS and WSTRESS values (i.e. the best performance). The results found are shown in Table 2. Optimized coefficients in Table 2 reduced STRESS value from 25.1 to 21.8 and WSTRESS value from 26.6 to 24.0. As we can see, Table 2 shows enough similar values for coefficients a1 and a2 but different values for coefficient a3, although in fact this is not very relevant because samples in our experiment have very low C*ab,45 values and the term including the coefficient a3 is almost negligible. The values for coefficients a4 and a6 in Table 2 are also enough divergent in the two optimizations. In particular, adoption of the value a4 = 0.01 found in STRESS optimization may be problematic because in the case of very achromatic samples with no flop (solid colors), the value of the weighting function SL may be very small, producing meaningless high values in AUDI2000 computations [Eq. (1)]. It can be also noted that values for optimized coefficients a5 and a6 in Table 2 are enough different to those in the original AUDI2000 formula. If an unique set of coefficients ai (i = 1,…,6) describing the results of our current experiment is required, we’d suggest values found minimizing WSTRESS (i.e. values in last row of Table 2). Using ai values in the last (fourth) row of Table 2 the next advantages can be mentioned: 1) the potential abnormal high values derived from very low a4 values in solid achromatic color pairs may be avoided; 2) WSTRESS value achieves the lowest value of 24.0 units indicating best performance; 3) STRESS value for the whole results of our experiment is 22.1 units, which is a good result, very similar to 21.8 units found as the lowest STRESS value using coefficients in the third row of Table 2.
Taking into account that color differences in our experiment covered a relatively wide range [Fig. 3(left)], we considered that an exponential function (ΔEAUDI2000)α may reduce the WSTRESS found with the AUDI2000 formula with optimized coefficients (i.e. coefficients in fourth row of Table 2), as suggested in previous literature [38,39]. A lowest WSTRESS value (best performance) of 21.8 units was found with α = 0.82, improving the earlier WSTRESS value of 24.0 found without the exponential function (i.e. assuming α = 1.0). It is interesting to note that this 21.8 WSTRESS value is lower than the inter-observer variability in our experiment (23.2 STRESS units), which means that this optimized non-linear formula predicts our visual results as close as an individual observer, as desired by useful color-difference formulas employed in industrial applications. Also it is worth to mention that F-tests  indicate that this modified AUDI2000 formula with optimized coefficients plus exponential function describes satisfactorily the whole visual results in the current experiment, being statistically significant better than the original AUDI2000 formula.
While the results of our current experiment can be fitted with a modified AUDI2000 formula, by the moment we are not proposing to replace current original AUDI2000 formula  before additional experiments are performed. Different modifications in our current experiment can be suggested leading to slightly different results to those shown in Table 1 and therefore to different values of the optimized coefficients. For example, we selected color pairs from spectrophotometric measurements at 45° but our spectroradiometric measurements showed that for 6 color pairs at specific angles (mainly −15° and 15°) the color differences were above the range provided by the gray scale used in our experiment. Missing these 6 color pairs in our analyses is unacceptable because they showed the medium-high lightness flop values we wanted to analyze. Removing just only some visual assessments in these 6 color pairs is against our experimental design considering all six standard visual angles. In a few words, we consider that the design of our experiment was correct and adapted to the available samples, but we recognize that it can be improved. In fact, the current experiment is now being replicated using a more appropriate set of color pairs, and also color pairs with greater size to allow the flop effects can be more evident. Hopefully our next results together with current ones can be used to propose an optimized AUDI2000 color-difference formula. By the moment, the results in this paper only indicate that the structure of the AUDI2000 formula is good to describe a visual color-different experiment using automotive solid and gonioapparent samples, while the optimal values of some of the coefficients in the weighting function for lightness in the AUDI2000 color-difference formula may be different to the original ones.
A visual data set with automotive samples showing flop effects has been developed. Detailed results in this data set (i.e. color coordinates of samples and visual differences) are available from the authors. While AUDI2000 was the best formula predicting our visual results, it seems that this formula can be improved. Further tests of AUDI2000 or other potential future color-difference formulas including flop effects are encouraged.
To all volunteer observers participating in our experiment. We are also very grateful to BYK Additive and Instruments (https://www.byk.com/en/instruments/products/color-measurement.html), for the loan of the byko-spectra effect light booth used in our experiments. Thanks also to the Color in Informatics and Media Technology (CIMET) Erasmus-Mundus Master (http://www.master-erasmusmundus-color.eu/); the results of this paper and references  and , constituted Juan Martínez-Garcia’s CIMET Master Thesis. This research was supported by the Ministry of Economy and Competitiveness (Spain), research projects DPI2011-30090-C02 and FIS2010-19839, with European Regional Development Fund.
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