Color-difference formulas modified by power functions provide results in better agreement with visually perceived color differences. Each of the modified color-difference formulas proposed here adds only one relevant parameter to the corresponding original color-difference formula. Results from 16 visual data sets and 11 color-difference formulas indicate that the modified formulas achieve an average decrease of 5.7 STRESS (Standardized Residual Sum of Squares) units with respect to the original formulas, signifying an improvement of 17.3%. In particular, for these 16 visual data sets, the average decrease for the current CIE/ISO recommended color-difference formula CIEDE2000 modified by an exponent 0.70 was 5.4 STRESS units (17.5%). The improvements of all modified color-difference formulas with respect to the original ones held for each of the 16 visual data sets and were statistically significant in most cases, particularly for all data sets with color differences close to the threshold. Results for 2 additional data sets with color pairs in the blue and black regions of the color space confirmed the usefulness of the proposed power functions. The main reason of the improvements found for the modified color-difference formulas with respect to the original color-difference formulas seems to be the compression provided by power functions.
© 2015 Optical Society of America
Color-difference evaluation is a classical field in color science . While research in this field has advanced the knowledge of the human visual system and the proposal of new color spaces, it can currently be stated that the development of successful color-difference formulas for industrial applications is the most important challenge. A color-difference formula can be defined as a mathematical equation providing a number ΔE from instrumental color measurements of two given color stimuli. The goal of any color-difference formula is to make ΔE values agree with color differences visually perceived by average human observers (ΔV), for color stimuli placed in any region of color space under any viewing conditions. In other words, color-difference formulas are meant to be useful in color-quality control and related applications, providing objective predictions of subjective color differences perceived by average observers. According to Kuehni , color-quality control is a technological tool that is becoming increasingly critical in world economic systems. If it is of sufficient accuracy, color-quality control can serve as an objective arbiter of pass/fail or other color-difference assessments, and can significantly speed up approval processes.
Color-difference formulas are currently used in many industries and applications such as automobiles [3, 4], printing [5, 6], textiles [7, 8], dentistry [9, 10], medical images [11, 12], food [13, 14], and agriculture [15, 16]. In recent years, significant advances have been made in the development of successful color-difference formulas, leading to the recent joint recommendation of the CIEDE2000 color-difference formula by the International Organization for Standardization (ISO) and the International Commission on Illumination (CIE) . However, it must be recognized that, with an average accuracy of around 65-75%, all modern color-difference formulas are unfortunately not very accurate in predicting perceived color differences [2, 18, 19]. That is, modern color-difference formulas need improvements in order to be more reliable in automatic quality control and industrial applications.
One aspect generally agreed upon by researchers in this field is that the performance of the human visual system is not the same for color differences with different magnitudes (i.e. threshold, small and large color differences). Thus, the two last CIE-recommended color-difference formulas, CIE94  and CIEDE2000 , have been recommended mainly for color differences within the range 0-5 CIELAB units. Poorer performances of most available color-difference formulas for very small and very large color differences have been reported in the literature [22, 23]. It is desirable for color-difference formulas to predict color differences within any range of magnitude, and the CIE is currently investigating the validity of the range of CIEDE2000 and the failures of color-difference formulas predicting color differences below 2 CIELAB units, through its Technical Committees 1-63 and 1-81, respectively .
The goal of the current paper is to show that simple power functions, or transformations such as ΔE’ = a ΔEb, where ΔE are the results from a given color-difference formula, may improve the performance of most currently available color-difference formulas. We will report the optimal values of coefficients a and b for 11 different color-difference formulas. These optimal values were those providing the best predictions of visually perceived color differences for a broad set of 16 visual data sets from different laboratories, involving a total of 9360 color pairs including threshold, small, and large color differences. In fact, Attridge and Pointer  already suggested these kinds of power functions in color-difference evaluation, but they considered only a reduced set of visual data and color-difference formulas.
It may be thought that the power functions we are proposing here are simply Steven’s power law , which suggests the existence of a power relationship between the magnitude of a physical stimulus and its perceived intensity or strength. Indeed, Steven’s power law is given by the equation S = a Ib, where S is the psychophysical variable related to the subjective magnitude of the sensation evoked by the physical stimulus with intensity I, b is an exponent depending on the type of stimulation, and a is a constant related to the type of stimulation and units used. However, it should be added that Steven’s power law and the power functions we are proposing here are essentially different. Steven’s law operates with two intrinsically different magnitudes (i.e. a physical magnitude and its corresponding sensorial magnitude), while our proposed transform uses two analogous magnitudes (i.e. two color differences). Therefore, the purpose of this paper is not to establish a connection with Steven’s power law, nor describe a new visual phenomenon. We try only to show that power functions are useful to improve the performance of current color-difference formulas, as desired by engineers and practitioners in color-quality control and many other color applications. As we will show, the improvement provided by power functions using only one relevant parameter (the exponent b) is a noteworthy result, because in most cases the ΔE’ values are in statistical terms significantly better correlated to visually perceived color differences than are the original ΔE values. The potential reasons of the improvements achieved by the proposed power functions will also be discussed.
2. Materials and methods
2.1. Visual data sets
We have considered 16 independent visual data sets of color differences, which may be grouped for certain analytical purposes into three different groups according to the average magnitude of their color differences: threshold color differences (TCD), small color differences (SCD), and large color differences (LCD). The main characteristics of each of these 16 experimental data sets are summarized in Table 1. The TCD group includes 5 experimental data sets [27–29] with average color differences in the range 0.55-1.10 CIELAB units, 4 of them developed in two different experiments [27, 28] carried out at Beijing Institute of Graphic Communication (BIGC), involving semi-gloss (SG), matt (M), and gloss (G) color pairs. The SCD group considers another data set developed at BIGC , plus the 4 experimental data sets employed at the development of the CIEDE2000 color-difference formula : RIT-DuPont , Leeds , BFD-P  and Witt . The average color differences in the SCD group fall within the range 1.44-3.04 CIELAB units. Finally, the LCD group includes 6 experimental data sets [35–41] with average color differences within the range 8.91-14.30 CIELAB units.
Overall, the 16 visual data sets considered here involve a very high number (9360) of color pairs, with color differences in a relatively wide range of magnitudes, from 0.04 to 35.81 CIELAB units. These 16 data sets were considered as proposed by their original authors, without using scale factors to put them on common scales, and, in general, their color pairs subtended visual fields higher than 4° and were spread throughout all the regions of the color space. To achieve general results, we omitted some specific color-difference data sets developed at the blue and black regions of color space [42, 43] in order to develop our current power functions, but they will be used for testing the performance of those functions. To offer an idea of the different magnitudes of the color pairs in the 16 visual data sets employed for the development of our power functions, Fig. 1 shows the percentages of color pairs for different ranges of CIELAB color differences in each data set. Additional details on these 16 data sets can be found in their corresponding references (see last column in Table 1).
2.2. Color-difference formulas
We have tested the performance of 11 color-difference formulas considering both the original ΔE formulas and their corresponding transformed formulas ΔE’ using power functions of the type ΔE’ = a ΔEb. Henceforth, the ΔE and ΔE’ values will be designated as the original and modified color-difference formulas, respectively. The 11 selected color-difference formulas are very well known in current research and applications in the field of color-difference evaluation: CIELAB , CMC , CIE94 , CIEDE2000 , DIN99d , ULAB , CAM02-LCD , CAM02-SCD , CAM02-UCS , OSA , and OSA-GP-Euclidean . It can be noted that the selected color-difference formulas pertain to three main different groups: CIELAB-based formulas [17, 20, 21, 45–47], CIECAM02-based formulas , and OSA-based formulas [49, 50]. Here the so-called parametric factors  in all tested color-difference formulas were assumed as kL = kC = kH = 1. Special attention will be paid to the results achieved by the CIEDE2000 color-difference-formula, because it is the current formula jointly recommended by ISO and CIE .
2.3. Performance tests
We used the STRESS index  to measure the performance of each color-difference formula with respect to each experimental visual data set. Low STRESS values, which are within the range 0-100, indicate better performance of a color-difference formula. STRESS values usually achieved by most advanced color-difference formulas are around 35-25 units, meaning that the average agreement between perceived and computed color differences is around 65-75% [2, 18, 19]. The STRESS index can be also used to ascertain whether or not two color-difference formulas significantly differ with respect to a given visual data set: the square of the ratio of the STRESS values from two different color-difference formulas A and B (in this order) is a parameter F to be compared with a specific confidence interval [FC ; 1/ FC], where FC is a critical value from a two-tailed F-distribution which depends on the assumed confidence level (95% in our case) and the number of color pairs . If the value of parameter F is lower than 1, it can be concluded that the color-difference formula A is better than B, F<FC indicating that the color-difference formula A is significantly better than B. Analogously, F values higher than 1 mean that the color-difference formula A is worse than B, F>(1/FC) indicating that the color-difference formula A is significantly worse than B.
The STRESS index can also be used to compute intra- and inter-observer variability in visual experiments . As is well known, visual assessments of color differences by real observers usually have high variability, even in the case of experiments carried out with rigorous methodologies under well established illuminating and viewing conditions. Clearly, such variability or uncertainty in visual assessments needs be taken into account. The STRESS values corresponding to observers’ uncertainty in rigorous experiments on color differences may be around 20-25 units [3, 27], slightly lower than those achieved by the best available color-difference formulas. This means that in many cases color-difference formulas require further refinement to be completely reliable for industrial applications. A color-difference formula can be considered successful when it achieves STRESS values below the ones corresponding to average observers’ uncertainty, because in this case the results predicted by such color-difference formula are comparable to those found by an average real observer.
Currently, the STRESS index is being widely used by researchers in color-difference evaluation, but other indices (e.g. the eclectic PF/3 index  or the Pearson’s correlation coefficient) and methods may also be useful to study the performance and merits of color-difference formulas .
Table 2 shows STRESS values for the 11 original color-difference formulas, ΔE, and each visual data set. Mean results for TCD, SCD, and LCD groups, as well as for all 16 visual data sets considered together (last row), are also shown in Table 2. Bold letters in Table 2 indicate the color-difference formula with the best predictions (lowest STRESS value) for each visual data set. Modified color-difference formulas, ΔE’, with ΔE’ = a ΔEb, were formulated by using all 16 visual data sets together, and the optimal results found for parameters a and b in each color-difference formula are shown in Table 3. Parameter a was fitted to put all color differences into the modified formulas on a common scale, the one corresponding to the BFD-P data set, and it has no influence on STRESS values . Parameter b was optimized to give the lowest STRESS values for the whole set of 16 visual data sets. Table 4 is analogous to Table 2, but for the 11 modified color-difference formulas, ΔE’, derived using the values provided in Table 3.
From last row in Table 2, we can see that the mean STRESS values for all 16 visual data sets are in the range 30.2-38.3, with an average value of 32.9. Also, in Table 2 the STRESS values for data sets in the TCD group are considerably higher than those for data sets in the SCD and LCD groups. Specifically, the average STRESS values for all color-difference formulas were 43.4, 30.5, and 26.2 in the TCD, SCD, and LCD groups, respectively. This means that the performance of all tested color-difference formulas is considerably worse for very small color-differences, as previously reported by other authors [22, 24]. In particular, for the color-difference formula CIEDE2000 (currently recommended as a standard by the ISO and CIE), the average STRESS values were 41.9, 25.5, and 26.0, for the TCD, SCD, and LCD groups, respectively, with an average STRESS value for all 16 visual data sets of 30.8.
Values of parameters b in Table 3 are in the range 0.55-0.85. For the CIEDE2000 color-difference formula the optimal value of parameter b was 0.70. Note that values of parameter b in all modified color-difference formulas are lower than 1.0, signifying that small/large values in the original color-difference formulas will be increased/decreased in the modified color-difference formulas. That is, all modified color-difference formulas proposed here compress the values provided by their corresponding original color-difference formulas.
The last row in Table 4 indicates that STRESS values for all 16 visual data sets in the modified color-difference formulas were in the range 25.4-29.6, with an average value of 27.2. Comparing these results with those previously reported in Table 2, we conclude that, for the 16 visual data sets, the improvement of the modified color-difference formulas with respect to the original formulas were in the range of 4.0-8.8 STRESS units (or 13.7%-23.0%), with an average improvement of 5.7 STRESS units (or 17.3%). Also, a similar degree of improvement was achieved for every individual data set. From Table 4, the average STRESS values for the modified color-difference formulas were 35.4, 25.8, and 21.5 in the TCD, SCD, and LCD groups, respectively. This means that the average improvements of the modified color-difference formulas with respect to the original ones for the TCD, SCD, and LCD groups were 8.0, 4.6, and 4.6 STRESS units (or 18.5%, 15.2%, and 17.6%), respectively. In particular, for the CIEDE2000 color-difference formula, the exponent 0.70 in the modified color-difference formula led to an overall average decrease of 5.4 STRESS units (i.e. an improvement of 17.5%).
Table 5 shows the square of the ratios of STRESS values found for the modified color-difference formulas (Table 4) with respect to those found for the original color-difference formulas (Table 2), in this order, which can be used to test whether or not the modified and original color-difference formulas are statistically different . It bears noting that all values in Table 5 are below 1.0, indicating that with the use of only one parameter (the exponent b) adjusted for all 16 visual data sets together, any modified color-difference formula improves the original one for each of the individual 16 visual data sets. Traditionally, it is found that the characteristics of the TCD, SCD, and LCD are different, so that parameters were changed to fit different groups of data, as done, for example in the development of CAM02-UCS, CAM02-SCD, and CAM02-LCD color-difference formulas . With the use of current power functions, this difference has been reduced, improving all three groups of data. More notably, we find that most values in Table 5 are below the corresponding FC values shown in the last column of Table 5. This means that the modified color-difference formulas were significantly better than the original color-difference formulas in most cases (the ones without underlined values in Table 5), and, in particular, for all 5 visual data sets included in the TCD group.
4. Discussion and conclusion
As mentioned earlier, our main goal was to show that simple transformations of color-difference formulas using power functions substantially improve the predictions of visually perceived color differences made by such formulas. As demonstrated in the previous section, this goal was achieved for a wide set of 16 visual data sets with a total of 9360 color pairs considering 11 different color-difference formulas, including the current ISO/CIE recommended color-difference formula CIEDE2000 , as well as most important color-difference formulas currently in use. On average, the improvement achieved by the different modified formulas was a decrease of 5.7 STRESS units (or 17.3%), which may be considered a moderate improvement, although probably important in color-quality control and applications such as those mentioned in references [3–16]. Most importantly, it was found (Table 5) that the modified color-difference formulas proposed here were significantly better than the original ones for all visual data sets with very small color differences (TCD group), as well as for most visual data sets in the SCD and LCD groups. Only for Witt , OSA , and Badu-P  visual data sets, the improvements achieved by most modified color-difference formulas were more moderate and not statistically significant. Unfortunately, we are unable to explain the reasons of the smaller improvements of modified color-difference formulas in these specific data sets.
The improvements achieved by our modified color-difference formulas (Table 3) also hold for visual data sets other than those used in their developments. As an example, two recent visual data sets were selected from North Carolina State University (USA) [42, 43], which involved color pairs in only two specific regions of color space (blue and black regions), where it should be expected that the performance of most color-difference formulas would not be very good. These data sets in the blue  and black  regions have a reduced number of 66 and 50 color pairs, respectively, with an average color difference of 2.3 CIELAB units (in both cases), which is in the range of the data sets we have previously considered within the SCD group (Table 1). Table 6 shows the results found for the original and modified color-difference formulas for these 2 data sets using as performance measurements both the STRESS  and PF/3  indices. Lower STRESS or PF/3 values (these latter ones not necessarily within the range 0-100) imply better performance of a color-difference formula, the ideal values being 0 in both cases.
The results in Table 6 show that all modified color-difference formulas improved the original ones, considering both the STRESS and PF/3 performance measurements. More specifically, on the average, the modified color-difference formulas improved the original ones by 3.5 (or 13.9%) and 5.2 (or 16.2%) STRESS units, for the blue  and black  data sets, respectively. These improvements are slightly lower than the average 5.7 STRESS units (or 17.3%) previously found for all 16 visual data sets [27–41] used to develop the modified color-difference formulas, but they are very consistent with the 4.6 STRESS units (or 15.2%) found for the visual data sets in the SCD group. In general, it is not possible to establish a simple comparison between the results found using the STRESS and PF/3 indices, but for the two data sets in Table 6 the improvements of the modified color-difference formulas with respect to the original ones seem considerably greater using the PF/3 index than using the STRESS index.
In addition to the practical usefulness of the improvements achieved by the modified color-difference formulas with respect to the original ones, in particular for very small color-difference data sets such as those in the TCD group, it bears examining the potential reasons leading to such improvement. According to our analyses, the improvement achieved by the modified color-difference formulas cannot be attributed to the use of the STRESS index (as shown, for example, by the PF/3 results given in Table 6), nor to specific characteristics of color spaces. It should be borne in mind that we found improvements for all our modified color-difference formulas, which are based on three very different color spaces: CIELAB , CIECAM02 , and OSA . It may also be considered that non-linear processes, such as the ones inherent to the use of power functions, are relatively frequent in different steps of color-vision models and equations defining color spaces [44, 49, 55]. Therefore, the improvements achieved by the power functions proposed here may indicate that the specific functions defining color spaces and associated color-difference formulas are not optimal and require further optimization. Research on this last point should focus on improvements of specific individual color spaces, but this lies beyond of the scope of the current study.
We think that the main cause explaining the improvement made by the power functions defining the modified color-difference formulas is the compression resulting from such formulas. Indeed, Fig. 2 shows the percentage change of the modified color-difference formulas with respect to the original ones for four modified color-difference formulas (see Table 3): CIELAB , CIEDE2000 , CAM02-UCS , and OSA-GP-Euclidean . As we can see, the modified color-difference formulas increase/decrease in a non-linear way the small/large color-difference values provided by the original color-difference formulas. In other words, the power functions result in compression, because small color differences increase while large color differences decrease. As shown in Fig. 2, the percentage increase produced by the modified color-difference formulas for very small color-difference values may be high, while for large color-difference values the percentage decrease appears as an asymptotic trend. The final overall result is that original color-difference values are compressed around a specific value, the point where each curve in Fig. 2 cuts the x-axis, which is dependent on each original color-difference formula. As an example of this effect, Fig. 3 shows a plot corresponding to the 890 color pairs in the BIGC-T1-G data set  considering the results found from the original and modified CIELAB color-difference formulas, which have been represented here by the shafts and heads of the arrows, respectively. From the CIELAB modified color-difference formula, ΔE’ = 1.26 ΔE0.55 (see Table 3), we see that for ΔE values below/above 1.67 the corresponding ΔE’ values increase/decrease in a non-linear way with respect to the original ΔE values, as indicated by the arrows pointing to the right/left side in Fig. 3. Therefore, the points corresponding to the modified CIELAB color-difference formula (i.e. the heads of the arrows in Fig. 3) are closer to the regression line passing through the origin than are the points corresponding to the original CIELAB color-difference formula (i.e. the shafts of the arrows), and, consequently, the STRESS values for the modified CIELAB formula decrease.
In visual experiments involving very small color differences such as the ones producing the data sets in the TCD group (Table 1), observers do have difficulty in scaling color differences, and they tend to avoid ΔV values close to zero [54, 56] reporting overestimated ΔV values, as indicated in previous works . The increase in ΔE values prompted by the modified color-difference formulas for very small color differences, illustrated by the arrows pointing to the right side in Fig. 3, follows this trend. Therefore this may be an explanation of the strong improvement achieved by the modified color-difference formulas for the TCD data sets. On the other hand, it is also known that the visual estimations in the range from moderate to large color differences tend to be slightly asymptotic , and the decrease of high original ΔE values achieved by the modified color-difference formulas (i.e. arrows such as the ones pointing to the left side in Fig. 3), also agrees with this trend.
In conclusion, modified color-difference formulas using appropriate power functions may significantly improve the performance of original color-difference formulas, in particular for color differences with very small (around threshold) size. This result could be useful for improving color-quality control and for many different applications such as those mentioned in references [3–16]. This paper provides tentative values for different modified color-difference formulas, including the standard color-difference formula currently proposed by the ISO and CIE, CIEDE2000. Optimal values for modified color-difference formulas used in specific applications may need appropriate optimizations. An example of a successful power function optimizing a color-difference formula currently used by the automotive industry is provided in reference .
This research was supported by National Natural Science Foundation of China (grant 61308081), the Top Young Talents (CIT&TCD201404127) and Young Talents of Beijing Municipal Commission of Education (YETP1465), and the Ministry of Economy and Competitiveness of Spain (project FIS2013-40661-P), with European Regional Development Fund support.
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