Parametric effects by using the strip-pair comparison method around red CIE color center

: The strip comparison method, based on the serial exploration method described by Torgerson [ Theory and Methods of Scaling ; Wiley & Sons (1958); Chap. 7], for the development of near-threshold color di � erence models was presented and validated with theoretical data by the authors in a previous work. In this study, we investigate parametric e � ects derived from the use of the strip comparison method on chromaticity-discrimination ellipses around the red CIE color center. The results obtained led to the conclusion that the strip comparison method has little e � ect on the parameters of the chromaticity-discrimination ellipses determined by the pair comparison method when pairs of patches in the strips are separated by a black line 0.5 mm thick or are separated by 3 mm spacing on a white background and also correlates well with the parameters reported by other authors using the pair comparison method at the threshold.


Introduction
Since the CIE committee 1.3 (colorimetry) established guidelines for the scientific community for research in the field of color-di erence assessment in a coordinated manner (Robertson [1]), many studies in the field of color-di erence models have been conducted [2][3][4][5][6][7][8][9][10][11][12] and added to previous studies developed by MacAdam, Wyszecki, and others [13][14][15][16].More recently, after the publishing of a second set of CIE guidelines for coordinated future work on industrial color-di erence evaluation (Maier [17]), Melgosa [18], chairman of the CIE committee 2.3, made a third call in 2007 to collect new datasets to complement those used to develop the CIEDE2000 [19,20] color-di erence formula in order to develop new uniform color spaces to which colordi erence formulas could be associated to improve their applicability in industrial environments.Since CIEDE2000, new work has been published on near-threshold color-di erence assessment [21][22][23][24], and, also, to study parametric e ects on color di erences [25][26][27][28][29].
The collection of new datasets is essential for the improvement of existing color-di erence models.However, if larger datasets are not currently available, it is due to the di culty of obtaining them.The psychometric tests needed to evaluate the sensation of color di erence usually require significant concentration from the observers, which often extends the duration of the tests, complicating the recruitment of volunteers.It is, therefore, interesting to develop new methods that will simplify and shorten the decision-making process during psychometric tests.
Most of the psychometric tests used, to date, to assess the sensation of color di erence are based on two methods: Gray Scale (GSM) and Pair Comparison (PCM).
This study proposes the use of the Strip Comparison Method (SCM), based on the serial exploration method described by Torgerson [30], the description and validation with theoretical data of which was presented in a previous paper [31].The method is based on printed strips containing pairs of patches arranged in rows or columns, such that the patches on one row or column of the strip are printed as close to color constant as possible, while the patches of the other row or column are printed so that, along the strip, the variation of the color di erence E ⇤ ab between pairs increases in approximately constant steps from a zero color di erence, at one end of the strip, to a selected maximum color di erence at the other end, or as close to this as possible.Additionally, the variations of L*, a*, and b* on each strip should conform to the pattern of directions, in the CIELAB color space, indicated in Fig. 1.As described by Brusola et al. [31], observers are asked to indicate the number of the pair of patches in each strip from which a slight color di erence is observed.Collected frequency data and measured color di erences of the strip patches are then processed to obtain the parameters of the chromaticity-discrimination ellipses.This study aims to determine the parametric e ects due to the use of SCM with respect to the results that would have been obtained using PCM around the CIE red (L*=44, a*=37, b*=23) color center.
In this paper we do not intend to validate the SCM, because it was already validated in [31] with perfect theoretical data, in the sense that when generating the frequency data simulating the observers' perception, as if they were really responding to an assumed underlying color-di erence model (CIE94, in the case of the article mentioned), SCM can predict the assumed model with the necessary precision, even with the presence of certain noise level.
We have chosen red CIE color to determine the parametric e ects by using SCM in this work, because, according to the results published by a good number of authors, is one of many that presents greatest discrepancies in terms of the color-discrimination ellipsoid coe cients, as shown in [29].Obviously, other studies should be carried out, to confirm if it is possible to generalize the results of this work to other color centers.
It is necessary to take into account the laboriousness of the analysis carried out, where more than 100 observations by 8 strips and by 3 separations in the case of SCM and 100 observations by 80 pairs of patches and by 3 separations in the case of PCM have had to be made, which means more than 26,400 observations, regardless of the time needed to prepare the color samples.

Generation of color samples
Control strips were printed with an Epson Stylus 7000 inkjet printer on 250 g/m 2 Epson Premium Semigloss photo paper.Printing was performed by Adobe Photoshop from a tif file, with 16-bit color depth per channel, generated directly with MATLAB routines programmed by the research team, following a procedure similar to that described by Brusola et al. [32].
Three sets of eight pairs of strips were printed, each consisting of ten pairs of patches whose color di erence E ⇤ ab increased from one pair to the next in approximately equal steps, up to a maximum di erence of E ⇤ ab =4.In each of the sets, each of the eight strips should correspond to the eight theoretical directions in the chromatic plane a*-b* shown in Fig. 1.
Three sets of strips were generated, each corresponding to a di erent separation between pairs of patches on each strip.A pair of patches was printed with no separation in the first set, with a black line 0.5 mm thick separation in the second set, and with a spacing of 3 mm on a white paper background in the third set, using the same spacing pattern used by Brusola et al. [29].Figure 2 shows the described separation pattern for vector directions #1 and #6 near the red CIE color center.Once the strips were printed, a measurement was taken after approximately 10 minutes of drying.The measurement was made with an Xrite i1-iO automatic spectrophotometer reading system.
The e ective plotter repeatability was estimated by using the parameter mean color di erence from the mean (MCDM) of a set of measurements as proposed by Hunt and Pointer [33], which was obtained from the measurement of the reference patches in every strip.For all strip sets, the MCDM parameter obtained was less than 0.2 E ⇤ ab units.With the same set of reference patches, the following values of standard deviation from CIELAB coordinates were obtained: L =0.11, a =0.20, and b =0.30.However, by calculating the standard deviation of a*, b*, and L* of color di erences in the first pair of patches of all strips printed (which theoretically should be equal) the following values were obtained: L =0.10, a =0.13, and b =0.14, which indicates that the printer is slightly more precise (precision) than exact (accuracy).The values obtained allow us to foresee a slight deviation in the printing directions on each strip from the desired values and a good grouping of the points along the strips, as confirmed in the results section.

Visual evaluation
Each set of color strips was evaluated 10 times by 15 observers with normal color vision, in accordance with the Farnsworth-Munsell 100-tone test.The observers were asked to indicate the number of the pair of patches on every strip from which they first began to perceive a barely noticeable color di erence (jnd, just noticeable color-di erence pair).Thus, according to the above procedure, just one frequency record per observer and per strip was obtained in each evaluation.
The strips were observed in a Verivide viewing cabinet, equipped with a MASTER TL-D 90 Graphica 18W/965 lamp, with a correlated color temperature of 6500 K and a color rendering index Ra = 98.The display background corresponded to a neutral gray with CIELAB coordinates L*=62, a*=0, b*=0.These same conditions were reported by Brusola et al. [29] when performing the evaluation by PCM.

Statistical model and validation of results
The statistical model for processing the data is the same as that described by Brusola et al. [31].The results obtained by SCM were compared with the results by PCM, after cutting the strips used into corresponding independent pairs.The PCM results were published in a previous work by Brusola et al. [29] to evaluate the parametric e ect of sample separation on chromaticity-discrimination ellipses around the red CIE color center.

Results
Figures 3-5 show the detail of the adjustment of the chromaticity-discrimination ellipses by SCM for pairs of patches without separation (Fig. 3), for pair of patches separated with a black line 0.5 mm thick (Fig. 4), and for pairs of patches with a spacing of 3 mm on a white background (Fig. 5).Tables 1-3 together present the results of applying PCM with the proposed SCM, where both followed a procedure similar to the one proposed by Alman, Berns, et al. [5,7] to obtain the T50 tolerances for the theoretical direction scheme indicated in Fig. 1.In the first column of the aforementioned tables, the identifier of the theoretical vector direction is indicated according  The T50 tolerance was obtained by adjusting the frequency data to a normal cumulative probability distribution for PCM and to a normal probability density function for SCM.The LCL and UCL columns correspond to the lower and upper limits of the confidence interval of T50, respectively.The S column corresponds to the standard deviation associated with the corresponding psychometric curve.The last column, Standardized Residual Sum of Squares (STRESS) index, corresponds to the quality indicator of the adjustment obtained in each direction proposed by García et al. [34].For the calculation of STRESS, the visual di erences have been determined from the frequencies observed for each pair by the PCM.For SCM, the visual a V: identification of vector direction according to scheme shown in Fig. 1; da* and db*: unitary vectors of the vector directions after fitting measured color di erence ( a*, b*) points on every strip to the line that passes through the origin; T50: median of the probability distribution (psychometric curve) fitted to the observed frequencies in every vector direction; S: standard deviation of the psychometric curve; STRESS: Standardized Residual Sum of Squares index proposed by García et al. [34].a V: identification of vector direction according to scheme shown in Fig. 1; da* and db*: unitary vectors of the vector directions after fitting measured color di erence ( a*, b*) points on every strip to the line that passes through the origin; T50: median of the probability distribution (psychometric curve) fitted to the observed frequencies in every vector direction; S: standard deviation of the psychometric curve; STRESS: Standardized Residual Sum of Squares index by García et al. [34].a V: identification of vector direction according to scheme shown in Fig. 1; da* and db*: unitary vectors of the vector directions after fitting measured color di erence ( a*, b*) points on every strip to the line that passes through the origin; T50: median of the probability distribution (psychometric curve) fitted to the observed frequencies in every vector direction; S: standard deviation of the psychometric curve; STRESS: Standardized Residual Sum of Squares index by García et al. [34].
di erences have been determined from the accumulated frequencies calculated from the observed ones, under the assumption that an observer should perceive, in each observation, a color di erence for those patches, from the one selected as jnd in each strip, whose E ⇤ ab measurement is higher than that of the jnd pair.
Table 4 shows a summary of the results obtained by the two methods with respect to STRESS, T50, and S. As shown in Fig. 6(a), the average of the STRESS values obtained by the strip method is similar to that obtained by the pair method, around 14, these values having been obtained by averaging the STRESS values in each of the directions and in the three cases of separation.This indicates a very similar degree of adjustment by both methods.4; continuous horizontal lines correspond to the mean values ( m, shown in Table 4) and dashed horizontal lines correspond to the median values.However, the ratio of the average of the T50 values obtained by PCM to the average of the T50 values obtained by SCM becomes 1.4 times higher.This shows a clear tendency (parametric e ect) of the strip method to increase the size of the chromaticity-discrimination ellipses with respect to those obtained by the pair method in this experiment, as can be observed too in Fig. 6(b).
Conversely, the relationship between the average S values, the standard deviation of the psychometric curve, associated to some extent with the degree of confusion of the observers to decide whether they perceive the color di erence or not, by SCM with respect to PCM becomes 0.84.As can be seen in Fig. 6(c) the reduction of S is only evident for samples separated 3 mm on white background.This indicates to some extent that the degree of confusion in responding to the corresponding psychometric test by the observers is lower by SCM than by PCM for the aforementioned case.
During the recording of experimental data, an approximate reduction of 50% of the assessment time was observed for the SCM with respect to the PCM, which could be explained by the fact that the pairs of patches near and below the threshold are the most time consuming, but just as by SCM the observer only has to stop and make an assessment for the set of patches near the threshold, that can take double time than the evaluation of one single pair of patches, by PCS the observer has to repeat the same laborious assessment for each of the pairs near and below the threshold, which in average can represent 25% of the pairs of patches in one strip.
Table 5 shows the parameters of the chromaticity-discrimination ellipses obtained for the di erent cases considered in this article.For both PCM and SCM, two fitting techniques were used, described in Table 5 as Bayesian or T50.The Bayesian method corresponds to the method described by Brusola et al. for PCM [35] and that described for SCM [36].The fitting technique, T50, is based on the calculation procedure used by Melgosa et al. [10] from the T50 tolerance values obtained in Tables 1-4 for both PCM and SCM.Resultant chromaticity discrimination ellipses are plotted in Fig. 7.As Table 5 and Fig. 7 show, the parameters of the ellipses obtained by the Bayesian fitting technique and by T50 are very similar when used for either PCM or SCM.However, the parametric e ect, as mentioned in the previous paragraph, is confirmed in the sense that the size factor (K G ) of the chromaticity-discrimination ellipses determined by SCM are larger than those determined by the pair method.However, the size factor reported by Huang et al. [22], K G = 2.4 approximately, for printed samples, and Xu et al. [21], K G = 3.36 approximately, for samples displayed on a computer screen, are much more like those obtained by SCM for near red CIE color centers at the threshold.5. Continuous lines correspond to chromaticity discrimination ellipses obtained by T50 method and dashed lines correspond to chromaticity discrimination ellipses obtained by Bayesian methods.the major axis with respect to + a*; Tilt ( ✓): angular di erence between the hue angle (h ⇤ ab ) of the color center and the angle of the major axis of the discrimination ellipses with respect to + a*, in this order; K G : size factor = p ⇡AB; S: standard deviation of the psychometric curve; 2 : twice the standard deviation of the posterior distribution of ✓ (equivalent to a credibility interval of 95% probability); s 0 : samples with no separation; b 05 : samples separated by a black line 0.5 mm thick; w 3 : samples separated 3 mm on white background.
Conversely, a certain degree of regularity of the tilt of the ellipses, ✓, between PCM and SCM is obtained for pairs of patches separated by a 0.5 mm black line or by a 3 mm white space, as can be seen in Figs.7(b) and 7(c).For these cases, tilt does not di er by more than 10°, in accordance with the order of magnitude of the 95% credibility intervals (shown in column 2 of Table 5, obtained only by Bayesian techniques).A large discrepancy of the tilt is observed between the PCM and SCM values for the case of pairs of patches with no separation, as can be seen in Fig. 7(a), where the di erence can be greater than 30°.
With respect to the values of S, the standard deviation of the psychometric curve, the tendency observed in Tables 1-4 for each direction again confirms that the overall values of S obtained by SCM are lower than those obtained by PCM, thus indicating a lower degree of confusion by SCM than PCM.
Table 6 shows the tolerance values that would minimize the classification error using di erent color-di erence formulas, for the datasets obtained by SCM, according to the method described by Berns [37].Classification errors are of the same order of magnitude, or less, than those reported in Table 9 by Brusola et al. [29] for PCM datasets, with an average classification error of approximately 6%.As indicated by Brusola et al. [29] for the PCM datasets, the SCM datasets show that the CAM02-SCD and CAM02-UCS formulas produce the best results, thus generating a lower error rate.
There is a high degree of consistency in terms of the percentage of wrong classification decisions between the weighted color di erence formulae (all except CIELAB) shown in Table 6, with the highest error in all formulae for samples separated 3 mm on white background, intermediate error for samples separated with a black line 0,5 mm thick and the lowest percentage of wrong classification decisions for samples with no separation.The discrimination tolerance is also quite consistent, corresponding the highest value for samples separated 3 mm on white background with respect to all the color di erence formulas and the lowest for samples no separation.The tolerance for samples separated with a black line 0,5 mm thick is maintained in an intermediate position and is equal to the tolerance for samples without separation in only two cases: CIE94 and CAM02-SCD.In this regard, we believe that the degree of precision of the printing system used, around E ⇤ ab = 0.2, may be the cause of the slight fluctuations observed in Figs.4(a), 4(b), and Table 6.

Fig. 1 .
Fig. 1.Theoretical di erence distribution pattern in the a*-b* plane for strips with 10 pairs of patches and E ⇤ ab =4 maximum di erence in last patch.Identification of direction vectors is in bold.

Fig. 2 .
Fig. 2. Proposed strips for red CIE (L*=44, a*=37, b*=23) color center along vector direction #1 ( L*=0, a*=-2.8,b*=-2.8)and vector direction #6 ( L*=0, a*=2.8, b*=-2.8)for three types of separation between pairs of patches: no separation (no sep.), black line 0.5 mm thick (b05) and separated 3 mm under white background (w3).Note: the authors cannot guarantee that the colorimetric values observed in the figure correspond to those used in the experiment due to the impossibility of controlling the color management workflows involved when readers are observing the figure displayed or printed in their own devices.

Fig. 3 .
Fig. 3. Chromaticity-discrimination ellipse obtained by SCM for pairs of patches without separation.

Fig. 4 .
Fig. 4. Chromaticity-discrimination ellipse resulting from SCM, pairs of patches separated by a black line 0.5 mm thick.

Fig. 5 .
Fig. 5. Chromaticity-discrimination ellipse resulting from SCM, pairs of patches separated by 3 mm on white background

Fig. 6 .
Fig. 6.Box and Whisker plot of the results shown in Table4; continuous horizontal lines correspond to the mean values ( m, shown in Table4) and dashed horizontal lines correspond to the median values.

Table 4 .0 s 0 b 05 b 05 w 3 w 3 s 0 s 0 b 05 b 05 w 3 w 3 s 0 s 0 b 05 b 05 w 3 w 3
Comparison of STRESS, T50, and S values between PCM and SCM a

Fig. 7 .
Fig. 7. Plot of the Chromaticity-discrimination ellipses shown in Table5.Continuous lines correspond to chromaticity discrimination ellipses obtained by T50 method and dashed lines correspond to chromaticity discrimination ellipses obtained by Bayesian methods.

Table 5 . Chromaticity ellipse parameters in a*-b* plane a
a g 11 , g 22 , and g 12 : metric coe cients of the ellipse; A and B: major and minor semi-axes, respectively; ✓: angle of