The precise measurement and specification of color, as well as the measurement of color differences between two color pairs of samples, are very important issues in Colorimetry, with applications in various fields such as automotive industry, textiles, agriculture, healthcare, etc. This relevance is especially pronounced in areas where color is not only an attribute but also adds significant value to the final product. Research carried out in this topic is mainly based on psychophysics experimentations on color differences perception. In this work, an exhaustive analysis of various color-difference databases has been carried out using data analysis techniques through fuzzy logic. The ultimate goal of this analysis has been to identify pairs of colors that are inconsistent compared to other pairs, in order to improve the quality and consistency of the initial databases. To achieve this, a methodology based on detecting discrepancies between visually perceived color differences and calculated color differences compared to the rest of the data has been implemented, using fuzzy logic methods. The result of this work is the identification, analysis, and elimination of pairs of colors considered inconsistent in various databases. This data cleansing significantly contributes to improving the quality of color-difference databases, which play a fundamental role in the development and evaluation of new color-difference formulas. In this way the results of these formulas align more precisely with the visual perception of color differences by the human visual system.
In the field of color differences, databases are essential, both for the development of new color
difference formulas and checking their performance [
With the color coordinates of each of the stimuli in the pair, a color difference between them can be computed using different mathematical formulas, which can be just the Euclidean distance or other more sophisticated formulas. The calculated color difference is generally called E, and it is desirable that V and E been as correlated as possible throughout the entire database.
The measurement of agreement between perceived and calculated color differences
constitutes another field of study within Colorimetry. Different statistical measures have been
proposed, and today the STRESS index [
As we have mentioned, the consistency of databases is important for their reliability. We
consider an inconsistency when two data, which must be close in a color space, and whose
distance between the colors of the pair is similar, have very different perceived or computed
color differences. The CIELAB color space is used in this work [
The objective of this work is to analyze the effects of the different parameters and variables involved in the definition of this consistency analysis method. Preliminary results are shown here. A deeper analysis in the database is desirable to eventually provided cleaned version of the database.
In [
In this work, a set of databases obtained from a series of psychophysical experiments will be considered. Specifically, four different databases with different structure and composition are studied. Each data of these databases consists of a pair of colors, specified by their color coordinates in a color space (i.e. CIELAB), the perceived color difference by a panel of observers and sometimes other significant data.
The considered databases are: •
•
The combined dataset was used in the development and performance testing of CIEDE2000
color-difference formula [
Developed by researcher Michael Pointer and Geoffrey G. Attridge, at United Kingdom. It has
1308 color pairs distributed across 27 color centers. For each one, approximately 48 points,
varying in L*, a*, and b* have been tested. The color differences, in CIELAB unit, range from 0.88
to 26.21, average of 8.91 and standard deviation of 4.47. Some details can be found in [
Developed by researcher Michal Vik in the Faculty of Textile Engineering of the University of Liberec, in Czech Republic. It has 284 color pairs around 9 color centers. The color differences, in CIELAB units, range from 0.05 to 13.72, average of 2.53 and standard deviation of 2.64.
The complete name of the database is RIT-DuPont Supra-Threshold Color-Tolerance
Individual Color-Difference Pair. This is the complete database of an extensive experiment
conducted at the Rochester Institute of Technology (RIT) in the USA. These original judgments
were transformed, by probit analysis, to 156 color pairs in the so-called RIT-DuPont dataset,
which is one of the four included in the COMData. Thus, it is the extensive original dataset, which
would be worthy analyses independently of the synthesized version inside COMDATA. It has 828
color pairs, distributed across 19 color centers. Each experimental data was assessed by a panel
of 50 observers, consequently this database is considered as one of the most reliable. The color
differences, in CIELAB units, range from 0.03 to 5.42, average of 1.50 and standard deviation of
0.64. A complete description of this database can be found in [
The figure of merit known as STRESS[
Assuming that a color difference formula is accurate enough, STRESS can be considered as a measurement of the consistency of certain dataset. Therefore, removing inconsistent data will imply a reduction in the value of STRESS.
As shown in the previous paper[
Each data of the dataset is compared with the rest of the data to inquire into its consistency. Thus, let us consider two data (i.e. two color-pairs), which are close in the color space. The two data are considered inconsistent if any of these cases applied: • •
The two pairs are very close in color space and have similar E values but very different V values.
The two pairs are very close in color space and have similar V values but very different E values.
Thus, the important variables are the distance between the color pairs and the comparison
between the perceived and computed color differences. All this reasoning is formulated by
means of vague linguistic terms; thus, fuzzy logic can be used for its numerical representation.
The two former cases, were implemented in two fuzzy rules, known as Fuzzy Rule 1 (FR1) and
Fuzzy Rule 2 (FR2), which detect inconsistencies in a color-differences dataset. The details can
be found in [
The first condition, that the two pairs are close, is related to the distance in the corresponding color space. In fuzzy logic, for an inference rule to compute certainty of the rule consequent it needs the antecedent to have not null certainty. Otherwise, no inference is carried out. The certainty of the antecedent for two color pairs is called fuzzy degree of neighborhood and it is defined for each pair considering the rest of pairs in the database. It takes into account the distance and the similarity of V between the two data.
Therefore, these fuzzy rules are applied for each color pair with respect to all other color pairs in the set to find out whether there exist inconsistencies. Eventually, all possible color pairs are compared. It may happen that for some color pairs there is no other enough similar color pairs to be compared to. This should not be confused with color pairs for which there indeed exist similar color pairs to be compared to but no inconsistence if observed for them. To distinguish between these two cases, we propose to compute what is called the number of fuzzy neighbors for each color pair. This is computed as the sum of the certainty of the antecedent of the fuzzy rule when computed for all other data. For the result of the fuzzy rule for a color pair to be considered representative, its number of fuzzy neighbors must be at least equal to 1. Otherwise, the procedure considers that there is not enough evidence to make conclusions about the color pair.
For the application of any of defined fuzzy rules it is necessary that V and E are on the same
scale, so a normalization of V is carried out following the procedure in [
Finally, these fuzzy rules provide a number, Iij in the interval [
Once the degree of inconsistency is computed, a threshold value must be defined in order to
consider an inconsistency or not. The threshold represents the measure of how similar two pairs
of colors need to be to be considered consistent. When the threshold is low, such as 0.05, a very
high similarity value between the pairs of colors is required for them to be considered consistent.
Therefore, it is more likely that inconsistent pairs will be detected. Authors in [
In the next step, in order to clean the database, it is decided which of the two color-pairs must be removed when the degree of inconsistency is higher than the threshold. The decision is to remove, between the two pairs, the one with higher difference between V and E.
Firstly, the distribution of experimental data based on the number of fuzzy neighbors is analyzed for the different databases.
Fig. 1 shows the clustering of the experimental data from the four databases based on the number of fuzzy neighbors (the total fuzzy degree of neighborhood sum). When the number of fuzzy neighbors is lower than 0.5, there is a significant lack of surrounding information to determine the quality of an experimental datum. In this scenario, the evaluation can be highly uncertain due to the scarcity of nearby or similar reference points. It is desirable a number of fuzzy neighbors at least equal to 1, or higher, to make informed decisions about their consistency of the data. In Fig. 1 we can see that most of the data in Poiter database has too low number of fuzzy neighbors, while in COM, RIT-Dupon-Ind and LCAM-WDC databases most of the data has enough fuzzy neighbors to do the comparison.
Once we know the number of fuzzy neighbors, the degree of inconsistency can be worked out between each combination of two data in each database. Fig. 2 shows the number of inconsistent pairs as a function of the threshold. These plots provide valuable insight into how the choice of threshold affects the identification of inconsistent experimental data.
As expected, the plots show an increase in the number of inconsistent pairs as the threshold decreases and vice versa. The number of inconsistent pairs against the thresholds has a tendency between linear and exponential depending on the database. In some databases, the number of inconsistent pairs reaches zero as the threshold increases, suggesting that the data is highly consistent within that threshold range. However, this point can vary depending on the specific dataset, highlighting the importance of customizing the threshold according to the context and requirements of the study. Accordingly, initially the most consistent database is the COMData. In general, some differences can be found between the two fuzzy rules, also depending on the database. A particular case is RIT-Dupont-Ind, where only RF2 obtains inconsistencies. The accuracy of this database makes that no inconsistencies at all are found with FR1, and only few with FR2.
The next step, after identifying the inconsistencies, is cleaning and optimizing the database. To check the efficiency of the process, the STRESS value is compared with the uncleaned/initial version. This methodology can give us the criterion to choose the threshold: A threshold can be admitted if/while the STRESS improves considerably when removing the corresponding inconsistent data. Obviously, the quality of the database will improve as more inconsistent data are removed. However, there is a critical point in this process. There comes a point when the experimental data remaining in the database after removal are equally consistent compared to those that were removed. At this point, the STRESS value stops improving and remains practically constant. This approach seeks to find a balance between improving the quality of the database by removing inconsistent data and preserving the original nature of the database. If too many data are removed, the database could lose its original representation and become a smaller, but highly consistent, set. Therefore, this criterion aims to find the balance at which the quality of the database is maximized without losing its initial integrity.
Fig. 3 shows the results for the COMData. The combination of the two fuzzy rules is the most effective approach, removing more data and achieving the lowest STRESS values. This result will be general for any database. In this case, thresholds between 0.55 and 0.50 get the lowest improvement in STRESS, being these values the desired balance.
In the case of Pointer, similar results are obtained, but FR1 is better than FR2. In this case, a threshold of 0.5 is a suitable candidate. This means that by removing experimental data below this threshold, a substantial improvement in the STRESS level is achieved, as it can be seen in Fig. 4. For FR2, less than 4.2% of the data, 6.0% for FR1, and for the combination less than 9.5% of the data would be removed.
For the LCAM-WDC database, as it is shown in Fig. 5, the results are not so lineal. Three different values of the threshold could be considered. The most conservative one is 0.95, achieving an important improvement in the STRESS value, by removing only 2 data (0.7%). This means that these 2 data are highly inconsistent in this dataset and could be a typo. Threshold must decrease until 0.75 to find more inconsistent data (3 data). Again, a good balance seems to be a threshold of 0.5, which removes 10 data (3.5%) and reduces the STRESS value in 1.2 points.
Fig. 6 shows the result of the RIT-Dupont-Ind database, which is a special case, as commented above. FR 1 is no able to detect inconsistencies, thus FR2 and the combination give the same results. This is a quite consistent dataset, since inconsistencies only appear for threshold values lower than 0.80 (3 data, 0.36%) and 0.55 (4 data, 0.48%). Even with the extremely low threshold value of 0.1, only 25 data (3%) can be considered as inconsistent.
As a result of this analysis, we have observed that applying fuzzy logic to color-differences databases provides substantial information about the consistency of them. Also, it is possible to identify and remove the inconsistencies up to a balance between improvement and reducing the number of data is achieved. These results indicate that optimal threshold values vary significantly depending on the database and the applied rule. Finding a balance is crucial, as overly low thresholds can lead to excessive removal of useful data, while overly high thresholds may fail to detect inconsistencies.