J. Eur. Opt. Society-Rapid Publ. 22, 45( 2026) 449
Table D1. Influence of GA parameters on the percentage of points required to reach 95 % gamut.
Crossover rate P c |
Mutation rate P m |
Modification rate m |
Percentage of points to reach 95 % gamut |
0.2 |
0.2 |
0.1 |
35 % |
0.2 |
0.2 |
0.2 |
44 % |
0.2 |
0.4 |
0.1 |
38 % |
0.2 |
0.4 |
0.2 |
46 % |
0.5 |
0.2 |
0.1 |
41 % |
0.5 |
0.2 |
0.2 |
45 % |
0.5 |
0.4 |
0.1 |
47 % |
0.5 |
0.4 |
0.2 |
48 % |
insight for Pareto front identification when the initial input set lies on a predefined grid. Likewise, the PSD metric offers limited relevance in our case, as the objective is to highlight configurations where the resulting color deviates significantly from the others, rather than simply maximizing dispersion in performance space. Same for the repeatability metrics which was overtaken by the parameter filtering during the measurement. Naturally, the criteria are not the same when tackling with best black( resp. white) research within the sample, what is more favored here is the closeness to the neutral axis and to minimize( resp. maximize) the lightness.
To compute chromaticity, the RGB channels value of the squares were converted to the CIE Lab color space, known for being perceptually uniform. Then it was converted to cylindrical coordinates LCh( Lightness, Chroma, Hue) where the chroma value is directly accessible.
The second criterion for color exploration is to determine every color’ s contribution to hue diversity. It is intended to encourage the correct classification of colors that extend the gamut the most in hue ranges where the gamut is restricted, so that these colors are more likely to be selected subsequently and the gamut can be explored around them. In the case of the sol-gel and SLICID sample, the initial gamuts are often eccentric, which tends to penalize certain colors, which could nevertheless be interesting, on this criterion of hue diversity. It was therefore decided not to apply it to the colors as measured, but to the colors obtained after the gamut grey axis alignment( 3), which enables them to be refocused around the grey axis. To compute this metric, the a * b * plane of the CIE Lab space is segmented into a random number of segments ranging from 4 to 72, with a random angular offset on the starting sector. Then in every sector the colors are ranked based on their chroma. For each different hue wheel, every color can have a different ranking. Every candidate is then characterized by its ranking frequency, appearing n times in rank 1, m times in rank 2,... The global ranking of the solutions is done using lexicographic ordering meaning number of rank 1 is given strict priority, then rank 2,...
For the search of black and white, the two criteria that were discussed for color exploration are discarded. For achromatic exploration, the chroma must be minimized as the colors closer to the grey axis should be prioritized. For the black colors, the lightness criteria in CIE Lab has to be minimized, for the white colors it should be maximized.
Followingtheframeworktotacklemulti-objectiveGA, the candidate solutions were sorted using the Non-Dominated Sorting Algorithm( NDSA), an algorithm that sorts entries according to the number of entries that are better than them in both criteria at the same time. The NDSA works by first identifying the pareto front, i. e. the set of solutions such that no other solutions constitute an improvement in both criteria, the pareto front is labeled as front 1 Figure D1a. Theneverypointthatwerenotpartof the pareto front go through the NDSA again to find the front 2 Figure D1b. The process continues until every point have been assigned to a front number Figure D1c. Once all points have been sorted, they are randomly selected using a Roulette-wheel selection, with selection probabilities depending on the front number, points belonging to lower order front having higher chance of being selected. The number of colors selected is fixed at about 20 % of the initial population size.
After the selection, offspring parameters were generated using uniform crossover( crossover rate of 0.2) and mutation( mutation rate of 0.2 and parameter modification rate of 0.1), applied independently. Crossover promotes recombination of promising solutions and mutation maintains diversity. In the following paragraph, we explore the impact of those parameters. The stop criteria to interrupt the iteration was when the gamut has note increased significantly in the past iterations.
The adjustment of the mutation and crossover parameters was performed by a postmortem experiment. Using convergence rate for different parameters reported in Table D1, defined by the proportion of the number of points required to reach 95 % of the gamut volume, done on a 14000 points exploration made on SLICID.