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parameters and the colors [ 26 ], however their reliance on large, sample-specific training datasets limits their rapid deployment across new materials.
2.2 Laser processing
Laser inscriptions were carried out with a nanosecond pulsed laser coupled to a scanner head. This setup, described in Appendix A, enabled the printing of micrometric square pixels under well-defined processing parameter sets. In all the experiments shown here, the only laser parameters used to explore the color space were only the laser power, the laser repetition rate, and the beam scanning speed. The laser fluence ranges between 40 mJ / cm 2 and 457 mJ / cm 2, the repetition rate ranges from 10 kHz to 600 kHz, the scanning speed from 50 mm / s to 2000 mm / s. Other parameters, such as the laser polarization, distance between hatched lines in a pixel, and defocus, could also generate different colors, but they were not varied during this study. The parameters are printed on a gridlike matrix where a given laser processing parameter set is selected for each square. The squares that are produced to record the laser-induced colors are made of 5 5 identical printed pixels, as exemplified in Figure 1.
2.3 Color measurement
Color images of the laser-processed samples were recorded using a standard high-resolution commercial RGB camera integrated into the experimental setup for in line acquisition. Different positions of the white light source and the camera relative to the sample enable the recording of images in different observation modes including transmission, front-side diffuse and specular reflection, as described in Appendix B. Since the optimization process operates in a closed loop where both the color measurements and the final image generation are performed using the same camera and illumination conditions, additional calibration of the camera or lighting was deemed unnecessary. Any biases would be consistent among all iterations and even for the final image printing. The color measurement is derived by calculating the mean value of the pixel color within an area centered on the square center that encompasses 60 % of the square size. To ensure repeatability, a filtering algorithm is implemented to discard squares whose color appears inhomogeneous( inscription defects such as scratches or color gradients for the same parameter indicating instabilities). This is achieved by first calculating the standard deviation value of the RGB channels within the square. If the calculated value exceeds a predefined threshold, established through empirical means, the square is filtered accordingly.
2.4 Optimization methods
Given the unpredictable nature of the color in relation to the laser parameters, the only option is to measure them after inscription. This makes an iterative approach suitable for optimizing the color gamut on any type of sample. This approach is well-suited to the nature of the experiment, as the measurements can be taken in line on the setup by adding an RGB camera. Thus, the optimization step is integrated between the first inscription and parameter selection to print the image. After every new inscription on the sample, an algorithm suggests new parameters to inscribe, aiming at improving the gamut volume( Fig. 2). This continues until an exit criterion is met, such as a limit on the number of inscriptions( iterations), achieving a certain gamut value, or a limited increase in gamut.
Both the GA and BO approaches share a common iterative experimental loop( as illustrated in Fig. 2):( 1) Inscription of an initial random set of parameters, selected using Latin hypercube sampling( LHS)( 2) measurement and filtering of the resulting colors,( 3) algorithmic suggestion of new parameters based on previous evaluations, and( 4) re-inscription. The process is repeated until the gamut volume improvement drops below 10 % over three consecutive iterations. The fundamental difference between the two methods lies in how the new parameters are suggested, which is detailed in the following subsections.
Using a GA constitutes a well-established approach for this complex problem. It is a robust method for global exploration of the parameter space without assumptions regarding the parameter-to-color function. However, GA does not fully exploit previously measured color information. Therefore, a BO that is more deterministic with the use of a surrogate model might increase the convergence speed.
2.4.1 Genetic algorithm
This approach draws extensively from Cucerca’ s work [ 30 ], adhering to the standard guidelines set forth in the GA framework. We included the hue diversity criterion as a metric for evaluation, which is a key takeaway for the algorithm to consider even the hues where the colors are the least saturated. The main difference was to scrap the DSD( Design Space Diversity) and PSD( Performance Space Diversity) metrics, as they were found to be ill-suited for the materials assessed in this study. Instead, an emphasis was placed on saturation and hue diversity. Another key difference is the replacement of the repeatability criterion by a more straightforward filtering process. Figure 3 schematizes the process:
A detailed description of the metrics, the multi-objective NDSA procedure, and parameterization choices are provided in Appendix D.
2.4.2 Bayesian optimization
The second approach employed Bayesian Optimization( BO) to guide the search for optimal parameters that would maximize the gamut. The proposed BO framework is:
Set-Based: the parameter optimization concerns the entire gamut range; each color is evaluated in comparison with the rest of the palette, i. e. the objective function leverages all preceding predictions to make each new evaluation.