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J. Eur. Opt. Society-Rapid Publ. 22, 45( 2026)
to predict relative rankings from metrics such as dL * range or a * b * hull area, after ANOVA / AIC selection. Since each basic computable metric is normalized using fixed bounds and several terms depend jointly on the image content and the gamut mapping( chromatic adaptation, gray-axis alignment, lightness / chroma soft-compression), the model is not constrained to return exactly 1 for the original image and small deviations around unity are expected. These images indicate that both optimization methods are useful in enhancing the perceptual fidelity to the original image by simply improving the gamut. BO optimization appears even better as expected from the larger increase in the gamut. The fourth row of Figure 6 shows the images recorded in diffuse reflection of the printed samples before and after GA and BO optimizations of the gamut. The image quality metric value is provided but should be taken carefully as the metric was designed with simulated images only. Except some defects on the samples, the experimental results confirm the predictions.
3.3 Summary of optimization performance
Based on the outcomes of both the post-mortem analysis and the live experiments, the BO approach proposed in the paper exhibits several advantages over the GA approach:
Simplified problem formulation: The BO approach operates with a single-objective formulation, whereas the GA approach, which requires reformulating the problem as a multi-objective optimization. The objective is defined as the improvement of the gamut volume alone, without complexifying it with criteria such as hue diversity, because BO does not persist in trying parameters in the same zone repeatedly. BO reduces the complexity of the optimization and the associated computational overhead.
Faster expansion of the color gamut: By focusing on enhancing the gamut volume, the method rapidly increases the number of colors that can be achieved with each inscription. This efficiency stems from the Bayesian optimization framework’ s ability to intelligently select the subsequent candidate points based on predicted improvement, as opposed to attempting to balance multiple competing objectives, which often slows convergence.
Better gamut coverage at fixed cost: Given a“ printing” budget( a fixed number of inscriptions), the method can produce a larger color gamut than GA, which struggles to efficiently explore the parameters and often results in redundant sampling in dense regions. The single-objective Bayesian optimization model naturally avoids this problem by targeting regions of the color space that maximize volume gain.
Effective without explicit diversity criteria: In GA, a hue diversity criterion is required to prevent the algorithm from repeatedly sampling in the same color regions. The Bayesian optimization approach balances exploration and exploitation to ensure coverage of new regions without the need for explicit diversity enforcement. This makes the method both faster and more effective, achieving broader gamut expansion with a reduced number of evaluations.
Ultimately, the proposed Bayesian optimization framework proves to be highly generalizable and can be applied to any material system for which color can be modified by laser processing. Its versatility has been demonstrated here on two distinct types of samples, including plasmonic random metasurfaces. In all cases, the framework successfully identified the laser processing parameters required to maximize the achievable color gamut in a given observation mode.
4 Conclusion
In this study, we introduced a Bayesian Optimization( BO) model for laser-induced color printing and compared it with the Genetic Algorithm( GA) method, which had not yet been applied to this type of sample. Both approaches resulted in a substantial expansion of the accessible color gamut. BO demonstrated faster convergence and a more systematic exploitation of the available data, leading to a larger gamut achieved. The perceptual comparison of generated images further confirmed the benefit of the optimization.
The proposed BO model, relying on a single and easily defined acquisition function, facilitates scalability to higher-dimensional problems involving multimodal colors. This framework can be directly extended to the optimization of a hyper-gamut spanning multiple viewing conditions, where the same principle applies. The algorithm not only enhances image quality but also favors a better control over the inscription process, enabling the simultaneous optimization of multiple constrained images on a single sample by adjusting their respective hypercolors.
As a next step, this approach could be extended toward image multiplexing applications, thereby facilitating simultaneous optimization across multiple output modes. Although this was not explored in the present work due to the lack of a setup allowing simultaneous color measurement in both modes, it remains a promising direction for future research. Another possible next step would be to return to a multi-objective Bayesian Optimization( MOBO) framework, introducing reproducibility as an additional optimization criterion to complement the current single-objective formulation.
Abbreviations
GA BO DSD PSD NDSA IQM
Genetic Algorithm Bayesian Optimization Design Space Diversity Performance Space Diversity Non-Dominated Sorting Algorithm Image Quality Metric