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J. Eur. Opt. Society-Rapid Publ. 22, 45( 2026)
Figure 2. Algorithmic flowchart of the iterative laser parameter optimization and image printing framework.
defined by a vector-valued mean function and a multi-output covariance kernel. The GP implementation is further detailed in Appendix F.
Using this GP we substitute the physical col( p) bya predictive probability distribution ðcjpÞ, the practical optimization step solved by the algorithm is therefore:
p n ¼ argmax p2X EV ðpÞ: ð4Þ
Let f gam be a function that associates to a point c of the color space the corresponding contribution to the added volume in the gamut gam:
f gam ðÞ¼Vol c ðConvexHullðgam [ fg c ÞÞ � VolðConvexHullðgamÞÞ: ð5Þ
The EV is meant to operate by computing the volume that every parameter is expected to bring by summing the added volume of any given possible color the parameter set could produce times the density of probability of said color occurring:
Z
EV ðpÞ ¼ PðcjpÞVðfg c [ C n Þdc: ð6Þ
R 3
While equation( 6) defines the theoretical integral over the entirety of R 3, computing this unbounded integral numerically is intractable. Because the probability density function ðcjp n Þ of a Gaussian Process decays exponentially away from its predicted mean, the numerical integration is practically restricted to a localized bounding box in the color space where the probability mass is computationally significant.
Figure 3. GA steps.
Within this truncated domain, the integral is approximated numerically using a discrete step size of( DL *, Da *, Db *) =( 0.1, 0.1, 0.1). This expected improvement evaluation is systematically performed across a dense grid of candidate parameters with a resolution of( DPower, DSpeed, DFreq) =( 1 %, 10 mm / s, 5 kHz). The candidate parameter p n that maximizes this numerical EV is ultimately selected for the next physical inscription.
The EV function is designed to consider the weight of any possible color within a given parameter range of colors available that lies outside of the already measured gamut.
Figure 4 shows how two different parameters will be evaluated by the algorithm: the parameter P 1 will most likelyproduceacolorthatisalreadyprintableasitismost likely inside the convex hull of the already measured colors gamut, P 2 and P 3 on the other hand have more chances to produce a color outside which would increase the volume.
The maximum value for the expected improvement across the whole parameters set space is selected to be