J. Eur. Opt. Society-Rapid Publ. 21, 10( 2025) 97
Algorithm 1. Bayesian optimization for fitting model with limited extent of points. 1: Input: Data points fðx i; y i Þg N i¼1, Search space X for( x min, x max) and model parameters a N max
, N max is the polynomial fit order and T as the number of iterations, 2: Output: Optimal ðx min; x max; a Þ 3: Initialize the surrogate model( e. g., Gaussian Process) 4: for t = 1toTdo
5: |
Propose ðx ðtÞ min; xðtÞ |
6: |
if x ðtÞ min xðtÞ max then |
max; a ðtÞ Þ by optimizing the acquisition function
9: |
Select subset fðx i; y i Þjx ðtÞ min x i x ðtÞ maxg |
10: |
if selected subset is empty then |
11: |
f( params) |
1 |
12: |
else |
|
13: |
Fit model G( x, y, a) using the selected subset |
14: |
Calculate mean squared error( MSE) for the model on the subset |
15: |
f( params) |
MSE |
16: |
end if |
|
17: |
end if |
|
18: Update the surrogate model with the new observation ðx ðtÞ min; xðtÞ
19: end for 20: Return ðx min; x max; a Þ corresponding to the lowest observed f( params)
max; a ðtÞ; f ðparamsÞÞ fit, refining the boundary to exclude irrelevant points. This process is repeated several times, continuously improving the fit by selecting the most relevant data points. Thus, the sideband noises are effectively filtered out, allowing the algorithm to accurately determine the true( x min, x max) as the boundaries of the posterior layers. After determining the boundaries of a cut, a cord line connects both ends. This line can be seen as the possible tilt reference.
Following the application of the PPC and Bayesian Optimization, the corneal and sub-structural layers are identified – a morphology as depicted in Figure 2 – which enables the definition of key geometries for further characterization. This foundational step allows for the automatic measurement of local thickness, which is essential for understanding the structural properties of the cornea.
The Euclidean distance between two corresponding points at two segments estimates the local thickness. Those pairs of points are found as the cross sections between a line, normal to the cord, that crosses two segments. Thus, the following thicknesses can be determined( see Figure 2),
See the Equation( 7) bottom of the page
where FT, CT, and LT refer to the flap, cap, and lenticule thickness.
Due to the variation around center, all LT-CT-FT values are calculated as an average within close proximity
(
FT ¼jjCðx c; y c Þ�Aðx a; y a Þjj LT ¼jjAðx a; y a Þ�Pðx p; y p Þjj; CT ¼jjAðx a; y a Þ�Cðx c; y c Þjj
with the center( 10 pixels) and represented as Average ± StdDev. A positioning standard deviation is also to be considered for reporting thickness measurements.
The lenticule and flap diameters( LD, FD) are measured as the extent of the left and right boundaries( i. e. the cord length).
Figure 3 illustrates the impact of our post-processing approach. It represents the post-processed version of Figure 1, obtained by applying the steps outlined in the Materials section, which are described in detail below.
To ensure optimal performance, the initial parameters of our algorithms were determined via a preliminary tuning. This trial involves a 5 % subset of the dataset from both lenticule and flap intrastromal cuts. These parameters were set to maximize performance across the entire dataset. Subsequent to fine-tuning on this subset, the algorithm operated unsupervised on the complete dataset. Thereby, the output variability is minimized. All parameters were derived from this initial optimization phase. As is customary in pre-tuning for unsupervised tasks, this subset was selected to optimize algorithm performance. Notably, the algorithm processes the remaining data in its entirety, unless incomplete cuts or faint traces are detected.
Furthermore, the proposed approach enables the determination of the optical zone. Characterizing transition zones would determine the quality of the created corneal interface. The design of the transition zone can influence residual aberrations, such as coma and spherical aberration [ 66 ].
2 flap 2 lenticule ð7Þ