96
J. Eur. Opt. Society-Rapid Publ. 21, 10( 2025)
By combining these two approaches, the Huber loss provides a hybrid solution. It behaves like L 2 for small differences, preserving fine details, while transitioning to L 1 for larger differences, ensuring robustness against noise and outliers. This balance is particularly advantageous in image processing tasks, where maintaining important features while reducing noise is critical.
The estimation of d mostly relies on the noise standard deviation of images( r) commonly set to a range of 1.5 r – 2.5 r, a heuristic that has been followed in our work. Thus, given average r = 5 pixels throughout the collected dataset, d = 10 pixels was set. The optimal value of h was empirically determined to be 10 which ensures balancing noise reduction and structural integrity.
The Sobel operator is used to find the approximate gradient of the image intensity function [ 58, 59 ]. It consists of two convolution kernels( or masks) that estimate the gradients in the horizontal( x) and vertical( y) directions. Given an input image I, the gradients in the x and y directions( denoted as I x and I y) are calculated using the convolution of the image with the Sobel kernels,
I i ¼ G i
I: i ¼ ðx; yÞ: ð4Þ
The maximum amplification of gradients through OCT images can be obtained with the Sobel’ skernel( G) inthe horizontal and vertical directions as follows,
�1 0 1 1 2 1
G x; G y ¼ �2 0 2
; 0 0 0 ð5Þ
�1 0 1
�1 �2 �1
where the combination resembles Robert Cross operator [ 60 ].
The above chain of processing leads to collect a 2D cloud of points encompassing the cornea, and intrastromal cuts. A vertical peak finder algorithm was used to specify points belonging to the topmost intensified feature that forms the cornea [ 61 ].
Additionally, fitting a parabolic polynomial classifier( PPC) to the entire cloud of points distinctly splits the cloud of points such that,
x i 2 Corneal Segment y i � PPCðx i Þ > ð6Þ x i 2 Intrastromal Cuts y i � PPCðx i Þ <
is a threshold to assure that points are in proximity to the desired regions. The PPC aims to minimize the fitting error, ultimately leading to a fit positioned between the cornea and the intrastromal structure. The epsilon parameter, therefore, helps determine the layer to which each peak belongs. was considered to be zero for splitting two layers, whether the cornea-intrastromal or anteriorposterior, as vertical upper and lower layers.
The R 2 score was used to identify whether the intrastromal points would determine a flap or lenticule cut. The R 2 of the PPC in flap cuts reaches a higher value compared to lenticule cuts. A lenticule substructure typically exhibits more vertical peaks at lateral positions compared to the flap. Therefore, when fitting a PPC through additional corneal peaks, the resulting model exhibits higher variance.
Consequently, the R 2 score for lenticules is typically at least 10 % lower than that for flap substructures.
A similar methodology can be applied to distinguish the anterior and posterior segments of a lenticule cut. However, the PPC approach is very sensitive to sparse noises, which refers to occasional outliers that occur at isolated points within the peak data. It may lead to extended flap or posterior segments since the classifier may cross the sidebands of anterior. Thus, applying the equation 6 would not solely suffice for a precise segmentation. Hereafter, the sideband noise( or sideband falsely detected peaks) refers to peaks that appear outside the boundaries of the lenticule or flap extension. These falsely recognized peaks can lead to overestimation of the diameter.
For flap cases, a solution to filter out falsely recognized points involves scanning the point density starting from the center and extending toward the sidebands. As the scan progresses, the point density reaches a threshold where it no longer exhibits significant variation compared to the earlier steps. This method effectively identifies and excludes false peaks detected in the sidebands, improving the accuracy of the layer recognition. This approach can be seen as a density-point scan routine for refining flap boundary identification.
For posterior segments of lenticules, a Bayesian Optimization( BO) approach was implemented [ 62 – 64 ]. Empirically, it was observed that the point-density scanning routine, while effective for flap cases, is not yet sufficiently accurate to handle lenticule structures.
A mean-squared-error( MSE) was set as the cost function to be minimized by shortening( if needed) the lateral extent of posterior segments. We have utilized GP- Minimized package from Scikit-optimize( open-source) library which performs sequential model-based optimization. GP-Minimized algorithm has certain advantages. However, it should be noted that GP-Minimized assumes a smooth and non-singular underlying function [ 65 ].
The pseudo-algorithm to filter out noises from posterior cuts can be summarized as Algorithm 4 where( x min, x max) specifies the range obtained by the initial PPC, a is the set of weights for the polynomial fit ontheposteriorsegment, and T is the maximum number of iterations, thereby the optimization reaches a monotonic behavior. The algorithm was initialized with a = 0andN max = 12. The initialization of a was made to provide a neutral starting condition, free from any prior assumptions or bias that could influence the optimization process. Additionally, it was anticipated that all layers could be accurately described by a 12th-order polynomial fit. This assumption was based on the need for a model that could capture the complex curvature and variations within the corneal and substructural layers.
Algorithm 1 iteratively filters out points outside a boundary while fitting polynomials. It works by proposing new candidate parameters, such as the bounds of the data and the polynomial fit order, fitting a model to the selected data points, and checking how well the model performs. The performance is evaluated using an objective function, typically the MSE, which is minimized during the process. Based on the performance, the algorithm updates its search strategy to focus on areas that are likely to lead to a better