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J. Eur. Opt. Society-Rapid Publ. 22, 25( 2026)
Figure
1. Mesh, Gaussian curvature, and thickness( mm) over the mesh of reference cornea A.
The piston term is just an arbitrary number – particularly, 10 – selected to avoid negative numbers in z, but, of course, without changing the shape. Following equation( 8), I computed the Gaussian curvature following the procedure described in Section. p2.1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. These functions are defined within a circular domain x 2 þ y 2 < 5 mm. Considering that real biometric data is affected by measurement noise, I added white Gaussian noise to the z values with the Matlab function awgn. The level of Gaussian noise was simulated through the Signal-to-Noise Ratio( SNR), whose units are dBs.
Figure 1 plots the surface mesh, the Gaussian curvature, and the thickness corresponding to these equations with the following parameters: R x = 7.79 mm, R y = 7.61 mm, Q x = �0.27, Q y = �0.27, h 0 = 0.2 mm, r x = 0.5, r y = 0.5, x 0 = 1.5mm, andy 0 = �1.5 mm and with a SNR = 90 dB. From now on, this cornea model will be referred to as test A.
As mentioned in Section 2.1, there is a trade-off between mesh density and noise level. To exemplify this fact, I computed the Gaussian curvature map for two noise levels: low noise( SNR = 90 dB) and moderate noise( SNR = 70 dB), and three levels of mesh sampling density determined by the number of mesh vertices: 5.025, 20.081, and 45.225 points. Figure 2 shows the results. Although for low noise, the more accurate Gaussian curvature estimation is achieved with high sampling density( 45.225 mesh points), for moderate noise, the best estimation requires fewer mesh sampling points( between 20.081 and 45.225). Therefore, the choice a priori of the optimal mesh density, to improve the SNR ratio, is not an easy task because it also depends on the geometry shape. Nevertheless, a rule of thumb is that the stronger the local curvature changes, the denser the mesh required.
4 Spectral decomposition
A first potential of the Laplace – Beltrami operator comes from its spectral decomposition obtained with equation( 5). As mentioned in the introduction, it can be used for mesh compression or smoothing through filtering. In both cases, the Laplace – Beltrami operator serves to separate the curvature shape features of the surface in question. The first eigenvectors of the spectral decomposition( associated with low eigenvalues) capture the low-curvature shape signal, whereas the following ones apprehend the high-curvature part of the shape.
Figure 3 shows the mesh reconstruction of surface test A with an SNR = 80 dB for different numbers of eigenvectors in the spectral decomposition, particularly: 5, 10, 25, 50, 75, 100, 200, and 300. The figure shows that only when more than approximately 50 eigenvectors are included does the“ bump” part of the keratoconus start to be revealed. Note that, as with any spectral method, some artifacts are introduced at the periphery. This problem, related to the