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one [ x f, y f, z f ] based on the matrix operator given by the equation:
2 3 2
3 2 3 x i R 11 R 12 R 13 t x x f
y i 6 7 4 z i 5 ¼ R 21 R 22 R 23 t y
6
7 4 R 31 R 32 R 33 t z 5 y f
6 7 4 z f 5; ð11Þ
1 0 0 0 1 1
where t x, t y and t z are translations along directions x, y and z; and R ij are entries of the Euler rotation matrix [ 34 ]. I implemented this with the help of the Matlab function rigidtform3d. m. Particularly, I modelled an eye fixation change provided by an Euler angle a = 45 ° and a translation defined by [ t x, t y, t z ] = [ 0.4, 0, 0 ]. Applying this transformation to the reference surface A, surface B is obtained.
In a second example, I modelled a uniform scaling transformation that isotropically enlarges or shrinks the ocular surface by a scale factor equally in all directions. This is implemented through a scaling matrix operator.
2 3 2 3 2 3 x i t 0 0 x f 6 7 6 7 6 7 4 y i 5 ¼ 4 0 t 0 5 4 y f 5; ð12Þ
0 0 t z i
where t is the scalar factor. Using a scaling dilatation t = 1.005 onto the surface A, generate surface C.
The meshes and Gaussian curvatures over them of surfaces A, B, and C are depicted in Figure 6. It illustrates the change in eye fixation between surface B and A, aswellas the uniform dilatation of mesh C with respect to A. Observing the Gaussian curvature map, its invariance with scaling shows the potential of this map with respect to plain meshes.
The shape-DNA distances between reference surface A, and surfaces B, C, are 8E – 13, and 3E – 13, respectively; quantities which are clearly negligible.
6 Conclusion
The large increase, in recent times, of the biometry capabilities applied to the anterior segment of the eye has led to a synchronous development of data processing techniques, which include compact and statistical surface representation [ 35, 36 ], image segmentation [ 37 ], or data interpolation / extrapolation [ 38, 39 ]. A similar process has occurred in reconstructing, modelling, or analyzing wavefronts [ 3, 40 ].
However, all these techniques, although indirectly dealing with curved surfaces, do not fully address the curvature shape information. In this sense, the Laplace – Beltrami spectral decomposition may emerge as a new tool offering this advantage. Of course, further exploration of the potential of the methodologies proposed in this paper requires testing them with real clinical data, such as optical coherence tomography measurements or highly dense wavefront sensor data.
As shown, the spectra are especially suitable for performing shape analysis through shape-DNA fingerprints. However, these fingerprints can be spatially localized, occupying different scales. When the objective is to perform a
z f global shape comparison, but at the same time compare local geometrical features, a possible approach is to pick up the spectra associated with selected eigenfunctions [ 20 ]; for instance, in the discriminative representation of brain morphology [ 41 ]. This is something that could be explored in the future.
In the mathematical part, Laplace – Beltrami spectral theory still offers challenging open questions. One of which is how to address boundary conditions in open bounded surfaces. Although in the present approach, I have not complied with boundary conditions, because the discrete Laplace – Beltrami operator based on the cotangent formula can be implemented without them. However, boundary conditions affect both the eigenvalues and their corresponding eigenfunctions [ 11 ]. Classical boundary conditions, namely homogeneous Dirichlet – constant values of the solution along the boundary – or Neumann – zero derivative of the solution with respect to the outward normal –, are not strictly valid on the human eye anatomical or wavefront surfaces. Therefore, I hypothesize that some improvements in the solutions – related to edge reconstruction errors as shown in Figure 3 – are expected if proper numerical considerations for the boundaries are included. I plan to research rigorous options related to more realistic boundary conditions – the more reasonable are non-homogeneous Dirichlet conditions –, as explored in the mathematical literature [ 42 ].
Finally, it is worth mentioning that machine learning techniques – which are also increasingly being used in visual optics biometry processing [ 37, 43 ]– are boosting spectral shape analysis in what has been named geometric deep learning [ 44 ].
Funding
This work was supported by grant PID2023-150166NB-I00 funded by MCIN / AEI / 10.13039 / 501100011033.
Conflicts of interest The author declares that he has no competing interests to report.
Data availability statement
The research data associated with this article are included within the article.
References
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