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boundary conditions in the Laplace – Beltrami will be discussed in Section 6.
Generally, the problem of choosing the optimal number of eigenvectors would depend on the shape structure and the level of measurement noise. Therefore, preliminary modelling of the shape features of interest would surely help in that selection.
5 Laplace – Beltrami for shape-DNA analysis of ocular surfaces
A substantial benefit of the Laplace – Beltrami spectra is their invariance to isometric changes. These are deformations that preserve the length of every arc on the surface, so they conserve the shape. The most common isometric changes are rigid motions( rotation or global translation) and similarity scaling [ 11 ]. On the one hand, rotation or global translation is a typical issue due to eye fixation changes among measurements, for instance, in optical coherence tomography [ 31 ]. On the other hand, different eyes may differ in size but not in ocular shape; in this case, the Laplace – Beltrami spectra would be simply scaled, as will be shown.
The isometric invariance of the Laplace – Beltrami spectrum enables its use to extract geometrical fingerprints characterizing ocular surfaces, which has been coined a shape-DNA [ 11 ]. This signature is derived from the Laplace – Beltrami spectra as computed through equation( 6).
The shape-DNA emerges akin to Fourier analysis as a spectral tool analysis, but contrary to the latter, a tool that captures better the shape geometry. Also, contrary to Fourier analysis, the surface mesh geometry determines different eigenbases of the Laplace – Beltrami spectrum, so the corresponding eigenvalues of different meshes cannot be directly compared; some type of normalization is required so that the shape-DNA is invariant to the surface’ ssize [ 7, 11 ]. Particularly, I propose scaling the Laplace – Beltrami spectra coefficients to the first coefficient: ^f ¼ ^f
^f ð1Þ
, a procedure that is appropriate when trying to identify a surface shape pattern [ 11 ].
Furthermore, to apply the shape-DNA for quantitative comparison purposes, or to identify surface-mesh shape patterns, a distance metric must be constructed [ 11 ]. Given two arbitrary shape-DNAs( normalized set of eigenvalues): k =( k 1, k 2,... k n) andl =( l 1, l 2,... l n), from now on it will be used the 2-norm distance, which is unitless because the above normalization, expressed as:
dðk; lÞ ¼ Xn i¼1 ðk i � l i Þ 2! 1 = 2
: ð10Þ
5.1 Shape-DNA similarity: continuous deformation
The shape-DNA offers a way to quantitatively track, with a single metric, the timing of ocular surface shape changes. This could be useful, for instance, as a clinical tool to analyze the time evolution of pathologies manifested by surface shape abnormalities. The underlying mathematical principle
Figure 4. Shape-DNA distance to reference cornea model A for different scalar functions – Gaussian curvature, mean curvature and thickness – as function of timing surface deformation: explicitly the h o parameter, but implicitly also r x and r x.
is that of similarity; namely, the shape-DNA fingerprint should depend continuously on shape deformations [ 11 ].
As an example, I propose an artificial model of keratoconus variation more or less compatible with the reported literature [ 32 ]. Starting with surface model A, I modelled a surface elevation change and thinning of the cornea with time manifested by a change of h o ranging from 0:2 to0:4; and, synchronous with this process, a spatial expansion of the local abnormality modelled by a change of r x and r y values from 0.5 to 2. Three different shape-DNA were generated: associated with the Gaussian curvature, the mean curvature, and with the cornea thickness.
I computed the shape-DNA distance( Eq.( 10)) between all these values and the initial reference value( h o = 0.2, r x = 1, andr y = 1), sampling the ranges in 100 steps. Results are plotted in Figure 4, revealing that the similarity principle is fulfilled, because the shape-DNA fingerprint depends smoothly on the continuous variation of the h o, r x and r y. However, it should be observed that this dependence is not necessarily linear or even monotonically increasing, as revealed by the Gaussian and mean curvature curvature fitting curves. Also, the relation depends strongly on the scalar function defined over the mesh, as revealed by the difference in the fitting curves of the Gaussian curvature and thickness. To complement this figure, Figure 5 shows the mesh, Gaussian curvature, and thickness for three representative stages of the previous timing model.
5.2 Shape-DNA similarity: stability
As mentioned before, the isometry invariance of the Laplace – Beltrami operator implies that the fingerprint should not differ between surfaces that can be isometrically mapped onto each other. The most common isometrical mappings are rigid motions and scalings. Additionally, as it will be shown in Figure 6, the Gaussian curvature has the same value at corresponding points of isometric surfaces( p. 177 [ 33 ]).