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them. Then, the Laplace – Beltrami operator emerges as the best option because it involves second-order derivatives, but it is linear within the surface geometry coordinates. Here, I use the so-called geometric mesh Laplacian, because it explicitly encodes the geometric information of the mesh [ 7 ]. Furthermore, selecting a curvature-based local shape index as the scalar function defined over the mesh leads to an optically meaningful Laplace – Beltrami spectrum [ 7, 20 ]. One of the potentials of the Laplace – Beltrami, for our purposes, stems from the fact that it characterizes the smoothness of a scalar function over a surface, grosso modo, the difference between the scalar at a point P and a weighted average in a small neighborhood of P( Sect. 6.2 [ 7 ]).
For certain applications, shape analysis focuses on capturing morphologically relevant shape descriptors, also called signatures or fingerprints. Such a signature has been called shape-DNA; so, from now on, I will follow this notation [ 11 ]. As it will be seen, employing a local curvature shape index scalar( for instance, the Gaussian curvature) notably increases the discrimination capacities of the Laplace – Beltrami spectrum.
To date, I am not aware that geometric mesh analysis has been applied in the visual optics literature, except for a paper in retina imaging [ 21 ]. I believe that there is a call to introduce these techniques, especially for analysing anterior chamber depth interfaces, ocular wavefront surfaces, or even ophthalmic lens surfaces. In this paper, I present a systematic approach to these techniques and provide some applications. A major hypothesis is that surface secondorder differential features, captured by the Laplace – Beltrami spectrum, can be employed to detect, compare, and study ocular surface shape features. The paper is organized as follows. First, in Section 2, I introduce all the basic mathematical aspects of the Laplace – Beltrami spectra decomposition theory. Next, in Section 3, I introduce a particular, but meaningful, geometric eye surface-thickness model that will be used in subsequent sections, particularly a keratoconus model. Subsequently, in Sections 4 and 5, I present two relevant applications of the theory: the spectral decomposition and the shape-DNA analysis, respectively. Finally, in Section 6, I summarize results and discuss their implications.
2 Laplace – Beltrami spectral decomposition theory
The Laplacian operator( D), i. e., the divergence of the gradient ¼ @ 2 þ @ 2
, measures how a function differs from its
@ 2 x @ 2 y average in a local neighborhood. The Laplace – Beltrami operator is a generalization of the Laplacian to functions defined over surfaces or manifolds in higher dimensions.
Let f 2 C 2 be a real-valued function( continuous second derivatives) defined over a smooth bounded surface S, locally represented by a parametrization: r( x, y)( x, y, z
( x, y)): S 2? R 3, beingS 2 a two-dimensional( circular or not) Cartesian parametric domain. Denoting r x ¼
� 1; 0; @ z
@ x ð 1; 0; zx Þ and r x ¼ 0; 1; @ z @ y
� 0; 1; z y, the
Riemannian metric tensor or first fundamental form matrix( symmetric) is:
g 11 g 12 g 21 g 22
r "
xr x r x r y
¼ ð1 þ z #
xÞ 2 z x z y; r y r x r y r y z x z y ð1 þ z y Þ 2
which determinant is g ¼ g 11 g 22 � g 2
12.
Now, f x, f y, f xx, f yy, f xy, denoting partial derivatives of f and considering relations between the elements of the Riemannian metric and those of the inverse matrix( p. 153 [ 22 ]), the Laplace – Beltrami operator over f is defined as [ 23 ]:
� � � g11 f yy � C 1
g ½f Š¼ g 12 f yx � C 1
21 f x � C 2 21 f y g
� g �
22 f xx � C 1 11 f x � C 2
11 f �
y � g12 f xy � C 1 g
22 f x � C 2 22 f y
12 f x � C 2 12 f y
ð1Þ
Here, C k ij are the so-called Christoffel symbols; their values as a function of partial derivatives of z have been derived elsewhere [ 24 ]:
C 1 11 ¼ z xx z x ð1 þ z 2 x þ z2 y Þ;
C 1 22 ¼ z yy z x ð1 þ z 2 x þ z2 y Þ;
C 2 12 ¼ z xy z y ð1 þ z 2 x þ z2 y Þ;
C1 12 ¼ z xy z x ð1 þ z 2 x þ z2 y Þ;
C2 11 ¼ z xx z y ð1 þ z 2 x þ h2 y Þ;
C2 22 ¼ z yy z y ð1 þ z 2 x þ z2 y Þ:
I note that the Laplace – Beltrami operator encodes the intrinsic geometry, i. e., that part of it not depending on the surface parameterization, so any other parametrization( e. g., polar parametrization) instead of the Cartesian one would provide the same geometrical structure. Also, it is worth mentioning that when the local curvature of the surface is quasi-flat( z x, z y 1), the Laplace – Beltrami operator is approximately equal to the Laplacian.
In discretized surfaces, the discrete Laplace operator replaces the continuous one. The surface is described by a triangular mesh M, described by a set of geometric vertices coordinates, a matrix V =( x 1,..., x n, y 1,..., y n, z 1,..., z n), and an ordering collection of those vertices through edges. Then, the discrete Laplace – Beltrami operator( D M) ofascalar function f at a vertex i can be estimated with the cotangent formula [ 7 ]:
M ½f Š i ¼ 1 X ðcot a ij þ cot b 2A ij Þðf j � f i Þ; i j ð2Þ
: