J. Eur. Opt. Society-Rapid Publ. 2026, 22, 25 Ó The Author( s), published by EDP Sciences, 2026 https:// doi. org / 10.1051 / jeos / 2026014 Available online at: https:// jeos. edpsciences. org
Journal of the European Optical Society-Rapid Publications RESEARCH ARTICLE
Laplace – Beltrami shape analysis of ocular or wavefront surfaces
Sergio Barbero * Instituto de Óptica( CSIC), Serrano 121, Madrid, Spain
Received 2 December 2025 / Accepted 8 February 2026
Abstract. Shape analysis techniques are widely used across biomedical fields for processing surface shape properties. The Laplace – Beltrami operator, as an extension of the Laplacian to surfaces, reveals geometrical properties related to surface curvature through its spectral decomposition, making it a promising tool in visual optics. I introduce Laplace – Beltrami shape analysis for ocular, or wavefront, surfaces of the human eye, described as meshes. I present a set of techniques and their numerical implementation for this purpose. Two particular tools, spectral decomposition and shape-DNA analysis, are explained in detail and illustrated with an example of application: a keratoconus cornea model.
Keywords: Laplace – Beltrami operator, Shape analysis, Ocular surfaces, Surface curvature.
1 Introduction
The progress in high-resolution and efficient optical metrology technologies – such as optical coherence tomography or intensity-based wavefront sensing [ 1 – 3 ] – applied to measuring either ocular interfaces or, indirectly, the wavefront surface has permeated ophthalmology practice. The highly dense sampling of these surfaces consequently leads, at least for some applications, to local surface representations such as splines [ 4, 5 ] or mesh surfaces [ 6 ], i. e., those constructed by a set of polygonal faces, typically triangles. Furthermore, these high-spatial-sampled surfaces encourage the application of spectral analysis to them. For instance, classical Fourier techniques have been proposed for analysing keratoconus eyes – i. e. an eye condition characterized by a thinning and bulging of the cornea degrading vision – through wavefront phase imaging [ 3 ]. Besides Fourier analysis, Zernike decomposition of optical surfaces is undoubtedly the most typical approach to analyze, or characterize, optical wavefront or surfaces. However, both Zernike and Fourier coefficients do not fully capture the shape information, especially the effect of curvature on the spectrum itself [ 7 ]. Indeed, this intrinsic limitation of conventional Zernike polynomials led to the proposal of some alternatives, such as the so-called Zernike curvature polynomials [ 8 ]. However, these are only valid when the surface gradient is small, so the wavefront curvature reduces to its Laplacian; an approximation that is not sustained in many practical cases.
By analogy to classical Fourier analysis, spectral methods over surfaces defined as meshes were introduced at the
* Corresponding author: sergio. barbero @ csic. es end of the last century for discrete geometry processing [ 7 ]. The goal of these methods is to capture the geometrical shape properties of the surface, which is referred to as shape analysis. Mesh surfaces are especially suitable for shape analysis thanks to the fact that discrete differential geometry techniques can be directly applied to them [ 9 ]. A pioneering work in spectral methods over meshes was that of Taubin [ 10 ], which defined the mesh vertex coordinates as a 3D signal and applied Laplacian operators. Another option is to define a scalar function over the mesh surface.
Surface spectral methods comprise three steps: 1) selecting an appropriate linear operator; 2) the eigendecomposition of that operator; and 3) employing the spectral eigenvalues and eigenvectors in a specific manner according to the processing purpose. In other words, spectral methods inspect a surface by examining the eigenvalue decomposition of a meaningful selected linear operator. Therefore, the application at hand determines the selection of the linear operator and how to employ the spectrum.
Surface spectral methods have been used intensively in a diversity of techniques [ 11 ], such as mesh compression [ 12, 13 ], segmentation [ 14 ], smoothing or denoising through filtering [ 15 ], symmetry detection [ 16, 17 ], surface reconstruction from point clouds [ 18 ], or watermarking [ 19 ], all of them relevant in image processing applications.
As mentioned before, the first and most crucial step in spectral methods is the selection of an appropriate linear operator. In optics, the curvature – in its various mathematical forms – is the most relevant geometrical property because it linearly depends on the optical power, the fundamental property in all imaging systems. In any form, the curvature involves second-order derivatives, so a mandatory choice for such an operator is that it should include
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