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Laplace–Beltrami Operator on Digital Surfaces Thomas Caissard, David Coeurjolly, Jacques-Olivier Lachaud, Tristan Roussillon

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Laplace–Beltrami Operator on Digital Surfaces Thomas Caissard, David Coeurjolly, Jacques-Olivier Lachaud, Tristan Roussillon Laplace–Beltrami Operator on Digital Surfaces Thomas Caissard, David Coeurjolly, Jacques-Olivier Lachaud, Tristan Roussillon To cite this version: Thomas Caissard, David Coeurjolly, Jacques-Olivier Lachaud, Tristan Roussillon. Laplace–Beltrami Operator on Digital Surfaces. Journal of Mathematical Imaging and Vision, Springer Verlag, 2019, 61 (3), pp.359-379. 10.1007/s10851-018-0839-4. hal-01717849v3 HAL Id: hal-01717849 https://hal.archives-ouvertes.fr/hal-01717849v3 Submitted on 28 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) Laplace–Beltrami Operator on Digital Surfaces Thomas Caissard · David Coeurjolly · Jacques-Olivier Lachaud · Tristan Roussillon Received: date / Accepted: date Abstract This article presents a novel discretization of the Laplace–Beltrami operator on digital surfaces.We adapt an existing convolution technique proposed by Belkin et al. [5] for triangular meshes to topological border of subsets of n Z . The core of the method relies on first-order estimation of measures associated with our discrete elements (such as length, area etc.). We show strong consistency (i.e. pointwise convergence) of the operator and compare it against various other discretizations. Keywords Laplace–Beltrami Operator · Digital Surface · Discrete Geometry · Differential Geometry Fig. 1 A digital surface of dimension two embedded in R3. 1 Introduction Computer graphics, and particularly the field of geometry It is used for example as a basis for Functional Maps [48] or processing, revolves around studying discrete embedded mesh compression [42]. Other applications are for example surfaces (in many cases 2D surfaces in 3D). The surface fairing, mesh smoothing, remeshing or feature Laplace–Beltrami operator (the Laplacian on a manifold) is extractions (see [42]). The operator is also related to a fundamental tool in geometry as it holds many properties diffusion and the heat equation on a surface and connected of the surface. Eigenfunctions of the operator form a natural to a large field of classical mathematics linking geometry of basis for square integrable functions on the manifold, in the manifold to properties of the heat flow (see for example same manner as Fourrier harmonics for functions on a circle. [56]). This work has been partly funded by C O M E D I C ANR-15-CE40- Many characterizations of discrete surfaces exist such 0006 research grant. We would like to thank the anonymous reviewers as triangular, quadrangular meshes (or more generally sim- for their detailed comments and suggestions for the manuscript. plicial complexes), points clouds, etc. Our model of surface Thomas Caissard · David Coeurjolly · Tristan Roussillon comes from the Digital Geometry theory [35], where the Univ de Lyon, CNRS, INSA-Lyon, LIRIS, UMR 5205, F-69621, France discrete structure is the topological boundary of a subset of d+ E-mail: [email protected] points in Z 1 called a digital surface (an example of this ob- E-mail: [email protected] E-mail: [email protected] ject is pictured in Fig. 1). Such surfaces can be constructed from mathematical modeling or from boundaries of parti- Jacques-Olivier Lachaud tions in volumetric images. Indeed, digital objects naturally Laboratoire de Mathematiques´ (LAMA), UMR 5127 CNRS, Universite´ arise in many material sciences or medical imaging applica- Savoie Mont Blanc, Chambery,´ France tions as tomographic volumetric acquisition devices usually E-mail: [email protected] generate regularly spaced data (e.g [29,22]). 2 Thomas Caissard et al. Our goal here is to present a discretization of the Laplace– A more versatile expression of discrete exterior calcu- Beltrami operator on digital surfaces which satisfies strong lus comes with Hirani’s thesis [33] and the monograph of consistency (i.e. pointwise convergence) with respect to the Desbrun, Hirani, Leok and Marsden [16]. Their primal-dual Laplace–Beltrami operator on the underlying manifold when construction does not impose the use of triangular meshes. the digital surface is the boundary of the digitization of a con- The discretization is not an approximation of the smooth tinuous object. As we demonstrate in our experiments, pre- calculus, but rather a discrete analog: vious works fail to efficiently estimate the Laplace–Beltrami operator on these specific surfaces. The main obstacle is the We do not prove that these definitions converge to the smooth counterparts. The definitions are chosen so as to make some fact that normal vectors to these surfaces do not converge to important theorems like the generalized Stokes’ theorem true the normal vectors of the underlying manifold, whatever the by definition, to preserve naturality with respect to pullbacks, sampling rate. and to ensure that operators are local. We adapt the operator of Belkin et al. [5] to our spe- [33,16] cific data. The method uses an accurate estimation of areas Metrics play a role in musical operators (flat and sharp associated with digital surface elements. This estimation is which convert vector field to k-forms and conversely) and achieved through a multigrid convergent digital normal esti- Hodge stars. Note that discrete exterior calculus coincides mator of Coeurjolly et al. [11]. This paper is a direct follow- with the cotangent scheme on triangular meshes when the up on [6] where we experimentally investigate applications Voronoi dual is used. such as heat diffusion or shape approximation through the In parallel, another discrete calculus emerges in the im- eigenvectors decomposition. We show strong consistency of age, graph, electric circuits and network analysis communi- the discrete operator, and compare it experimentally with var- ties, summed up in Grady and Polimeni’s book [24]. Met- ious other discretizations adapted on digital surfaces. rics are also incorporated, although without the relation with the ambient space. This feature was desired since people frequently wish to analyze data without any knowledge of Discretization schemes overview. The Laplacian being a sec- an embedding. Authors then show how classical filtering ond order differential operator, a discrete calculus framework procedures and (discrete versions of) energy models (e.g. is required to define this operator on embedded combinato- Mumford-Shah, Total Variation) fit well within this frame- rial structures such as meshes or digital surfaces. The first work. elements of discrete calculus may be traced back to Regge A much-alike discrete calculus on “chainlets” appears in calculus [55] for quantum physics, where discrete domains geometric measure theory, for the mathematical analysis of are modeled with adjacent tetrahedra and metrics are only general compact shapes like fractals [25,26]. The exterior determined by edge lengths. The discrete Laplacian has also derivative, a Hodge star and Laplace–Beltrami are defined been present in spectral analysis of graphs since the 1950s. there for very general spaces. However computational as- Then, with the development of geometric acquisition devices pects are unclear. We can also mention a complex analysis and modeling techniques, interest grew toward a calculus approach to discrete calculus for 2D digital surfaces [44,45] working on meshes and more generally simplicial complexes. with applications to digital surface parametrization and tex- Early works include the widely studied cotangent formula ture mapping [8]. [58,59,17,50,23,43,46,49] for various applications, which Operators for point clouds can be found in [4] and more may be derived directly from standard finite element method recently in [54]. A discretization on polygonal surfaces was (e.g. see [42]). proposed by Alexa and Wardetzky [2]. As digital surfaces Discrete exterior calculus was then developed in the com- being specific quadrangulated polygonal surface, such ap- putational mathematics and geometry processing community, proach perfectly fits with our data. However, we show in with a particular focus on triangulated meshes. The “German the experiments such polygonal Laplace–Beltrami operator school” of discrete calculus developed an exact 2D calcu- gives inconsistent results. Indeed, digital surface quads are lus which generalizes the cotangent Laplacian, and is based axis-aligned quads and do not capture the metric of the un- on (conforming and non-conforming) finite elements [51], derlying continuous object properly. Note that Alexa and thus obtaining expected theoretical results such as Stokes’ Wardetzky polygonal operator matches with the cotangent theorem and Hodge decomposition. Its applications range one on triangular meshes. Other operators on polyhedral sur- widely: exact integration allows accurate remeshing via L2 faces can be found in [31,64,32] (see discussion below). projection, shape morphing by prescribing first-order data on the surface, etc. This theory provides a sound base for Convergence and consistency of the operator. We clarify no- actual computation, with one important limitation: the neces- tions of convergences for operators. Suppose that you want to sity to only use triangles (and, furthermore, triangles with solve the equation Au = f where A : X ! Y is a bounded lin- good aspect ratios, for positive Laplacian). ear operator between two Banach spaces and f 2 Y is given. Laplace–Beltrami Operator on Digital Surfaces 3 Suppose also that you have an approximate Ae of A and fe This setting is called the weak consistency and is related to of f and that ue is the solution of Ae ue = fe .(e.g. we can the weak form of the Laplace–Beltrami operator (see Wardet- consider e as the grid step for example). We say that the zky’s thesis for a proper definition [64]).
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