Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunctions

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14327

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In this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using Laplace-Beltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first non-trivial eigenfunctions of the Laplace-Beltrami Operator. Our approach requires a topological condition on those eigenfunctions, whose nodal domains must be 2. We show the efficiency of the approach through numerical experiments performed on cortical surface meshes.

Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunctions

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In this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using Laplace-Beltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first non-trivial eigenfunctions of the Laplace-Beltrami Operator. Our approach requires a topological condition on those eigenfunctions, whose nodal domains must be 2. We show the efficiency of the approach through numerical experiments performed on cortical surface meshes.

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Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunctions Julien Lefèvre, Guillaume Auzias LSIS, CNRS UMR 7296 Institut des Neurosciences de la Timone, CNRS UMR 7289 Aix-Marseille Université (AMU) 2nd Conference on Geometric Science of Information October 30, 2015 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 1 / 23 1 Context: the shape of the brain 2 Background on harmonic analysis for manifold 3 Nodal lines of eigenfunctions 4 Small Steps toward a proof 5 Results Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 2 / 23 1 - Context: the shape of the brain - The brain is a highly folded shape. - Each brain hemisphere can be represented as a (genus-0) surface. - Several methods to find a (spherical) parameterization of surfaces =⇒ comparison of shapes (curvature) or functional activities. Gu et al, IEEE TMI, 2004 Auzias et al, IEEE TMI, 2013 Lombaert et al, IPMI, 2013 Gu et al, Jour. Scient. Comp., 2014 - Oscillating structure of cortical folding pattern =⇒ Fourier-like analysis = eigenfunctions of Laplace-Beltrami operator. Germanaud, Lefèvre et al, Neuroimage, 2012 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 3 / 23 1 Context: the shape of the brain 2 Background on harmonic analysis for manifold 3 Nodal lines of eigenfunctions 4 Small Steps toward a proof 5 Results Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 4 / 23 2 - Background on harmonic analysis for manifold Generalization of Fourier analysis Fourier modes on [0, 1] are solutions of f = −λf with boundary conditions (Dirichlet, Neumann...). Definition On a 2-Riemannian manifold (surface) M, with a local coordinate system x : p ∈ M → R2 the Laplace-Beltrami operator is defined as: ∆Mf (x) = 1 det g n i=1 ∂i det g n j=1 gi,j ∂j f (x) (1) where g is the (local) metric tensor. Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 5 / 23 2 - Background on harmonic analysis for manifold Spectral properties On L2(M) = {u : M → R / M u2 < +∞} equiped with < u, v >= M uv, −∆u is a semi-definite positive operator. There exists λ0 = 0 ≤ λ1 ≤ ... and φ0, φ1, ... an orthonormal basis of L2(M) satisfying: −∆φi = λi φi . (1) M. Berger, A Panoramic View of Riemannian Geometry, 2002 φ1 φ100 φ1000 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 5 / 23 2 - Background on harmonic analysis for manifold Where is the information ? Spectrum λ0 ≤ λ1 ≤ ... has been used to discriminate shapes Niethammer et al. MICCAI, 2007 Wachtinger et al. MICCAI, 2014 But it cannot always provide adapted descriptors. See the famous question by Kac, "Can we hear the shape of a drum ?", with the counter example: Gordon, Webb, Wolpert, Bull. Amer. Math. Soc., 1992 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 5 / 23 1 Context: the shape of the brain 2 Background on harmonic analysis for manifold 3 Nodal lines of eigenfunctions 4 Small Steps toward a proof 5 Results Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 6 / 23 3 - Nodal lines of eigenfunctions There is also a spatial information. Chladni’s plates. Spherical harmonics Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 7 / 23 3 - Nodal lines of eigenfunctions Definition Given an eigenfunction Φ of the Laplace-Beltrami Operator, we call nodal set the set of points N(Φ) where Φ vanishes. The nodal domains correspond to the connected components of the complementary of the nodal set. Theorem (Courant’s nodal domain theorem) The number of nodal domains for the n-th eigenfunction is inferior or equal to n + 1 (Neuman boundary conditions). Theorem (S.H. Cheng 1976) Except on a closed set of points, the nodal set of an eigenfunction Φ is a C∞ submanifold, i.e. a line in our applications. Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 8 / 23 3 - Nodal lines of eigenfunctions Three first non-trivial eigenfunctions and corresponding nodal sets. It looks like first spherical harmonics, 3 great circles are visible. Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 9 / 23 3 - Nodal lines of eigenfunctions M −→ R3 −→ S2 p −→ Φ1(p), Φ2(p), Φ3(p) −→ Φ1(p),Φ2(p),Φ3(p) √ Φ1(p)2+Φ2(p)2+Φ3(p)2 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 10 / 23 3 - Nodal lines of eigenfunctions A natural conjecture Let M be a genus zero surface in R3. Let Φ1, Φ2 and Φ3 be three orthogonal eigenfunctions of the Laplace-Beltrami operator. We assume they have only two nodal domains. Then the mapping Φ : M −→ S2 p −→ Φ1(p)2 + Φ2(p)2 + Φ3(p)2 −1 Φ1(p), Φ2(p), Φ3(p) is well defined and it is a C∞ diffeomorphism. A similar embedding was proposed by Pierre Bérard in SN for N sufficiently large. Bérard. Séminaire de théorie spectrale et géométrie, 1984 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 11 / 23 3 - Nodal lines of eigenfunctions Important remark The number of nodal domains must be 2. For elongated shapes, the bounds in Courant’s nodal theorem are reached. Φ2 Φ3 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 12 / 23 1 Context: the shape of the brain 2 Background on harmonic analysis for manifold 3 Nodal lines of eigenfunctions 4 Small Steps toward a proof 5 Results Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 13 / 23 4 - Small Steps toward a proof Our initial strategy was: to prove first that intersection points of two nodal sets exist, thanks to global arguments. to characterize those intersections in terms of the angle between the two isolines (= the gradient of the eigenfunctions), by using local results on eigenfunctions. Results known for auto-intersection of nodal sets (equiangular system). Cheng Comment. Math. Helvetic., 1976 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 14 / 23 4 - Small Steps toward a proof Proposition Let M be a genus zero surface in R3. We consider two eigenfunctions Φ and Ψ with only two nodal domains and different associated eigenvalues. Then their nodal sets have at least one intersection point. Proof by contradiction: D1 = {p|Φ(p) > 0, Ψ(p) > 0} D2 = {p|Φ(p) > 0, Ψ(p) < 0} D3 = {p|Φ(p) < 0, Ψ(p) < 0} By using Green’s formula and properties of eigenvalues on λ D2 ΦΨ and λ D2 ΦΨ, one obtains D2 ΦΨ = 0, a contradiction. Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 15 / 23 4 - Small Steps toward a proof An interesting flaw Injectivity of Φ by using properties of F : p → Φ1(p), Φ2(p), Φ3(p) . Link between Laplacian of coordinates and mean curvature: ∆F(p) = 2H(p)N(p) combined with ∆F = (−λ1Φ1, −λ2Φ2, −λ3Φ3) But first formula holds if ∆ is the Laplace-Beltrami operator of F(M) ! Second conjecture F(M) is a genus-zero surface whose mean curvature has a constant sign. Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 16 / 23 1 Context: the shape of the brain 2 Background on harmonic analysis for manifold 3 Nodal lines of eigenfunctions 4 Small Steps toward a proof 5 Results Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 17 / 23 5 - Results Data 138 meshes of cortical surfaces (OASIS database): 106 to 167 kvertices. Fast Matlab implementation of Finite Element Methods : 3.76s to 6.71s. Nodal domains Experimentally the three first eigenfunctions have always 2 nodal domains Diffeomorphic property Flipped triangle faces: 0.008% to 7.01% (average: 0.29 ± 0.7%) Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 18 / 23 5 - Results Reproducibility 6 intersection points, their representation in 3D (Talairach space) and the angles between nodal lines (erratum: π/2 − angle). Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 19 / 23 5 - Results Distorsions Angular error and length error computed on each mesh triangle before and after the mapping. Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 20 / 23 6 - Conclusion Topological property on nodal domains =⇒ spherical parameterization. Fast method suitable to initialize a spherical mapping. Extension for manifolds of higher dimensions: Upper bound for embedding dimension with L.B.O eigenfunctions Bates. Appl. Comput. Harmon. Anal., 2014 Can one find n manifolds such as : - n + 1 first eigenfunctions have 2 nodal domains - it defines a canonical diffeomorphism toward the sphere Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 21 / 23 7 - Acknowledgments Thank you ! Guillaume Auzias ANR JCJC "Modegy" Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 22 / 23 Julien Lefèvre (LSIS, AMU) Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunOctober 30, 2015 23 / 23