Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry

By Bruno Lévy. Shape Modeling International Conference, 2006

513 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computer Graphics and Computer-Aided Design

View Paper: https://www.semanticscholar.org/paper/e14ac3aee3a897f74dfa8f316ae9871bfc86cf1c

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In Computational Mechanics

In Computer Graphics and Computer-Aided Design

Citations in the corpus

Multimodal Manifold Analysis by Simultaneous Diagonalization of Laplacians (2015, unknown): “Other applications heavily relying on the properties of the Laplacian include spectral graph partitioning [5], spectral hashing [6], spectral correspondence , image segmentation [7], and spectral shape analysis [8], [9], [10], [11].”
Learning class‐specific descriptors for deformable shapes using localized spectral convolutional networks (2015, methodology): “Spectral descriptor try to exploit the geometry arising from the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the surface [BBG94, CL06, Lév06]; popular methods include shapeDNA [RWP06], global point signature (GPS) [Rus07], heat kernel signatures (HKS) [SOG09,GBAL09], wave kern”
Laplace-Beltrami eigenfunctions for deformation invariant shape representation (2007, unknown): “Lévy’s main point in [Lév06] is that this basis is the one ; the expansion coefﬁcients provide a canonical parametrization of functions deﬁned on the surface, and all geometry processing should be carried out in this coefﬁcient domain.”
Intrinsic shape context descriptors for deformable shapes (2012, methodology): “Diffusion geometry offers an intuitive interpretation of many shape properties in terms of spatial frequencies and allows to use standard harmonic analysis tools [26].”
Scale-invariant heat kernel signatures for non-rigid shape recognition (2010, methodology): “…and eigenfunctions of the Laplace- Beltrami operator discretized according to 11 are computed by solving the generalized eigendecomposition problem [19] AΦ = ΛLΦ, (12) where Λ is the (k+1)×(k+1) diagonal matrix of the smallest eigenvalues λ 0 , ..., λ k , and Φ is an N × (k + 1) matrix of…”
Mesh parameterization: theory and practice (2007, background): “Several reviews on this topic are available [Gotsman, 2003; Lévy, 2006], and we also give some web references on the following webpage http: //alice.”
ShapeNet: Convolutional Neural Networks on Non-Euclidean Manifolds (2015, background): “The pioneering works of [B´erard et al. 1994; Coifman and Lafon 2006; L´evy 2006; Rustamov 2007] on diffusion and spectral geometry have led to the emergence of intrinsic spectral shape descriptors that are invariant to isometric deformations of the shape and are dense , i.e. can be computed at…”
A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching (2010, background): “By solving the generalized eigendecomposition problem (Lévy 2006) Aφ = λLφ, (15) k smallest eigenvalues λ 0 , . . . , λ k and corresponding eigenfunctions φ 0 , . . . , φ k : ˆ X → R of the discretized Laplace- Beltrami operator are computed.”
Spatial adaption of robot trajectories based on laplacian trajectory editing (2015, background): “The Laplace–Beltrami operator is well-known in the computer graphics community where it is applied to deform (Botsch and Kobbelt 2004a; Sorkine and Cohen-Or 2004); classify (Luxburg 2007; Reuter et al. 2009) and compress (Karni and Gotsman 2000; Levy 2006) triangular surfaces meshes.”
Geodesic Convolutional Neural Networks on Riemannian Manifolds (2015, background): “The works [11] [28] on diffusion and spectral geometry have led to the emergence of intrinsic spectral shape descriptors that are dense and isometry-invariant by construction.”
Reeb graphs for shape analysis and applications (2008, background): “We think that one of the most promising research direction is the use of Laplacian eigenfunctions, that are intrinsic to the shape and encapsulate the idea of orthogonality of shape description [76].”