Eurographics Symposium on Geometry Processing (2007) Discrete Laplace Operators: No Free Lunch

By A. Belyaev, Michael Garland, Max Wardetzky, Saurabh Mathur, Felix Kälberer, Eitan Grinspun

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Geometric Deep Learning: Going beyond Euclidean data (2016, background): “This simplest discretization, however, does not capture correctly the geometry of the underlying continuous manifold (for example, the graph Laplacian would typically not converge to the continuous Laplacian operator of the manifold with the increase of the sampling density [68]).”
A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching (2010, background): “It can be shown (Wardetzky et al. 2008) that this discretization preserves many important properties of the continuous Laplace-Beltrami operator, such as positive semi-definiteness, symmetry, and locality, and in addition it is numerically consistent, i.e., converges to the continuous…”
Recent Trends, Applications, and Perspectives in 3D Shape Similarity Assessment (2016, methodology): “Unfortunately,Wardet-zky et al. in [WMKG07] showed that for a general mesh, it is theoretically impossible to satisfy all properties of the Laplace-Beltrami operator at the same time, and thus the ideal discretization does not exist.”
New perspectives on polygonal and polyhedral finite element methods (2014, background): “The many generalizations, connections, and applications of barycentric coordinates in computer graphics have been extensive [32, 33, 34, 35, 36, 37, 38].”
Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, background): “An interesting study of possible Laplacians from a axiomatic development can be found in [47].”
Technical Section: Discrete Laplace-Beltrami operators for shape analysis and segmentation (2009, methodology): “Since the Laplace-Beltrami operator is self-adjoint and semipositive definite [51], it admits an orthonormal eigensystem B :={(�i, i)}i, that is a basis of the space of square integrable function, with � i = �i i, �0≤ �1≤ ...,�i≤ �i+1...≤ +∞. For a detailed discussion on the main properties of the L”
$\Gamma$-convergence of graph Ginzburg-Landau functionals (2012, background): “j d j u i p d i (cf. [41]). This falls outside our current framework and hence we will not consider it here.15 Some discussion of the pros and cons of dierent graph Laplacians can be found in e.g. [56, 58]. (2) The Dirichlet energy is given by 1 2 kruk2 E = 1 4 P i;j Im ! ij(u i u j) 2, independent”
Scene Flow from Depth and Color Images (2011, background): “cotangent weights, often used in computer graphics [16] with uniform sampling.”
Coupled quasi‐harmonic bases (2012, background): “Several papers have studied consistent discretizations of the Laplace-Beltrami operator [PP93,MDSB03,WMKG08] The inﬂuential paper of Taubin [Tau95] drew the analogy between the classical signal processing theory and manifold harmonics, showing that standard tools in signal processing such as…”
Scale-invariant heat kernel signatures for non-rigid shape recognition (2010, background): “This discretization preserves many important properties of the continuous Laplace-Beltrami operator, such as positive semi-definiteness, symmetry, and locality, and in addition it is numerically consistent [43].”
Spatial adaption of robot trajectories based on laplacian trajectory editing (2015, background): “The second condition is necessary as combinatorial mesh Laplacians do not take into account the geometry of the graph and thus rely on a regular mesh structure ( for a good approximation (Taubin 1995; Botsch and Kobbelt 2004b; Wardetzky et al. 2007).”