Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

By Mark Meyer, Mathieu Desbrun, Peter Schröder, A. Barr. 2002

2,225 citations recorded by the source corpus.

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

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In Computational Mechanics

In Computer Graphics and Computer-Aided Design

Citations in the corpus

Learning shape correspondence with anisotropic convolutional neural networks (2016, methodology): “Isotropic Laplacians were computed using the cotangent formula [Mac49, Duf59, PP93, MDSB03]; anisotropic Laplacians were computed according to (30).”
Deep Functional Maps: Structured Prediction for Dense Shape Correspondence (2017, methodology): “The Laplace-Beltrami operator ∆ is discretized as a symmetric n× n matrix L = A−1W using a classical linear FEM scheme [30], where Figure 2.”
On Linear Variational Surface Deformation Methods (2008, methodology): “as proposed in [18], [45], and [51] and employed, for”
A novel cubic-order algorithm for approximating principal direction vectors (2004, methodology): “(3) The normalized average of the adjacent edges, each weighted by the sum of the cotangents of the angles opposite the edge in the two triangles sharing the edge [Meyer et al. 2003].”
Mathematical framework for large-scale brain network modeling in The Virtual Brain (2015, positive): “In fact one of the original motivations for further regularizing the surfaces meshes generated by the algorithms implemented in packages such as FreeSurfer was difficulties that were encountered, while attempting to use the commonly used cotangent approximation for the discrete Laplace–Beltrami oper”
Ridge-valley lines on meshes via implicit surface fitting (2004, methodology): “The two main approaches to curvature tensor estimation consist of using (cid:2)nite differences [Meyer et al. 2003; Cohen-Steiner and Morvan 2003] and polynomial (cid:2)tting [Cazals and Pouget 2003; Goldfeather and Interrante 2004].”
Functional maps (2012, methodology): “…the point-to-point maps provided by the methods of Kim et al. [2011] and Sahillioglu and Yemez [2011], and converted them into a functional representation using the basis provided by the first 100 eigenfunctions of the cotangent weight scheme [Meyer et al. 2002] of the LaplaceBeltrami operator.”
Point Cloud Skeletons via Laplacian Based Contraction (2010, positive): “Many alternatives are available for the choice of the weighting scheme, such as combinatorial weights [44], mean value weights [47] or cotangent weights [48].”
Intrinsic shape context descriptors for deformable shapes (2012, methodology): “We used the cotangent weight scheme [30] to compute the ﬁrst 300 eigenvalues and eigenvectors of the Laplace-Beltrami operator on each shape.”
A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching (2010, methodology): “In this paper, we used the popular cotangent weight scheme (Meyer et al. 2003), in which w ij = cot α ij + cot β ij (α ij and β ij are the two angles opposite to the edge between vertices i and j in the two triangles sharing the edge) for j in the 1- ring neighborhood of vertex i and zero…”
Spatial adaption of robot trajectories based on laplacian trajectory editing (2015, background): “Another class are geometric mesh Laplacians, explicitly taking into account the underlying Riemannian geometry (Strichartz 1983; Meyer et al. 2002; Dierkes et al. 2010).”
Geometric triangulations and discrete Laplacians on manifolds (2005, methodology): “Laplace operators on graphs and on piecewise Euclidean manifolds have been studied in many different contexts, for instance [6], [10], [11], [20], [21], [25], [26], [27], [35], [42].”
Interactive localized liquid motion editing (2013, methodology): “For mesh-based control particles, we replace N (Li) by the one-ring neighborhood of the mesh vertex Li, and set wij to the cotan weights used in the usual discrete Laplacian [Meyer et al. 2002].”