Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids

By Martin Reuter, Franz-Erich Wolter, Niklas Peinecke. Comput. Aided Des., 2006

884 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computer Graphics and Computer-Aided Design, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/1d539f77408f3aee8228d873501edf4e8ad74923

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In Computational Mechanics

In Computer Graphics and Computer-Aided Design

In Computer Vision and Pattern Recognition

Citations in the corpus

Globally Approximate Gaussian Processes for Big Data With Application to Data-Driven Metamaterials Design (2019, methodology): “For a real-valued function f de ﬁ ned on a Riemannian manifold [61], the Helmholtz equation reads as follows: where the Laplace – Beltrami operator Δ is de ﬁ ned as follows: The eigenvalues of the Helmholtz equation are the LB spectrum and denoted as follows: Our unit cells reside in a 2D planar…”
A spectral notion of Gromov–Wasserstein distance and related methods (2011, methodology): “in the remarkable [7], where, in the context of shape retrieval and comparison, the authors propose to use a subset of the collection of all eigenvalues (spectrum) of the Laplace–Beltrami operator of a shape as its signature.”
One Point Isometric Matching with the Heat Kernel (2010, neutral): “…distortion of the interpolated dense maps for one ﬁxed landmark correspondence using the Laplace-Beltrami operator discretization of Reuter et al. [RWP06] called Shape-DNA (using linear FEM), the method of Belkin et al. [BSW08] called MeshLP and the cotangent weight scheme of Meyer et al.…”
A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching (2010, methodology): “For example, Reuter et al. (2006), Ruggeri et al. (2009) consider the spectra (eigenvalues) of the Laplace-Beltrami operator and use this as bending invariant signature for shape recognition.”
Technical Section: Discrete Laplace-Beltrami operators for shape analysis and segmentation (2009, methodology): “, tetrahedra or voxel meshes) and higher order computations by choosing a set F:1⁄4fjigi of higher order form functions while introducing new nodes within each triangle; for more details on the cubic finite elements, we refer the reader to [49,46].”
Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry (2006, methodology): “polynomials), and use the divergence formula to transform the equation into a generalized eigenvalue problem, as done in [24].”
Learning Spectral Descriptors for Deformable Shape Correspondence (2014, methodology): “Based on this belief, Reuter et al. [27] proposed to use truncated sequences of the Laplacian eigenvalues as isometryinvariant shape descriptors, dubbed by the authors as shape DNA.”
Learning class‐specific descriptors for deformable shapes using localized spectral convolutional networks (2015, methodology): “…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 kernel signatures (WKS) [ASC11], and heat kernel maps…”
Efficient Globally Optimal 2D-to-3D Deformable Shape Matching (2016, methodology): “As baselines for our comparisons we use the spectral retrieval method of [20] and a pure region-based retrieval technique using segments computed with [24].”
Partial Similarity of Objects, or How to Compare a Centaur to a Horse (2009, background): “…shapes is an important direction emerging in the last decade in the pattern recognition community and arising in applications of face recognition (Bronstein et al. 2003, 2005), shape watermarking (Reuter et al. 2006), texture mapping and morphing (Bronstein et al. 2006b, 2007a), to mention a few.”