Robust fairing via conformal curvature flow

Crane, Keenan; Pinkall, Ulrich; Schröder, Peter

doi:10.1145/2461912.2461986

Cited by 114 publications

(114 citation statements)

References 25 publications

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“…However, we found that self‐intersections are often removed by cMCF early on, even for high genus shapes (see Figure 13). This is in contrast to the conformal Willmore flow [CPS13]. Our experiments show that this method removes all intersections at a much later state (see Figure 14).…”

Section: Experiments and Resultsmentioning

confidence: 76%

Consistent Volumetric Discretizations Inside Self‐Intersecting Surfaces

Sacht

Jacobson

Panozzo

et al. 2013

Computer Graphics Forum

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Input triangle mesh cMCF flow Intrinsic reverse flow Output tetrahedral mesh matching input Figure 1: The triangle mesh of the Hand forms a closed surface, but contains nearly 2000 intersecting triangle pairs. Our method flows the surface according to conformalized mean-curvature flow (cMCF) until all self-intersections are removed. Then we reverse the flow so that shape intrinsics are restored but self-intersections are avoided. Finally we can tet-mesh inside this surface and map the mesh so that it matches the original surface. We may then solve PDEs, such as this biharmonic function. AbstractDecades of research have culminated in a robust geometry processing pipeline for surfaces. Most steps in this pipeline, like deformation, smoothing, subdivision and decimation, may create self-intersections. Volumetric processing of solid shapes then becomes difficult, because obtaining a correct volumetric discretization is impossible: existing tet-meshing methods require watertight input. We propose an algorithm that produces a tetrahedral mesh that overlaps itself consistently with the self-intersections in the input surface. This enables volumetric processing on self-intersecting models. We leverage conformalized mean-curvature flow, which removes self-intersections, and define an intrinsically similar reverse flow, which prevents them. We tetrahedralize the resulting surface and map the mesh inside the original surface. We demonstrate the effectiveness of our method with applications to automatic skinning weight computation, physically based simulation and geodesic distance computation.

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Section: Experiments and Resultsmentioning

confidence: 76%

Consistent Volumetric Discretizations Inside Self‐Intersecting Surfaces

Sacht

Jacobson

Panozzo

et al. 2013

Computer Graphics Forum

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“…For domains with boundary Bohle & Pinkall [BP13] showed that prescribing binormal boundary conditions preserves the self‐adjointness and ellipticity of the extrinsic Dirac operator (see [CPS13, Section 6.3] for a discretization); a similar argument may be possible for the relative Dirac operator. However, the shape of the boundary can cause trouble in spectral geometry processing particularly in the case of partial matching of surface patches [RCB∗17].…”

Section: Boundary Conditionsmentioning

confidence: 99%

A Dirac Operator for Extrinsic Shape Analysis

Liu

Jacobson

Crane

2017

Computer Graphics Forum

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The eigenfunctions and eigenvalues of the Laplace‐Beltrami operator have proven to be a powerful tool for digital geometry processing, providing a description of geometry that is essentially independent of coordinates or the choice of discretization. However, since Laplace‐Beltrami is purely intrinsic it struggles to capture important phenomena such as extrinsic bending, sharp edges, and fine surface texture. We introduce a new extrinsic differential operator called the relative Dirac operator, leading to a family of operators with a continuous trade‐off between intrinsic and extrinsic features. Previous operators are either fully or partially intrinsic. In contrast, the proposed family spans the entire spectrum: from completely intrinsic (depending only on the metric) to completely extrinsic (depending only on the Gauss map). By adding an infinite potential well to this (or any) operator we can also robustly handle surface patches with irregular boundary. We explore use of these operators for a variety of shape analysis tasks, and study their performance relative to operators previously found in the geometry processing literature.

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“…Geometric surface flows have various applications within computer graphics [Eckstein et al 2007;Pan et al 2012;Crane et al 2013]. We consider multimaterial normal and mean curvature flows.…”

Section: Geometric Flowsmentioning

confidence: 99%

Multimaterial mesh-based surface tracking

Batty

Grinspun

2014

ACM Trans. Graph.

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Figure 1: Two-Droplet Collision: Using our multimaterial mesh-based surface tracker, two immiscible liquid droplets with different materials but identical physical properties impact symmetrically in zero gravity under strong surface tension. The collision merges the droplets so that a new interface separates the two liquids, and a non-manifold triple-curve is created where the two liquids meet the ambient air. AbstractWe present a triangle mesh-based technique for tracking the evolution of three-dimensional multimaterial interfaces undergoing complex deformations. It is the first non-manifold triangle mesh tracking method to simultaneously maintain intersection-free meshes and support the proposed broad set of multimaterial remeshing and topological operations. We represent the interface as a nonmanifold triangle mesh with material labels assigned to each halfface to distinguish volumetric regions. Starting from proposed application-dependent vertex velocities, we deform the mesh, seeking a non-intersecting, watertight solution. This goal necessitates development of various collision-safe, label-aware non-manifold mesh operations: multimaterial mesh improvement; T1 and T2 processes, topological transitions arising in foam dynamics and multiphase flows; and multimaterial merging, in which a new interface is created between colliding materials. We demonstrate the robustness and effectiveness of our approach on a range of scenarios including geometric flows and multiphase fluid animation.

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Robust fairing via conformal curvature flow

Cited by 114 publications

References 25 publications

Consistent Volumetric Discretizations Inside Self‐Intersecting Surfaces

Consistent Volumetric Discretizations Inside Self‐Intersecting Surfaces

A Dirac Operator for Extrinsic Shape Analysis

Multimaterial mesh-based surface tracking

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