Journal ArticleDOI
Bounded distortion mapping spaces for triangular meshes
Yaron Lipman
- Vol. 31, Iss: 4, pp 108
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TLDR
It is shown how common geometric processing objective functionals can be restricted to these new spaces, rather than to the entire space of piecewise linear mappings, to provide a bounded distortion version of popular algorithms.Abstract:
The problem of mapping triangular meshes into the plane is fundamental in geometric modeling, where planar deformations and surface parameterizations are two prominent examples. Current methods for triangular mesh mappings cannot, in general, control the worst case distortion of all triangles nor guarantee injectivity.This paper introduces a constructive definition of generic convex spaces of piecewise linear mappings with guarantees on the maximal conformal distortion, as-well as local and global injectivity of their maps. It is shown how common geometric processing objective functionals can be restricted to these new spaces, rather than to the entire space of piecewise linear mappings, to provide a bounded distortion version of popular algorithms.read more
Citations
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Journal ArticleDOI
Quad-Mesh Generation and Processing: A Survey
David Bommes,Bruno Levy,Nico Pietroni,Enrico Puppo,Cláudio T. Silva,Marco Tarini,Denis Zorin +6 more
TL;DR: The advantages and problems of techniques operating on quadrilateral meshes are discussed, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrisation and remeshing.
Journal ArticleDOI
Integer-grid maps for reliable quad meshing
TL;DR: A novel convex Mixed-Integer Quadratic Programming (MIQP) formulation which ensures by construction that the resulting map is within the class of so called Integer-Grid Maps that are guaranteed to imply a quad mesh.
Journal ArticleDOI
Bijective parameterization with free boundaries
Jason Smith,Scott Schaefer +1 more
TL;DR: This work presents a fully automatic method for generating guaranteed bijective surface parameterizations from triangulated 3D surfaces partitioned into charts by using a distortion metric that prevents local folds of triangles in the parameterization and a barrier function that prevents intersection of the chart boundaries.
Journal ArticleDOI
Locally injective mappings
TL;DR: An algorithm is presented that modifies any deformation energy to guarantee a locally injective mapping, i.e., without inverted elements, and it is demonstrated that enforcing local injectivity increases fidelity of the results in applications such as shape deformation and parameterization.
Journal ArticleDOI
Robust field-aligned global parametrization
TL;DR: A robust method for computing locally bijective global parametrizations aligned with a given cross- field using robust cross-field integral line tracing and demonstrates that the algorithm succeeds on a test data set of over a hundred meshes.
References
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Book
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Journal ArticleDOI
Principal warps: thin-plate splines and the decomposition of deformations
TL;DR: The decomposition of deformations by principal warps is demonstrated and the method is extended to deal with curving edges between landmarks to aid the extraction of features for analysis, comparison, and diagnosis of biological and medical images.
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Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator
TL;DR: Triangle as discussed by the authors is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunayer refinement algorithm for quality mesh generation, and it is shown that the problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is impossible for some PSLGs.
Proceedings ArticleDOI
Multiresolution analysis of arbitrary meshes
TL;DR: A method for overcoming the subdivision connectivity restriction, meaning that completely arbitrary meshes can now be converted to multiresolution form, is presented, based on the approximation of an arbitrary initial mesh M by a mesh MJ that has subdivision connectivity and is guaranteed to be within a specified tolerance.
Journal ArticleDOI
Computing Discrete Minimal Surfaces and Their Conjugates
Ulrich Pinkall,Konrad Polthier +1 more
TL;DR: A new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3 is presented and an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map is presented.