Meshless parameterization and surface reconstruction

Floater, Michael S.; Reimers, Martin

doi:10.1016/s0167-8396(01)00013-9

Cited by 136 publications

(75 citation statements)

References 6 publications

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“…Gortler et al, extending results of Gu and Yau (2002), proved that two independent solutions can be used as the x-and y-coordinates of an embedding onto a torus. Floater and Reimers (2001) observed that Tutte's method can also be used to reconstruct surfaces with boundary of genus zero and Tewari et al (2006) extended the observation to closed surfaces of genus one, as follows. Construct the k-nearest neighbor graph G k of P and then set up the equations introduced above.…”

Section: Application To Surface Reconstructionmentioning

confidence: 99%

Cycle bases of graphs and sampled manifolds

Gotsman

Kaligosi

Mehlhorn

et al. 2007

Computer Aided Geometric Design

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Section: Application To Surface Reconstructionmentioning

confidence: 99%

Cycle bases of graphs and sampled manifolds

Gotsman

Kaligosi

Mehlhorn

et al. 2007

Computer Aided Geometric Design

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“…Thus meshless parameterization can be used for surface reconstruction. The method, described in [13], begins by first dividing the set X N into two disjoint subsets: X I , the set of interior points, and X B , the set of boundary points. Moreover the boundary points must be ordered.…”

Section: Parameterization Of Unorganized Pointsmentioning

confidence: 99%

“…Moreover the boundary points must be ordered. A method was outlined in [13] for first identifying boundary points and subsequently ordering them using a univariate analog of meshless parameterization. After that the method is identical to the convex combination mapping method of Section 3, the only difference being that we now have to find suitable neighbourhoods N xi for the points x i ∈ X I .…”

Section: Parameterization Of Unorganized Pointsmentioning

confidence: 99%

“…After that the method is identical to the convex combination mapping method of Section 3, the only difference being that we now have to find suitable neighbourhoods N xi for the points x i ∈ X I . Several choices of such neighbourhoods were proposed in [13] but a simple and effective choice is to take the 'd nearest neighbours', in other words, we let N x i be the set of the d points x j closest to x i , illustrated in Figure 19. Going on several numerical tests, it appears that setting d = 10 or d = 20 is adequate for all but the most extreme data sets.…”

Section: Parameterization Of Unorganized Pointsmentioning

confidence: 99%

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Parameterization of Triangulations and Unorganized Points

Floater

Hormann

2002

Mathematics and Visualization

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Abstract.A parameterization of a surface is a one-to-one mapping from a parameter domain to the surface. This paper studies the construction of piecewise linear parameterizations of triangulations and of discrete parameterizations of sets of unorganized points. Such parameterizations are useful tools for the approximation of triangulations by smoother surfaces as well as for the construction of triangulations from unorganized points. Parameterization can also be used as the starting point for multiresolution analysis of triangulations, through remeshing by triangulations with subdivision connectivity.

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“…al introduced the dual contour method for reconstruction [8]. Floater and Reimers reconstructed surfaces based on parameterizations [9]. Surface reconstruction has been applied to reverse engineering [10], geometric modelling [11], mesh optimization and simplification [12] and many other important applications.…”

Section: Introductionmentioning

confidence: 99%

Geometric Accuracy Analysis for Discrete Surface Approximation

Dai¹,

Luo²,

Yau

et al. 2006

Geometric Modeling and Processing - GMP 2006

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In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surfaces. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density. This paper gives explicit formulae to the bounds of Hausdorff distance, normal distance and Riemannian metric distortion between the smooth surface and the discrete mesh in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formulae can be directly applied to design sampling density for data acquisitions and surface reconstructions. Furthermore, we prove that the meshes induced from the Delaunay triangulations of the dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields. The Riemannian metrics and the Laplace-Beltrami operators on the meshes are also convergent to those on the smooth surfaces. These theoretical results lay down the foundation for a broad class of reconstruction and approximation algorithms in geometric modeling and processing.Practical algorithms for approximating surface Delaunay triangulations are introduced based on global conformal surface parameterizations and planar Delaunay triangulations. Thorough experiments are conducted to support the theoretical results.

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Meshless parameterization and surface reconstruction

Cited by 136 publications

References 6 publications

Cycle bases of graphs and sampled manifolds

Cycle bases of graphs and sampled manifolds

Parameterization of Triangulations and Unorganized Points

Geometric Accuracy Analysis for Discrete Surface Approximation

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