Dual-Primal Mesh Optimization for Polygonized Implicit Surfaces With Sharp Features

[+] Author and Article Information
Yutaka Ohtake

Computer Graphics Group, Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany

Alexander G. Belyaev

Computer Graphics Group, Max–Planck–Institut für Informatik, 66123 Saabrücken, GermanyUniversity of Aizu, Aizu-Wakamatsu 965-8580, Japan

J. Comput. Inf. Sci. Eng 2(4), 277-284 (Mar 26, 2003) (8 pages) doi:10.1115/1.1559153 History: Received September 01, 2002; Revised January 01, 2003; Online March 26, 2003
Copyright © 2002 by ASME
Topics: Optimization
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Left: initial 303 Marching Cubes mesh, 2.9 K triangles. Middle: mesh optimized without adaptive subdivision (2.9 K triangles), see Section 2 for details. Right: mesh optimization with adaptive subdivision was used (20K triangles), see Section 3 and Table 1 for details.
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Top-left: an initial low-resolution Marching Cubes mesh of a model with sharp features. Top-right: mesh is optimized by the method developed in this paper. Bottom: magnified view of a part of the optimized mesh; sharp features are very well reconstructed.
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Left: Schwarz’s polygonal cylinder. Right: Cylinder is approximated by a triangle mesh tangent to the cylinder.
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Left: A curve is approximated by a polyline whose vertices lie on the curve. Right: The same curve is approximated by a polyline tangent to the curve at inner edge vertices.
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Top-left: initial (primal) Marching Cubes mesh. Top-right: optimized dual mesh whose vertices are placed onto the implicit surface. Bottom: optimized primal mesh.
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(a) An initial mesh usually delivers a poor approximation of a given implicit surface, so the dual mesh is considered. (a), (b) The vertices of the dual mesh are projected onto the implicit surface and form an optimized dual mesh. (c) the planes tangent to the implicit surface at the vertices of the modified dual mesh are determined. (d) For each vertex of the primal mesh, its optimal position is found by minimizing the sum of squared distances from the vertex to the tangent planes at the neighboring vertices of the modified dual mesh.
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Left: a cross-section of the subtraction of sphere 152−x2−y2−z2=0 from cube (1−|x/12|=0)∩(1−|y/12|=0)∩(1−|z/12|=0). Right: the normalized vector field at the right corner part of the left image.
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Left: bidirectional searching was not used. Right: bidirectional searching was used.
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Instead of P we need P̃ as the projection of centroid C. Thus bidirectional searching is used.
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Geometric meaning of Edist.
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If τ is not large enough sharp features are not well reconstructed. Choosing too large values of τ leads to numerical instability.
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Left: initial mesh produced by Marching Cubes. Right: uniform mesh produced by three iterations of the double dual resampling process (2), (3) with equal weights (c=0 in (4)).
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Left: equal weights are used. Right: curvature-dependent weights are used.
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Meshes optimized by the method proposed in Section 2 after three preliminary rounds of the vertex resampling procedure. Left: c=0 produces a uniform mesh which is not dense enough to catch sharp features. Right: adaptive remeshing with c=2 leads to a good reconstruction of sharp features.
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Top-left: initial Marching Cubes mesh. Top-right: the optimized mesh. Bottom: influence of the subdivision parameter ε; a magnified view of a spike part of the mesh optimized with ε=10−3 (two left images) and with ε=10−4 (two right images).
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Top: optimized without adaptive subdivision. Bottom: optimized with adaptive subdivision.
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Left: Doraemon polygonized by Marching Cubes, 64 K triangles, 1003 grid was used. Middle: optimized mesh, 75 K triangles. Right: 90 percent-decimated mesh, 7.5 K triangles.
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Left: Initial Marching Cubes mesh (27 K triangles). Middle: dynamic mesh optimization (27 K triangles, stabilization is achieved after about 20 sec.). Right: the Marching Cubes mesh optimized by the proposed method; adaptive subdivision was not used (1 sec.). Sharp features are very well reconstructed.
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Left: 1003 Marching Cubes mesh (40 K triangles). Right: optimized 503 Marching Cubes mesh (34 K triangles).
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Schwarz’s polygonal cylinder from Fig. 3 is optimized with one round of subdivision (left) and two rounds of subdivision (right).
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Optimized mesh for a curvilinear dodecahedron with sharp edges and corners.
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Middle: a spike part generated initially as an implicit surface. Left: a quad mesh produced by DCoHD 17. Right: a triangle mesh generated by the method developed in this paper.




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