Download - Mesh Reduction with error control
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Mesh Reduction with error control
Scott BuffaJeranfer Bermudez
Alex PeerCSE 872
Implementation of (Klein, 2001)
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Goals Mesh simplification
› Hausdorff distance› Constrained 3D Delaunay triangulation
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Other solutions Coplanar facets merging [HH92, MSS94] Mesh decimation: [SZL92] Mesh optimization: [HDD93] Point coalescence: [RB93] Re-tiling: [Tur92] Multiresolution retiling: [EDD 95]
Drawback is no common way to measure the error between original and simplified meshes.
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Hausdorff distance Measures how far two subsets of space
are from each other› ›
› To get the maximum of the minimum distances betweentwo sets
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Delaunay Triangulation A triangulation such that no point is
inside a circumcircle of any triangle.› Maximizes the minimum angle of all angles
of triangles
Invented by Boris Delaunay, 1934
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Delaunay Triangulation cont.
Given three points:
› Radius:
› Circumsphere center : Where:
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Delaunay Triangulation cont.
Example:
› Points:
› Radius r = 3.605
› Center PC: = .5 = 0 = .5
017
,011
,051
321 PPPP1
P2 P3
y
x
Pc
017
5.011
0051
5.034
CP
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The Algorithm Overview
› Remove vertices and re-triangulate the resulting holes
› Stops when no vertices can be removed without exceeding the Hausdorff distance.
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The Algorithm cont. Iterative approach Determining the vertex to remove
› Hausdorff distance› Distance function› triangulation
Remove vertex› If we remove a vertex v from the
triangulation, its adjacent triangles are removed and the remaining hole is re-triangulated.
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The Algorithm cont. Hausdorff Distance
› For every vertex, simulate removal› Rank the vertices by the error that would
be introduced (hausdorff)› Remove vertex that is at top of list› Update error for vertices› Repeat removal and update until none can
be removed without exceeding maximum error tolerance
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The Algorithm cont. Delaunay Triangulation
› Find how vertex relates to triangle› If new vertex lies within triangle, split
triangle. Otherwise, connect to nearest points
› Use spheres to generate new faces
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What didn’t go well Constrained 3D Delaunay Triangulation
for face generation› Why did’t go well?
No examples. Poor documentation in 3D Needed ordered vertex list Overlapping faces
› Resorted to simple fan algorithm Generating list of vertices with proper
winding order
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Demo
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Results Simplified Mesh Original Mesh
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Results cont.
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Results cont.
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Results cont.
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Questions?