Mesh Reduction with error control

Scott BuffaJeranfer Bermudez

Alex PeerCSE 872

Implementation of (Klein, 2001)

Goals Mesh simplification

› Hausdorff distance› Constrained 3D Delaunay triangulation

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.

Hausdorff distance Measures how far two subsets of space

are from each other› ›

› To get the maximum of the minimum distances betweentwo sets

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

Delaunay Triangulation cont.

Given three points:

› Radius:

› Circumsphere center : Where:

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

The Algorithm Overview

› Remove vertices and re-triangulate the resulting holes

› Stops when no vertices can be removed without exceeding the Hausdorff distance.

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.

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

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

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

Demo

Results Simplified Mesh Original Mesh

Results cont.

Results cont.

Results cont.

Questions?