Sample Shuffling for Quality Hierarchic Surface Meshing

Surface Meshing

Sample Decimation

Surface Reconstruction

Foot data

No decimation

• Medial axis

• Local feature size f(p)

• -sampling

• d(p)/f(p)

Local feature size and samplingAmenta-Bern-Eppstein

Reconstruction

• Functional approach• Tangent plane [HDeDDMS92]

• Natural Neighbors [BC00]

• Voronoi/Delaunay filtering

• Alpha shapes [EM94]• Crust [AB98]

• Cocone [ACDL00]

Cocones

• Compute cocones

• Filter triangles whose duals intersect cocones

• Extract manifold

Space spanned by vectors making angle /8 with horizontal

Approximating density• Need an approximation to Restricted Voronoi on S

• Need an approximation to local feature sizes

Radius and height

• radius r(p): distance from p to pº.

• height h(p): min distance to the poles

• C(p,space spanned by vectors making angles <= with horizontal.

• pº : point at max distance from p

Deletion and Insertion

• Vertex p is deleted if there is nearby sample point

p)/h(p) < '.

• Insert p° if deletion of p destroys density

r(p)/h(p) >

Shuffling

Reconstruction (Dey-Giesen)

Cocone(P,

Compute VP;

for each pP

if pB compute T of triangles with

duals intersecting C(p endif

endfor; Extract manifold;

end

B:= Boundary(P)

Main Theorem

Theorem 1: For sufficiently small an sample P of S can be shuffled to Q s.t. a surface mesh M can be computed from Q with

• M is homeomorphic to S

• |M-S| = O(f(p) for some p on S

• each triangle has aspect ratio O(

Synthetic data (Parbol)

No decimation,

8K pts

pts

pts

Synthetic data (Hyperbol)

No decimation,

8K pts

Synthetic data (Parcyl)

No decimation,

6K pts

1K pts

0.7K pts

Experimental Data

Rocker data

No decimation,

40K pts

pts

pts

Rocker data

pts

pts

Experimental data

Hip data

No decim,

265k pts

pts

pts

126K pts

pts

Conclusions

• Introduced sample shuffling

• Achieves sample decimation retaining features

• Achieves quality meshing

• What about both coarsening and refining?

• How to take care of the boundaries?

• How to take care of noise?

• Softwares:

www.cis.ohio-state.edu/~tamaldey/cocone.html