Localized%20Delaunay%20Refinement%20For%20Piecewise-Smooth%20Complexes

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Localized Delaunay Refinement For
Piecewise-Smooth Complexes

• Andrew G. SlattonJoint work with Tamal K. Dey
• The Ohio State University
• Department of Computer Science and Engineering

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The Problem

• Input Piecewise-smooth complex (PSC) D
• Output Triangular mesh approximating D
• Constraint Use localized Delaunay framework
• Generate many local sub-meshes
• Ignore mesh structure at global level

(mostly)
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Delaunay Refinement

• Large meshes ? Memory thrashing
• Localized framework avoids this

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Sharp Features

• Preserve via
• Cheng, Dey, Ramos 2008

protecting balls
weighted points
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Localization Protection

• Localization Dey, Levine, Slatton 2010
• Divide sample in octree
• Refine one node at a time
• Protection Cheng, Dey, Shewchuk 2012
• Preserve sharp features via protecting balls
• Refine protecting balls that are too large

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Difficulties

• An assimilation of previous works
• So wheres the challenge?
• Is ball refinement local?
• Yes! this must be proven
• What is its radius of operation?
• Max distance at which we insert/delete samples
• Need this to initialize a local triangulation

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Node Processing

• Split
• When P?gt ?
• Refine

P?
gathering distance
dg
N?
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Refinement Criteria

• Ball connectivity
• Patch vertices
• Disk
• Size

?
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Point Insertion

• Inter-point distance LB ? Termination
• Must avoid arbitrarily close insertions

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Reprocessing

• Refining ?
• Do we affect other local meshes?

dg
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Ball Refinement

• Refining b
• Remove contiguous set of balls containing b
• Cover exposed segment with smaller balls
• Remove zero-weighted points inside new balls
• Radius of operation dg
• Refining b??, all affected points lie in P?UN?

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Potential Complications

• Why is radius of operation important?
• dg too large increased overhead
• Too small
• May preclude lower bound on inter-ball distance
• May allow zero-weighted points to lie inside a
  ball
• Would falsify some lemmas leading to termination

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Local Ball Refinement Theorem

• Proven by showing radius of operation dg

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Guarantees

• Termination
• Subcomplex of restricted Delaunay Del(P )D
• Each point in output lies close to D
• For sufficiently small ?
• Output homeomorphic to input

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Results
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Software Results
http//www.cse.ohio-state.edu/tamaldey/locpsc.htm
l
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Thank You!

• Questions?