Post on 18-Dec-2015
Computing Medial Axis and Curve Skeleton from Voronoi
Diagrams
Tamal K. Dey
Department of Computer Science and EngineeringThe Ohio State University
Joint work with Wulue Zhao, Jian Sun
http://web.cse.ohio-state.edu/~tamaldey/medialaxis.htm
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Medial Axis for a CAD modelhttp://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm
CAD model
Point Sampling Medial Axis
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Medial axis approximation for smooth models
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• Amenta-Bern 98: Pole and Pole Vector
• Tangent Polygon
• Umbrella Up
Voronoi Based Medial Axis
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Filtering conditions
• Medial axis point m• Medial angle θ• Angle and Ratio
Conditions
Our goal: : approximate the medial axis as a approximate the medial axis as a subset of Voronoi facets.subset of Voronoi facets.
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Angle Condition
• Angle Condition [θ ]:
pqσ,tnpUσ
max
2
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‘Only Angle Condition’ Results
= 18 degrees
= 3 degrees = 32 degrees
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‘Only Angle Condition’ Results
= 15 degrees
= 20 degrees
= 30 degrees
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Ratio Condition
• Ratio Condition []:
R
qpmin
pU
||||
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‘Only Ratio Condition’ Results
= 2
= 4
= 8
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‘Only Ratio Condition’ Results
= 2
= 4
= 6
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Medial axis approximation for smooth models
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Theorem
• Let F be the subcomplex computed by MEDIAL. As approaches zero:• Each point in F converges to a medial
axis point. • Each point in the medial axis is
converged upon by a point in F.
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Experimental Results
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Medial AxisMedial Axis
Medial Axis from a CAD model
CAD model
Point Sampling
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Medial AxisMedial Axis
Medial Axis from a CAD modelhttp://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm
CAD model
Point Sampling
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Further work
• Only Ratio condition provides theoretical convergence:• Noisy sample
• [Chazal-Lieutier] Topology guarantee.
Curve-skeletons with Medial Geodesic Function
Joint work with J. Sun 2006
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Curve Skeleton
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• 1D representation of 3D shapes, called curve-skeleton, useful in some applications• Geometric modeling, computer vision, data analysis, etc
• Reduce dimensionality• Build simpler algorithms
• Desirable properties [Cornea et al. 05]
• centered, preserving topology, stable, etc
• Issues• No formal definition enjoying most of the desirable properties• Existing algorithms often application specific
Motivation (D.-Sun 2006)
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• Medial axis: set of centers of maximal inscribed balls
• The stratified structure [Giblin-Kimia04]: generically, the medial axis of a surface consists of five types of points based on the number of tangential contacts.
• M2: inscribed ball with two contacts, form sheets
• M3: inscribed ball with three contacts, form curves
• Others:
Medial axis
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Medial geodesic function (MGF)
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Properties of MGF
• Property 1 (proved): f is continuous everywhere and smooth almost everywhere. The singularity of f has measure zero in M2.
• Property 2 (observed): There is no local minimum of f in M2.
• Property 3 (observed): At each singular point x of f there are more than one shortest geodesic paths between ax and bx.
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Defining curve-skeletons
• Sk2=SkM2: set of singular points of MGF on M2 (negative divergence of Grad f.
• Sk3=SkM3: extending the view of divergence
• A point of other three types is on the curve-skeleton if it is the limit point of Sk2 U Sk3
• Sk=Cl(Sk2 U Sk3)
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Examples
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Shape eccentricity and computing tubular regions
• Eccentricity: e(E)=g(E) / c(E)
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Conclusions
• Voronoi based approximation algorithms• Scale and density independent• Fine tuning is limited• Provable guarantees
• Software• Medial:
www.cse.ohio-state.edu/~tamaldey/cocone.html• Cskel: www.cse.ohio-state.edu/~tamaldey/cskel.html
Thank you!