Computing Medial Axis and Curve
Skeleton from Voronoi Diagrams
Tamal K. Dey
Department of Computer Science and Engineering
The Ohio State University
Joint work with Wulue Zhao, Jian Sun
http://web.cse.ohio-state.edu/~tamaldey/medialaxis.htm
Medial Axis for a CAD model
http://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm
CAD model
Point Sampling
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Medial Axis
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Medial axis approximation for
smooth models
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Voronoi Based Medial Axis
• Amenta-Bern 98: Pole
and Pole Vector
• Tangent Polygon
• Umbrella Up
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Filtering conditions
Our goal: approximate the medial axis as a
subset of Voronoi facets.
• Medial axis point m
• Medial angle θ
• Angle and Ratio
Conditions
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Angle Condition
• Angle Condition [θ ]:
nσ,t pq
maxσU p
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

2

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‘Only Angle Condition’ Results
 = 3 degrees
 = 18 degrees
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 = 32 degrees
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‘Only Angle Condition’ Results
 = 15 degrees
 = 20 degrees
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 = 30 degrees
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Ratio Condition
• Ratio Condition []:
min U p
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|| p  q ||

R

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‘Only Ratio Condition’ Results
=2
=4
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=8
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‘Only Ratio Condition’ Results
=2
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=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 Axis from a CAD model
CAD model
Point Sampling
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Medial Axis
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Medial Axis from a CAD model
http://web.cse.ohio-state.edu/~tamaldey/medialaxis_CADobject.htm
CAD model
Point Sampling
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Medial Axis
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Further work
• Only Ratio condition provides
theoretical convergence:
• Noisy sample
• [Chazal-Lieutier] Topology guarantee.
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Curve-skeletons with Medial
Geodesic Function
Joint work with J. Sun 2006
Curve Skeleton
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Motivation
(D.-Sun 2006)
• 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
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Medial axis
• 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:
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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
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Thank you!