Computing Shapes and Their
Features from Point Samples
Tamal K. Dey
The Ohio State University
Problems
 Surface reconstruction (Cocone)
 Medial axis (Medial)
 Shape segmentation and matching (SegMatch)
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Surface Reconstruction
Point Cloud
`
Surface Reconstruction
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Voronoi based algorithms
1.
Alpha-shapes (Edelsbrunner, Mucke 94)
2.
Crust (Amenta, Bern 98)
3.
Natural Neighbors (Boissonnat, Cazals 00)
4.
Cocone (Amenta, Choi, Dey, Leekha, 00)
5.
Tight Cocone (Dey, Goswami, 02)
6.
Power Crust (Amenta, Choi, Kolluri 01)
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Medial Axis
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Local Feature Size
[Amenta-Bern-Eppstein 98]
f(x)
 f(x) is the distance
to medial axis
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-Sampling
[ABE98]
x
 Each x has a sample
within f(x) distance
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Voronoi/Delaunay
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Normal and Voronoi Cells(3D)
[Amenta-Bern SoCG98]
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Poles
P+
P-
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Normal Lemma
P+
 The angle between the pole vector
vp
np
vp and the normal np is O().
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Cocone Algorithm
[Amenta-Choi-Dey-Leekha SoCG00]
 Simplified/improved the Crust
 Only single Voronoi computation
 Analysis is simpler
 No normal filtering step
 Proof of homeomorphism
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Cocone
 vp= p+ - p is the pole vector
 Space spanned by vectors
within the Voronoi cell making
angle > 3/8 with vp or -vp
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Cocone Algorithm
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Cocone Guarantees
Theorem:
Any point x  S is within O()f(x) distance from a point in
the output. Conversely, any point of output surface has a
point x  S within O()f(x) distance.
Theorem:
The output surface computed by Cocone from an  -sample
is homeomorphic to the sampled surface for sufficiently
small .
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Undersampling
[Dey-Giesen SoCG01]
 Boundaries
 Small features
 Non-smoothness
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Boundaries
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Small Features
 High curvature regions are often undersampled
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Data Set Engine
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Nonsmoothness
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Watertight Surfaces
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Tight Cocone
[Dey-Goswami SM03]
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Tight COCONE Principle
 Compute the Delaunay triangulation of the input point set.
 Use COCONE along with detection of undersampling to get an initial
surface with undersampled regions identified.
 Stitch the holes from the existing Delaunay triangles without inserting any
new point.
 Effectively, the output surface bounds one or more solids.
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Result
 Sharp corners and edges of AutoPart can be reconstructed.
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Timings
PIII, 933Mhz, 512MB
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Noisy Data – Ram Head
Front view
Rear view
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Example movie file
Mannequin
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Bunny data
• Bunny
Point data
Tight Cocone
Robust Cocone
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Medial axis from point sample
Dey-Zhao SM02
• [Hoffman-Dutta 90],[Culver-Keyser-Manocha 99],[Giblin-Kimia
00], [Foskey-Lin-Manocha 03]
• Voronoi based
[Attali-Montanvert-Lachaud 01]
• Power shape : guarantees topology, uses power diagram
[Amenta-Choi-Kolluri 01]
• Medial : Approximates the medial axis as a Voronoi subcomplex
and has converegence guarantee.
[Dey-Zhao 02]
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Medial Axis
• Medial Ball
• Medial Axis
•  -Sampling
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Geometric Definitions
•
•
•
•
•
Delaunay Triangulation
Voronoi Diagram
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


2

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Ratio Condition
• Ratio Condition []:
min U p
|| p  q ||

R

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Algorithm
MEDIAL ( P )
1 Compute VP and DP ;
2 F  ;
3 for each p  P
4
Compute U p ;
5
for each Delaunay edge pq  U p
6
if pq satisfies Angle Condition
7
F :  F  Dual pq
8
endif
9
endfor
10 endfor
11 Output cloure( F )
8  or Ratio Condtion 8
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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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Experimental Results
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Experimental Results
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Computation Time
• Pentium PC
• 933 MHz CPU
• 512 MB memory
• CGAL 2.3
• C++
• O1 optimization
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Medial Axis from a CAD model
CAD model
Point Sampling
Medial Axis
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Medial Axis from a CAD model
CAD model
Point Sampling
Medial Axis
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Example movie file
Anchor Medial
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Segmentation and matching
•
•
•
•
•
Siddiqui-Shokoufandeh-Dickinson-Zucker 99 (Shock graphs)
Hilaga-Shinagawa-Kohmura-Kunni 01 (Reeb graph)
Osada-Funkhouser-Chazelle-Dobkin 01 (Shape distribution)
Bespalov-Shokoufandeh-Regli-Sun 03(spectral decomposition)
Dey-Giesen-Goswami 03 (Morse theory)
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Segmentation and matching
Dey-Giesen-Goswami 03
• Segment a shape into `features’
• Match two shapes based on the
segmentation
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Feature definition
Continuous
Flow
Discretization
Discrete flow
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Anchor set
• Height fuinction:
 : Shape
h( x)  inf p p  x
2
for all x  R d
• Anchor set: A( x)  arg min p p  x
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Driver and critical points
• Anchor Hull : H(x) is convex hull of A(x)
• Driver : d(x) is the closest point on the anchor hull
• Critical points
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Flow
• Vector field v :
x  d ( x) if x is regular and 0 otherwise
v( x) 
x  d ( x)
• Flow  induced by v
Fix points of  are the critical points of h
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Features
• F(x) = closure(S(x)) for a maximum x
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Flow by discrete set
• Driver d(x): closest point on  dual to the Voronoi object
containing x
• Vector field:
x  d ( x)
v( x) 
x  d ( x)
• This also induces a flow 
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Stable manifolds
• Gabriel edges are stable manifolds of saddles
• Stable manifolds of maxima are shaded
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Stable manifolds
• Feature F(x) = closure(S(x)) for a maximum x
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Stable manifolds in 3D
•
Stable manifolds are not subcomplexes of Delaunay
• We approximate the stable manifolds with Delaunay simplices
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~
Algorithm for F ( x)
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Merging
• Small perturbations create insignificant features
• Sampling artifacts introduce more segmentations
• Merge stable manifolds
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Results (2D)
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Results (3D)
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Results (3D)
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Matching CAD models
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Conclusions
 Noisy samples: Reconstruction and segmentation
 Improving segmentation and matching for CAD
models (requires understanding of nonsmoothness)
 Software available from
http://www.cis.ohio-state.edu/~tamaldey/cocone.html
http://www.cis.ohio-state.edu/~tamaldey/segmatch.html
 Acknowledgement: NSF, DARPA, ARO, CGAL
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