tamal k. dey the ohio state university computing shapes and their features from point samples
Post on 22-Dec-2015
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2/52Department of Computer and Information Science
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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f(x) is the
distance
to medial axis
Local Feature Size[Amenta-Bern-Eppstein 98]
f(x)
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Each x has a sample
within f(x) distance
-Sampling[ABE98]
x
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Normal and Voronoi Cells(3D) [Amenta-Bern SoCG98]
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Normal Lemma
The angle between the pole vector
vp and the normal np is O().
P+
P-
np
vp
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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 Guarantees
Theorem:
Any point x is within O(f(x) distance from a point in the output. Conversely, any point of output surface has a point x 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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Small Features
High curvature regions are often undersampled
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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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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
• 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,tnpUσmax
2
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Ratio Condition
• Ratio Condition []:
R
qpmin
pU
||||
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Algorithm
)cloure(Output 11endfor10
endfor9endif8
Dual78 Condtion or Ratio Condition Angle satisfies if6
edgeDelaunayeachfor5; Compute4
eachfor3;2
;and Compute1
8
F
pqF:Fpq
UpqUPp
FDV
P
p
p
PP
)(M EDIAL
)cloure(Output 11endfor10
endfor9endif8
Dual78 Condtion or Ratio Condition Angle satisfies if6
edgeDelaunayeachfor5; Compute4
eachfor3;2
;and Compute1
8
F
pqF:Fpq
UpqUPp
FDV
P
p
p
PP
)(M EDIAL
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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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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 AxisMedial Axis
Medial Axis from a CAD model
CAD model
Point Sampling
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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 matchingDey-Giesen-Goswami 03
• Segment a shape into `features’• Match two shapes based on the
segmentation
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Feature definition
Flow
Continuous
Discrete flow
Discretization
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Anchor set
Shape :d
p xxpxh R allfor inf)(2
xpxA p minarg)(• Anchor set:
• Height fuinction:
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Driver and critical points
• Driver : d(x) is the closest point on the anchor hull
• Critical points
• Anchor Hull : H(x) is convex hull of A(x)
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Flow
• Vector field v :
)(
)()(
xdx
xdxxv
if x is regular and 0 otherwise
• Flow induced by v
Fix points of are the critical points of h
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Flow by discrete set• Driver d(x): closest point on dual to the Voronoi
object containing x
• Vector field:
• This also induces a flow
)(
)()(
xdx
xdxxv
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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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Merging• Small perturbations create insignificant features• Sampling artifacts introduce more segmentations
• Merge stable manifolds
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Conclusions
Noisy samples: Reconstruction and segmentation
Improving segmentation and matching for CAD
models (requires understanding of non-
smoothness)
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