Tamal K. Dey The Ohio State University Computing Shapes and Their Features from Point Samples.

Tamal K. Dey The Ohio State University

Computing Shapes and Their Features from Point Samples

2/52Department of Computer and Information Science

Problems

Surface reconstruction (Cocone)

Medial axis (Medial)

Shape segmentation and matching (SegMatch)


Surface Reconstruction

`

Point Cloud

Surface Reconstruction


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)


Medial Axis


f(x) is the

distance

to medial axis

Local Feature Size[Amenta-Bern-Eppstein 98]

f(x)


Each x has a sample

within f(x) distance

-Sampling[ABE98]

x


Voronoi/Delaunay


Normal and Voronoi Cells(3D) [Amenta-Bern SoCG98]


PolesP+

P-


Normal Lemma

The angle between the pole vector

vp and the normal np is O().

P+

P-

np

vp


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


Cocone

vp= p+ - p is the pole vector

Space spanned by vectors

within the Voronoi cell making

angle > 3/8 with vp or -vp


Cocone Algorithm


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 .


Undersampling [Dey-Giesen SoCG01]

Boundaries

Small features

Non-smoothness


Boundaries


Small Features

High curvature regions are often undersampled


Data Set Engine


Nonsmoothness


Watertight Surfaces


Tight Cocone [Dey-Goswami SM03]


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.


Result Sharp corners and edges of AutoPart can be reconstructed.


Timings

PIII, 933Mhz, 512MB


Noisy Data – Ram Head

Front view Rear view


Example movie file

Mannequin


Bunny data

• Bunny

Point data Tight Cocone Robust Cocone


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]


Medial Axis

• Medial Ball• Medial Axis -Sampling


Geometric Definitions

• Delaunay Triangulation

• Voronoi Diagram • Pole and Pole Vector• Tangent Polygon • Umbrella Up


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.


Angle Condition

• Angle Condition [θ ]:

pqσ,tnpUσmax

2


Ratio Condition

• Ratio Condition []:

R

qpmin

pU

||||


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


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.


Experimental Results






Computation Time

• Pentium PC • 933 MHz CPU• 512 MB memory

• CGAL 2.3• C++

• O1 optimization


Medial Axis from a CAD model

CAD model

Point Sampling Medial Axis


Medial AxisMedial Axis

Medial Axis from a CAD model

CAD model

Point Sampling


Example movie file

Anchor Medial


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)


Segmentation and matchingDey-Giesen-Goswami 03

• Segment a shape into `features’• Match two shapes based on the

segmentation


Feature definition

Flow

Continuous

Discrete flow

Discretization


Anchor set

Shape :d

p xxpxh R allfor inf)(2

xpxA p minarg)(• Anchor set:

• Height fuinction:


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)


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


Features

• F(x) = closure(S(x)) for a maximum x


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


Stable manifolds

• Gabriel edges are stable manifolds of saddles

• Stable manifolds of maxima are shaded


Stable manifolds

• Feature F(x) = closure(S(x)) for a maximum x


Stable manifolds in 3D• Stable manifolds are not subcomplexes of Delaunay• We approximate the stable manifolds with Delaunay simplices


Algorithm for )(~

xF )(~

xF


Merging• Small perturbations create insignificant features• Sampling artifacts introduce more segmentations

• Merge stable manifolds


Results (2D)


Results (3D)


Results (3D)


Matching CAD models


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

Tamal K. Dey The Ohio State University Computing Shapes and Their Features from Point Samples.

Documents

department of computer

information science

p p

information science

undersampled slide

fx fx slide

cocone amenta

matching segmatch slide