Shape Reconstruction from Shape Reconstruction from Samples with CoconeSamples with Cocone

Tamal K. Dey

Dept. of CIS

Ohio State University

A point cloud and reconstruction

Surface meshing from sample

A point set from satelite imaging

A reconstruction with and A reconstruction with and without noisewithout noise

Why Sample Based Modeling?

• Sampling is easy and convenient with advanced technology

• Automatization (no manual intervention for meshing)

• Uniform approach for variety of inputs (laser scanner, probe digitizer, MRI,scientific simulations)

• Robust algorithms are available

Challenges

• Nonuniform data

• Boundaries

• Undersampling

• Large data

• Noise

Nonuniform data

Boundaries

Undersampling

Large data

3.4 million points3.4 million points

Cocone

• Cocone meets the challenges

• It guarantees geometrically close surface with same topological type

• Detects boundaries

• Detects undersampling

• Handles large data (Supercocone)

• Watertight surface (Tight Cocone)

Sampling (ABE98)

Each x has a sample within f(x)

f(x) is the distance to medial axis

Voronoi/Delaunay

Surface and Voronoi Diagram

• Restricted Voronoi

• Restricted Delaunay

• skinny Voronoi cell

• poles

Cocone algorithm

• Cocone

Space spanned by vectors making angle /8 with horizontal

Radius, height and neighbors• p is the farthest point from p in the cocone.

•radius r(p): p radius of cocone

• height h(p): min distance to the poles

• cocone neighbors Np

Flatness condition

• Vertex p is flat if

1. Ratio condition: r(p) h(p)

2. Normal condition: v(p),v(q) q with pNq

Boundary detection

Boundary(P,,) Compute the set R of flat vertices;

while pR and pNq with qR and r(p)h(p) and v(p),v(q) R:=Rp; endwhile return P\Rend

Detected Boundary Samples

Detected Boundary Samples

Undersampling repaired

Holes are created

Tight Cocone

Guarantee: A water tight surface no Guarantee: A water tight surface no matter how the input is.matter how the input is.

Tight Cocone output

Holes are created

Hole filling

Time

Time

Large Data• Delaunay takes space and time

• Exact computation is necessary. Doubles the time.

Floating point Exact arithmetic

Large Data (Supercocone)

•Octree subdivision

Cracks• Cracks appear in surface computed from octree boxes

Surface matching

David’s Head

2 mil points, 93 minutes

Lucy25

3.5 million points, 198 mints

Shape of arbitrary dimension

Tangent and Normal Polytopes

• T(p) = V(p)T(p)

• N(p) = V(p)N(p)

Experiments

Sample Decimation

Original

40K points

= 0.4

8K points

= 0.33

12K points

Rocker

0.33

11K points

Original

35K points

Bunny

0.4

7K points

0.33

11K points

Original

35K points

Bunny

0.4

7K points

0.33

11K points

Original

35K points

Triangle Aspect Ratio

Medial axis

Medial axis

Noise

Outliers Cleaned

Noise (Local)

This is a challenge unsolved. Perturbation by very tiny amount is tolerated by Cocone.

Boundaries

Engineering Medical

Geometric Models

Sports Drug design

Undersampling for Nonsmoothness

Modeling by Parts

Simplification

• Sample decimation vs. model decimation

Guarantees• Topology preserved, no self intersection, feature dependent

13751 tri 3100 tri

Multiresolution

15766 tri 10202 tri 7102 tri

Model Analysis

• Feature line detection

• Detection of dimensionality

Mixed Dimensions

Model Reconstruction after Data Segmentation

Conclusions• SBGM with Del/Vor diagrams has great potential

• Challenges are

• Boundaries

• Nonsmoothness

• Noise

• Large data

• Robust simplification

• Robust feature detection