Shape Reconstruction from Samples with Cocone

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Shape Reconstruction from Samples with Cocone

• Tamal K. Dey
• Dept. of CIS
• Ohio State University

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A point cloud and reconstruction
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Surface meshing from sample
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A point set from satelite imaging
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A reconstruction with and without noise
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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

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Challenges

• Nonuniform data
• Boundaries
• Undersampling
• Large data
• Noise

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Nonuniform data
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Boundaries
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Undersampling
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Large data
3.4 million points
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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)

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e-Sampling (ABE98)
f(x) is the distance to medial axis
Each x has a sample within ef(x)
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Voronoi/Delaunay
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Surface and Voronoi Diagram

• Restricted Voronoi
• Restricted Delaunay
• skinny Voronoi cell
• poles

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Cocone algorithm

• Cocone

Space spanned by vectors making angle q? ?/8 with
horizontal
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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

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Flatness condition

• Vertex p is flat if

1. Ratio condition r(p) ? ? h(p)
2. Normal condition ?v(p),v(q) ? ? ?q with p?Nq
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Boundary detection
Boundary(P,?,?) Compute the set R of flat
vertices while ?p?R and p?Nq with q?R
and r(p)??h(p) and ?v(p),v(q) ??
RR?p endwhile return P\R end
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Detected Boundary Samples
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Detected Boundary Samples
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Undersampling repaired
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Holes are created
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Tight Cocone
Guarantee A water tight surface no matter how
the input is.
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Tight Cocone output
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Holes are created
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Hole filling
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Time
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Time
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Large Data

• Delaunay takes space and time
• Exact computation is necessary. Doubles the time.

Floating point
Exact arithmetic
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Large Data (Supercocone)

• Octree subdivision

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Cracks

• Cracks appear in surface computed from octree
  boxes

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Surface matching
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Davids Head

2 mil points, 93 minutes
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Lucy25
3.5 million points, 198 mints
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Shape of arbitrary dimension
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Tangent and Normal Polytopes

• T?(p) V(p)?T(p)
• N?(p) V(p)?N(p)

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Experiments
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Sample Decimation
Original 40K points

• 0.4
• 8K points

• 0.33
• 12K points

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Rocker

• ? 0.33
• 11K points

Original 35K points
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Bunny

• ? 0.4
• 7K points

• ? 0.33
• 11K points

Original 35K points
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Bunny

• ? 0.4
• 7K points

• ? 0.33
• 11K points

Original 35K points
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Triangle Aspect Ratio
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Medial axis
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Medial axis
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Noise
Cleaned
Outliers
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Noise (Local)
This is a challenge unsolved. Perturbation by
very tiny amount is tolerated by Cocone.
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Boundaries
Engineering
Medical
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Geometric Models
Sports
Drug design
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Geometric Models
Entertainment
Mathematical
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Meshing
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Boundary Detection
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Data set Engine
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Undersampling for Nonsmoothness
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Modeling by Parts
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Simplification

• Sample decimation vs. model decimation

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Guarantees

• Topology preserved, no self intersection,
  feature dependent

13751 tri
3100 tri
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Multiresolution
7102 tri
15766 tri
10202 tri
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Model Analysis

• Feature line detection
• Detection of dimensionality

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Mixed Dimensions
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Model Reconstruction after Data Segmentation
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Conclusions

• SBGM with Del/Vor diagrams has great potential
• Challenges are
• Boundaries
• Nonsmoothness
• Noise
• Large data
• Robust simplification
• Robust feature detection