# Detecting Undersampling in Surface Reconstruction: Theory and ...

Shape Reconstruction from Samples 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 without 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 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) f(x) is the distance to medial axis

Each x has a sample within f(x)

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\R end

Detected Boundary Samples Detected Boundary Samples

Undersampling repaired Holes are created Tight Cocone

Guarantee: A water tight surface no 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 Davids 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 = 0.33

= 0.4 40K points

12K points 8K points Rocker

Original 0.33

35K points 11K points Bunny

Original 0.33

0.4 35K points 11K points

7K points Bunny

Original 0.33 0.4

35K points 11K points

7K 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

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