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) 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 N p

Flatness condition Vertex p is flat if 1. Ratio condition: r(p)   h(p) 2. Normal condition:  v(p),v(q)    q with p  N q

Boundary detection Boundary (P, ,  ) Compute the set R of flat vertices; while  p  R and p  N q with q  R and r(p)  h(p) and  v(p),v(q)  R:=R  p; endwhile return P\R end

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

Large Data Delaunay takes space and time Exact computation is necessary. Doubles the time. Floating pointExact 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

Lucy 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   = K points

Rocker   K points Original 35K points

Bunny   0.4 7K points   K points Original 35K points

Bunny   0.4 7K points   K points Original 35K points

Triangle Aspect Ratio

Medial axis

Noise Outliers Cleaned

Noise (Local) This is a challenge unsolved. Perturbation by very tiny amount is tolerated by Cocone.

Boundaries EngineeringMedical

Geometric Models SportsDrug design

Geometric Models Entertainment Mathematical

Meshing

Boundary Detection

Data set Engine

Undersampling for Nonsmoothness

Modeling by Parts

Simplification Sample decimation vs. model decimation

Guarantees Topology preserved, no self intersection, feature dependent tri3100 tri

Multiresolution tri10202 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