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 Davids 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