Topology Matching For Fully Automatic Similarity Matching of 3D Shapes Masaki Hilaga Yoshihisa Shinagawa Taku Kohmura Tosiyasu L. Kunii

Shape Matching Problem Similarity between 3D objects Metric near- invariants Rigid transformations Surface simplification Noise Fast

Technique (1) Construct Multiresolution Reeb Graph (MRG) normalized geodesic distance Geodesic distance function Multiresolution Reeb Graph

Technique (2) MRG matching algorithm for similarity queries Finds most similar regions Most similar regions on two frogsMatching nodes of two MRGs

Reeb Graph Same as in Chand’s presentation Can use any function 

Geodesic distance function Integral of geodesic distances (v) =  p g(v,p) dS Normalize  n (v) = ((v) – min()) / min()

Geodesic Approximation Approximate integral Sample Simplify distance Use Dijkstra’s

Multiresolution Reeb Graph Binary discretization Preserve parent-child relationships Exploit them for matching

Matching process Calculate similarity Match nodes Find pairs with maximal similarity Preserve multires hierarchy topology Sum up similarity

Matching Process RS Match if:

Matching Process RS Match if: Same height range

Matching Process RS Match if: Same height range Parents match

Matching Process RS Match if: Same height range Parents match

Matching Process RS Match if: Same height range Parents match Match on graph path

Results Invariants satisfied fairly well Between pairs, similarity  0.94 Across pairs, similarity  0.76

Results Database, 7 levels of MRG Similarity calculated in tens of milliseconds Database searched in average ~10 seconds

Critique Subjectively good matching Meet invariance criteria Approximation of geodesic distance Reeb graph discretization All models in DB must have same parameters Similarity metric