Topology Matching For Fully Automatic Similarity

Matching of 3D Shapes

Masaki HilagaYoshihisa ShinagawaTaku KohmuraTosiyasu 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

distanceGeodesic 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

R S Match if:

Matching Process

R S Match if:

Same height range

Matching Process

R S Match if:

Same height range

Parents match

Matching Process

R S Match if:

Same height range

Parents match

Matching Process

R S 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