Topology Matching For Fully Automatic Similarity Matching of 3D Shapes

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Topology Matching For Fully Automatic Similarity
Matching of 3D Shapes

• Masaki Hilaga
• Yoshihisa Shinagawa
• Taku Kohmura
• Tosiyasu L. Kunii

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Shape Matching Problem

• Similarity between 3D objects
• Metric near-invariants
• Rigid transformations
• Surface simplification
• Noise
• Fast

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Technique (1)

• Construct Multiresolution Reeb Graph (MRG)
• normalized geodesic distance

Geodesic distance function
Multiresolution Reeb Graph
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Technique (2)

• MRG matching algorithm for similarity queries
• Finds most similar regions

Most similar regions on two frogs
Matching nodes of two MRGs
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Reeb Graph

• Same as in Chands presentation
• Can use any function ?

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Geodesic distance function

• Integral of geodesic distances
• ?(v) ?p g(v,p) dS
• Normalize
• ?n(v) (?(v) min(?)) / min(?)

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Geodesic Approximation

• Approximate integral
• Sample
• Simplify distance
• Use Dijkstras

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Multiresolution Reeb Graph

• Binary discretization
• Preserve parent-child relationships
• Exploit them for matching

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Matching process

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

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Matching Process
R
S
Match if
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Matching Process
R
S
Match if Same height range
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Matching Process
R
S
Match if Same height range Parents match
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Matching Process
R
S
Match if Same height range Parents match
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Matching Process
R
S
Match if Same height range Parents match Match
on graph path
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Results

• Invariants satisfied fairly well
• Between pairs, similarity ? 0.94
• Across pairs, similarity ? 0.76

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Results

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

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Critique

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