1 Introduction to Topological Shape Modeling Part I Overview: What is topology?
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Transcript of 1 Introduction to Topological Shape Modeling Part I Overview: What is topology?
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What is Topology? Pliable geometry?!
Identifies shapes if they are equivalent under smooth deformation
Deformation without object splitting and merging
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What can Topology do? Roughly classify a variety of
shapes Works as a upper layer in hierarchical
representation of shapes
Classification based on the number of torus holes
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Examples Connectivity
Graphs Shape structure
Decomposition into Cells Shape embedding in space
Knots and links
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Shape Structure Decomposing a shape into
topological entities
Topological structure of a torusVertex Edge Edge
Face peak
pass
pass
pit
Morse theory
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Embedding in Space Objects have restrictions in space.
Different between unknotted and knotted circles
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How does the topology classify shapes? Prepare special equivalence
relations Geometry: equal(=) Topology: ???
Find quotient space based on the equivalence relation
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Grouping Numbers If we use equal(=) for grouping…
2 43 73 6 95
Too detailed to understand the global distribution
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Grouping Numbers If we classify into even and odd…
(If we compare remainders when the nubmers is divided by 2.)
642 53 73 9Even numbers:
The remainder is 0when divided by 2
Odd numbers:The remainder is 1when divided by 2
Only two groups!!
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Grouping Numbers If we compare remainders when
the numbers are divided by 3
4 2 5 76 93 3The remainder is 0when divided by 3
The remainder is 1When divided by 3
The remainder is 2When divided by 3
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Grouping Shapes What is an equivalence relation for
shapes?
Equivelent? Equivalent?
Topology provides good equivalence relationsfor rough shape classification.
Equivalent?
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What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more…
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What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more…
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What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more…
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What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more…
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Tracing Isosurface Transitions Topological volume skeleton
Splitting and merging of isosurfaces
Volume skeleton tree (VST)
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Volume Analysis
Topological analysis of volume
Transfer function DesignBased on topological analysis
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Embedding-dependentRendering
TF by default VST-based Embedding-dependent
Visualizing complicated inner structure
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What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more…