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Introduction toTopological Shape Modeling

Part IOverview: 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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Upper layer

What can Topology do?

1 hole 2 holesno hole 3 holes

Close surfaces

…

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Examples Connectivity

Graphs Shape structure

Decomposition into Cells Shape embedding in space

Knots and links

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Connectivity Isomorphism between graphs

All graphs are isomorphic.4KComplete graph

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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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Grouping Shapes Equivalent if they can change into

each other without splitting and merging

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Grouping Shapes Answer is as follows:

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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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Topological Surface Design

peak

pass

pass

pit

Upper layerin hierarchicalrepresentation

Solid Modeling 1997 20

Topological Surface Design

Solid Modeling 1997 21

Examples

Torus

Solid Modeling 1997 22

Examples:

Toy dog Letters

Solid Modeling 1997 23

Examples

Double-layered swirl

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What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more…

Eurographics 1995 25

Terrain Surface Analysis Rendered images

Mt. Fuji Lake Ashi

Eurographics 1995 26

Terrain Surface Analysis Peaks, passes, pits, and contours

Mt. Fuji Lake Ashi

Eurographics 1995 27

Terrain Surface Analysis Ridge and ravine lines

Mt. Fuji Lake Ashi

Eurographics 1995 28

Terrain Surface Analysis Surface networks

Mt. Fuji Lake Ashi

Eurographics 1995 29

Terrain Surface Analysis Reeb graphs (Contour trees)

Mt. Fuji Lake Ashi

Eurographics 1995 30

Terrain Surface Analysis Reeb graphs (Contour trees)

Mt. Fuji Lake Ashi

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Wireframe representation

Surface Analysis

Topological skeleton(Reeb graph)

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Surface Analysis

Reeb graphs (Topological skeletons)

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Reeb graph (Topological skeleton)

Surface Analysis

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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…

Computer and Graphics 2001 39From a human head to a tiger head

Morphing = Surface + Time

Computers and Graphics 2001 40From a bunny to a cat

Morphing = Surface + Time

Pacific Graphics 2001 41

Topological Evolution?

Need to specify the topology in evolution!!

Pacific Graphics 2001 42

“8”-“0”-“V”-“11”-“H”-“B”-“A”

Topological Curve Morphing

Pacific Graphics 2001 44

Morphing design

From torus to sphere by cutting

Pacific Graphics 2001 45

The opening to a void within a solid is closed.

Results