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Introduction to Topological Shape Modeling

Introduction to Topological Shape Modeling. Part I Overview: What is topology?. What is Topology?. Pliable geometry?! Identifies shapes if they are equivalent under smooth deformation. Deformation without object splitting and merging. What can Topology do?.

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Introduction to Topological Shape Modeling

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  1. Introduction toTopological Shape Modeling Part I Overview: What is topology?

  2. What is Topology? • Pliable geometry?! • Identifies shapes if they are equivalent under smooth deformation Deformation without object splitting and merging

  3. 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

  4. Upper layer no hole 1 hole 2 holes 3 holes … What can Topology do? Close surfaces

  5. Examples • Connectivity • Graphs • Shape structure • Decomposition into Cells • Shape embedding in space • Knots and links

  6. Complete graph Connectivity • Isomorphism between graphs All graphs are isomorphic.

  7. Face peak pass pass pit Vertex Edge Edge Shape Structure • Decomposing a shape into topological entities Morse theory Topological structure of a torus

  8. Embedding in Space • Objects have restrictions in space. Different between unknotted and knotted circles

  9. How does the topology classify shapes? • Prepare special equivalence relations • Geometry: equal(=) • Topology: ??? • Find quotient space based on the equivalence relation

  10. Grouping Numbers • If we use equal(=) for grouping… 2 3 3 4 5 6 7 9 Too detailed to understand the global distribution

  11. 2 4 6 3 3 5 7 9 Grouping Numbers • If we classify into even and odd…(If we compare remainders when the nubmers is divided by 2.) Odd numbers: The remainder is 1when divided by 2 Even numbers:The remainder is 0when divided by 2 Only two groups!!

  12. 3 3 6 9 2 5 7 Grouping Numbers • If we compare remainders when the numbers are divided by 3 4 The remainder is 2 When divided by 3 The remainder is 0when divided by 3 The remainder is 1 When divided by 3

  13. Grouping Shapes • What is an equivalence relation for shapes? Equivelent? Equivalent? Topology provides good equivalence relationsfor rough shape classification. Equivalent?

  14. Grouping Shapes • Equivalent if they can change into each other without splitting and merging

  15. Grouping Shapes • Answer is as follows:

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

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

  18. Topological Surface Design peak pass Upper layer in hierarchicalrepresentation pass pit

  19. Topological Surface Design Solid Modeling 1997

  20. Examples Torus Solid Modeling 1997

  21. Examples: Toy dog Letters Solid Modeling 1997

  22. Examples Double-layered swirl Solid Modeling 1997

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

  24. Terrain Surface Analysis • Rendered images Mt. Fuji Lake Ashi Eurographics 1995

  25. Terrain Surface Analysis • Peaks, passes, pits, and contours Mt. Fuji Lake Ashi Eurographics 1995

  26. Terrain Surface Analysis • Ridge and ravine lines Mt. Fuji Lake Ashi Eurographics 1995

  27. Terrain Surface Analysis • Surface networks Mt. Fuji Lake Ashi Eurographics 1995

  28. Terrain Surface Analysis • Reeb graphs (Contour trees) Mt. Fuji Lake Ashi Eurographics 1995

  29. Terrain Surface Analysis • Reeb graphs (Contour trees) Mt. Fuji Lake Ashi Eurographics 1995

  30. Surface Analysis Wireframe representation Topological skeleton (Reeb graph)

  31. Surface Analysis Reeb graphs (Topological skeletons)

  32. Surface Analysis Reeb graph (Topological skeleton)

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

  34. Tracing Isosurface Transitions • Topological volume skeleton • Splitting and merging of isosurfaces Volume skeleton tree (VST)

  35. Volume Analysis Transfer function Design Based on topological analysis Topological analysis of volume

  36. Embedding-dependentRendering • Visualizing complicated inner structure Embedding- dependent TF by default VST-based

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

  38. Morphing = Surface + Time From a human head to a tiger head Computer and Graphics 2001

  39. Morphing = Surface + Time From a bunny to a cat Computers and Graphics 2001

  40. Topological Evolution? Need to specify the topology in evolution!! Pacific Graphics 2001

  41. Topological Curve Morphing “8”-“0”-“V”-“11”-“H”-“B”-“A” Pacific Graphics 2001

  42. Morphing design From torus to sphere by cutting Pacific Graphics 2001

  43. Results The opening to a void within a solid is closed. Pacific Graphics 2001

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