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Martin Isenburg University of North Carolina at Chapel Hill

Connectivity Shapes. Martin Isenburg University of North Carolina at Chapel Hill. Stefan Gumhold University of Tübingen. Craig Gotsman Technion - Israel Institute of Technology. Introduction. Overview. Shape from Connectivity Connectivity from Shape Hierarchical Methods Applications

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Martin Isenburg University of North Carolina at Chapel Hill

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  1. Connectivity Shapes Martin Isenburg University of North Carolina at Chapel Hill Stefan Gumhold University of Tübingen Craig Gotsman Technion - Israel Instituteof Technology

  2. Introduction

  3. Overview • Shape from Connectivity • Connectivity from Shape • Hierarchical Methods • Applications • Graph Drawing • Compression • Connectivity Creatures • Discussion

  4. Shape from Connectivity

  5. Shape from Connectivity

  6. Connectivity Shape Given a connectivity graph C = ( V, E ) consisting of a list verticesV = ( v1 ,v2 ,... ,vn ) and a set undirected edgesE = { e1 ,e2 ,... ,em } :ej = ( i1 ,i2 ) The connectivity shape CS ( C ) of C is alist of vectors ( x1 ,x2 ,x3 ,... ,xn ) :xi  R3that satisfy some“natural” property.

  7. Some “Natural” Property “all edges have unit length”  Equilibrium state of spring system. The connectivity shape is the solution to a set of m equations of the form ||xi - xj || = 1  ( i , j ) E The number of unknowns is determined by Euler’s relation m = n + f + 2g - 1

  8. Spring Energy ES Minimize ES = (|| xi - xj || - 1 )2 ( i , j ) E

  9. Roughness Energy ER ER = L( xi )2

  10. Final equation

  11. Family of Connectivity Shapes

  12. opt = argmax Volume( CS( C, ))   [0,1] Optimal Smoothing opt

  13. Iterative Solver

  14. Modified Spring Energy E’S E’S = (|| xi - xj ||2- 1 )2 ( i , j ) E

  15. Connectivity from Shape

  16. Connectivity from Shape

  17. Meshing / Re-meshing objective: generate a faithful approximation of a given shape, but use only edges of unit length we customized Turk method

  18. Smoothing Parameter dev

  19. Example Run

  20. Hierarchical Methods

  21. Hierarchical Methods

  22. Constructing the Hierarchy

  23. Applications

  24. Mesh Compression

  25. Connectivity Creatures

  26. End

  27. Bloopers

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