Vector Field Based Shape Deformations

1. ACMSIGGRAPH 2006 Vector Field Based Shape Deformations Wolfram von Funck / Holger Theisel / Hans-Peter Seidel MPI Informatik Computer Graphics Lab. SoHyeon Jeong 2007/04/16

2. Contents • Introduction • Constructing the vector field • Modeling metaphors • Implementational Details • Evalutation and Comparison

3. 1. Introduction

4. Shape Deformation Original shape New Deformed shape Transformation with Constraints • Performance • Detail & Feature preservation • Volume preservation • Avoidance of self-intersections • Deformation metapor

5. Deformation Metaphors • Free movement of certain handles • Singh and Fiume 1998 • Bendels and Klein 2003 • Pauly et al. 2003 • 9-dof object • Botsch and Kobbelt 2004 • Two Handed metaphor • Llamas et al. 2003 • New Metaphor • Implicit tools ?

6. Modeling Metaphors: Implicit Tools • Idea • Use simple implicit objects as deformation tools

7. Deformation Approaches • Mapping problem • Finding a mapping transformation between the original and the new deformed shape • Finding path problem • Finding continous path that certain point should follow

8. Finding Paths Problem • Integration of vector field at time T • Similar to “flow of fluid” [Foster and Fedkiw 2001] scaling Translation Rotation

9. The Main Idea to Solve • Constructing vector fields that produce useful deformation • Computing deformation by integrating using vector fields • Flexible • Variety of different deformations • Translations & rotations • Simple • Fast computation • Interactivity & large mesh deformation

10. Properties of Vector Fields • Divergence-free • C1 continuity • Time-dependent path integration • Volume preservation[Davis 1967] • Smooth deformation • No self-interaction[Theisel et al. 2005] Simple local properties of vector Global and local properties of shape deformation

11. 2. Constructing the Vector Fields v

12. Piecewise Region Field • Inner region • Well-defined region • Outer region • No deformation • Intermediate region • Blending between Inner & Outer region • Divergence-free • C1continuity

13. Piecewise Region Field • Region Field : • Inner region • Intermediate region • Outer region

14. Terms Scalar field Gradient Co-gradient Divergence 2D 3D

15. Constructing the Deformation Vector Field V • Constructing a divergence-free vector field • 2D • Co-gradient field of a scalar field [Davis 1967] : • 3D • Cross product of gradients of two scalar fields

16. Constructing the Deformation Vector Field V (3D) • Define scalar field in terms of region field • Construct divergence-free field using defined scalar fields : Berstein polynomials

17. Blending intermediate region • Inner & outer region should be connected smoothly • It requires C1continuity • Scalar fields : C2 continuity  Vector field : C1continuity e(x) 0

18. 1 1 1 1 1 0 0 0 0 0 -1 -1 -1 -1 -1 Blending: 2D Example outer region v = 0 intermediate region inner region v constant

19. 2D Example • Region Field

20. Special Deformations - Translation • Translation vector field : A constant vector field • The center point c : to determine DOF : The center point

21. Special Deformation - Rotation • Rotation vector field : linear vector field v and r • A center point : • An Axis : • Ristrected as a cylinder

22. Vector Field • Translation • Rotation

23. 3. Modeling Metaphor

24. Deformation Cycle • Usually • r(x) : the distance to a certain point • c : the center of the inner region • u, w, a : determined by interactive input device(mouse) • Integration • If tool moves , the integration inside the inner region moves the points by • The step size of the path line integration is chosen so that the path line follows the path of the tool Define region field r(x) with ri, ro and c Define scalar field e(x), f(x)with orthogonal vector u, w, a center c and an axis a Update v Integrate point of the shape with v

25. Implicit Tools • Point tools • Points in the inner regionat the beginning followthe movement of the tool • Other points never enter the inner region  no self-intersection • Line tools

26. Deformation Paint • The tool is moved along a path on the surface • : the location of the point on the shape at a certain time • , = choosen interactively

27. Moving Point Sets • Multiple isolated point set the shape • : Smooth approximated distance function to this point set • , : interactively choosen • : Barycenter of all points

28. Collision Tools • An arbitrary closed tool shape for which a repeated collistion detection with the deformed shape • Find collision region using Bounding box hierarchy • Setting • Collision detected points : • r = smooth approximated distance function along with ri = 0 • Inner region is constant • It follows the path of the input device

29. Collision Tools : Example 1

30. Collision Tools : Example 2

31. Twisting & Bending • Linear and quadratic vector field • Twsiting • : linear • : direction of the twisting axis • : on the twising axis

32. Twisting & Bending • Bending • Using a rotation • ,

33. Twisting & Bending twisting bending

34. Feature Preservation • Details on the surface are preserved during deformation

35. 4. Implementation

36. Integration with adaptive stepsize • Best tradeoff between speed and accuracy [Nielson et al. 1997] • 4th order Runge-Kutta integration with adaptive stepsize

37. Remeshing • Large deformation causes unpleasing artifacts • Undersample • Volume changing • Idea • Remeshing both the original and deformed object • New vertices undergo same deformation as the original vertices  It Requires remeshing

38. Remeshing M : original mesh, M’ : deformed mesh, P : deformation path • M and P are stored • All edges of the M’ are tested for refinement • length(edge) > threshold • Angle between the normals of the end-vertices is large  Edge split on both M and M’ • New vertices of M are deformed using P • Diffusion of the vertices • Guarantee a uniform distribution of the vertices • Vertices moves to the barycenter of its 1-ring • Vertex is projected back onto the surface of the undiffused mesh • Repeated a fixed number of steps • Decimation step • length(edge) < threshold • Small angle • Perform step 3 again for collapsed points  collapsed

39. 5. Evaluation and Comparision

40. Visual Quality • The twisting of a box [Proposed] [Yu et al 2004] [Lipman et al. 2005] [Zhou et al. 2005]

41. Visual Quality • Bending a sylinder [Laplacian surface] [Poisson Mesh] [Proposed] [Zhou et al. 2005] [Botsch and Kobbelt 2004]

42. Speed • Factors • Vertex # in inner, intermediate region • Vertex # in intermediate effects more than vertex # of inner region • Modeling metaphor • Region field r • Simple r gives a higher performance • Collision detection step in shape stamping • Deformation is highly parallelizable using GPU • 4th order Runge-Kutta path line integration of points • Read-back of the computed points drops performces • But still 10 times faster than CPU

43. Speed • Implementation Environment • AMD Opteron 152(2.6 GHz) • 2GB RAM • GeForce 6800 GT GPU

44. Accuracy • Accuracy in volume • Discrete surface points produces slight changes of the vlume • But is tolerable

45. 6. Conclusion and Future Work

46. Conclusion • Alternative approach to shape deformation • Time-dependent divergence-free vector field • volume-preserving • Self-intersection • Sharp features • Realtime deformation • Accuracy in volume preserving is high

47. Future Work • Preformance can futher be increased • Multi-processor parallelization of the integration • Integration of vertices is carried out independently • Applcation to point-based shape representation • Does not rely on any connectivity information of the mesh • Modeling metaphor can be extended • Full and zero deformation can be marked explicitly on the surfaces