Shape Preserving Deformation David Levin Tel-Aviv University Afrigraph 2009

1. Shape Preserving DeformationDavid Levin Tel-Aviv UniversityAfrigraph 2009 Based on joint works with Yaron Lipman andDaniel Cohen-Or Tel-Aviv University

2. Table of Contents • Representation and manipulation of discrete surfaces. • Rigid motion invariant discrete surface representation. • Moving Frames for Surface Deformation. • Volume representations and Green coordinates.

3. Representation and manipulation of discrete surfaces. • In this part we address the deformation problem: • Approach: use special surface representations. • Rotation invariant surface representation. • Moving frames. • Volume representation.

4. Linear Rotation-invariant Coordinates for Meshes SIGGRAPH 2005, special issue of ACM Transactions on Graphics. • Shape is invariant to rigid-motions: • We wish to find a rigid-motion invariant shape representation for discrete surfaces (meshes): • Local support. • Easy to transform from other representations. • Easy to reconstruct.

5. Moving Frames on surfaces • Moving frames on surfaces, generalization of Frenet-frame: • Now we have two parameters.

6. Tangential form: Rotation invariant Surface representation On Discrete Surfaces (Meshes): place a frame at each vertex (In a rigid-motion invariant manner).

7. Rotation invariant Surface representation On Discrete Surfaces (Meshes)

8. Rotation invariant Surface representation Discrete surface (linear!) equations

9. Theorem 1: The set of coefficients define a unique discrete surface up to rigid motion. Rotation invariant Surface representation Rigid motion invariant representation. 1. Solve for the frames (Linear Least Squares) 2. Solve for the vertices position. (Integration – Linear Least Squares)

10. Applications • Deformation (prescribing frames on the surface):

11. Linear blending of rotation-invariant Coords. Linear blending of world Coords. Applications • Morphing (linear blending of the coordinates) – can handle large rotations.

12. Applications • Morphing (linear blending of the coordinates) – can handle large rotations. can handle large rotations

13. Applications

14. Moving Frames for Surface DeformationACM Transactions on Graphics (2007). • In previous method we look for new orientations of local frames. • The deformation energy can be quantified in a more precise manner. • We define: Geometric distance which reflects resemblance between curves and as such should ignorerigid motions. • Use a classical rigid motion invariant representation: Curvature and torsion to define:

15. Theorem: Geometric Deformation of Curves • Reduction: Instead for looking for f, we search for R:

16. Geometric Deformation of Curves • Reduction: Instead for looking for f, we search for R: Dirichlet energy

17. Geometric Deformation of Curves • Critical solutions of Dirichlet integral are called harmonic mappings. • We have (still for curves): Shape is preserved if: • Frenet Frames where rotated by harmonic quantity. • The Frenet Frames rotation map is geodesic curve on SO(3). • But we know what are geodesics on SO(3): one parameter subgroup of SO(3)

18. Geometric Deformation of Curves • Algorithm:

19. H the matrix of (Gauss map) in local frame, Geometric Distance and Optimal Rotation Field [Lipman et al. 2007] • Let us seek optimal deformation . • Maintain the first fundamental form (isometries) and minimize the second fundamental form. Define a geometric distance between isometric surfaces:

20. Geometric Distance and Optimal Rotation Field [Lipman et al. 2007] • Similarly to the curve case we look for mapping R. • Analogous to the curve case we have and therefore The shape is preserved if the rotation map is harmonic

21. Rotation angle Rotation axis 0 And when one rotation axis is involved we can assume Algorithm • To Minimize we choose a parameterization of SO(3): • Orthogonal parameterization • Conformal parameterization:

22. Discretization • Choose your favorite Laplace-Beltrami discretization (E.g. cot weight). • Solve for the rotations (with constraints). Two cases: • One rotation axis: exist linearsolution– the angle of rotation should be harmonic function of the mesh. • More than one rotation angle: A non-linearproblem. However linear approximation is enough. • Integrate: use the first discrete surface equation.

23. Results Rotations solved In SO(3) Discrete surface equations

24. Results

25. Green Coordinates Yaron Lipman, David Levin, Daniel Cohen-Or Tel Aviv University

26. Related work • Free-form (space) deformations. • Lattice-based (Sederberg & Parry 86, Coquillart 90, …) • Curve-/handle-based (Singh & Fiume 98, Botsch et al. 05, …) • Cage-based (Mean Value Coordinates) (Floater 03, Ju et al. 05, Langer et al. 06, Joshi et al. 07, Lipman et al. 07, Langer et al.08,… ) • Vector Field constructions (Angelidis 04, Von Funck 06,…)

27. Joshi et.al. 2007 Generalized Barycentric Coordinates • Recent generalization to barycentric coordinates allow defining space deformations using a flexible control polyhedra [Floater 03, Ju et al 05, Joshi et al. 07]: • Advantages: Fast, simple, robust, not limited to surfaces. • Disadvantage (in our context): Do not preserve shape.

28. Shape Preserving Free-Form deformation • What is the class of space mappings we look for? (preserve shape) • Conformal mappings are shape preserving: • Angle preserving. • Conformal mappings induce local similarity transformations :

29. Shape Preserving Free-Form deformation • Two problems: • Conformal mappings are hard to compute. • Schwarz-Christoffel mapping – computationally hard [ T.A.Driscoll L.N.Trefethen 02]. • Conformal mappings exist only in 2D.

30. Shape Preserving Free-Form deformation • What class of space mappings can be produced using the standard Free-from deformation formulation? • Each axis is treated independently. • Affine invariance. Scalar affine weights (coordinates) Constant points (w.r.t. p) No hope for conformal mappings in this formulation of free-form deformations

31. Result break Conformal space deformation Harmonic coordinates

32. Shape Preserving Free-Form deformation • Idea: use the normals to blend between the coordinates…

33. Shape Preserving Free-Form deformation • Then the deformation is defined by:

34. Green Coordinates • Question: what are and ? • Green’s third identity encodes a similar relation: • A harmonic function can be written as

35. Green Coordinates • Use the coordinate functions for u:

36. Green Coordinates • Use the coordinate functions for u:

37. Green Coordinates • Theorem 1: • The mapping for d=2 is conformal for all . Green Coordinates Might go out of the cage Harmonic Coordinates

38. 2D Green Coordinates 3D Green Coordinates Green Coordinates • Theorem 2: • The coordinate functions have closed form expressions for d=2,3.

39. Results Green Coordinates Harmonic coordinates

40. Results • Observation (Conjecture): The mapping is quasi-conformal for d=3. Mean-Value Coordinates Green Coordinates Harmonic Coordinates Distortion histogram

41. Results Mean Value Coordinates Green Coordinates

42. Results Mean Value coordinates Green Coordinates

43. Results Mean Value coordinates Green Coordinates

44. Results Green Coordinates Mean Value Coordinates

45. Green Coordinates • Theorem 3: • The extension through every simplicial face is unique in the sense of analytic continuation in 2D and real-analytic continuation in 3D. • The formula for the unique extension is simply where is an affine map.

46. Employment of partial cages Mean-Value Coordinates Green Coordinates

47. Thanks for listening…