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ECCV Tutorial Mesh Processing Discrete Exterior Calculus

ECCV Tutorial Mesh Processing Discrete Exterior Calculus. Bruno Lévy INRIA - ALICE. Motivations Global parameterization methods. [Ray et.al]. [Gu & Yau]. [Tong et.al]. Differential Geometrie to the rescue: Differential Atlas. Work in coordinate charts. 2.  x.  x.  x.  u.  v.

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ECCV Tutorial Mesh Processing Discrete Exterior Calculus

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  1. ECCV TutorialMesh ProcessingDiscrete Exterior Calculus Bruno Lévy INRIA - ALICE

  2. MotivationsGlobal parameterization methods [Ray et.al] [Gu & Yau] [Tong et.al]

  3. Differential Geometrie to the rescue:Differential Atlas Work in coordinate charts ...

  4. 2 x x  x u v u IP= x 2 x x u v v Frustration with coordinate charts Lots of computations ! Handling charts is a hassle

  5. Exterior Calculus to the rescue ... • Invented by Elie Cartan • Pinkall & Polthier • Gu & Yau • Desbrun & Schroeder, DEC, DDG • Joy of EC (Google search it)

  6. (EC version) The Fundamental Theorem (classic)

  7. (EC version) The Fundamental Theorem dw = exterior derivative of w ddw = 0 w is a k-form 0-form : functions 1-forms : vector fields 2-forms : functions integrated on surfaces  : integration domain ; ∂ = border of 

  8. (fundamental theorem) The Fundamental TheoremExample 1 (Divergence theorem) Ostrogradsky-Gauss

  9. (the plain old fundamental theorem) (fundamental theorem) a- b+ a b ∂  The Fundamental TheoremExample 2

  10. Importance of BordersHomology: C1 and C2 are the border of smthg.

  11. Importance of BordersHomology: C1 and C2 are the border of smthg.

  12. Homology basis [Closed loops / Borderism] like a vector basis for linear spaces ... ... each loop can be decomposed as a sum of those

  13. Duality The fundamental theorem (again !) alternative notation :

  14. More Duality

  15. Co-homology [Closed forms / Exact forms] Consequence: their integral on any closed curve  match:

  16. [Gu and Yau's] method

  17. [Gu and Yau] in a nutshell • Capture object's topology (homology) • Construct function space (co-homology) • Optimize function (2g coordinates to find) • Integrate homolorphic complex potential (i.e. from gradients to coordinates)

  18. Streamlines and beyond Anisotropic Polygonal Remeshing [Alliez et.al 03] can we do a continuous version of this ?

  19. Streamlines and beyondGlobal contouring, [Ni et.al], [Dong et.al]

  20. cos(q) sin(q) q U = Periodic Global Parameterization

  21. Periodic Global Parameterization Optimizes alignment with curvature tensor Demo

  22. Modified Tutte condition[Steiner & Fischer] "Translational" (affine) Differential manifold

  23. Modified Tutte condition[Tong et. al] "Translational + rot90" (complex) Differential manifold

  24. Discrete Exterior Calculus (DEC) • Discretize equations on a mesh • Simple • Rigorous • [Harrisson], [Mercat], [Hirani], [Arnold], [Desbrun] • Based on k-forms

  25. OverviewThis tutorial 1. Introduction 2. Differential Geometry on Meshes Mesh Parameterization 3. Functions on Meshes Exterior Calculus -------- 10h30 - 10h50: Coffee Break ----------------- Discrete Exterior Calculus 4. Spectral Mesh Processing 5. Numerics

  26. Exterior CalculusReminder • Functions over arbitrary manifolds • k-forms (functions, vector fields) • exterior derivative • border operator

  27. (fundamental theorem) The Fundamental TheoremReminder 1 (Divergence theorem) Ostrogradsky-Gauss

  28. (the plain old fundamental theorem) (fundamental theorem) a- b+ a b ∂  The Fundamental TheoremReminder 2

  29. k-forms mesh dual mesh

  30. 0-forms -4.6 5.1 -7.5 3.5 3.5 -2.7 0.2 5

  31. 1-forms -4.6 -7.5 5.1 -5.3 0.3 -4.8 8.1 -2.7 3.5 0.2 3.5 5

  32. 2-forms 2.1 3.1 3.4 -5.9 4.7 3.2 6.2 -1.4 -1.6 3.8 2.7

  33. dual 0-forms 6.2 0.6 -1.6 1.5 2.7 -4.7 -7.1 -0.1 -2.2 -4.6 0.5 1.6 -4.3 -6.6 -5.2 -3.3 -4.6

  34. dual 1-forms 4.3 -6 -4.6 3.5 -4.6 5.6 2.2 3.6 0.1 5.5 -4.6 -1.2 -2.6 -0.6 0.6 1.2

  35. dual 2-forms 3.2 -5.9 2.1 6.2 3.1 3.4 -1.6 3.8 2.7

  36. Hodge star *0 i mesh dual mesh

  37. Hodge star *1 β i *ij mesh dual mesh j β’

  38. Exterior derivative d Oriented connectivity of the mesh: l i k j

  39. DEC Laplacian In DEC the Laplacian is *0-1dT*1d 0-form (function) f

  40. DEC Laplacian In DEC the Laplacian is *0-1dT*1d 1-form (gradient) df 0-form (function) f d (fj-fi)

  41. DEC Laplacian In DEC the Laplacian is *0-1dT*1d 1-form (gradient) df 0-form (function) f d (cot(β)+cot(β’)) (fj-fi) *1 dual 1-form (cogradient) *df

  42. DEC Laplacian In DEC the Laplacian is *0-1dT*1d 1-form (gradient) df 0-form (function) f d (cot(β)+cot(β’)) (fj-fi) Σ *1 dT dual 0-form (integrated laplacian) d*df dual 1-form (cogradient) *df

  43. DEC Laplacian In DEC the Laplacian is *0-1dT*1d 0-form (function) f 0-form (pointwise laplacian) *-1d*df 1-form (gradient) df d (cot(β)+cot(β’)) (fj-fi) Σ*i *0-1 *1 |*i| dT dual 0-form (integrated laplacian) d*df dual 1-form (cogradient) *df

  44. DEC Laplacian i b a j aij = 2 (cotan a + cotan b) / (AiAj)

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