1. ECCV Tutorial Mesh Processing Discrete Exterior Calculus
Bruno Lévy
INRIA - ALICE

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

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

4. Frustration with coordinate charts
IP =
 x
 u 2
 v
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. The Fundamental Theorem
(classic)
(EC version)

7. The Fundamental Theorem
(EC version)
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. The Fundamental Theorem Example 1
(Divergence theorem)
Ostrogradsky-Gauss

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

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

11. Importance of Borders Homology: 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 beyond Global contouring, [Ni et.al], [Dong et.al]

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

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. Overview This tutorial
1. Introduction
2. Differential Geometry on Meshes
Mesh Parameterization
3. Functions on Meshes
Exterior Calculus
h h50: Coffee Break
Discrete Exterior Calculus
4. Spectral Mesh Processing
5. Numerics

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

27. The Fundamental Theorem Reminder 1
(Divergence theorem)
Ostrogradsky-Gauss

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

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
from to
term
0-forms
dual 2-forms
|*i| i
mesh
dual mesh

37. |*ij|/|ij| = cot(β)+cot(β’)
Hodge star *1
from to
term
1-forms
dual 1-forms
|*ij|/|ij| = cot(β)+cot(β’) β i
*ij j
mesh
dual mesh β’

38. Exterior derivative d from to term 0-forms 1-forms df (ij) = fi - fj d
Oriented connectivity of the mesh: d i j k l ij -1 +1 jk ki il lj f fi fj fk fl l i k j

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

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

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

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

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