1. Fast Marching on Triangulated Domains
Computer Science Department
Technion-Israel Institute of Technology
Fast Marching on Triangulated Domains
Ron Kimmel
Geometric Image Processing Lab

2. Brief Historical Review
Upwind schemes: Godunov 59
Level sets: Osher & Sethian 88
Viscosity SFS: Rouy & Tourin 92, (Osher & Rudin)
Level sets SFS: Kimmel & Bruckstein 92
Continuous morphology: Brockett & Maragos 92,Sapiro et al. 93
Minimal geodesics: Kimmel, Amir & Bruckstein 93
Fast marching method: Sethian 95
Fast optimal path: Tsitsiklis 95
Level sets on triangulated domains:Barth & Sethian 98
Fast marching on triangulated domains: Kimmel & Sethian 98
Applications based on joint works with: Elad, Kiryati, Zigelman

3. 1D Distance: Example 1 T(x) x x0 Find distance T(x), given T(x0)=0.
Solution: T(x)=|x-x0|.
except at x0.
T(x) x x0

4. 1D Distance: Example 2 T(x) x x0 x1
Find the distance T(x), given T(x0)=T(x1)=0
Solution:
T(x) = min{|x-x0|,|x-x1|},
Again, ,
except x0,x1 and (x0+x1)/2.
T(x) x x0 x1

5. 1D Eikonal Equation T(x) x x0 x1
with boundary conditions T(x0)=T(x1)=0.
Goal: Compute T that satisfies the equation `the best'.
T(x) x x0 x1

6. Numerical Approximation
Restrict , where h= grid spacing.
Possible solutions for are
T(x) x x0 x1

7. Approximation II T(x) x x0 x1 Updated i has always
`upwind' from where the `wind blows'
T(x) x x0 x1

8. Update Procedure T T T Set , and T(x0)=T(x1)=0.
REPEAT UNTIL convergence,
FOR each i T i T
i+1 T
i-1
i-1 i
i+1 h

9. Update Order What is the optimal order of updates?
Solution I: Scan the line successively left to right. N scans, i.e. O(N )
Solution II: Left to right followed by right to left. Two scans are sufficient. (Danielson`s distance map 1980)
Solution III: Start from x0, update its neighboring points, accept updated values, and update their neighbors, etc. 2 1 2 3 1 2

10. Weighted Domains x x0 x1 Local weight , Arclength
Goal: distance function characterized by:
By the chain rule:
The Eikonal equation is
T(x)
F(x) x x0 x1

11. 2D Rectangular Grids Isotropic inhomogeneous domains
Weighted arclength:
the weight is
Goal: Compute the distance T(x,y) from p0
where

12. Upwind Approximation in 2Dij T
i+1,j T
i,j-1
i+1,j T
i,j-1
i+1,j T
i-1,j ij
i-1,j
i,j+1

13. 2D Approximation T T Initialization: given initial value or Update:
Fitting a tilted plane with gradient , and two values anchored at the relevant neighboring grid points. T 1
i+1,j T 2
i,j-1 ij
i-1,j
i,j+1

14. Computational Complexity
T is systematically constructed from smaller to larger T values.
Update of a heap element is O(log N).
Thus, upper bound of the total is O(N log N).

15. Shortest Path on Flat Domains
Why do graph search based algorithms (like Dijkstra's) fail?

16. Edge Integration Cohen-Kimmel, IJCV, 1997.
Solve the 2D Eikonal equation
given T(p)=0
Minimal geodesic w.r.t.

17. Shape from Shading Rouy-Tourin SIAM-NU 1992,
Kimmel-Bruckstein CVIU 1994,
Kimmel-Sethian JMIV 2001.
Solve the 2D Eikonal equation
where
Minimal geodesic w.r.t.

18. Path Planning 3 DOF Solve the Eikonal Eq. in 3D {x,y,j}-CS
given T(x0,y0,j0)=0,
Minimal geodesic w.r.t.

19. Path Planning 3 DOF

20. Path Planning 4 DOF Solve the Eikonal Eq. in 4D
Minimal geodesic w.r.t.

21. Update Acute Angle Given ABC, update C. Consistency and monotonicity:
Update only `from within the triangle' h in ABC
Find t=EC that satisfies the gradient approximation
(t-u)/h= F. c c

22. Update Procedure We end up with: t must satisfy u<t, and h in ABC.
The update procedure is
IF (u<t) AND (a cos q < b(t-u)/t < a/cosq)
THEN T(C) = min {T(C),t+T(A)};
ELSE T(C)= min {T(C),bF+T(A),aF+T(B)}. u C B A

23. Obtuse Problems This front first meets B, next A, and only then C.
A is `supported’ by a single point.
The supported section of incoming fronts is a limited section.

24. Solution by splitting Initialization step!
Extend this section and link the vertex to one within the extended section.
Recursive unfolding: Unfold until a new vertex Q is found.
Initialization step!

25. Recursive Unfolding: Complexity
e = length of longest edge
The extended section maximal area
is bounded by a<= e /(2a ).
The minimal area of any unfolded triangle
is bounded below a >= (h a ) q /2,
The number of unfolded triangles before Q is found is bounded by
m<= a /a = e /(q h a ).
max 2
max
min 2
min
min
min
min 2 2 3
max
min
max
min
min
min

26. 1st Order Accuracy The accuracy for acute triangles is O(e )
Accuracy for the obtuse case
O(e /(p-q ))
max
max
max

27. Minimal Geodesics

28. Minimal Geodesics

29. Linear Interpolation
ODE ‘back tracking’

30. Quadratic Interpolation

31. Voronoi Diagrams and Offsets
Given n points, { p D, j 0,..,n-1}
Voronoi region:
G = {p D| d(p,p ) < d(p,p ), V j = i}. j i i j

32. Geodesic Voronoi Diagrams and Geodesic Offsets

33. Geodesic Voronoi Diagrams and Geodesic Offsets

34. Marching Triangles
The intersection set of two functions is linearly interpolated via `marching triangle'

35. Voronoi Diagrams and Offsets on Weighted Curved Domains

36. Voronoi Diagrams and Offsets on Weighted Curved Domains

37. Cheap and Fast 3D Scanner
PC + video frame grabber.
Video camera.
Laser line pointer.
Joint with G. Zigelman
motivated by simple shape from
structure light methods, like
Bouguet-Perona 99, Klette et al. 98
Lego Mindstorms rotates the laser (E. Gordon)
A frame grabber built at the Technion by Y Grinberg

38. Cheap and Fast 3D Scanner

39. Detection and Reconstruction

40. Examples of Decimation
Decimation - 3% of vertices
Sub-grid sampling

41. Results

42. Results

43. Texture Mapping Environment mapping: Blinn, Newell (76).
Environment mapping: Greene, Bier and Sloan (86).
Free-form surfaces: Arad and Elber (97).
Polyhedral surfaces: Floater (96, 98), Levy and Mallet (98).
Multi-dimensional scaling: Schwartz, Shaw and Wolfson (89).

44. Difficulties Need for user intervention. Local and global distortions.
Restrictive boundary conditions.
High computational complexity.

45. Flattening via MDS Compute geodesic distances between pairs of points.
Construct a square distance matrix of geodesic distances^2.
Find the coordinates in the plane via multi-dimensional scaling.
The simplest is `classical scaling’.
Use the flattened coordinates for texturing the surface, while preserving the texture features.
Zigelman, Kimmel, Kiryati,
IEEE T. on Visualization and Computer Graphics (in press).

46. Flattening

47. Flattening

48. Distances - comparison

49. Texture Mapping

50. Texture Mapping

51. Bending Invariant Signatures

52. Bending Invariant Signatures?
Elad, Kimmel, CVPR’2001

53. Bending Invariant Signatures?
Elad, Kimmel, CVPR’2001

54. Bending Invariant Signatures?
Elad, Kimmel, CVPR’2001

55. Bending Invariant Signatures
Elad, Kimmel, CVPR’2001

56. Bending Invariant Signatures
Elad, Kimmel, CVPR’2001

57. Bending Invariant Signatures3
Original surfaces Canonical surfaces in R
Elad, Kimmel, CVPR’2001

58. Bending Invariant Clustering
2nd moments based MDS for clustering
Original surfaces Canonical forms D D D
0.8 A A E
0.8 E
0.7 A
0.7 D E
0.6 E B
0.6 E E E
0.5 C C C
0.5 C D C C
0.4 B C
0.4 D D E D
0.3 F A
0.3 B B B B
0.2 F B B C
0.2 A A A A
0.1 F
0.1 F 1 1 1
0.8 1
0.8
0.8
0.6
*A=human body
0.8 F F F
0.6
0.6
0.4
0.6 F
0.4
0.4
0.4
0.2
0.2
*B=hand
0.2
0.2
*C=paper
*D=hat
*E=dog
*F=giraffe
Elad, Kimmel, CVPR’2001

59. More Applications semi-manual re-triangulation segmentation
halftoning in 3D
Adi, Kimmel 2002

60. Conclusions
Applications of Fast Marching Method on rectangular grids: Path planning, edge integration, shape from shading.
O(N) consistent method for weighted geodesic distance: ‘Fast marching on triangulated domains’.
Applications: Minimal geodesics, geodesic offsets, geodesic Voronoi diagrams, surface flattening, texture mapping, bending invariant signatures and clustering of surfaces, triangulation, and semi-manual segmentation.