Slides for a course on mesh processing.

1. Active Contours
www.numerical-tours.com
Gabriel Peyré
CEREMADE, Université Paris-Dauphine

2. Overview
• Parametric Edge-based Active
Contours
• Implicit Edge-based Active Contours
• Region-based Active Contours
2

3. Parametric Active Contours
Local minimum: argmin E( ) = L( ) + ⇥R( )
Data Regularization
Boundary conditions: fidelity
x0
– Open curve: (0) = x0 and (1) = x1 .
– Closed curve: (0) = (1).
x1
3

4. Parametric Active Contours
Local minimum: argmin E( ) = L( ) + ⇥R( )
Data Regularization
Boundary conditions: fidelity
x0
– Open curve: (0) = x0 and (1) = x1 .
– Closed curve: (0) = (1).
Snakes energy: (depends on parameterization) x1
1 1
L( ) = W ( (t))|| (t)||dt, R( ) = || (t)|| + µ|| (t)||dt
0 0
Image f Weight W (x) Curve 3

5. Geodesic Active Contours
Geodesic active contours: (intrinsic) Replace W by W + ,
1
E( ) = L( ) = W ( (t))|| (t)||dt
0
(local) minimum of the weighted length L.
local geodesic (not minimal path).
Weight W (x) Curve 4

6. Curve Evolution
Family of curves { s (t)}s>0 minimizing E( s ).
s
Do not confound:
t: abscise along the curve.
s+ds
s: artificial “time” of evolution.
5

7. Curve Evolution
Family of curves { s (t)}s>0 minimizing E( s ).
s
Do not confound:
t: abscise along the curve.
s+ds
s: artificial “time” of evolution.
Local minimum of: min E( )
d
Minimization flow: s = E( s )
ds
5

8. Curve Evolution
Family of curves { s (t)}s>0 minimizing E( s ).
s
Do not confound:
t: abscise along the curve.
s+ds
s: artificial “time” of evolution.
Local minimum of: min E( )
d
Minimization flow: s = E( s )
ds
Warning: the set of curves is not a vector space.
1
Inner product at : µ, ⇥⇥ = µ(t), ⇥(t)⇥|| (t)||dt
0
Riemannian manifold of infinite dimension.
5

9. Curve Evolution
Family of curves { s (t)}s>0 minimizing E( s ).
s
Do not confound:
t: abscise along the curve.
s+ds
s: artificial “time” of evolution.
Local minimum of: min E( )
d
Minimization flow: s = E( s )
ds
Warning: the set of curves is not a vector space.
1
Inner product at : µ, ⇥⇥ = µ(t), ⇥(t)⇥|| (t)||dt
0
Riemannian manifold of infinite dimension.
Numerical implementation: (k+1)
= (k)
⇥k E( (k)
)
5

10. Intrinsic Curve Evolutions
E( ) only depends on { (t) t [0, 1]}.
Intrinsic energy E: evolution along the normal
s
d
⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ns
ds
speed normal
s (t)
Normal: ns (t) =
|| s (t)||
1
Curvature: ⇥s (t) = ns (t), s (t)⇥
|| s (t)||2
6

11. Mean Curvature Motion
1
No data-fidelity: E( ) = || (t)||dt
0
d
Curve-shortening flow. s = E( s )
ds
1
(t)
E( + ⇥) = E( ) + , ⇥ (t)⇥dt + O(||⇥||)
0 || (t)||
1 d (t)
⌅E( ) : t ⇤⇥
|| (t)|| dt || (t)|| 0
Mean-curvature motion:
s
d
s (t) = ⇥s (t)ns (t)
ds
Speed: (x, n, ⇥) = ⇥
7

12. Discretization
Discretization: = { (i)}N 1
i=0 R2 , with (N ) = (0).
⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)||
k
8

13. Discretization
Discretization: = { (i)}N 1
i=0 R2 , with (N ) = (0).
⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)||
Discrete energy: E( ) = i || (i) (i + 1)||
k
8

14. Discretization
Discretization: = { (i)}N 1
i=0 R2 , with (N ) = (0).
⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)||
Discrete energy: E( ) = i || (i) (i + 1)||
Gradient descent flow: k+1 = k ⇥k E( k )
k
E( k )
8

15. Discretization
Discretization: = { (i)}N 1
i=0 R2 , with (N ) = (0).
⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)||
Discrete energy: E( ) = i || (i) (i + 1)||
Gradient descent flow: k+1 = k ⇥k E( k )
k
1
Discrete gradient: ⇥E( ) = ⇥ N ⇥( )
||⇥ ||
(⇥ )(i) = (i + 1) (i)
E( k )
(⇥ )(i) = (i 1) (i)
(i)
(N )(i) =
|| (i)||
8

16. Geodesic Active Contours Motion
Weighted length:
1
E( ) = L( ) = W ( (t))|| (t)||dt
0
Evolution:
d
⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) 0
ds
(x, n, ⇥) = W (x)⇥ W (x), n
attraction toward areas
where W is small.
s
finite di erences discretization.
Weight W (x)

17. Open vs. Closed Curves
0
s
x0
s
0
x1
Weight W (x) Image f (x)

18. Global Minimum with Fast Marching
Geodesic distance map:
Ux0 (x1 ) = min L( )
(0)=x0 , (1)=x1
Global minimum: Ux0 (x1 ) = L( ) Image f Metric W (x)
Distance Ux0 (x) Geodesic curve (t)

19. Global Minimum with Fast Marching
Geodesic distance map:
Ux0 (x1 ) = min L( )
(0)=x0 , (1)=x1
Global minimum: Ux0 (x1 ) = L( ) Image f Metric W (x)
Fast O(N log(N )) algorithm:
– Compute Ux0 with Fast Marching.
– Solve EDO: Distance Ux0 (x) Geodesic curve (t)
d
(t) = Ux0 ( (t))
dt
(0) = x1

20. Overview
• Parametric Edge-based Active Contours
• ImplicitEdge-based Active
Contours
• Region-based Active Contor
12

21. Level Sets
Level-set curve representation:
s (x) 0
{ s (t) t [0, 1]} = x R ⇥s (x) = 0 .
2
Example: circle of radius r s (x) = ||x x0 || s
Example: square of radius r s (x) = ||x x0 || s
s (x) 0

22. Level Sets
Level-set curve representation:
s (x) 0
{ s (t) t [0, 1]} = x R ⇥s (x) = 0 .
2
Example: circle of radius r s (x) = ||x x0 || s
Example: square of radius r s (x) = ||x x0 || s
s (x) 0
Union of domains: s = min( 1
s, s)
2
Intersection of domains: s = max( 1
s, s)
2
s (x) 0
s = min( 1
s, s)
2

23. Level Sets
Level-set curve representation:
s (x) 0
{ s (t) t [0, 1]} = x R ⇥s (x) = 0 .
2
Example: circle of radius r s (x) = ||x x0 || s
Example: square of radius r s (x) = ||x x0 || s
s (x) 0
Union of domains: s = min( 1
s, s)
2
Intersection of domains: s = max( 1
s, s)
2
s (x) 0
Popular choice: (signed) distance to a curve
⇥s (x) = ± min || s (t) x||
t
infinite number of mappings s s.
s = min( 1
s, s)
2

24. Level Sets Evolution
Dictionary parameteric implicit:
Position: x= s (t)
s (x)
Normal: ns (t) =
|| s (x)||
⇥s
Curvature: s (x) = div (x)
|| ⇥s ||
14

25. Level Sets Evolution
Dictionary parameteric implicit:
Position: x= s (t)
s (x)
Normal: ns (t) =
|| s (x)||
⇥s
Curvature: s (x) = div (x)
|| ⇥s ||
Evolution PDE:
d
⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t)
ds
d ⇥s (x) ⇥s
⇥s (x) = || ⇥s (x)|| ⇥s (x), , div (x) .
ds || ⇥s (x)|| || ⇥s ||
All level sets evolves together.
14

26. Proof
d
Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( )
ds
Definition of level-sets: t, ⇥s ( s (t)) = 0
15

27. Proof
d
Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( )
ds
Definition of level-sets: t, ⇥s ( s (t)) = 0
Deriving with respect to t:
⇤ s ⇤⇥s
⇥s (x), (t) + (x) = 0 for x = s (t) ( )
⇤s ⇤s
15

28. Proof
d
Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( )
ds
Definition of level-sets: t, ⇥s ( s (t)) = 0
Deriving with respect to t:
⇤ s ⇤⇥s
⇥s (x), (t) + (x) = 0 for x = s (t) ( )
⇤s ⇤s
⌅⇤s
( )+( ): (x) = (x, ns (t), ⇥s (t)) ⇤s (x), ns (t)
⌅s
15

29. Proof
d
Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( )
ds
Definition of level-sets: t, ⇥s ( s (t)) = 0
Deriving with respect to t:
⇤ s ⇤⇥s
⇥s (x), (t) + (x) = 0 for x = s (t) ( )
⇤s ⇤s
⌅⇤s
( )+( ): (x) = (x, ns (t), ⇥s (t)) ⇤s (x), ns (t)
⌅s
s (x)
=
|| s (x)||
For all x on the curve,
d ⇥s (x) ⇥s
⇥s (x) = || ⇥s (x)|| ⇥s (x), , div (x) .
ds || ⇥s (x)|| || ⇥s ||
15

30. Implicit Geodesic Active Contours
Evolution PDE:
d s
s = || s ||div W .
ds || s ||
Comparison with explicit active contours:
: 2D instead of 1D equation.
+ : allows topology change.
16

31. Implicit Geodesic Active Contours
Evolution PDE:
d s
s = || s ||div W .
ds || s ||
Comparison with explicit active contours:
: 2D instead of 1D equation.
+ : allows topology change.
Re-initialization: ⇥s (x) = ± min || s (t) x||
t
Eikonal equation: || s || =1 with ⇥s ( s (t)) = 0
16

32. Multiple Fluids Dynamics
See Ron Fedkiw homepage. http://physbam.stanford.edu/ fedkiw/
Multiple gaz:
Fluid/air interface:
17

33. Overview
• Parametric Edge-based Active Contours
• Implicit Edge-based Active Contours
• Region-based Active Contours
18

34. Energy Depending on Region
Optimal segmentation [0, 1]2 = c
:
min L1 ( ) + L2 ( c
) + R( ) R( ) = | |
Data Regularization
fidelity
Chan-Vese binary model: L1 ( ) = |I(x) c1 |2 dx
More general models
19

35. Energy Depending on Region
Optimal segmentation [0, 1]2 = c
:
min L1 ( ) + L2 ( c
) + R( ) R( ) = | |
Data Regularization
fidelity
Chan-Vese binary model: L1 ( ) = |I(x) c1 |2 dx
More general models
Level set implementation: = {x (x) > 0}
2 x H(x)
Smoothed Heaviside: H (x) = atan
⇥ x
L1 ( ) ⇥ L( ) = H ( (x))||I(x) c1 ||2 dx
R( ) R( ) = ||⇥(H )(x)||dx
19

36. Descent Schemes
For a given c = (c1 , c2 ) R2 :
min Ec ( ) = H ( (x))||I(x) c1 ||2 +
⇥
H ( ⇥(x))||I(x) c2 ||2 + ||⇥(H ⇥)(x)||dx
20

37. Descent Schemes
For a given c = (c1 , c2 ) R2 :
min Ec ( ) = H ( (x))||I(x) c1 ||2 +
⇥
H ( ⇥(x))||I(x) c2 ||2 + ||⇥(H ⇥)(x)||dx
Descent with respect to :
⇥(k+1) = ⇥(k) k Ec (⇥(k) )
Ec ( ) = H ( (x))G(x)
⇥⇥
G(x) = ||I(x) 2
c1 || ||I(x) 2
c2 || div (x)
||⇥⇥||
20

38. Descent Schemes
For a given c = (c1 , c2 ) R2 :
min Ec ( ) = H ( (x))||I(x) c1 ||2 +
⇥
H ( ⇥(x))||I(x) c2 ||2 + ||⇥(H ⇥)(x)||dx
Descent with respect to :
⇥(k+1) = ⇥(k) k Ec (⇥(k) )
Ec ( ) = H ( (x))G(x)
⇥⇥
G(x) = ||I(x) 2
c1 || ||I(x) 2
c2 || div (x)
||⇥⇥||
Limit 0: ⇥Ec (⇥) { =0} (x)||⇥⇥(x)||G(x)
Numerically, use Ec ( ) = || (x)||G(x)
20

39. Update of c
Joint minimization: min Ec ( )
,c1 ,c2
21

40. Update of c
Joint minimization: min Ec ( )
,c1 ,c2
Update of : ⇥(k+1) = ⇥(k) k Ec(k) (⇥(k) )
21

41. Update of c
Joint minimization: min Ec ( )
,c1 ,c2
Update of : ⇥(k+1) = ⇥(k) k Ec(k) (⇥(k) )
Update of (c1 , c2 ):
(k+1) (k+1)
(c1 , c2 ) = argmin Ec ( (k)
)
c1 ,c2
(k+1)
c1 = c( (k) ) I(x)H( (x))dx
c( ) =
(k+1)
c2 = c( (k) ) H( (x))dx
21

42. Example of Evolution
Use de-localized initialization.
(0)
22

43. Conclusion
Curve evolution
Energy minimization
Parametric vs. level set representation.
Dictionary to translate
– curve properties.
– energy gradients.
Edge based vs. region based energies.
23