5. Parametric Surfaces
Parameterized surface: u ∈ R2 → ϕ(u) ∈ M.
u1 ∂ϕ
u2 ϕ ∂u1
γ ¯
γ
γ
¯
γ ∂ϕ
∂u2
Curve in parameter domain: t ∈ [0, 1] → γ(t) ∈ D.
def.
Geometric realization: γ (t) = ϕ(γ(t)) ∈ M.
¯
3
6. Parametric Surfaces
Parameterized surface: u ∈ R2 → ϕ(u) ∈ M.
u1 ∂ϕ
u2 ϕ ∂u1
γ ¯
γ
γ
¯
γ ∂ϕ
∂u2
Curve in parameter domain: t ∈ [0, 1] → γ(t) ∈ D.
def.
Geometric realization: γ (t) = ϕ(γ(t)) ∈ M.
¯
For an embedded manifold M ⊂ Rn :
∂ϕ ∂ϕ
First fundamental form: Iϕ = , .
∂ui ∂uj i,j=1,2
Length of a curve
1 1
def.
L(γ) = ||¯ (t)||dt =
γ γ (t)Iγ(t) γ (t)dt.
0 0 3
7. Riemannian Manifold
Riemannian manifold: M ⊂ Rn (locally)
Riemannian metric: H(x) ∈ Rn×n , symmetric, positive definite.
1
def. T
Length of a curve γ(t) ∈ M: L(γ) = γ (t) H(γ(t))γ (t)dt.
0
4
8. Riemannian Manifold
Riemannian manifold: M ⊂ Rn (locally)
Riemannian metric: H(x) ∈ Rn×n , symmetric, positive definite.
1
def. T
Length of a curve γ(t) ∈ M: L(γ) = γ (t) H(γ(t))γ (t)dt.
0
Euclidean space: M = Rn , H(x) = Idn .
W (x)
4
9. Riemannian Manifold
Riemannian manifold: M ⊂ Rn (locally)
Riemannian metric: H(x) ∈ Rn×n , symmetric, positive definite.
1
def. T
Length of a curve γ(t) ∈ M: L(γ) = γ (t) H(γ(t))γ (t)dt.
0
Euclidean space: M = Rn , H(x) = Idn .
2-D shape: M ⊂ R2 , H(x) = Id2 .
W (x)
4
10. Riemannian Manifold
Riemannian manifold: M ⊂ Rn (locally)
Riemannian metric: H(x) ∈ Rn×n , symmetric, positive definite.
1
def. T
Length of a curve γ(t) ∈ M: L(γ) = γ (t) H(γ(t))γ (t)dt.
0
Euclidean space: M = Rn , H(x) = Idn .
2-D shape: M ⊂ R2 , H(x) = Id2 .
Isotropic metric: H(x) = W (x)2 Idn .
W (x)
4
11. Riemannian Manifold
Riemannian manifold: M ⊂ Rn (locally)
Riemannian metric: H(x) ∈ Rn×n , symmetric, positive definite.
1
def. T
Length of a curve γ(t) ∈ M: L(γ) = γ (t) H(γ(t))γ (t)dt.
0
Euclidean space: M = Rn , H(x) = Idn .
2-D shape: M ⊂ R2 , H(x) = Id2 .
Isotropic metric: H(x) = W (x)2 Idn .
Image processing: image I, W (x)2 = (ε + ||∇I(x)||)−1 .
W (x)
4
12. Riemannian Manifold
Riemannian manifold: M ⊂ Rn (locally)
Riemannian metric: H(x) ∈ Rn×n , symmetric, positive definite.
1
def. T
Length of a curve γ(t) ∈ M: L(γ) = γ (t) H(γ(t))γ (t)dt.
0
Euclidean space: M = Rn , H(x) = Idn .
2-D shape: M ⊂ R2 , H(x) = Id2 .
Isotropic metric: H(x) = W (x)2 Idn .
Image processing: image I, W (x)2 = (ε + ||∇I(x)||)−1 .
Parametric surface: H(x) = Ix (1st fundamental form).
W (x)
4
13. Riemannian Manifold
Riemannian manifold: M ⊂ Rn (locally)
Riemannian metric: H(x) ∈ Rn×n , symmetric, positive definite.
1
def. T
Length of a curve γ(t) ∈ M: L(γ) = γ (t) H(γ(t))γ (t)dt.
0
Euclidean space: M = Rn , H(x) = Idn .
2-D shape: M ⊂ R2 , H(x) = Id2 .
Isotropic metric: H(x) = W (x)2 Idn .
Image processing: image I, W (x)2 = (ε + ||∇I(x)||)−1 .
Parametric surface: H(x) = Ix (1st fundamental form).
DTI imaging: M = [0, 1]3 , H(x)=diffusion tensor.
W (x)
4
14. Geodesic Distances
Geodesic distance metric over M ⊂ Rn
dM (x, y) = min L(γ)
γ(0)=x,γ(1)=y
Geodesic curve: γ(t) such that L(γ) = dM (x, y).
def.
Distance map to a starting1057 x0 ∈ M: Ux0 (x) = dM (x0 , x).
2 ECCV-08 submission ID point
metric
geodesics
Euclidean Shape Isotropic Anisotropic Surface 5
15. Anisotropy and Geodesics
Tensor eigen-decomposition:
T T
H(x) = λ1 (x)e1 (x)e1 (x) + λ2 (x)e2 (x)e2 (x) with 0 λ1 λ2 ,
{η η ∗ H(x)η 1}
e2 (x)
λ2 (x)
1
−2
x e1 (x)
1
M λ1 (x) −2
6
16. Anisotropy and Geodesics
Tensor eigen-decomposition:
T T
H(x) = λ1 (x)e1 (x)e1 (x) + λ2 (x)e2 (x)e2 (x) with 0 λ1 λ2 ,
{η η ∗ H(x)η 1}
e2 (x)
λ2 (x)
1
−2
x e1 (x)
1
M λ1 (x) −2
Geodesics tend to follow e1 (x).
6
17. Anisotropy and Geodesics
Tensor eigen-decomposition:
T T
H(x) = λ1 (x)e1 (x)e1 (x) + λ2 (x)e2 (x)e2 (x) with 0 λ1 λ2 ,
{η η ∗ H(x)η 1}
4 ECCV-08 submission ID 1057
e2 (x)
λ2 (x)
1
−2
x e1 (x)
Figure 2 shows examples of geodesic curves computed from a single starting
1
λ (x) −2
MS = {x1 } in the center of the image Ω = [0,11]2 and a set of points on the
point
boundary of Ω. The geodesics are computed for a metric H(x) whose anisotropy
α(x) (defined in equation (2)) is to follow e1 (x).making the Riemannian space
Geodesics tend increasing, thus
progressively closer to the Euclidean space. λ1 (x) − λ2 (x)
Local anisotropy of the metric: α(x) = ∈ [0, 1]
λ1 (x) + λ2 (x)
Image f
Image f α = .1
α = .95 α = .2
α = .7 α = .5
α = .5 α = 10
α= 6
18. Overview
• Riemannian Data Modelling
•Numerical Computation of
Geodesics
• Geodesic Image Segmentation
• Geodesic Shape Representation
• Geodesic Meshing
• Inverse Problems with Geodesic Fidelity 7
19. Eikonal Equation and Viscosity Solution
Distance map: U (x) = d(x0 , x)
Theorem: U is the unique viscosity solution of
||∇U (x)||H(x)−1 = 1 with U (x0 ) = 0
√
where ||v||A = v ∗ Av
8
20. Eikonal Equation and Viscosity Solution
Distance map: U (x) = d(x0 , x)
Theorem: U is the unique viscosity solution of
||∇U (x)||H(x)−1 = 1 with U (x0 ) = 0
√
where ||v||A = v ∗ Av
Geodesic curve γ between x1 and x0 solves
γ(0) = x1
γ (t) = −ηt H(γ(t))
−1
∇Ux0 (γ(t)) with
ηt 0
8
21. Eikonal Equation and Viscosity Solution
Distance map: U (x) = d(x0 , x)
Theorem: U is the unique viscosity solution of
||∇U (x)||H(x)−1 = 1 with U (x0 ) = 0
√
where ||v||A = v ∗ Av
Geodesic curve γ between x1 and x0 solves
γ(0) = x1
γ (t) = −ηt H(γ(t))
−1
∇Ux0 (γ(t)) with
ηt 0
Example: isotropic metric H(x) = W (x)2 Idn ,
||∇U (x)|| = W (x) and γ (t) = −ηt ∇U (γ(t))
8
22. Discretization γ x0
Control (derivative-free) formulation:
B(x) y
U (x) = d(x0 , x) is the unique solution of
U (x) = Γ(U )(x) = min U (x) + d(x, y) x
y∈B(x)
9
23. Discretization γ x0
Control (derivative-free) formulation:
B(x) y
U (x) = d(x0 , x) is the unique solution of
U (x) = Γ(U )(x) = min U (x) + d(x, y) x
y∈B(x)
Manifold discretization: triangular mesh.
U discretization: linear finite elements.
B(x)
H discretization: constant on each triangle. xi
xk
xj
9
24. Discretization γ x0
Control (derivative-free) formulation:
B(x) y
U (x) = d(x0 , x) is the unique solution of
U (x) = Γ(U )(x) = min U (x) + d(x, y) x
y∈B(x)
Manifold discretization: triangular mesh.
U discretization: linear finite elements.
B(x)
H discretization: constant on each triangle. xi
xk
Ui = Γ(U )i = min Vi,j,k
f =(i,j,k) xj
Vi,j,k = min tUj + (1 − t)Uk xi
0t1 xk
+||tUj + (1 − t)Uk − Ui ||Hijk
γ
txj + (1 − t)xk
xj 9
25. Discretization γ x0
Control (derivative-free) formulation:
B(x) y
U (x) = d(x0 , x) is the unique solution of
U (x) = Γ(U )(x) = min U (x) + d(x, y) x
y∈B(x)
Manifold discretization: triangular mesh.
U discretization: linear finite elements.
B(x)
H discretization: constant on each triangle. xi
xk
Ui = Γ(U )i = min Vi,j,k
f =(i,j,k) xj
Vi,j,k = min tUj + (1 − t)Uk xi
0t1 xk
+||tUj + (1 − t)Uk − Ui ||Hijk
γ
→ explicit solution (solving quadratic equation).
txj + (1 − t)xk
→ on regular grid: equivalent to upwind FD. xj 9
26. Numerical Schemes
Fixed point equation: U = Γ(U )
Γ is monotone: U V =⇒ Γ(U ) Γ(V )
Γ is L∞ contractant: ||Γ(U ) − Γ(V )||∞ ||U − V ||∞
Iterative schemes: Jacobi, Gauss-Seidel, accelerations.
[Borneman and Rasch 2006]
10
27. Numerical Schemes
Fixed point equation: U = Γ(U )
Γ is monotone: U V =⇒ Γ(U ) Γ(V )
Γ is L∞ contractant: ||Γ(U ) − Γ(V )||∞ ||U − V ||∞
Iterative schemes: Jacobi, Gauss-Seidel, accelerations.
[Borneman and Rasch 2006]
Causality condition: ∀ j ∼ i, Γ(U )i Uj
→ The value of Ui depends on {Uj }j with Uj Ui .
→ Compute Γ(U )i using an optimal ordering.
→ Front propagation, O(N log(N )) operations.
10
28. Numerical Schemes
Fixed point equation: U = Γ(U )
Γ is monotone: U V =⇒ Γ(U ) Γ(V )
Γ is L∞ contractant: ||Γ(U ) − Γ(V )||∞ ||U − V ||∞
Iterative schemes: Jacobi, Gauss-Seidel, accelerations.
[Borneman and Rasch 2006]
Causality condition: ∀ j ∼ i, Γ(U )i Uj
→ The value of Ui depends on {Uj }j with Uj Ui .
→ Compute Γ(U )i using an optimal ordering.
→ Front propagation, O(N log(N )) operations.
Holds for: - Isotropic H(x) = W (x)2 Idn , square grid. Good
- Surface (first fundamental form),.
triangulation with no obtuse angles. Good Bad
10
29. Front Propagation
Front ∂Ft , Ft = {i Ui t}
∂Ft
x0
State Si ∈ {Computed, F ront, F ar}
Algorithm: Far → Front → Computed.
1) Select front point with minimum Ui
Iteration
2) Move from Front to Computed .
3) Update Uj = Γ(U )j for neighbors and
11
40. 2D Shapes
2D shape: connected, closed compact set S ⊂ R2 .
Piecewise-smooth boundary ∂S.
Geodesic distance in S for uniform metric: 1
def. def.
dS (x, y) = min L(γ) where L(γ) = |γ (t)|dt,
γ∈P(x,y) 0
Shape S
Geodesics
19
41. Distribution of Geodesic Distances
Distribution of distances 80
60
to a point x: {dM (x, y)}y∈M
40
20
0
80
60
40
20
0
80
60
40
20
0
20
42. Distribution of Geodesic Distances
Distribution of distances 80
60
to a point x: {dM (x, y)}y∈M
40
20
0
80
60
Extract a statistical measure
40
20
0
a0 (x) = min dM (x, y).
80
60
40
y 20
0
a1 (x) = median dM (x, y).
y
a2 (x) = max dM (x, y).
y
x x x
Min Median Max 20
43. Distribution of Geodesic Distances
Distribution of distances 80
60
to a point x: {dM (x, y)}y∈M
40
20
0
80
60
Extract a statistical measure
40
20
0
a0 (x) = min dM (x, y).
80
60
40
y 20
0
a1 (x) = median dM (x, y).
y
a2 (x) = max dM (x, y). a2
y
a(x)
x x x
a1
a0
Min Median Max 20
44. Benging Invariant 2D Database
[Ling Jacobs, PAMI 2007]
Our method
(min,med,max)
100 1D
100
4D
Average Precision
80
max only 80
Average Recall
60 [Ion et al. 2008] 60
40 40
20 1D 20
4D
0 0
0 10 20 30 40 0 20 40 60 80 100
Image Rank Average Recall
→ State of the art retrieval rates on this database.
21
45. Perspective: Textured Shapes
Take into account a texture f (x) on the shape.
Compute a saliency field W (x), e.g. edge detector.
1
def.
Compute weighted curve lengths: L(γ) = W (γ(t))||γ (t)||dt.
0
Euclidean
Image f (x)
Weighted
||∇f (x)|| Max Min 22
46. Overview
• Riemannian Data Modelling
• Numerical Computations of Geodesics
• Geodesic Image Segmentation
• Geodesic Shape Representation
•Geodesic Meshing
• Inverse Problems with Geodesic Fidelity 23
47. Meshing Images, Shapes and Surfaces
Vertices V = {vi }M .
Triangulation (V, F): i=1
Faces F ⊂ {1, . . . , M }3 .
M
Image approximation: fM = λ m ϕm
m=1
λ = argmin ||f − µm ϕm ||
µ
m
ϕm (vi ) = m
δi is affine on each face of F.
24
48. Meshing Images, Shapes and Surfaces
Vertices V = {vi }M .
Triangulation (V, F): i=1
Faces F ⊂ {1, . . . , M }3 .
M
Image approximation: fM = λ m ϕm
m=1
λ = argmin ||f − µm ϕm ||
µ
m
ϕm (vi ) = m
δi is affine on each face of F.
There exists (V, F) such that ||f − fM || Cf M −2
Optimal (V, F): NP-hard.
24
49. Meshing Images, Shapes and Surfaces
Vertices V = {vi }M .
Triangulation (V, F): i=1
Faces F ⊂ {1, . . . , M }3 .
M
Image approximation: fM = λ m ϕm
m=1
λ = argmin ||f − µm ϕm ||
µ
m
ϕm (vi ) = m
δi is affine on each face of F.
There exists (V, F) such that ||f − fM || Cf M −2
Optimal (V, F): NP-hard.
Domain meshing:
Conforming to complicated boundary.
Capturing PDE solutions:
Boundary layers, chocs . . .
24
50. Riemannian Sizing Field
Sampling {xi }i∈I of a manifold.
Distance conforming: ε
∀ xi ↔ xj , d(xi , xj ) ≈ ε e1 (x)
1
∼ λ1 (x) −2 e2 (x)
Triangulation conforming: x
∆ =( xi ↔ xj ↔ xk ) ⊂ x ||x − x∆ ||T (x∆ ) η 1
∼ λ2 (x)− 2
Building triangulation
⇐⇒
Ellipsoid packing
⇐⇒
Global integration of
local sizing field
25
52. Geodesic Sampling
Sampling {xi }i∈I of a manifold.
Farthest point algorithm: [Peyr´, Cohen, 2006]
e
xk+1 = argmax min d(xi , x)
x 0ik
Metric Sampling
53. Geodesic Sampling
Sampling {xi }i∈I of a manifold.
Farthest point algorithm: [Peyr´, Cohen, 2006]
e
xk+1 = argmax min d(xi , x)
x 0ik
Geodesic Voronoi: Metric Sampling
Ci = {x ∀ j = i, d(xi , x) d(xj , x)}
Voronoi
54. Geodesic Sampling
Sampling {xi }i∈I of a manifold.
Farthest point algorithm: [Peyr´, Cohen, 2006]
e
xk+1 = argmax min d(xi , x)
x 0ik
Geodesic Voronoi: Metric Sampling
Ci = {x ∀ j = i, d(xi , x) d(xj , x)}
Geodesic Delaunay connectivity:
(xi ↔ xj ) ⇔ (Ci ∩ Cj = ∅)
→ geodesic Delaunay refinement. Voronoi Delaunay
→ distance conforming. → triangulation conforming if the metric is “gradded”.
58. Approximation Driven Meshing
Linear approximation fM with M linear elements.
Minimize approximation error ||f − fM ||Lp .
L∞ optimal metrics for smooth functions:
Images: T (x) = |H(x)| (Hessian)
Surfaces: T (x) = |C(x)| (curvature tensor)
Isotropic Anisotropic
59. Approximation Driven Meshing
Linear approximation fM with M linear elements.
Minimize approximation error ||f − fM ||Lp .
L∞ optimal metrics for smooth functions:
Images: T (x) = |H(x)| (Hessian)
Surfaces: T (x) = |C(x)| (curvature tensor)
Isotropic Anisotropic
For edges and textures: → use structure tensor.
[Peyr´ et al, 2008]
e
Anisotropic triangulation JPEG2000
60. Approximation Driven Meshing
Linear approximation fM with M linear elements.
Minimize approximation error ||f − fM ||Lp .
L∞ optimal metrics for smooth functions:
Images: T (x) = |H(x)| (Hessian)
Surfaces: T (x) = |C(x)| (curvature tensor)
Isotropic Anisotropic
For edges and textures: → use structure tensor.
[Peyr´ et al, 2008]
e
Anisotropic triangulation JPEG2000
→ extension to handle
boundary approximation.
[Peyr´ et al, 2008]
e
61. Overview
• Riemannian Data Modelling
• Numerical Computations of Geodesics
• Geodesic Image Segmentation
• Geodesic Shape Representation
• Geodesic Meshing
•Inverse Problems with Geodesic
Fidelity
with G.Carlier, F. Santambrogio, F. Benmansour
29
62. Variational Minimization with Metrics
Metric T (x) = W (x)2 Idd .
Geodesic distance: dW (x, y) = min W (γ(t))||γ (t)||dt
γ(0)=x,γ(1)=y
W → dW (x, y) is concave.
30
63. Variational Minimization with Metrics
Metric T (x) = W (x)2 Idd .
Geodesic distance: dW (x, y) = min W (γ(t))||γ (t)||dt
γ(0)=x,γ(1)=y
W → dW (x, y) is concave.
Variational problem: min Ei,j (dW (xi , xj ))2 + R(W )
W ∈C
i,j
C: admissible metrics.
R: regularization (smoothness).
Ei,j : interaction functional.
30
64. Variational Minimization with Metrics
Metric T (x) = W (x)2 Idd .
Geodesic distance: dW (x, y) = min W (γ(t))||γ (t)||dt
γ(0)=x,γ(1)=y
W → dW (x, y) is concave.
Variational problem: min Ei,j (dW (xi , xj ))2 + R(W )
W ∈C
i,j
C: admissible metrics.
R: regularization (smoothness).
Ei,j : interaction functional.
Example: shape optimization,
Eij (d) = −ρi,j d convex
traffic congestion,
Eij (d) = (d − di,j )2 non
seismic imaging, . . . convex
30
65. Variational Minimization with Metrics
Metric T (x) = W (x)2 Idd .
Geodesic distance: dW (x, y) = min W (γ(t))||γ (t)||dt
γ(0)=x,γ(1)=y
W → dW (x, y) is concave.
Variational problem: min Ei,j (dW (xi , xj ))2 + R(W )
W ∈C
i,j
C: admissible metrics.
R: regularization (smoothness).
Ei,j : interaction functional.
Example: shape optimization,
Eij (d) = −ρi,j d convex
traffic congestion,
Eij (d) = (d − di,j )2 non
seismic imaging, . . . convex
Compute the gradient of W → dW (x, y).
30
66. Gradient with Respect to the Metric
If γ is unique, this shows that ξ → dξε (xs , xt ) is differentiable at ξ, and that its
δξ (xs , xt ) is a measure supported along the curve γ. In the case where this geode
unique, this quantity may fail to be differentiable. Yet, the map ξ → dξ (xs , xt ) i
concave (as an infimum of linear quantities in ξ) and for each geodesic we get an
Formal derivation: super-differential through Equation + εZ, ∇dW (x, y) + o(ε)
of the dW +εZ (x, y) = dW (x, y) (1.9).
1
The extraction of geodesics is quite unstable, especially for metrics such that x
Z, ∇dW (x, y) = toby many curves robust manner to the minimumthe geodesic(xs , xt ).
are connected
Z(γ (t))dt length γ : the gradient of distance dξ distance
unclear how discretize in a
of close
geodesic x → y
0
from the continuous definition (1.9). We propose in this paper an alternative
where δξ (xs , xt ) is defined unambiguously as a subgradient of a discretized geod
tance. Furthermore, this discrete subgradient is computed with a fast Subgradien
ing algorithm.
Figure 1 shows two examples of subgradients, computed with the algorithm
in Section 3. Near a degenerate configuration, we can see that the subgradient
might be located around several minimal curves.
xs 0.7 2
xs
0.6 1.8
0.5
1.6
0.4
1.4
0.3
1.2
0.2
1
0.1
xt
∇dW (x, y)
0 0.8
W (x)
31
Figure 1: On the left, δξ (xs , xt ) and some of its iso-levels for ξ = 1. In the midd
67. Gradient with Respect to the Metric
If γ is unique, this shows that ξ → dξε (xs ,ξε (xsisxdifferentiable at ξ, at ξ, that its grad
If γ is unique, this shows that ξ → dξ xt ) s , t ) is differentiable and and that its
ε t
δξ (xsδξ (xsisxa)measure supported alongalongcurvecurve γ. Incase where this geodesic is
, xt ) s, t is a measure supported the the γ. In the the case where this geode
ξ t
unique, this quantity may fail to beto be differentiable. the map ξ → dξ (xsdξ (xsisxt ) i
unique, this quantity may fail differentiable. Yet, Yet, the map ξ → , xt ) s , any
ξ t
concave (as an infimum of linearlinear quantities in ξ)for each geodesic we get an elem
concave (as an infimum of quantities in ξ) and and for each geodesic we get an
Formal derivation:the super-differential through Equation + εZ, ∇dW (x, y) + o(ε)
dW +εZ (x, y) = dW (x, y)
of the super-differential through Equation (1.9).(1.9).
of
of
The extraction 1 geodesics is quite quite unstable, especially for metrics that xs anx
The extraction of geodesics is unstable, especially for metrics such such that
are connected by many curves of length close close to the minimum distancesdξ (xs ,Ittis t
are connected by many curves of length to the minimum distance dξ (x , xt ).s x t).
Z, ∇dW (x, y) = discretize in (t))dtmannerγ :gradient of theofgeodesic distance dire
unclear how
Z(γ a robust geodesic xthe geodesic distance
unclear how to to discretize in a robust manner the gradient
the → y ξ distanc
from fromcontinuous 0 definition (1.9).(1.9). propose in this paper an alternative meth
the the continuous definition We We propose in this paper an alternative
where δξ (xsδξ (xsisxdefined unambiguously as a subgradient of a discretized geodesic
where , xt ) s, t ) is defined unambiguously as a subgradient of a discretized geod
ξ t geo
Problem: W tance. W (x, y) non discrete subgradientnot unique. with fast Subgradien
→ dFurthermore, this smooth ifsubgradient is computed
tance. Furthermore, this discrete γ is computed with a fastaSubgradient Ma
ing algorithm.
ing algorithm.
y) is concave. two examples of compute sup-differetials.
W → dW (x, Figure 1 shows two examples of subgradients, computed with withalgorithm deta
Figure 1 shows subgradients, computed the the algorithm
in Section 3. Near Near a degenerate configuration, wesee that the subgradient δξ (xs
in Section 3. a degenerate configuration, we can can see that the subgradient
might be located around several minimal curves.
might be located around several minimal curves.
xs xs 0.7 0.7
0.7 2
xs x s
2
2
0.6 0.6
0.6 1.8 1.8
1.8
0.5 0.5
0.5
1.6 1.6
1.6
0.4 0.4
0.4
1.4 1.4
1.4
0.3 0.3
0.3
1.2 1.2
1.2
0.2 0.2
0.2
1 1
1
0.1 0.1
0.1
xt xt0 0.8
xt
0 0.8
∇dW (x, y) ∇dW (x, y)
0 0.8
W (x) W (x)
Figure 1: On the left, δξ (xsδξ (xsand) somesome of its iso-levels for1. = 1. Inmiddle, a
Figure 1: On the left, , xt ) s, xt and of its iso-levels for ξ = ξ In the the midd
ξ t 31
69. Subgradient Marching
Discretized geodesic distance computed by FM: Ui ≈ dW (x0 , xi )
Theorem: W ∈ RN → Ui ∈ R is concave.
Fast marching update: Ui ← u solution of xi
xk
u = Γ(U )i ∈ R solution of:
(u − Uj )2 + (u − Uk )2 = h2 Wi2 xj
32
70. Subgradient Marching
Discretized geodesic distance computed by FM: Ui ≈ dW (x0 , xi )
Theorem: W ∈ RN → Ui ∈ R is concave.
Fast marching update: Ui ← u solution of xi
xk
u = Γ(U )i ∈ R solution of:
(u − Uj )2 + (u − Uk )2 = h2 Wi2 xj
Gradient update: ∇Ui ≈ ∇dW (x0 , xi )
h2 Wi δi + αj ∇Uj + αk ∇Uk αj = Ui − Uj
∇Ui ←
αj + αk δi (s) = δ(i − s) (Dirac)
32
71. Subgradient Marching
Discretized geodesic distance computed by FM: Ui ≈ dW (x0 , xi )
Theorem: W ∈ RN → Ui ∈ R is concave.
Fast marching update: Ui ← u solution of xi
xk
u = Γ(U )i ∈ R solution of:
(u − Uj )2 + (u − Uk )2 = h2 Wi2 xj
Gradient update: ∇Ui ≈ ∇dW (x0 , xi )
h2 Wi δi + αj ∇Uj + αk ∇Uk αj = Ui − Uj
∇Ui ←
αj + αk δi (s) = δ(i − s) (Dirac)
Theorem: ∇Ui ∈ RN is a sup-gradient of W → Ui
Complexity: O(N 2 log(N )) operations to compute all (∇Ui )i ∈ RN ×N .
32
72. Landscape Design
max ρi,j dW (xi , xj )
∈C
W
C= W W (x)dx = λ, a W b
Ω
73. Landscape Design
max ρi,j dW (xi , xj )
∈C
W
C= W W (x)dx = λ, a W b
Ω
Sub-gradient descent:
W (+1) = ProjC W () + η ρi,j ∇dW (xi , xj )
i,j
Convergence: k = 100
η = +∞, η k+∞
2 = 300 k = 500
Figure 9: Iterations ξ (k) computed for a domain Ω with a hole and with P = 5 landmarks.
Extension of the model. It is possible to modify the energy E defined in (4.3) to mix
differently the distances between the points {xs }s . One can for instance minimize
Emin (ξ) = − min dξ (xs , xt ).
t=s
s
This functional is the opposite of the minimum of concave functions, and hence Emin is
a convex function. The maximization of the energy Emin forces each landmark to be
maximally distant from its closest neighbors.
The subgradient of Emin is computed as
74. Landscape Design
max ρi,j dW (xi , xj )
∈C
W
C= W W (x)dx = λ, a W b
Ω
Sub-gradient descent:
W (+1) = ProjC W () + η ρi,j ∇dW (xi , xj )
i,j
Convergence: k = 100
η = +∞, η k+∞
2 = 300 k = 500
Figure 9: Iterations ξ (k) computed for a domain Ω with a hole and with P = 5 landmarks.
Extension of the model. It is possible to modify the energy E defined in (4.3) to mix
max/min generalization: between the points {xs }s . One can for instance minimize
differently the distances
max min dW (xi , xj ) Emin (ξ) = − min dξ (xs , xt ).
t=s
W ∈C j=i s
i
This functional is the opposite of the minimum of concave functions, and hence Emin is
a convex function. The maximization of the energy Emin forces each landmark to be
maximally distant from its closest neighbors.
The subgradient k =E100 = 100 = 100
of min is computed as
k k k = 300 = 300 = 300
k k k = 500 = 500 =
k k 500
75. Traffic Congestion
Sources {xi }i and destinations {yj }j . y1
y2
Traffic ratio: xi → yj : ρi,j 0
Traffic plan: distribution Q on the set of paths γ.
Q {γ γ(0) = xi , γ(1) = yj } = ρi,j
x1
x2
34
76. Traffic Congestion
Sources {xi }i and destinations {yj }j . y1
y2
Traffic ratio: xi → yj : ρi,j 0
Traffic plan: distribution Q on the set of paths γ.
Q {γ γ(0) = xi , γ(1) = yj } = ρi,j Bε (x)
x1
Traffic intensity: 1
1Bε (γ(t))|γ (t)|dt dQ(γ) x2
γ 0
iQ (x) = lim
ε→0 |Bε (x)|
34
77. Traffic Congestion
Sources {xi }i and destinations {yj }j . y1
y2
Traffic ratio: xi → yj : ρi,j 0
Traffic plan: distribution Q on the set of paths γ.
Q {γ γ(0) = xi , γ(1) = yj } = ρi,j Bε (x)
x1
Traffic intensity: 1
1Bε (γ(t))|γ (t)|dt dQ(γ) x2
γ 0
iQ (x) = lim
ε→0 |Bε (x)|
Congested metric: WQ (x) = ϕ(iQ (x)).
34
78. Traffic Congestion
Sources {xi }i and destinations {yj }j . y1
y2
Traffic ratio: xi → yj : ρi,j 0
Traffic plan: distribution Q on the set of paths γ.
Q {γ γ(0) = xi , γ(1) = yj } = ρi,j Bε (x)
x1
Traffic intensity: 1
1Bε (γ(t))|γ (t)|dt dQ(γ) x2
γ 0
iQ (x) = lim
ε→0 |Bε (x)|
Congested metric: WQ (x) = ϕ(iQ (x)).
Wardrop equilibria: Q is distributed on geodesics for WQ .
Q γ LWQ (γ) = dWQ (γ(0), γ(1)) = 1
1
LW (γ) = W (γ(t))|γ (t)|dt
where 0
dW (x, y) = min LW (γ)
γ(0)=x,γ(1)=y 34