Geodesic Method in Computer Vision and Graphics

Geodesic Methods
in Computer Vision
and Graphics

Gabriel Peyré
www.numerical-tours.com

Overview
•Riemannian Data Modelling
• Numerical Computations of Geodesics
• Geodesic Image Segmentation
• Geodesic Shape Representation
• Geodesic Meshing
• Inverse Problems with Geodesic Fidelity 2

Parametric Surfaces
Parameterized surface: u ∈ R2 → ϕ(u) ∈ M.

u1 ∂ϕ
u2 ϕ ∂u1

∂ϕ
∂u2

3

Parametric Surfaces

u1 ∂ϕ
u2 ϕ ∂u1

γ γ
∂ϕ
∂u2

Curve in parameter domain: t ∈ [0, 1] → γ(t) ∈ D.

3

Parametric Surfaces

u1 ∂ϕ
u2 ϕ ∂u1
γ ¯
γ
γ
¯
γ ∂ϕ
∂u2

def.
Geometric realization: γ (t) = ϕ(γ(t)) ∈ M.
¯

3

Parametric Surfaces

u1 ∂ϕ
u2 ϕ ∂u1
γ ¯
γ
γ
¯
γ ∂ϕ
∂u2

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

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

Riemannian Manifold
1
def. T
0
Euclidean space: M = Rn , H(x) = Idn .

W (x)

4

Riemannian Manifold
1
def. T
0
2-D shape: M ⊂ R2 , H(x) = Id2 .

W (x)

4

Riemannian Manifold
1
def. T
0
2-D shape: M ⊂ R2 , H(x) = Id2 .
Isotropic metric: H(x) = W (x)2 Idn .

W (x)

4

Riemannian Manifold
1
def. T
0
2-D shape: M ⊂ R2 , H(x) = Id2 .
Image processing: image I, W (x)2 = (ε + ||∇I(x)||)−1 .

W (x)

4

Riemannian Manifold
1
def. T
0
2-D shape: M ⊂ R2 , H(x) = Id2 .
Parametric surface: H(x) = Ix (1st fundamental form).

W (x)

4

Riemannian Manifold
1
def. T
0
2-D shape: M ⊂ R2 , H(x) = Id2 .
Parametric surface: H(x) = Ix (1st fundamental form).
DTI imaging: M = [0, 1]3 , H(x)=diﬀusion tensor.

W (x)

4

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

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

T T
{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

T T
{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

Overview
• Riemannian Data Modelling
•Numerical Computation of
Geodesics


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


||∇U (x)||H(x)−1 = 1 with U (x0 ) = 0
√

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


||∇U (x)||H(x)−1 = 1 with U (x0 ) = 0
√

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

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

B(x) y
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

B(x) y
U (x) = Γ(U )(x) = min U (x) + d(x, y) x
y∈B(x)

B(x)
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

B(x) y
U (x) = Γ(U )(x) = min U (x) + d(x, y) x
y∈B(x)

B(x)
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

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

Numerical Schemes

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

Numerical Schemes

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

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

Fast Marching on an Image

12

Fast Marching on Shapes and Surfaces

13

Overview
•Geodesic Image Segmentation

Isotropic Metric Design
Image-based potential: H(x) = W (x)2 Id2 , W (x) = (ε + |f (x) − c|)α

Image f Metric W (x) Distance Ux0 (x) Geodesic curve γ(t)

15

Isotropic Metric Design
Image-based potential: H(x) = W (x)2 Id2 , W (x) = (ε + |f (x) − c|)α

Image f Metric W (x) Distance Ux0 (x) Geodesic curve γ(t)

Gradient-based potential: W (x) = (ε + ||∇x f ||)−α

Image f Metric W (x) U{x0 ,x1 } Geodesics 15

Isotropic Metric Design: Vessels
˜
Remove background: f = Gσ f − f , σ ≈vessel width.

f ˜
f ˜
W = (ε + max(f , 0))−α

16

Isotropic Metric Design: Vessels
˜
Remove background: f = Gσ f − f , σ ≈vessel width.

f ˜
f ˜
W = (ε + max(f , 0))−α

3D Volumetric datasets:

16

Overview
•Geodesic Shape Representation

Bending Invariant Recognition
Shape articulations:

[Zoopraxiscope, 1876]

18

Bending Invariant Recognition
Shape articulations:

[Zoopraxiscope, 1876]

Surface bendings:
˜
x1

˜
x2
M
[Elad, Kimmel, 2003]. [Bronstein et al., 2005].

18

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

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


60

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


60

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

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

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

Overview
•Geodesic Meshing

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

Faces F ⊂ {1, . . . , M }3 .
M

m=1
µ
m
ϕm (vi ) = m

There exists (V, F) such that ||f − fM || Cf M −2
Optimal (V, F): NP-hard.

24

Faces F ⊂ {1, . . . , M }3 .
M

m=1
µ
m
ϕm (vi ) = m

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

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

Geodesic Sampling

Metric Sampling

Geodesic Sampling
Farthest point algorithm: [Peyr´, Cohen, 2006]
e
xk+1 = argmax min d(xi , x)
x 0ik

Metric Sampling

Geodesic Sampling
e
x 0ik

Geodesic Voronoi: Metric Sampling
Ci = {x ∀ j = i, d(xi , x) d(xj , x)}

Voronoi

Geodesic Sampling
e
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”.

Adaptive Meshing

# samples

Adaptive Meshing

# samples

Texture Metric Uniform Adaptive

Approximation Driven Meshing
Linear approximation fM with M linear elements.
Minimize approximation error ||f − fM ||Lp .

Isotropic

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

For edges and textures: → use structure tensor.
e

Anisotropic triangulation JPEG2000

→ extension to handle
boundary approximation.
e

Overview
•Inverse Problems with Geodesic
Fidelity
with G.Carlier, F. Santambrogio, F. Benmansour
29

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

γ(0)=x,γ(1)=y


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

γ(0)=x,γ(1)=y


W ∈C
i,j

Example: shape optimization,
Eij (d) = −ρi,j d convex
traﬃc congestion,
Eij (d) = (d − di,j )2 non
seismic imaging, . . . convex

30

γ(0)=x,γ(1)=y


W ∈C
i,j

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

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

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

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

Subgradient Marching
Discretized geodesic distance computed by FM: Ui ≈ dW (x0 , xi )

Theorem: W ∈ RN → Ui ∈ R is concave.

32



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



xk
(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



xk
(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

Landscape Design

max ρi,j dW (xi , xj )
∈C
W

C= W W (x)dx = λ, a W b
Ω

Landscape Design

∈C
W

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

Landscape Design

∈C
W

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

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

Trafﬁc Congestion
y2
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

Trafﬁc Congestion
y2
Q {γ γ(0) = xi , γ(1) = yj } = ρi,j Bε (x)
x1
γ 0
iQ (x) = lim
ε→0 |Bε (x)|

Congested metric: WQ (x) = ϕ(iQ (x)).

34

Trafﬁc Congestion
y2
Q {γ γ(0) = xi , γ(1) = yj } = ρi,j Bε (x)
x1
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

Convex Formulation
Congested metric W = ϕ(iQ ) solution of:

min ϕ(W (x))dx −
¯ ρi,j dW (xi , yj )
W 0
√
i,j
w3
ϕ → ϕ explicit, example: ϕ(i) = i −→ ϕ(w) =
¯ ¯
3

35

Convex Formulation
Congested metric W = ϕ(iQ ) solution of:

min ϕ(W (x))dx −
¯ ρi,j dW (xi , yj )
W 0
√
i,j
w3
ϕ → ϕ explicit, example: ϕ(i) = i −→ ϕ(w) =
¯ ¯
3
Unique solution if ϕ strictly convex.
¯
Algorithm: sub-gradient descent.
Γ-convergence: discrete solution → continuous solution.
x
x s 1s 1 x
x t 1t 1 0.5
0.5

0
0

−0.5
−0.5

−1
−1

−1.5
−1.5

x
x s 2s 2 x
x t 2t 2 −2
−2
35

Traveltime Tomography
Seismic imaging: emitters {xi }i , receivers {yj }j .
Wave propagation: initial conditions localized around xi
Geometric optics approximation: ﬁrst hitting time = dW (xi , yj ).

Emitters {xi }i Receivers {yj }j

W0 (unknown) W0 (unknown)
ξ0 ξ
36

Traveltime Tomography
Seismic imaging: emitters {xi }i , receivers {yj }j .
Wave propagation: initial conditions localized around xi
Geometric optics approximation: ﬁrst hitting time = dW (xi , yj ).
Measurements: di,j = dW0 (xi , yj ) + noise

Non-convex recovery: min |dW (xi , yj ) − di,j |2 + λ ||∇W (x)||2 dx
W
i,j

Emitters {xi }i Receivers {yj }j

W0 (unknown) W (recovered) W0 (unknown)
ξξ 0
W (recovered)
ξξ
0
36

Conclusion
Riemannian tensors encode geometric features.
→ Size, orientation, anisotropy.

37

Conclusion

Using geodesic curves: image segmentation.
Using geodesic distance: image and surface meshing

37

Conclusion

Using geodesic curves: image segmentation.
Using geodesic distance: image and surface meshing

xs 1

Variational minimization:
optimization of the metric.
→ inverse problems,
→ surfaces optimization. x
37s 2

Geodesic Method in Computer Vision and Graphics

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