Mesh Processing Course : Geodesics

Geodesic Data Processing

www.numerical-tours.com

Gabriel Peyré
CEREMADE, Université Paris-Dauphine

Local vs. Global Processing
Local Processing Global Processing
Differential Computations Geodesic Computations

Surface ﬁltering Front Propagation on Meshes

Fourier on Meshes Surface Remeshing
2

Overview
• Metrics and Riemannian Surfaces.
• Geodesic Computation - Iterative Scheme
• Geodesic Computation - Fast Marching
• Shape Recognition with Geodesic Statistics
• Geodesic Meshing
3

Parametric Surfaces
Parameterized surface: u ⇥ R2 ⇤ (u) ⇥ M.

u1 ⇥
u2 ⇥u1

⇥
⇥u2

4

Parametric Surfaces

u1 ⇥
u2 ⇥u1

⇥
⇥u2

Curve in parameter domain: t ⇥ [0, 1] ⇤ (t) ⇥ D.

4

Parametric Surfaces

u1 ⇥
u2 ⇥u1
¯
¯ ⇥
⇥u2

def.
Geometric realization: ¯ (t) = ⇥( (t)) M.

4

Parametric Surfaces

u1 ⇥
u2 ⇥u1
¯
¯ ⇥
⇥u2

def.
Geometric realization: ¯ (t) = ⇥( (t)) M.

For an embedded manifold M Rn : ⇥
⇥ ⇥
First fundamental form: I = , ⇥ .
⇥ui ⇥uj i,j=1,2
Length of a curve
1 1 ⇥
def.
L( ) = ||¯ (t)||dt = (t)I (t) (t)dt.
0 0 4

Isometric and Conformal
Surface not homeomorphic to a disk:

M is locally isometric to the plane: I = Id.
Exemple: M =cylinder.

Isometric and Conformal
Surface not homeomorphic to a disk:

M is locally isometric to the plane: I = Id.
Exemple: M =cylinder.
⇥ is conformal: I (u) = (u)Id.
Exemple: stereographic mapping plane sphere.

Riemannian Manifold
Riemannian manifold: M Rn (locally)
Riemannian metric: H(x) Rn , symmetric, positive deﬁnite.
n
1⇥
def. T
Length of a curve (t) M: L( ) = (t) H( (t)) (t)dt.
0

6

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

W (x)

6

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

W (x)

6

Riemannian Manifold
n
1⇥
def. T
0
Isotropic metric: H(x) = W (x)2 Idn .

W (x)

6

Riemannian Manifold
n
1⇥
def. T
0
Image processing: image I, W (x)2 = ( + || I(x)||) 1
.

W (x)

6

Riemannian Manifold
n
1⇥
def. T
0
.
Parametric surface: H(x) = Ix (1st fundamental form).

W (x)

6

Riemannian Manifold
n
1⇥
def. T
0
.
Parametric surface: H(x) = Ix (1st fundamental form).
DTI imaging: M = [0, 1]3 , H(x)=di usion tensor.

W (x)

6

Geodesic Distances
Geodesic distance metric over M Rn
dM (x, y) = min L( )
(0)=x, (1)=y

Geodesic curve: (t) such that L( ) = dM (x, y).
def.
Distance map to a starting point x0 M: Ux0 (x) = dM (x0 , x).

7

Geodesic Distances
(0)=x, (1)=y

def.
metric
geodesics

Euclidean 7

Geodesic Distances
(0)=x, (1)=y

def.
metric
geodesics

Euclidean Shape 7

Geodesic Distances
(0)=x, (1)=y

def.
metric
geodesics

Euclidean Shape Isotropic 7

Geodesic Distances
(0)=x, (1)=y

def.
metric
geodesics

Euclidean Shape Isotropic Anisotropic 7

Geodesic Distances
(0)=x, (1)=y

def.
metric
geodesics

Euclidean Shape Isotropic Anisotropic Surface 7

Anisotropy and Geodesics
Tensor eigen-decomposition:
T T
H(x) = 1 (x)e1 (x)e1 (x) + 2 (x)e2 (x)e2 (x) with 0 < 1 2,
{ H(x) 1}
e2 (x)
2 (x)
1
2 x e1 (x)
1
M 1 (x) 2

8

T T
{ H(x) 1}
e2 (x)
2 (x)
1
2 x e1 (x)
1
M 1 (x) 2

Geodesics tend to follow e1 (x).

8

T T
{ H(x) 1}
4 ECCV-08 submission ID 1057
e2 (x)
2 (x)
1
2 x e1 (x)
Figure 2 shows examples of geodesic curves computed from a single starting
1
point (x)
MS = {x1 } in the center of the image = [0,11]2 and a set of points on the
2

boundary of . The geodesics are computed for a metric H(x) whose anisotropy
⇥(x) (deﬁned in equation (2)) is to follow e1 (x).making the Riemannian space
Geodesics tend increasing, thus
progressively closer to the Euclidean space. ⇥1 (x) ⇥2 (x)
Local anisotropy of the metric: (x) = [0, 1]
⇥1 (x) + ⇥2 (x)

Image f
Image f = .1
= .95 = .2
= .7 = .5
= .5 = 10
= 8

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

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

9

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

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

Gradient-based potential: W (x) = ( + || x f ||)

Image f Metric W (x) U{x0 ,x1 } Geodesics 9

Isotropic Metric Design: Vessels
˜
Remove background: f = G ⇥ f f, ⇥vessel width.

f ˜
f ˜
W = ( + max(f , 0))

10

Isotropic Metric Design: Vessels
˜
Remove background: f = G ⇥ f f, ⇥vessel width.

f ˜
f ˜
W = ( + max(f , 0))

3D Volumetric datasets:

10

Overview
• Geodesic Computation - Iterative
Scheme

• Geodesic Meshing 11

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

12


|| U (x)||H(x) 1 = 1 with U (x0 ) = 0
where ||v||A = v Av

Geodesic curve between x1 and x0 solves
(0) = x1
(t) = ⇥t H( (t)) 1
Ux0 ( (t)) with
t >0

12


|| U (x)||H(x) 1 = 1 with U (x0 ) = 0
where ||v||A = v Av

Geodesic curve between x1 and x0 solves
(0) = x1
(t) = ⇥t H( (t)) 1
Ux0 ( (t)) with
t >0

Example: isotropic metric H(x) = W (x)2 Idn ,

|| U (x)|| = W (x) and (t) = ⇥t U ( (t))
12

Simpliﬁed Proof
1
U (x) = min L( ) = 0
H( (t)) (t), (t) dt
:x0 x
||⇤V (x)||2 1 = H 1
(x)⇤V (x), ⇤V (x)⇥ = 1,
V solving H
V (x0 ) = 0.

Simpliﬁed Proof
1
U (x) = min L( ) = 0
H( (t)) (t), (t) dt
:x0 x
||⇤V (x)||2 1 = H 1
(x)⇤V (x), ⇤V (x)⇥ = 1,
V solving H
V (x0 ) = 0.

U V Let : x0 x be any smooth curve.
If V is smooth on :
C.S.
, ⇤V ⇥ = H 1/2 , H 1/2
⇤V ⇥ ||H 1/2 ||||H 1/2
⇤V ||
=1

Simpliﬁed Proof
1
U (x) = min L( ) = 0
H( (t)) (t), (t) dt
:x0 x
||⇤V (x)||2 1 = H 1
(x)⇤V (x), ⇤V (x)⇥ = 1,
V solving H
V (x0 ) = 0.

U V Let : x0 x be any smooth curve.
If V is smooth on :
C.S.
, ⇤V ⇥ = H 1/2 , H 1/2
⇤V ⇥ ||H 1/2 ||||H 1/2
⇤V ||
=1
1 1
L( ) = 0
||H 1/2 || 0
⇥ , ⌅V ⇤ = V ( (1)) V ( (0)) = V (x)
=0
= U (x) = min L( ) V (x)

Simpliﬁed Proof (cont.)
U V Let x be arbitrary.
Deﬁne: (t) = H 1
( (t)) V ( (t))
(0) = x
x

x0

Deﬁne: (t) = H 1
( (t)) V ( (t))
(0) = x
x
If V is smooth on ([0, tmax )), then
dV ( (t))
= (t), V ( (t)) = 1
dt x0
= (tmax ) = x0

Deﬁne: (t) = H 1
( (t)) V ( (t))
(0) = x
x
If V is smooth on ([0, tmax )), then
dV ( (t))
= (t), V ( (t)) = 1
dt x0
= (tmax ) = x0
One has: H , = H V, V = 1
1
=1
tmax tmax
U (x) L( ) = 0
H , = 0 H ,
tmax
= 0
, V = V ( (tmax )) + V ( (0)) = V (x)
=0 13
14

Discretization
x0
Control (derivative-free) formulation:
B(x) y
U (x) = d(x0 , x) is the unique solution of
U (x) = (U )(x) = min U (y) + d(x, y) x
y B(x)

15

Discretization
x0
B(x) y
U (x) = (U )(x) = min U (y) + 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

15

Discretization
x0
B(x) y
U (x) = (U )(x) = min U (y) + 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 = (U )i = min Vi,j,k
f =(i,j,k) xj
Vi,j,k = min tUj + (1 t)Uk xi
0 t 1 xk
+||txj + (1 t)xk xi ||Hijk
explicit solution (solving quadratic equation).
txj + (1 t)xk
on regular grid: equivalent to upwind FD. xj 15

Update Step on a triangulation
(U )i = min Vi,j,k
f =(i,j,k)

Discrete Eikonal equation:
Vi,j,k = min tUj + (1 t)Uk +||txj + (1 t)xk xi ||Hijk
0 t 1

xi
xk

xj
16

(U )i = min Vi,j,k
f =(i,j,k)

0 t 1

Distance function in (i, j, k): U (x) = x xi , g + d

Unknowns: gradient = Vi,j,k

xi
xk

xj g
16

(U )i = min Vi,j,k
f =(i,j,k)

0 t 1

Distance function in (i, j, k): U (x) = x xi , g + d

Unknowns: gradient = Vi,j,k
Notations:
Hi,j,k = w2 Id3 (for simpliﬁty)
xi
X = (xj xi , xk xi ) R d 2

xk
u = (Uj , Uk ) R2
I = (1, 1) R2
xj g
S = (X X) 1
R 2 2
16

Update Step on a triangulation (cont.)
Find g = X , R2 and d = Vi,j,k . xi
X g + dI = u = = S(u dI) xk

xj 0

17


|| U (xi )|| = ||g|| = w xj 0

17


|| U (xi )|| = ||g|| = w xj 0

Quadratic equation:
a = SI, I
||XS(u dI)||2 = w2 b = SI, u
= c = Su, u w2
d 2
2bd + c = 0

17

0
X g + dI = u = = S(u dI) 1 xk

|| U (xi )|| = ||g|| = w xj 0

Quadratic equation:
a = SI, I
||XS(u dI)||2 = w2 b = SI, u
= c = Su, u w2
d 2
2bd + c = 0
b+
Admissible solution: d= = b2 ac
a

d if 0 dj = Uj + Wi ||xi xj ||
(ui ) =
min(dj , dk ) otherwise.
17

Numerical Schemes
Fixed point equation: U = (U )
is monotone: U V = (U ) (V )

Iterative schemes: U (0) = 0, U ( +1)
= (U ( ) ) || (U ( ) ) U ( ) ||
= U( +1)
U( )
C<+
U( )
U solving (U ) = U

U( )

18

Numerical Schemes
Fixed point equation: U = (U )
is monotone: U V = (U ) (V )

Iterative schemes: U (0) = 0, U ( +1)
= (U ( ) ) || (U ( ) ) U ( ) ||
= U( +1)
U( )
C<+
U( )
U solving (U ) = U
Minimal path extraction:
( +1)
= ( )
⇥ H( ( )
) 1
U( ( )
)
U( )

18

Numerical Examples on Meshes

19

Discretization Errors
For a mesh with N points: U [N ] RN solution of (U [N ] ) = U [N ]

Continuous geodesic distance U (x).

Linear interpolation: ˜
U [N ] (x) =
[N ]
Ui i (x)
i

Uniform convergence: ˜
||U [N ] U ||
N +
⇥ 0

20

Discretization Errors
For a mesh with N points: U [N ] RN solution of (U [N ] ) = U [N ]

Continuous geodesic distance U (x).

Linear interpolation: ˜
U [N ] (x) =
[N ]
Ui i (x)
i

Uniform convergence: ˜
||U [N ] U ||
N +
⇥ 0

1
Numerical evaluation: |UiN U (xi )|2
N i

20

Overview
21

Causal Updates
Causality condition: j i, (U )i Uj
The value of Ui depends on {Uj }j with Uj Ui .
Compute (U )i using an optimal ordering.
Front propagation, O(N log(N )) operations.

22

Causal Updates
Causality condition: j i, (U )i Uj xi,j+1
Compute (U )i using an optimal ordering. xi,j xi+1,j

Isotropic H(x) = W (x)2 Id, square grid.
u = (U )i is the solution of
max(u Ui 1,j , u Ui+1,j , 0)2 +
max(u Ui,j 1, u Ui,j+1 , 0)2 = h2 Wi,j
2

(upwind derivatives)

22

Causal Updates
Causality condition: j i, (U )i Uj xi,j+1
Compute (U )i using an optimal ordering. xi,j xi+1,j

Isotropic H(x) = W (x)2 Id, square grid.
u = (U )i is the solution of Good
max(u Ui 1,j , u Ui+1,j , 0)2 +
max(u Ui,j 1, u Ui,j+1 , 0)2 = h2 Wi,j
2
xi Bad
xk
(upwind derivatives)
xj
Surface (ﬁrst fundamental form)
triangulation with no obtuse angles.
Bad Good 22

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 = (U )j for neighbors and
23

Fast Marching on an Image

24

Fast Marching on Shapes and Surfaces

25

Volumetric Datasets

26

Propagation in 3D

27

Overview
• Shape Recognition with Geodesic
Statistics

• Geodesic Meshing 28

Bending Invariant Recognition
Shape articulations:

[Zoopraxiscope, 1876]

29

Bending Invariant Recognition
Shape articulations:

[Zoopraxiscope, 1876]

Surface bendings:
˜
x1

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

29

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( ) where L( ) = | (t)|dt,
⇥P(x,y) 0
Shape S
Geodesics

30

Distribution of Geodesic Distances
Distribution of distances 80

60

to a point x: {dM (x, y)}y
40

M 20

0

80
60
40
20
0

80
60
40
20
0

31


60

40

M 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 31


60

40

M 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 31

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.
32

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( ) = W ( (t))|| (t)||dt.
0

Euclidean
Image f (x)

Weighted
|| f (x)|| Max Min 33

Overview
34

Meshing Images, Shapes and Surfaces
Vertices V = {vi }M .
Triangulation (V, F): i=1
Faces F {1, . . . , M }3 .
M

Image approximation: fM = m ⇥m
m=1
= argmin ||f µm ⇥m ||
µ
m
⇥m (vi ) = m
i is a ne on each face of F.

35

Faces F {1, . . . , M }3 .
M

m=1
µ
m
⇥m (vi ) = m

There exists (V, F) such that ||f fM || Cf M 2

Optimal (V, F): NP-hard.

35

Faces F {1, . . . , M }3 .
M

m=1
µ
m
⇥m (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 . . .
35

Riemannian Sizing Field
Sampling {xi }i I of a manifold.

Distance conforming:
⇤ xi ⇥ xj , d(xi , xj ) e1 (x)
1
1 (x) 2 e2 (x)
Triangulation conforming: x
⇥
=( xi ⇤ xj ⇤ xk ) ⇥ x ||x x ||T (x ) 2 (x)
1
2

Building triangulation
⇥
Ellipsoid packing
⇥
Global integration of
local sizing ﬁeld
36

Geodesic Sampling

Metric Sampling

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

Metric Sampling

Geodesic Sampling
e
x 0 i k

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

Voronoi

Geodesic Sampling
e
x 0 i k

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

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

Computing geodesic distance:
iterative vs. propagation.

40

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

Computing geodesic distance:
iterative vs. propagation.

Using geodesic curves: image segmentation.

Using geodesic distance: image and surface meshing

40

Mesh Processing Course : Geodesics

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