Mesh Processing Course : Geodesics

Geodesic Data Processing

Gabriel PeyréCEREMADE, Université Paris-Dauphine

www.numerical-tours.com

Local vs. Global Processing

Local Processing Differential Computations

Global Processing Geodesic Computations

Surface filtering

Fourier on Meshes

Front Propagation on Meshes

Surface Remeshing

Overview

•Metrics and Riemannian Surfaces.

• Geodesic Computation - Iterative Scheme

• Geodesic Computation - Fast Marching

• Shape Recognition with Geodesic Statistics

• Geodesic Meshing3

Parametric Surfaces

Parameterized surface: u ⇥ R2 ⇤� �(u) ⇥M.

u2 �⇥�

⇥�

Parametric Surfaces

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

u2 �⇥�

⇥�

� �

Parametric Surfaces

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

u2 �⇥�

⇥�

��

� �

Parametric Surfaces

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

For an embedded manifoldM � Rn:First fundamental form: I� =

�� ⇥�

⇥ui,

⇥�

⇥uj⇥⇥

i,j=1,2

u2 �⇥�

⇥�

L(�) def.=� 1

0||��(t)||dt =

⇥��(t)I�(t)��(t)dt.

Length of a curve

��

� �

Isometric and Conformal

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

Surface not homeomorphic to a disk:

Isometric and Conformal

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

⇥ is conformal: I�(u) = �(u)Id.Exemple: stereographic mapping plane�sphere.

Surface not homeomorphic to a disk:

Riemannian Manifold

Length of a curve �(t) �M: L(�) def.=� 1

⇥��(t)TH(�(t))��(t)dt.

Riemannian manifold: M � Rn (locally)Riemannian metric: H(x) � Rn�n, symmetric, positive definite.

Riemannian Manifold

⇥��(t)TH(�(t))��(t)dt.

Euclidean space: M = Rn, H(x) = Idn.

Riemannian Manifold

⇥��(t)TH(�(t))��(t)dt.

Euclidean space: M = Rn, H(x) = Idn.2-D shape: M � R2, H(x) = Id2.

Riemannian Manifold

⇥��(t)TH(�(t))��(t)dt.

Isotropic metric: H(x) = W (x)2Idn.

Riemannian Manifold

⇥��(t)TH(�(t))��(t)dt.

Image processing: image I, W (x)2 = (� + ||�I(x)||)�1.

Riemannian Manifold

⇥��(t)TH(�(t))��(t)dt.

Parametric surface: H(x) = Ix (1st fundamental form).Image processing: image I, W (x)2 = (� + ||�I(x)||)�1.

Riemannian Manifold

⇥��(t)TH(�(t))��(t)dt.

Parametric surface: H(x) = Ix (1st fundamental form).Image processing: image I, W (x)2 = (� + ||�I(x)||)�1.

DTI imaging: M = [0, 1]3, H(x)=di�usion tensor.

Geodesic Distances

Geodesic distance metric overM � Rn

Geodesic curve: �(t) such that L(�) = dM(x, y).

Distance map to a starting point x0 �M: Ux0(x) def.= dM(x0, x).

dM(x, y) = min�(0)=x,�(1)=y

L(�)

Geodesic Distances

dM(x, y) = min�(0)=x,�(1)=y

L(�)

Euclidean

Geodesic Distances

dM(x, y) = min�(0)=x,�(1)=y

L(�)

Euclidean Shape

Geodesic Distances

dM(x, y) = min�(0)=x,�(1)=y

L(�)

Euclidean Shape Isotropic

Geodesic Distances

dM(x, y) = min�(0)=x,�(1)=y

L(�)

Euclidean Shape Isotropic Anisotropic

Geodesic Distances

dM(x, y) = min�(0)=x,�(1)=y

L(�)

Euclidean Shape Isotropic Anisotropic Surface

Anisotropy and Geodesics

H(x) = �1(x)e1(x)e1(x)T + �2(x)e2(x)e2(x)T with 0 < �1 � �2,Tensor eigen-decomposition:

x e1(x)

�2(x)�12

�1(x)�12

{� \ ��H(x)� � 1}

x e1(x)

�2(x)�12

�1(x)�12

{� \ ��H(x)� � 1}

Geodesics tend to follow e1(x).

Local anisotropy of the metric:

4 ECCV-08 submission ID 1057

Figure 2 shows examples of geodesic curves computed from a single starting77 77

point S = {x1} in the center of the image � = [0, 1]2 and a set of points on the78 78

boundary of �. The geodesics are computed for a metric H(x) whose anisotropy79 79

⇥(x) (defined in equation (2)) is increasing, thus making the Riemannian space80 80

progressively closer to the Euclidean space.81 81

Image f � = .1 � = .2 � = .5 � = 1

Fig. 2. Examples of geodesics for a tensor metric with an increasing anisotropy � (seeequation (2) for a definition of this parameter). The tensor field H(x) is computed fromthe structure tensor of f as defined in equation (8), its eigenvalues fields ⇥i(x) are thenmodified to impose the anisotropy �.

For the particular case of an isotropic metric H(x) = W (x)Idx, the geodesic82 82

distance and the shortest path satisfy83 83

||⇥xUS || = W (x) and ⇤�(t) = � ⇥xUS||⇥xUS ||

This corresponds to the Eikonal equation, that has been used to compute mini-84 84

mal paths weighted by W [1].85 85

1.3 Numerical Computations of Geodesic Distances86 86

In order to make all the previous definitions e�ective in practical situations,87 87

one needs a fast algorithm to compute the geodesic distance map US . The Fast88 88

Marching algorithm, introduced by Sethian [2] is a numerical procedure to e⇥-89 89

ciently solve in O(n log(n)) operations the discretization of equation (6) in the90 90

isotropic case. Several extensions of the Fast Marching have been proposed in91 91

order to solve equation (4) for a generic metric, see for instance Kimmel and92 92

Sethian [3] for triangulated meshes and Spira and Kimel [4], Bronstein et al. [5]93 93

for parametric manifolds.94 94

We use the Fast Marching method developed by Prados et al. [6], which is a95 95

numerical scheme to compute the geodesic distance over a generic parameteric96 96

Riemannian manifold in 2D and 3D in O(n log(n)) operations. As any Fast97 97

Marching method, it computes the distance US by progressively propagating a98 98

front, starting from the initial points in S. Figure 3 shows an example of Fast99 99

Marching computation with an anisotropic metric. The front propagates faster100 100

along the direction of the texture. This is because the Riemannian tensor is101 101

computed following equation (8) in order for the principal direction e1 to align102 102

with the texture patterns.103 103

Image f � = .5 � = 0� = .95 � = .7

�(x) =⇥1(x)� ⇥2(x)⇥1(x) + ⇥2(x)

� [0, 1]

x e1(x)

�2(x)�12

�1(x)�12

{� \ ��H(x)� � 1}

Geodesics tend to follow e1(x).

Isotropic Metric Design

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

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

Isotropic Metric Design

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

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

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

Geodesics

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

Isotropic Metric Design: Vessels

f f W = (� + max(f , 0))��

Remove background: f = G� ⇥ f � f , � ⇥vessel width.

Isotropic Metric Design: Vessels

f f W = (� + max(f , 0))��

3D Volumetric datasets:

Remove background: f = G� ⇥ f � f , � ⇥vessel width.

Overview• Metrics and Riemannian Surfaces.

•Geodesic Computation - Iterative Scheme

Eikonal Equation and Viscosity Solution

U(x) = d(x0, x)Distance map:

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 � between x1 and x0 solves

��(t) = �⇥t H(�(t))�1�Ux0(�(t))�(0) = x1

�t > 0with

where ||v||A =�

v�Av

Geodesic curve � between x1 and x0 solves

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

��(t) = �⇥t H(�(t))�1�Ux0(�(t))�(0) = x1

�t > 0

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

where ||v||A =�

v�Av

Simplified Proof

V solving�

||⇤V (x)||2H�1 = �H�1(x)⇤V (x), ⇤V (x)⇥ = 1,V (x0) = 0.

U(x) = min�:x0�x

L(�) =� 10

��H(�(t))��(t), ��(t)�dt

Simplified Proof

V solving�

||⇤V (x)||2H�1 = �H�1(x)⇤V (x), ⇤V (x)⇥ = 1,V (x0) = 0.

U � V

��, ⇤V ⇥ = �H1/2��, H�1/2⇤V ⇥ � ||H1/2��||||H�1/2⇤V ||C.S.

If V is smooth on �:

Let � : x0 � x be any smooth curve.

L(�) =� 10

��H(�(t))��(t), ��(t)�dt

Simplified Proof

V solving�

||⇤V (x)||2H�1 = �H�1(x)⇤V (x), ⇤V (x)⇥ = 1,V (x0) = 0.

U � V

��, ⇤V ⇥ = �H1/2��, H�1/2⇤V ⇥ � ||H1/2��||||H�1/2⇤V ||

=� U(x) = min�

L(�) � V (x)

L(�) =� 10 ||H1/2��|| �

� 10 ⇥�

�, ⌅V ⇤ = V (�(1))� V (�(0)) = V (x)

If V is smooth on �:

Let � : x0 � x be any smooth curve.

L(�) =� 10

��H(�(t))��(t), ��(t)�dt

Simplified Proof (cont.)

U � VDefine: ��(t) = �H�1(�(t))�V (�(t))

x�(0) = x

Let x be arbitrary.

U � VDefine: ��(t) = �H�1(�(t))�V (�(t))

x�(0) = x

Let x be arbitrary.

dV (�(t))dt

= ��(t), �V (�(t))� = �1

If V is smooth on �([0, tmax)), then

=� �(tmax) = x0

U � VDefine:

��(t) = �H�1(�(t))�V (�(t))

�H��, �� = �H�1�V, �V � = 1

x�(0) = x

Let x be arbitrary.

dV (�(t))dt

= ��(t), �V (�(t))� = �1

If V is smooth on �([0, tmax)), then

=� �(tmax) = x0

= �� tmax

0 ��, �V � = �V (�(tmax)) + V (�(0)) = V (x)

One has:

U(x) � L(�) =� tmax

��H��, �� =

� tmax

0 �H��, ��

Discretization

x0�B(x)Control (derivative-free) formulation:

U(x) = d(x0, x) is the unique solution ofy

U(x) = �(U)(x) = miny�B(x)

U(y) + d(x, y)

Discretization

x0�B(x)

Control (derivative-free) formulation:U(x) = d(x0, x) is the unique solution of

Manifold discretization: triangular mesh.

U discretization: linear finite elements.

H discretization: constant on each triangle.

U(x) = �(U)(x) = miny�B(x)

U(y) + d(x, y)

Discretization

x0�B(x)

Control (derivative-free) formulation:U(x) = d(x0, x) is the unique solution of

Manifold discretization: triangular mesh.

U discretization: linear finite elements.

H discretization: constant on each triangle.

Ui = �(U)i = minf=(i,j,k)

Vi,j,k

Vi,j,k = min0�t�1

tUj + (1� t)Uk

� on regular grid: equivalent to upwind FD.

� explicit solution (solving quadratic equation).

txj + (1� t)xk

+||txj + (1� t)xk � xi||Hijk

U(x) = �(U)(x) = miny�B(x)

U(y) + d(x, y)

Update Step on a triangulation

tUj + (1� t)Uk

�(U)i = minf=(i,j,k)

Vi,j,k

Discrete Eikonal equation:

tUj + (1� t)Uk

Distance function in (i, j, k):

Vi,j,k

Unknowns:

= Vi,j,k

U(x) = �x � xi, g� + d

gradient

tUj + (1� t)Uk

Distance function in (i, j, k):

Vi,j,k

X = (xj � xi, xk � xi) � Rd�2

u = (Uj , Uk) � R2

S = (X�X)�1 � R2�2

I = (1, 1) � R2

Notations:Unknowns:

= Vi,j,k

U(x) = �x � xi, g� + d

gradient

Hi,j,k = w2Id3 (for simplifity)

Update Step on a triangulation (cont.)

� � 0

X�g + dI = u =� � = S(u� dI)

Find g = X�, � � R2 and d = Vi,j,k.

� � 0

X�g + dI = u =� � = S(u� dI)

||�U(xi)|| = ||g|| = w

��

� � 0

X�g + dI = u =�

d2 � 2bd + c = 0

��

a = �SI, I�b = �SI, u�c = �Su, u� � w2=�

� = S(u� dI)

Quadratic equation:

||�U(xi)|| = ||g|| = w

��

||XS(u� dI)||2 = w2

� = b2 � acd =b +�

aAdmissible solution:

dj = Uj + Wi||xi � xj ||�(ui) =�

d if � � 0min(dj , dk) otherwise.

xk�1 � 0

� � 0

X�g + dI = u =�

d2 � 2bd + c = 0

��

a = �SI, I�b = �SI, u�c = �Su, u� � w2=�

� = S(u� dI)

Quadratic equation:

||�U(xi)|| = ||g|| = w

��

||XS(u� dI)||2 = w2

Numerical Schemes

Fixed point equation: U = �(U)� is monotone: U � V =� �(U) � �(V )

U (�+1) = �(U (�))

U (�) � U solving �(U) = U

U (0) = 0,

U (�)

Iterative schemes: ||�(U (�))� U (�)||=� U (�+1) � U (�) � C < +�

Numerical Schemes

Fixed point equation: U = �(U)� is monotone: U � V =� �(U) � �(V )

U (�+1) = �(U (�))

U (�) � U solving �(U) = U

U (0) = 0,

U (�)

Iterative schemes: ||�(U (�))� U (�)||

Minimal path extraction:

�(�+1) = �(�) � ⇥� H(�(�))�1�U(�(�))

=� U (�+1) � U (�) � C < +�

Numerical Examples on Meshes

Discretization Errors

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

Linear interpolation: U [N ](x) =�

U [N ]i �i(x)

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

Continuous geodesic distance U(x).

Discretization Errors

|UNi � U(xi)|2

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

Linear interpolation: U [N ](x) =�

U [N ]i �i(x)

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

Numerical evaluation:

Continuous geodesic distance U(x).

Overview

• Metrics and Riemannian Surfaces.

•Geodesic Computation - Fast Marching

Causal Updates

� j � i, �(U)i � UjCausality condition:

� 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.

Causal Updates

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 = h2W 2

(upwind derivatives)

Isotropic H(x) = W (x)2Id, square grid.

xi+1,j

xi,j+1

Causal Updates

triangulation with no obtuse angles.

Goodu = �(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 = h2W 2

(upwind derivatives)

Isotropic H(x) = W (x)2Id, square grid.

Surface (first fundamental form)

GoodBad

xi+1,j

xi,j+1

Front Propagation

Algorithm: Far � Front � Computed.

2) Move from Front to Computed .

nFront �Ft, Ft = {i \ Ui � t}

State Si � {Computed, Front, Far}

3) Update Uj = �(U)j for neighbors

1) Select front point with minimum Ui

Fast Marching on an Image

Fast Marching on Shapes and Surfaces

Volumetric Datasets

Propagation in 3D

Overview• Metrics and Riemannian Surfaces.

•Shape Recognition with Geodesic Statistics

Bending Invariant Recognition

[Zoopraxiscope, 1876]

Shape articulations:

Bending Invariant Recognition

[Zoopraxiscope, 1876]

Shape articulations:

Surface bendings:

[Elad, Kimmel, 2003]. [Bronstein et al., 2005].

2D Shapes

2D shape: connected, closed compact set S � R2.Piecewise-smooth boundary �S.

Geodesic distance in S for uniform metric:dS(x, y) def.= min

�⇥P(x,y)L(�) where L(�) def.=

0|��(t)|dt,

Distribution of Geodesic Distances

Distribution of distancesto a point x: {dM(x, y)}y�M

Distribution of distances

Extract a statistical measure

to a point x: {dM(x, y)}y�M

Min Median Max

a0(x) = miny

dM(x, y).

a1(x) = mediany

dM(x, y).

a2(x) = maxy

dM(x, y).

Distribution of distances

Extract a statistical measure

to a point x: {dM(x, y)}y�M

Min Median Max

a0(x) = miny

dM(x, y).

a1(x) = mediany

dM(x, y).

a2(x) = maxy

dM(x, y).a(x)

Benging Invariant 2D Database

0 20 40 60 80 1000

Average Recall

0 10 20 30 400

Image Rank

[Ling & Jacobs, PAMI 2007]

(min,med,max)Our method

max only[Ion et al. 2008]

� State of the art retrieval rates on this database.

Perspective: Textured Shapes

33Max Min||�f(x)||

Image f(x)

Euclidean

Weighted

Take into account a texture f(x) on the shape.

Compute a saliency field W (x), e.g. edge detector.

Compute weighted curve lengths: L(�) def.=� 1

0W (�(t))||��(t)||dt.

Overview

• Metrics and Riemannian Surfaces.

•Geodesic Meshing34

Meshing Images, Shapes and Surfaces

Vertices V = {vi}Mi=1.

Faces F � {1, . . . ,M}3.Triangulation (V,F):

fM =M�

�m⇥m

� = argminµ

||f ��

µm⇥m||

Image approximation:

⇥m(vi) = �mi is a�ne on each face of F .

fM =M�

�m⇥m

� = argminµ

||f ��

µm⇥m||

||f � fM || � CfM�2

There exists (V,F) such thatOptimal (V,F): NP-hard.

fM =M�

�m⇥m

� = argminµ

||f ��

µm⇥m||

||f � fM || � CfM�2

There exists (V,F) such thatOptimal (V,F): NP-hard.

Domain meshing:

Conforming to complicated boundary.Capturing PDE solutions:Boundary layers, chocs . . .

Riemannian Sizing Field

Distance conforming:

Sampling {xi}i�I of a manifold.

Triangulation conforming:

⇤xi ⇥ xj , d(xi, xj) � �

� =( xi ⇤ xj ⇤ xk) ⇥�x \ ||x� x�||T (x�) � �

e1(x)e2(x)� �1(x)�

� �2(x)�12

�⇥Building triangulation

Ellipsoid packing

Global integration oflocal sizing field

�⇥

Geodesic Sampling

Metric Sampling

xk+1 = argmaxx

min0�i�k

d(xi, x)

Geodesic Sampling

Metric Sampling

Farthest point algorithm: [Peyre, Cohen, 2006]

xk+1 = argmaxx

min0�i�k

d(xi, x)

Geodesic Sampling

Metric

Voronoi

Sampling

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

xk+1 = argmaxx

min0�i�k

d(xi, x)

� distance conforming. � triangulation conforming if the metric is “gradded”.

� geodesic Delaunay refinement.

Geodesic Sampling

Metric

DelaunayVoronoi

Sampling

Geodesic Delaunay connectivity:

(xi � xj)⇥ (Ci ⇧ Cj ⇤= ⌅)

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

Adaptive Meshing

# samples

Texture Metric Uniform Adaptive

Adaptive Meshing

# samples

Minimize approximation error ||f � fM ||Lp .

Isotropic

Approximation Driven MeshingLinear approximation fM with M linear elements.

Images: T (x) = |H(x)| (Hessian)Surfaces: T (x) = |C(x)| (curvature tensor)

Isotropic Anisotropic

L� optimal metrics for smooth functions:

Anisotropic triangulation JPEG2000

For edges and textures: � use structure tensor.[Peyre et al, 2008]

boundary approximation.

Anisotropic triangulation JPEG2000

For edges and textures: � use structure tensor.

� extension to handle

[Peyre et al, 2008]

ConclusionRiemannian tensors encode geometric features.� Size, orientation, anisotropy.

Computing geodesic distance:iterative vs. propagation.

ConclusionRiemannian tensors encode geometric features.� Size, orientation, anisotropy.

Using geodesic curves: image segmentation.

Using geodesic distance: image and surface meshing

Computing geodesic distance:iterative vs. propagation.

Mesh Processing Course : Geodesics

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