Mesh Processing Course : Mesh Parameterization

Parameterization and Flattening

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

Mesh Parameterization - Overview

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ma ture
pp
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M R3

parameterization

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Mesh Parameterization - Overview

tex
ma ture
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R3

re-
M

s am
1

pli
parameterization

ng

zoom
D R2 2

Overview
• Dirichlet Energy on Meshes
• Harmonic Parameterization
• Spectral Flattening
• Barycentric Coordinates for Warping
• Geodesic Flattening
• High Dimensional Data Analysis 3

Local Averaging
>0 if j Vi ,
Local operator: W = (wij )i,j V where wij =
0 otherwise.
(W f )i = wij fj .
(i,j) E

Examples: for i j,
wij = 1 wij = 1
||xj xi ||2 wij = cot( ij ) + cot(⇥ij )
combinatorial distance conformal
(explanations later)

˜ wij
Local averaging operator W = (wij )i,j
˜ V : ⇥ (i, j) E, wij =
˜ .
(i,j) E wij
˜
W =D 1
W with D = diagi (di ) where di = wij .
(i,j)⇥E
˜
Averaging: W 1 = 1.
4

Voronoi and Dual Mesh
Deﬁnition for a planar triangulation M of a mesh M R2 .

Voronoi for vertices: ⇧ i ⇤ V, Ei = {x ⇤ M ⇧ j ⌅= i, ||x xi || ⇥ ||x xj ||}

Voronoi for edges: ⌅ e = (i, j) ⇥ E, Ee = {x ⇥ M ⌅ e ⇤= e, d(x, e) d(x, e )}

Partition of the mesh: M = Ei = Ee .
i V e E

Ai
i i

cf A(ij)
cf
j j

Dual mesh 1:3 subdivided mesh 5

Approximating Integrals on Meshes
Approximation of integrals on vertices and edges:
⇥
f (x)dx Ai f (xi ) Ae f ([xi , xj ]).
M i V e=(i,j) E
i

T heorem : ⇥ e = (i, j) E, A(ij)
cf
1
Ae = Area(Ee ) = ||xi xj ||2 (cot( ij ) + cot(⇥ij )) j
2

Approximating Integrals on Meshes
Approximation of integrals on vertices and edges:
⇥
f (x)dx Ai f (xi ) Ae f ([xi , xj ]).
M i V e=(i,j) E
i

T heorem : ⇥ e = (i, j) E, A(ij)
cf
1
Ae = Area(Ee ) = ||xi xj ||2 (cot( ij ) + cot(⇥ij )) j
2
A
Proof: + + =
2
||AB||
A(ABO) = ||AB|| h = ||AB|| tan( )
2 O
h
2
||AB|| ⇤
A(ABO) = tan ( + ⇥) C
2 2
B

Cotangent Weights
Sobolev norm (Dirichlet energy): J(f ) = || f (x)||2 dx
M

Cotangent Weights ⇧ (i, j) ⇤ E, wij = 1.
Distance weights: they depends both on the geometry and the to
require faces information,
Sobolev norm (Dirichlet energy): J(f ) = 2 1
|| f (x)|| dx
⇧ (i, j) ⇤ E, wij =
||xj xi ||2
.
M
Conformal weights: they depends on the full geometrical realiza
require the face information
Approximation of Dirichelet energy: ⇧ (i, j) ⇤ E, wij = cot( ij ) + cot(⇥
⇥
|f (xj ) f (xi )|2
Figure 1.2 shows the geometrical meaning of the angles ij and
||⇤x f || dx ⇥
2
Ae |(Gf )e | = 2
Ae
ij = ⇥(xi , xj , xk1 )
||xj xand ⇥ij = ⇥(xi
2
M i ||
e E (i,j) E
where (i, j, k1 ) ⇤ F and (i, j, k2 ) ⇤ F are the two faces adjacent
= wij |fnext ) f (xiexplanation of these celebrated cotangent
in the
(xj section the )|2
(i,j) E

where wij = cot( ij ) + cot(⇥ij ).
xi

ij
xk1
ij
xk2
xj 7

Cotangent Weights ⇧ (i, j) ⇤ E, wij = 1.
Distance weights: they depends both on the geometry and the to
require faces information,
Sobolev norm (Dirichlet energy): J(f ) = 2 1
|| f (x)|| dx
⇧ (i, j) ⇤ E, wij =
||xj xi ||2
.
M
Conformal weights: they depends on the full geometrical realiza
require the face information
Approximation of Dirichelet energy: ⇧ (i, j) ⇤ E, wij = cot( ij ) + cot(⇥
⇥
|f (xj ) f (xi )|2
Figure 1.2 shows the geometrical meaning of the angles ij and
||⇤x f || dx ⇥
2
Ae |(Gf )e | = 2
Ae
ij = ⇥(xi , xj , xk1 )
||xj xand ⇥ij = ⇥(xi
2
M i ||
e E (i,j) E
where (i, j, k1 ) ⇤ F and (i, j, k2 ) ⇤ F are the two faces adjacent
= wij |fnext ) f (xiexplanation of these celebrated cotangent
in the
(xj section the )|2
(i,j) E

where wij = cot( ij ) + cot(⇥ij ).
xi

xk1
T heorem : wij > 0 ⇥ ij + ⇥ij < ⇤ ij
ij
xk2
xj 7

Overview

Mesh Parameterization
Parameterization: bijection : M ⇤ D ⇥ R2 .
Hypothesis: =( 1, 2) is smooth, minimizes
min0 wi,j (| 1 (i) 1 (j)|
2
+| 2 (i) 2 (j)| )
2

(i,j) E

With boundary conditions 0
: ⇥ xi ⇥M, (i) = 0
(i) ⇥D.

3D space (x,y,z)
2D parameter domain (u,v)

boundary
boundary 9

min0 wi,j (| 1 (i) 1 (j)|
2
+| 2 (i) 2 (j)| )
2

(i,j) E

: ⇥ xi ⇥M, (i) = 0
(i) ⇥D.

⇥ i / ⇥M, (L 1 )(i) = (L 2 )(i)
=0
Optimality conditions: ⇥ i ⇥M, (i) = 0 (i) ⇥D.
⇥ sparse linear system to solve. 3D space (x,y,z)

boundary
boundary 9

min0 wi,j (| 1 (i) 1 (j)|
2
+| 2 (i) 2 (j)| )
2

(i,j) E

: ⇥ xi ⇥M, (i) = 0
(i) ⇥D.

⇥ i / ⇥M, (L 1 )(i) = (L 2 )(i)
=0
Optimality conditions: ⇥ i ⇥M, (i) = 0 (i) ⇥D.
⇥ sparse linear system to solve. 3D space (x,y,z)
Remark: each point is the
average of its neighbors:

⇥ i / ⇥M, (i) = wi,j (j).
˜
(i,j) E

Theorem: (Tutte) if i, j, wij > 0,
then is a bijection.
boundary
boundary 9

Examples of Parameterization

Combinatorial
Conformal

Mesh

10

Examples of Parameterization

Combinatorial
Conformal

Mesh

11

Application to Remeshing

parameterization

re-sampling
1

zoom

P. Alliez et al., Isotropic Surface Remeshing, 2003. 12

Application to Texture Mapping

texture g(u)
pa
ra
m
et
er
iza
tio
n

color g( (x))
13

Mesh Parameterization #1
: S0 ⇥ S ⇥x S0 ⇥S0 , =0
S0 ⇥ S S

S0

14

: S0 ⇥ S ⇥x S0 ⇥S0 , =0
S0 ⇥ S S
Discretization of :
W = make_sparse(n,n);
for i=1:3
i1 = mod(i-1,3)+1; i2 = mod(i ,3)+1; i3 = mod(i+1,3)+1;
pp = vertex(:,faces(i2,:)) - vertex(:,faces(i1,:));
qq = vertex(:,faces(i3,:)) - vertex(:,faces(i1,:));
% normalize the vectors
pp = pp ./ repmat( sqrt(sum(pp.^2,1)), [3 1] );
qq = qq ./ repmat( sqrt(sum(qq.^2,1)), [3 1] );
% compute angles
a = 1 ./ tan( acos(sum(pp.*qq,1)) );
a = max(a, 1e-2); % avoid degeneracy
W = W + make_sparse(faces(i2,:),faces(i3,:), a, n, n );
end
S0

14

: S0 ⇥ S ⇥x S0 ⇥S0 , =0
S0 ⇥ S S
Discretization of :
for i=1:3
% compute angles
end
S0
Formation of the linear system:
D = spdiags(full( sum(W,1) ), 0, n,n);
L = D - W; L1 = L; L1(boundary,:) = 0;
L1(boundary + (boundary-1)*n) = 1;

14

: S0 ⇥ S ⇥x S0 ⇥S0 , =0
S0 ⇥ S S
Discretization of :
for i=1:3
% compute angles
end
S0
Formation of the linear system:
D = spdiags(full( sum(W,1) ), 0, n,n);
L = D - W; L1 = L; L1(boundary,:) = 0;
L1(boundary + (boundary-1)*n) = 1;

Formation of the RHS and resolution:
Rx = zeros(n,1); Rx(boundary) = x0;
Ry = zeros(n,1); Ry(boundary) = y0;
x = L1 Rx; y = L1 Ry;
14

Geometry image: re-sample X/Y/Z coordinates of on a grid.
store the surface as a color (R/G/B) image.

Exercise: perform the linear interpolation of the parameterization.

Exercise: display the geometry image using a checkboard texture.

15

Exercise: Locate the position of the eyes / the mouth
in the texture and on the mesh.

Exercise: Compute an a ne transformation to re-align the texture.

16

Mesh Deformations
Initial position: xi R3 .
Displacement of anchors:
i I, xi xi = xi + i R3
I xi
Linear deformation: xi
xi = xi + (i)
i I, (i) = i

i / I, (i) = 0

17

Mesh Deformations
Initial position: xi R3 . % modify Laplacian
L1 = L; L1(I,:) = 0;

Displacement of anchors: L1(I + (I-1)*n) = 1;
% displace vertices
vertex = vertex + ( L1 Delta0' )';
i I, xi xi = xi + i R 3

I xi
xi = xi + (i)
i I, (i) = i

i / I, (i) = 0

17

Mesh Deformations
Initial position: xi R3 . % modify Laplacian
L1 = L; L1(I,:) = 0;

Displacement of anchors: L1(I + (I-1)*n) = 1;
% displace vertices
vertex = vertex + ( L1 Delta0' )';
i I, xi xi = xi + i R 3

I xi
xi = xi + (i)
i I, (i) = i

i / I, (i) = 0
˜
ni xi
Non-linear deformation:
xi = xi + i
˜ ˜
xi
coarse details
scale
˜ ˜
xi
Extrusion along normals: xi = xi + i, ˜
n i ni
17

Overview

Mesh Flattening
No boundary condition, minimize:
⇥ || 1 ||= 1,
minn ||G 1 || =
2
wi,j | 1 (i) 1 (j)|
2
with
1 ⇥R
i j
⇥ 1 , 1⇤ = 0.

⇧
⌅ ||
⇤ 2 ||= 1,
minn ||G 2 ||
2
= wi,j | 2 (i) 2 (j)|
2
with ⇥ 2 , 1 ⇤ = 0,
2 ⇥R ⌅
⇥
2 , 1⇤ = 0.
i j ⇥

19

Mesh Flattening
No boundary condition, minimize:
⇥ || 1 ||= 1,
minn ||G 1 || =
2
wi,j | 1 (i) 1 (j)|
2
with
1 ⇥R
i j
⇥ 1 , 1⇤ = 0.

⇧
⌅ ||
⇤ 2 ||= 1,
minn ||G 2 ||
2
= wi,j | 2 (i) 2 (j)|
2
with ⇥ 2 , 1 ⇤ = 0,
2 ⇥R ⌅
⇥
2 , 1⇤ = 0.
i j ⇥
Theorem: ⇥i = i+1 L⇥i ,

where 0 =0 1 2 ... n 1 are eigenvalues of L = G⇥ G.

( 1 (i), 2 (i)) R2

combinatorial conformal 19

Proof
Spectral decomposition: L=G G=D W =U U
= diag( i ) where 0 = 1 < 2 ... n

U = (ui )n orthonormal basis of Rn .
i=1 u1 = 1

20

Proof
= diag( i ) where 0 = 1 < 2 ... n

i=1 u1 = 1
n
E(⇥) = ||G⇥||2 = i | ⇥, ui ⇥|2
i=1 n
If , 1 = 0, then E(⇥) = i ai where ai = | ⇥, ui ⇥|2
i=2

20

Proof
= diag( i ) where 0 = 1 < 2 ... n

i=1 u1 = 1
n
E(⇥) = ||G⇥||2 = i | ⇥, ui ⇥|2
i=1 n
If , 1 = 0, then E(⇥) = i ai where ai = | ⇥, ui ⇥|2
i=2
n
Constrained minimization: Pnmin i ai
i=2 ai =1
i=2
linear program: minimum reached at a = i.
±u2 = argmin E( )
, 1⇥=0,|| || 20

Flattening Examples
Main issue: No guarantee of being valid (bijective).

combinatorial conformal
21

Overview

Barycentric Coordinates
x1
x
x3

x2
˜ i (x) = A(x, xi+1 , xi+2 )
A(x1 , x2 , x3 )

23

x
x1 1
x2
x x
x3

x2 x3
˜ i (x) = A(x, xi+1 , xi+2 )
A(x1 , x2 , x3 )
i (x)
Barycentric coordinates: { i (x)}i I Normalized: ˜ i (x) =
j j (x)
Positivity: i (x) 0.
˜
Interpolation: ⇥i (xj ) = i,j

Reproduction of a ne functions: ˜ i (x)xi = x
i I

23

x
x1 1
x2
x x
x3

x2 x3
˜ i (x) = A(x, xi+1 , xi+2 )
A(x1 , x2 , x3 )
i (x)
j j (x)
˜

i I

Application: interpolation of data {fi }i f (x) = ˜ i (x)fi
I
i I

23

x
x1 1
x2 xi
x x
x3

x2 x3 xj
˜ i (x) = A(x, xi+1 , xi+2 )
A(x1 , x2 , x3 )
i (x)
j j (x)
˜

i I

Application: interpolation of data {fi }i f (x) = ˜ i (x)fi
I
i I
Application: mesh parameterization: wi,j = i (xj )
23

Mean Value Coordinates
Conformal Laplacian weights:
⇥i (x) = cotan( i (x)) + cotan( ˜i (x)) xi+1
not necessarily positive.
xi i i
Mean-value coordinates: x
˜i
tan( i (x)/2) + tan(˜ i (x)/2) ˜i
⇥i (x) =
||x xi ||
valid coordinates. xi
extend to non-convex coordinates (oriented angles).

˜ 1 (x) ˜ 2 (x)

24

Barycentric Coordinates for Warping
Cage C: polygon with vertices {xi }i I .
Data points: {yj }j J C. Example: textured grid, 3D model, etc.
Initialization: data anchoring, compute
j J, i I, i,j = i (yj ).

Satisﬁes: yi = i,j xi
i I

Cage warping: xi xi
Data warping: yj yj = i,j xi
i I

xi y
xi yi i

25

Harmonic Coordinates
Mean value coordinates:

“non-physical” behavior,
passes “through” the cage.

Harmonic mapping:
⇥ x C, i (x) = 0.

Boundary conditions:
⇥x ⇥C, i (x) = i (x).
0
Mean value Harmonic

26

Warping Comparison

Initial shape Mean value Harmonic

27

Overview

Geodesic Distances 1
def.
Length of a curve (t) M: L( ) = W ( (t))|| (t)||dt.
0
Geodesic distance: dM (x, y) = min L( )
(0)=x, (1)=y

Geodesic curve : dM (x, y) = L( )

Euclidean Shape Isotropic W = 1 Surface 29

Computation of Geodesic Distances
Distance map to a point: Ux0 (x) = dM (x0 , x).
Ux0
Non-linear PDE: || Ux0 (x)|| = W (x)
(viscosity) Ux0 (x0 ) = 0, x0
Upwind ﬁnite di erences approximation.
Fast Marching: front propagation in O(N log(N )) operations.

30

Manifold Flattening
Input manifold M, dM geodesic distance on M.

˜
Input geodesic distance matrix: D = (dM (˜i , xj )2 )i,j , for xi
x ˜ ˜ M.

Flattening: ﬁnd X = (xi )p ⇤ Rn
i=1
p
such that ||xi xj || ⇥ dM (˜i , xj ).
x ˜

˜
x1 x1
x1
˜
x1

x2
x2 ˜
x2
˜
x2
M R3
M R2
Surface parameterization Bending invariant

31

Stress Minimization
Geodesic stress: S(X) = |||xi xj || di,j |2 , di,j = dM (˜i , xj )
x ˜
i,j
1 ()
SMACOF algorithm: X ( +1)
= X B(X ( ) )
N di,j
where B(X)i,j =
||xi xj ||
Non-convex functional : X ( )
X local minimizer of S.

32

Projection on Distance Matrices
D(X)i,j = ||xi xj ||2
min |||xi xj ||2 d2 |2 = ||D(X)
i,j D||2 , Di,j = d2
X1=0 i,j
i,j

||xi xj ||2 = ||xi ||2 + ||xj ||2 2⇥xi , xj ⇤
=⇥ D(X) = d1T + 1d 2X T X where d = (||xi ||2 )i ⇤ Rn

33

D(X)i,j = ||xi xj ||2
min |||xi xj ||2 d2 |2 = ||D(X)
i,j D||2 , Di,j = d2
X1=0 i,j
i,j

||xi xj ||2 = ||xi ||2 + ||xj ||2 2⇥xi , xj ⇤

Centering matrix: J = Idn 11T /N
1 JX = X
For centered points: JD(X)J = X T X
2 J1 = 0

33

D(X)i,j = ||xi xj ||2
min |||xi xj ||2 d2 |2 = ||D(X)
i,j D||2 , Di,j = d2
X1=0 i,j
i,j

||xi xj ||2 = ||xi ||2 + ||xj ||2 2⇥xi , xj ⇤

Centering matrix: J = Idn 11T /N
1 JX = X
For centered points: JD(X)J = X T X
2 J1 = 0
Replace ||D(X) D|| by
min || J(D(X) D)J/2|| = ||X T X + JDJ/2||
X

Explicit solution: diagonalize 1
2 JDJ = U UT i i 1

= diag( 0 , . . . , k 1 ),
k
X = k Uk T
Uk = (u0 , . . . , uk 1 ) ,
33

Isomap vs. Laplacian
Flattening: f = (f1 , f2 ) R2 .
Laplacian: local smoothness: fi = argmin ||Gf || subj. to ||f || = 1.
⇥ (f1 , f2 ) eigenvectors (#2,#3) of L = GT G.
Isomap: global constraints: ||f (x) f (y)|| ⇥ dM (x, y).
⇥ (f1 , f2 ) eigenvectors (#1,#2) of J(dM (xi , xj )2 )ij J.
Bijective
Not bijective

Mesh Lapl. combin. Lapl. conformal Isomap 34

Bending Invariants of Surfaces
Bending invariants: [Elad, Kimmel, 2003].
Surface M, Isomap dimension reduction: x ⇥ M ⇤ IM (x) ⇥ R3 .

M

IM

[Elad, Kimmel, 2003]. 35

Geodesic isometry :M M : dM (x, y) = dM ( (x), (y)).
Theorem: up to rigid motion, IM is invariant to geodesic isometries:
IM (x) = v + U IM ( (x)) where U O(3) and v R3 .

M

IM

[Elad, Kimmel, 2003]. 35

Geodesic isometry :M M : dM (x, y) = dM ( (x), (y)).
Theorem: up to rigid motion, IM is invariant to geodesic isometries:
IM (x) = v + U IM ( (x)) where U O(3) and v R3 .

M

IM

[Elad, Kimmel, 2003]. [Bronstein et al., 2005]. 35

Bending Invariants

M

IM
36

Face Recognition

Rigid similarity Non-rigid similarity Alex

A. M. Bronstein et al., IJCV, 2005
37

Overview

High Dimensional Data Sets

39

Graph and Geodesics

40

Isomap Dimension Reduction

41

Isomap vs PCA Flattening

42

Laplacian Spectral Dimension Reduction

43

Parameterization of Image Datasets

44

Library of Images

45

When Does it Works?

46

Local patches in images

47

Mesh Processing Course : Mesh Parameterization

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