Gabriel Peyré
January 01, 2011
770

# Mesh Processing Course: Parameterization and Flattening

January 01, 2011

## Transcript

1. Parameterization and Flattening
Gabriel Peyré
www.numerical-tours.com

2. parameterization
Mesh Parameterization - Overview
2
texture
mapping
M R3

3. parameterization
D R2
Mesh Parameterization - Overview
2
1
re-sampling
zoom
texture
mapping
M R3

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

5. Local Averaging
4
Local operator: W = (wij
)
i,j V
where wij
= > 0 if j Vi,
0 otherwise.
(Wf)
i
=
(i,j) E
wijfj.
Examples: for i j,
wij
= 1
combinatorial
wij
= 1
||xj xi
||2
distance
wij
= cot(
ij
) + cot(⇥ij
)
conformal
Local averaging operator ˜
W = ( ˜
wij
)
i,j V
: ⇥ (i, j) E, ˜
wij
= wij
(i,j) E
wij
.
˜
W = D 1W with D = diag
i
(di
) where di
=
(i,j)⇥E
wij.
Averaging: ˜
W1 = 1.
(explanations later)

6. Voronoi and Dual Mesh
5
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 =
i V
Ei
=
e E
Ee.
i
j
Ai
cf
i
j
A(ij)
cf
Dual mesh 1:3 subdivided mesh

7. Approximating Integrals on Meshes
Approximation of integrals on vertices and edges:

M
f(x)dx
i V
Ai f(xi
)
e=(i,j) E
Ae f([xi, xj
]).
Theorem : ⇥ e = (i, j) E,
Ae
= Area(Ee
) =
1
2
||xi xj
||2 (cot(
ij
) + cot(⇥ij
))
i
j
A(ij)
cf

8. Approximating Integrals on Meshes
Approximation of integrals on vertices and edges:

M
f(x)dx
i V
Ai f(xi
)
e=(i,j) E
Ae f([xi, xj
]).
Theorem : ⇥ e = (i, j) E,
Ae
= Area(Ee
) =
1
2
||xi xj
||2 (cot(
ij
) + cot(⇥ij
))
i
j
A(ij)
cf
A
B
C
O
h
+ + =
2
A(ABO) = ||AB|| h = ||AB||
||AB||
2
tan( )
A(ABO) =
||AB||2
2
tan ⇤
2
( + ⇥)
Proof:

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

10. ⇧ (i, j) ⇤ E, wij
= 1.
Distance weights: they depends both on the geometry and the to
require faces information,
⇧ (i, j) ⇤ E, wij
=
1
||xj xi
||2
.
Conformal weights: they depends on the full geometrical realiza
require the face information
⇧ (i, j) ⇤ E, wij
= cot(
ij
) + cot(⇥
Figure 1.2 shows the geometrical meaning of the angles
ij
and
ij
= ⇥(xi, xj, xk1
) and ⇥ij
= ⇥(xi
where (i, j, k1
) ⇤ F and (i, j, k2
) ⇤ F are the two faces adjacent
in the next section the explanation of these celebrated cotangent
xi
xj
xk1
xk2
ij
ij
Cotangent Weights
7
Approximation of Dirichelet energy:
where wij
= cot(
ij
) + cot(⇥ij
).

M
||⇤xf||2dx ⇥
e E
Ae
|(Gf)
e
|2 =
(i,j) E
Ae
|f(xj
) f(xi
)|2
||xj xi
||2
=
(i,j) E
wij
|f(xj
) f(xi
)|2
Sobolev norm (Dirichlet energy): J(f) =
M
|| f(x)||2dx

11. ⇧ (i, j) ⇤ E, wij
= 1.
Distance weights: they depends both on the geometry and the to
require faces information,
⇧ (i, j) ⇤ E, wij
=
1
||xj xi
||2
.
Conformal weights: they depends on the full geometrical realiza
require the face information
⇧ (i, j) ⇤ E, wij
= cot(
ij
) + cot(⇥
Figure 1.2 shows the geometrical meaning of the angles
ij
and
ij
= ⇥(xi, xj, xk1
) and ⇥ij
= ⇥(xi
where (i, j, k1
) ⇤ F and (i, j, k2
) ⇤ F are the two faces adjacent
in the next section the explanation of these celebrated cotangent
xi
xj
xk1
xk2
ij
ij
Cotangent Weights
7
Approximation of Dirichelet energy:
Theorem : wij > 0 ⇥ ij
+ ⇥ij < ⇤
where wij
= cot(
ij
) + cot(⇥ij
).

M
||⇤xf||2dx ⇥
e E
Ae
|(Gf)
e
|2 =
(i,j) E
Ae
|f(xj
) f(xi
)|2
||xj xi
||2
=
(i,j) E
wij
|f(xj
) f(xi
)|2
Sobolev norm (Dirichlet energy): J(f) =
M
|| f(x)||2dx

12. Overview
• Dirichlet Energy on Meshes
•Harmonic Parameterization
• Spectral Flattening
• Barycentric Coordinates for Warping
• Geodesic Flattening
• High Dimensional Data Analysis
8

13. Mesh Parameterization
3D space (x,y,z)
2D parameter domain (u,v)
boundary
boundary 9
Parameterization: bijection : M ⇤ D ⇥ R2.
Hypothesis: = (
1, 2
) is smooth, minimizes
⇥ xi ⇥M, (i) = 0(i) ⇥D.
min
0
(i,j) E
wi,j
(| 1
(i)
1
(j)|2 + | 2
(i)
2
(j)|2)
With boundary conditions 0:

14. Mesh Parameterization
3D space (x,y,z)
2D parameter domain (u,v)
boundary
boundary 9
Parameterization: bijection : M ⇤ D ⇥ R2.
⇥ sparse linear system to solve.
Hypothesis: = (
1, 2
) is smooth, minimizes
⇥ xi ⇥M, (i) = 0(i) ⇥D.
min
0
(i,j) E
wi,j
(| 1
(i)
1
(j)|2 + | 2
(i)
2
(j)|2)
⇥ i / ⇥M, (L 1
)(i) = (L 2
)(i) = 0
⇥ i ⇥M, (i) = 0(i) ⇥D.
Optimality conditions:
With boundary conditions 0:

15. Mesh Parameterization
3D space (x,y,z)
2D parameter domain (u,v)
boundary
boundary 9
Parameterization: bijection : M ⇤ D ⇥ R2.
⇥ sparse linear system to solve.
Theorem: (Tutte) if i, j, wij > 0,
then is a bijection.
⇥ i / ⇥M, (i) =
(i,j) E
˜
wi,j
(j).
Hypothesis: = (
1, 2
) is smooth, minimizes
⇥ xi ⇥M, (i) = 0(i) ⇥D.
min
0
(i,j) E
wi,j
(| 1
(i)
1
(j)|2 + | 2
(i)
2
(j)|2)
⇥ i / ⇥M, (L 1
)(i) = (L 2
)(i) = 0
⇥ i ⇥M, (i) = 0(i) ⇥D.
Optimality conditions:
Remark: each point is the
average of its neighbors:
With boundary conditions 0:

16. Examples of Parameterization
Combinatorial
Conformal
Mesh
10

17. Examples of Parameterization
Combinatorial
Conformal
Mesh
11

18. Application to Remeshing
12
parameterization
1
zoom
re-sampling
P. Alliez et al., Isotropic Surface Remeshing, 2003.

19. Application to Texture Mapping
13
param
eterization
texture g(u)
color g( (x))

20. Mesh Parameterization #1
14
S
S0
: S0
⇥ S
S0
⇥ S
⇥ x S0
\⇥S0, = 0

21. Mesh Parameterization #1
14
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 );
W = W + make_sparse(faces(i3,:),faces(i2,:), a, n, n );
end
S
S0
: S0
⇥ S
S0
⇥ S
⇥ x S0
\⇥S0, = 0
Discretization of :

22. Mesh Parameterization #1
14
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 );
W = W + make_sparse(faces(i3,:),faces(i2,:), a, n, n );
end
D = spdiags(full( sum(W,1) ), 0, n,n);
L = D - W; L1 = L; L1(boundary,:) = 0;
L1(boundary + (boundary-1)*n) = 1;
S
S0
: S0
⇥ S
S0
⇥ S
⇥ x S0
\⇥S0, = 0
Formation of the linear system:
Discretization of :

23. Mesh Parameterization #1
14
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 );
W = W + make_sparse(faces(i3,:),faces(i2,:), a, n, n );
end
D = spdiags(full( sum(W,1) ), 0, n,n);
L = D - W; L1 = L; L1(boundary,:) = 0;
L1(boundary + (boundary-1)*n) = 1;
Rx = zeros(n,1); Rx(boundary) = x0;
Ry = zeros(n,1); Ry(boundary) = y0;
x = L1 \ Rx; y = L1 \ Ry;
S
S0
: S0
⇥ S
S0
⇥ S
⇥ x S0
\⇥S0, = 0
Formation of the linear system:
Formation of the RHS and resolution:
Discretization of :

24. Mesh Parameterization #2
15
Exercise: perform the linear interpolation of the parameterization.
Exercise: display the geometry image using a checkboard texture.
Geometry image: re-sample X/Y/Z coordinates of on a grid.
store the surface as a color (R/G/B) image.

25. Mesh Parameterization #3
16
Exercise: Locate the position of the eyes / the mouth
Exercise: Compute an a ne transformation to re-align the texture.
in the texture and on the mesh.

26. Mesh Deformations
17
Initial position: xi
R3.
Displacement of anchors:
i I, xi xi
= xi
+
i
R3
Linear deformation:
xi
xi
I
i I, (i) =
i
i / I, (i) = 0
xi
= xi
+ (i)

27. Mesh Deformations
17
Initial position: xi
R3.
Displacement of anchors:
i I, xi xi
= xi
+
i
R3
Linear deformation:
xi
xi
I
% modify Laplacian
L1 = L; L1(I,:) = 0;
L1(I + (I-1)*n) = 1;
% displace vertices
vertex = vertex + ( L1 \ Delta0' )';
i I, (i) =
i
i / I, (i) = 0
xi
= xi
+ (i)

28. Mesh Deformations
17
Initial position: xi
R3.
Displacement of anchors:
i I, xi xi
= xi
+
i
R3
Linear deformation:
xi
xi
xi
Non-linear deformation:
xi
= ˜
xi
+
i
˜
xi
coarse
scale
details
˜
xi
˜
xi
Linear deformation:
xi
= xi
+
i, ni
˜
ni
Extrusion along normals:
˜
ni
I
% modify Laplacian
L1 = L; L1(I,:) = 0;
L1(I + (I-1)*n) = 1;
% displace vertices
vertex = vertex + ( L1 \ Delta0' )';
i I, (i) =
i
i / I, (i) = 0
xi
= xi
+ (i)

29. Overview
• Dirichlet Energy on Meshes
• Harmonic Parameterization
•Spectral Flattening
• Barycentric Coordinates for Warping
• Geodesic Flattening
• High Dimensional Data Analysis
18

30. Mesh Flattening
19
min
1⇥Rn
||G 1
||2 =

i j
wi,j
| 1
(i)
1
(j)|2 with
|| 1
|| = 1,
⇥ 1, 1⇤ = 0.
min
2⇥Rn
||G 2
||2 =

i j
wi,j
| 2
(i)
2
(j)|2 with

|| 2
|| = 1,
⇥ 2, 1
⇤ = 0,
⇥ 2, 1⇤ = 0.
No boundary condition, minimize:

31. Mesh Flattening
19
min
1⇥Rn
||G 1
||2 =

i j
wi,j
| 1
(i)
1
(j)|2 with
|| 1
|| = 1,
⇥ 1, 1⇤ = 0.
min
2⇥Rn
||G 2
||2 =

i j
wi,j
| 2
(i)
2
(j)|2 with

|| 2
|| = 1,
⇥ 2, 1
⇤ = 0,
⇥ 2, 1⇤ = 0.
(
1
(i), 2
(i)) R2
Theorem: ⇥i
=
i+1L⇥i
,
where
0
= 0
1 2 . . . n 1
are eigenvalues of L = G⇥G.
No boundary condition, minimize:
conformal
combinatorial

32. Proof
20
L = G G = D W = U U
Spectral decomposition:
= diag(
i
) where 0 =
1 < 2 . . . n
U = (ui
)n
i=1
orthonormal basis of Rn. u1
= 1

33. Proof
20
L = G G = D W = U U
Spectral decomposition:
= diag(
i
) where 0 =
1 < 2 . . . n
U = (ui
)n
i=1
orthonormal basis of Rn.
If , 1 = 0, then
u1
= 1
E(⇥) = ||G⇥||2 =
n
i=1
i
| ⇥, ui
⇥|2
E(⇥) =
n
i=2
iai
where ai
= | ⇥, ui
⇥|2

34. Proof
20
L = G G = D W = U U
Spectral decomposition:
= diag(
i
) where 0 =
1 < 2 . . . n
U = (ui
)n
i=1
orthonormal basis of Rn.
If , 1 = 0, then
u1
= 1
E(⇥) = ||G⇥||2 =
n
i=1
i
| ⇥, ui
⇥|2
Constrained minimization:
linear program: minimum reached at a =
i
.
min
P
n
i=2
ai=1
n
i=2
iai
±u2
= argmin
, 1⇥=0,|| ||
E( )
E(⇥) =
n
i=2
iai
where ai
= | ⇥, ui
⇥|2

35. Flattening Examples
21
Main issue: No guarantee of being valid (bijective).
conformal
combinatorial

36. Overview
• Dirichlet Energy on Meshes
• Harmonic Parameterization
• Spectral Flattening
•Barycentric Coordinates for Warping
• Geodesic Flattening
• High Dimensional Data Analysis
22

37. Barycentric Coordinates
23
x1
x2
x3
x
˜
i
(x) = A(x, xi+1, xi+2
)
A(x1, x2, x3
)

38. Barycentric Coordinates
23
x1
x2
x3
x1
x3
x2
x x
Barycentric coordinates: { i
(x)}i I
Positivity:
i
(x) 0.
Interpolation:
Reproduction of a ne functions:
˜
i
(x) = A(x, xi+1, xi+2
)
A(x1, x2, x3
)
Normalized: ˜
i
(x) = i
(x)
j j
(x)
i I
˜
i
(x)xi
= x
˜
⇥i
(xj
) =
i,j

39. Barycentric Coordinates
23
x1
x2
x3
x1
x3
x2
x x
Barycentric coordinates: { i
(x)}i I
Positivity:
i
(x) 0.
Interpolation:
Reproduction of a ne functions:
Application: interpolation of data {fi
}i I
˜
i
(x) = A(x, xi+1, xi+2
)
A(x1, x2, x3
)
Normalized: ˜
i
(x) = i
(x)
j j
(x)
i I
˜
i
(x)xi
= x
f(x) =
i I
˜
i
(x)fi
˜
⇥i
(xj
) =
i,j

40. Barycentric Coordinates
23
x1
x2
x3
x1
x3
x2
x x
Barycentric coordinates: { i
(x)}i I
Positivity:
i
(x) 0.
Interpolation:
Reproduction of a ne functions:
Application: interpolation of data {fi
}i I
Application: mesh parameterization:
xi
xj
˜
i
(x) = A(x, xi+1, xi+2
)
A(x1, x2, x3
)
Normalized: ˜
i
(x) = i
(x)
j j
(x)
i I
˜
i
(x)xi
= x
f(x) =
i I
˜
i
(x)fi
wi,j
=
i
(xj
)
˜
⇥i
(xj
) =
i,j

41. Mean Value Coordinates
24
x
xi+1
xi
xi
i
˜
i ˜
i
i
Conformal Laplacian weights:
⇥i
(x) = cotan(
i
(x)) + cotan(˜
i
(x))
Mean-value coordinates:
not necessarily positive.
valid coordinates.
extend to non-convex coordinates (oriented angles).
⇥i
(x) =
tan(
i
(x)/2) + tan(˜
i
(x)/2)
||x xi
||
˜
1
(x) ˜
2
(x)

42. Barycentric Coordinates for Warping
25
Example: textured grid, 3D model, etc.
Data points: {yj
}j J
C.
Cage C: polygon with vertices {xi
}i I
.
Initialization: data anchoring, compute
j J, i I, i,j
=
i
(yj
).
Data warping: yj yj
=
i I
i,jxi
Satisﬁes: yi
=
i I
i,jxi
Cage warping: xi xi
xi
xi
yi
yi

43. Harmonic Coordinates
26
Mean value Harmonic
Mean value coordinates:
“non-physical” behavior,
passes “through” the cage.
⇥ x C, i
(x) = 0.
⇥ x ⇥C, i
(x) = 0
i
(x).
Harmonic mapping:
Boundary conditions:

44. Warping Comparison
27
Mean value Harmonic
Initial shape

45. Overview
• Dirichlet Energy on Meshes
• Harmonic Parameterization
• Spectral Flattening
• Barycentric Coordinates for Warping
•Geodesic Flattening
• High Dimensional Data Analysis
28

46. Geodesic Distances
29
Length of a curve (t) M: L( ) def.
=
1
0
W( (t))|| (t)||dt.
Geodesic distance: dM
(x, y) = min
(0)=x, (1)=y
L( )
Euclidean Shape Isotropic W = 1 Surface
dM
(x, y) = L( )
Geodesic curve :

47. Computation of Geodesic Distances
30
Non-linear PDE:
Distance map to a point: Ux0
(x) = dM
(x0, x).
Ux0
x0
|| Ux0
(x)|| = W(x)
Ux0
(x0
) = 0,
(viscosity)
Upwind ﬁnite di erences approximation.
Fast Marching: front propagation in O(N log(N)) operations.

48. Manifold Flattening
31
Input manifold M, dM
geodesic distance on M.
Input geodesic distance matrix: ˜
D = (dM

xi, ˜
xj
)2)
i,j
, for ˜
xi
M.
x1
˜
x1
x2
˜
x2
M R3
M R2
˜
x1
˜
x2
x1
x2
Flattening: ﬁnd X = (xi
)p
i=1
⇤ Rn p such that ||xi xj
|| ⇥ dM

xi, ˜
xj
).
Surface parameterization Bending invariant

49. Stress Minimization
32
Geodesic stress: di,j
= dM

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

50. Projection on Distance Matrices
33
D(X)
i,j
= ||xi xj
||2
Di,j
= d2
i,j
||xi xj
||2 = ||xi
||2 + ||xj
||2 2⇥xi, xj

min
X1=0
i,j
|||xi xj
||2 d2
i,j
|2 = ||D(X) D||2,
=⇥ D(X) = d1T + 1d 2XTX where d = (||xi
||2)
i
⇤ Rn

51. Projection on Distance Matrices
33
D(X)
i,j
= ||xi xj
||2
Di,j
= d2
i,j
||xi xj
||2 = ||xi
||2 + ||xj
||2 2⇥xi, xj

min
X1=0
i,j
|||xi xj
||2 d2
i,j
|2 = ||D(X) D||2,
=⇥ D(X) = d1T + 1d 2XTX where d = (||xi
||2)
i
⇤ Rn
For centered points:
Centering matrix: J = Id
n
11T /N
1
2JD(X)J = XTX
JX = X
J1 = 0

52. Projection on Distance Matrices
33
D(X)
i,j
= ||xi xj
||2
Di,j
= d2
i,j
||xi xj
||2 = ||xi
||2 + ||xj
||2 2⇥xi, xj

min
X1=0
i,j
|||xi xj
||2 d2
i,j
|2 = ||D(X) D||2,
=⇥ D(X) = d1T + 1d 2XTX where d = (||xi
||2)
i
⇤ Rn
For centered points:
Centering matrix: J = Id
n
11T /N
1
2JD(X)J = XTX
JX = X
J1 = 0
Replace ||D(X) D|| by
Explicit solution: diagonalize 1
2
JDJ = U UT
i i 1
k
= diag(
0, . . . , k 1
),
Uk
= (u0, . . . , uk 1
)T,
X =
kUk
min
X
|| J(D(X) D)J/2|| = ||XTX + JDJ/2||

53. Isomap vs. Laplacian
34
Flattening: f = (f1, f2
) R2.
Isomap: global constraints: ||f(x) f(y)|| ⇥ dM
(x, y).
⇥ (f1, f2
) eigenvectors (#2,#3) of L = GTG.
Laplacian: local smoothness: fi
= argmin ||Gf|| subj. to ||f|| = 1.
Mesh Lapl. combin. Lapl. conformal Isomap
Bijective
Not bijective
⇥ (f1, f2
) eigenvectors (#1,#2) of J(dM
(xi, xj
)2)
ijJ.

54. Bending Invariants of Surfaces
35
Surface M, Isomap dimension reduction: x ⇥ M ⇤ IM
(x) ⇥ R3.
M
IM

55. Bending Invariants of Surfaces
35
Surface M, Isomap dimension reduction: x ⇥ M ⇤ IM
(x) ⇥ R3.
Geodesic isometry : M M : dM
(x, y) = dM
( (x), (y)).
Theorem: up to rigid motion, IM
is invariant to geodesic isometries:
IM
(x) = v + UIM
( (x)) where U O(3) and v R3.
M
IM

56. Bending Invariants of Surfaces
35
Surface M, Isomap dimension reduction: x ⇥ M ⇤ IM
(x) ⇥ R3.
Geodesic isometry : M M : dM
(x, y) = dM
( (x), (y)).
Theorem: up to rigid motion, IM
is invariant to geodesic isometries:
IM
(x) = v + UIM
( (x)) where U O(3) and v R3.
[Elad, Kimmel, 2003]. [Bronstein et al., 2005].
M
IM

57. Bending Invariants
36
IM
M

58. Face Recognition
37
Rigid similarity Non-rigid similarity Alex
A. M. Bronstein et al., IJCV, 2005

59. Overview
• Dirichlet Energy on Meshes
• Harmonic Parameterization
• Spectral Flattening
• Barycentric Coordinates for Warping
• Geodesic Flattening
•High Dimensional Data Analysis
38

60. High Dimensional Data Sets
39

61. Graph and Geodesics
40

62. Isomap Dimension Reduction
41

63. Isomap vs PCA Flattening
42

64. Laplacian Spectral Dimension Reduction
43

65. Parameterization of Image Datasets
44

66. Library of Images
45

67. When Does it Works?
46

68. Local patches in images
47