texture mapping using surface flattening via multi-dimensional scaling g.zigelman, r.kimmel,...
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Texture Mapping using Surface Flattening via Multi-Dimensional Scaling
G.Zigelman, R.Kimmel, N.Kiryati
IEEE Transactions on Visualization and Computer Graphics
2002
2
Multidimensional scaling (MDS)
The idea: compute the pairwise geodesic distances between the vertices of the mesh:
Now, find n points in R2, so that their distance matrix is as close as possible to M.
2dist ( , )n n
M
i jx x
q1
q2
3
MDS – the math details
We look for X’,
such that || M’ – M || is as small as possible, where
M’ is the Euclidean distances matrix for points xi’.
| |
| |
d nX R
1 nx x
22dist ( , ) n nM R
i j i jx x x x
4
MDS – the math details
Ideally, we want:
2
2 2
,
|| || || || 2 ,
M M
M
M
i j
i j i j
i j i j
x x
x x x x
x x x x
2 2 2
|| || || || || ||
|| || || || || ||
|| || || || || ||
1 1 1
n n n
x x x
x x x
x x x
2
1 2
1 2
|| || || || || ||
|| || || || || ||
|| || || || || ||
1 n
n
n
x x x
x x x
x x x
| |
| |
1
1 n
n
x
x x
x
TX X want to get rid of these
5
MDS – the math details
Trick: use the “magic matrix” J :1 1
1 1 1
1 1
1
1
1
n n
n n n
n n n n
J
0a a a J
0
b
bJ
b
6
MDS – the math details
Cleaning the system:
2
2 2 2 1 2
1 2
|| || || || || || || || || || || ||
|| || || || || || || || || || || ||2
|| || || || || || || || || || || ||
TX X M
1 1 1 1 n
n
n n n n
x x x x x x
x x x x x x
x x x x x x
J J
12
2
:
T
T
X X JMJ
X X JMJ B
TX X B
7
How to find X’
We will use the spectral decomposition of B:
1| | | |
| | | |
T
T
n
X X B
1 n 1 nv v v v
1 1| | | | | |
| | | | | |
| | | | | |
| | | | | |
TT
Tn nd d
n n
X X
1 d 1 dv v v v v v
n d
d d
TX X
8
How to find X’
So we find X’ by throwing away the last nd eigenvalues
1
d
X
1
d
v
v
d n
2arg min T
LXX X X B
2
2
,ijL
i j
A A
9
Flattening results (Zigelman et al.)
10
Flattening results (Zigelman et al.)
11
Flattening results (Zigelman et al.)
The end