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