1. Texture Mapping using Surface Flattening via Multi-Dimensional Scaling G.Zigelman, R.Kimmel, N.Kiryati IEEE Transactions on Visualization and Computer Graphics 2002

2. 2 Multidimensional scaling (MDS) The idea: compute the pairwise geodesic distances between the vertices of the mesh: Now, find n points in R 2, so that their distance matrix is as close as possible to M. q1q1 q2q2

3. 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 x i ’.

4. 4 MDS – the math details Ideally, we want: want to get rid of these

5. 5 MDS – the math details Trick: use the “magic matrix” J :

6. 6 MDS – the math details Cleaning the system:

7. 7 How to find X’ We will use the spectral decomposition of B :

8. 8 How to find X’ So we find X’ by throwing away the last n  d eigenvalues

9. 9 Flattening results (Zigelman et al.)

10. 10 Flattening results (Zigelman et al.)

11. 11 Flattening results (Zigelman et al.)

12. The end