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Texture Mapping using Surface Flattening via Multi-Dimensional Scaling

Texture Mapping using Surface Flattening via Multi-Dimensional Scaling. G.Zigelman, R.Kimmel, N.Kiryati IEEE Transactions on Visualization and Computer Graphics 2002. Multidimensional scaling (MDS). The idea: compute the pairwise geodesic distances between the vertices of the mesh:

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Texture Mapping using Surface Flattening via Multi-Dimensional Scaling

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  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. 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. q2 q1

  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’.

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

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

  6. MDS – the math details Cleaning the system:

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

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

  9. Flattening results (Zigelman et al.)

  10. Flattening results (Zigelman et al.)

  11. Flattening results (Zigelman et al.)

  12. The end

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