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3D Geometry for Computer Graphics

3D Geometry for Computer Graphics. Class 7. The plan today. Review of SVD basics Connection PCA Applications: Eigenfaces Animation compression. The geometry of linear transformations. A. A linear transform A always takes hyper-spheres to hyper-ellipses. A.

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3D Geometry for Computer Graphics

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  1. 3D Geometry forComputer Graphics Class 7

  2. The plan today • Review of SVD basics • Connection PCA • Applications: • Eigenfaces • Animation compression

  3. The geometry of linear transformations A • A linear transform A always takes hyper-spheres to hyper-ellipses. A

  4. The geometry of linear transformations A • Thus, one good way to understand what A does is to find which vectors are mapped to the “main axes” of the ellipsoid. A

  5. Spectral decomposition • If we are lucky: A = V  VT, V orthogonal • The eigenvectors of A are the axes of the ellipse A

  6. Spectral decomposition • The standard basis: y x Rotation VT Au = V  VTu

  7. Spectral decomposition • Scaling: x’’ = 1 x’, y’’ =2 y’ Scale  Au = V  VTu

  8. Spectral decomposition • Rotate back: Rotate V Au = V  VTu

  9. General linear transformations: SVD • In general A will also contain rotations, not just scales: 1 1 1 2 A

  10. SVD • The standard basis: y x Rotation VT Au = U  VTu

  11. Spectral decomposition • Scaling: x’’ = 1x’, y’’ =2y’ Scale  Au = U  VTu

  12. Spectral decomposition • Rotate again: Rotate U Au = U  VTu

  13. SVD more formally • SVD exists for any matrix • Formal definition: • For square matrices A  Rnn, there exist orthogonal matrices U, V Rnn and a diagonal matrix , such that all the diagonal values i of  are non-negative and =

  14. Reduced SVD • For rectangular matrices, we have two forms of SVD. The reduced SVD looks like this: • The columns of U are orthonormal • Cheaper form for computation and storage Mn Mn nn nn =

  15. Full SVD • We can complete U to a full orthogonal matrix and pad  by zeros accordingly Mn MM Mn nn =

  16. SVD is the “working horse” of linear algebra • There are numerical algorithms to compute SVD. Once you have it, you have many things: • Matrix inverse  can solve square linear systems • Numerical rank of a matrix • Can solve least-squares systems • PCA • Many more…

  17. SVD is the “working horse” of linear algebra • Must remember: SVD is expensive! • For a nn matrix, SVD costs O(n3). • For sparse matrices could sometimes be cheaper

  18. Shape matching • We have two objects in correspondence • Want to find the rigid transformation that aligns them

  19. Shape matching • When the objects are aligned, the lengths of the connecting lines are small.

  20. Shape matching – formalization • Align two point sets • Find a translation vector t and rotation matrix R so that:

  21. Summary of rigid alignment = H is 22 or 33 matrix! • Translate the input points to the centroids: • Compute the “covariance matrix” • Compute the SVD of H: • The optimal rotation is • The translation vector is

  22. SVD for animation compression Chicken animation

  23. 3D animations • Each frame is a 3D model (mesh) • Connectivity – mesh faces

  24. 3D animations • Each frame is a 3D model (mesh) • Connectivity – mesh faces • Geometry – 3D coordinates of the vertices

  25. 3D animations • Connectivity is usually constant (at least on large segments of the animation) • The geometry changes in each frame  vast amount of data, huge filesize! 13 seconds, 3000 vertices/frame, 26 MB

  26. Animation compression by dimensionality reduction • The geometry of each frame is a vector in R3N space (N = #vertices) 3N  f

  27. Animation compression by dimensionality reduction • Find a few vectors of R3N that will best represent our frame vectors! VTff ff U 3Nf VT =

  28. Animation compression by dimensionality reduction VT • The first principal components are the important ones u1 u2 u3 =

  29. Animation compression by dimensionality reduction • Approximate each frame by linear combination of the first principal components • The more components we use, the better the approximation • Usually, the number of components needed is much smaller than f. u3 u1 u2 1 + 2 + 3 =

  30. Animation compression by dimensionality reduction ui • Compressed representation: • The chosen principal component vectors • Coefficients i for each frame Animation with only 2 principal components Animation with 20 out of 400 principal components

  31. Eigenfaces • Same principal components analysis can be applied to images

  32. Eigenfaces • Each image is a vector in R250300 • Want to find the principal axes – vectors that best represent the input database of images

  33. Reconstruction with a few vectors • Represent each image by the first few (n) principal components

  34. Face recognition • Given a new image of a face, w R250300 • Represent w using the first n PCA vectors: • Now find an image in the database whose representation in the PCA basis is the closest: The angle between w and w’ is the smallest w w’ w w’

  35. Non-linear dimensionality reduction • More sophisticated methods can discover non-linear structures in the face datasets Isomap, Science, Dec. 2000

  36. See you next time

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