1 / 36

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.

mhorner
Download Presentation

3D Geometry for Computer Graphics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related