1. 3D Geometry for Computer Graphics
Class 7

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

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

4. The geometry of linear transformations
Thus, one good way to understand what A does is to find which vectors are mapped to the “main axes” of the ellipsoid. A 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’’ = 1 x’, y’’ = 2 y’
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
Translate the input points to the centroids:
Compute the “covariance matrix”
Compute the SVD of H:
The optimal rotation is
The translation vector is =
H is 22 or 33 matrix!

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!
U 3Nf
 ff
VT ff VT =

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

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. u1 u2 u3 = 1
+ 2
+ 3

30. Animation compression by dimensionality reduction
Compressed representation:
The chosen principal component vectors
Coefficients i for each frame ui
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