1. SVD(Singular Value Decomposition) and Its Applications
Joon Jae Lee

2. The plan today Singular Value Decomposition Basic intuition
Formal definition
Applications

3. LCD Defect Detection
Defect types of TFT-LCD : Point, Line, Scratch, Region

4. Shading Scanners give us raw point cloud data
How to compute normals to shade the surface?
normal

5. Hole filling

6. Eigenfaces
Same principal components analysis can be applied to images

7. Video Copy Detection same image content different source
same video source different image content
AVI and MPEG face image
Hue histogram
Hue histogram

8. 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

9. Geometric analysis of linear transformations
We want to know what a linear transformation A does
Need some simple and “comprehendible” representation of the matrix of A.
Let’s look what A does to some vectors
Since A(v) = A(v), it’s enough to look at vectors v of unit length A

10. The geometry of linear transformations
A linear (non-singular) transform A always takes hyper-spheres to hyper-ellipses. A A

11. 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

12. Geometric analysis of linear transformations
If we are lucky: A = V  VT, V orthogonal (true if A is symmetric)
The eigenvectors of A are the axes of the ellipse A

13. Symmetric matrix: eigen decomposition
In this case A is just a scaling matrix. The eigen decomposition of A tells us which orthogonal axes it scales, and by how much: A 1 2 1

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

15. General linear transformations: SVD1 1 1 2 A
orthonormal
orthonormal

16. 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 =

17. SVD more formally
The diagonal values of  (1, …, n) are called the singular values. It is accustomed to sort them: 1  2 …  n
The columns of U (u1, …, un) are called the left singular vectors. They are the axes of the ellipsoid.
The columns of V (v1, …, vn) are called the right singular vectors. They are the preimages of the axes of the ellipsoid. =

18. 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…

19. Matrix inverse and solving linear systems
So, to solve

20. Matrix rank The rank of A is the number of non-zero singular values n1 2 n m =

21. Numerical rank
If there are very small singular values, then A is close to being singular. We can set a threshold t, so that numeric_rank(A) = #{i| i > t}
If rank(A) < n then A is singular. It maps the entire space Rn onto some subspace, like a plane (so A is some sort of projection).

22. Solving least-squares systems
We tried to solve Ax=b when A was rectangular:
Seeking solutions in least-squares sense: A x b =

23. Solving least-squares systems
We proved that when A is full-rank, (normal equations).
So: 1 1
12 = n n
n2

24. Solving least-squares systems
Substituting in the normal equations:
1/12 1
1/1 =
1/n2 n
1/n

25. Pseudoinverse The matrix we found is called the pseudoinverse of A.
Definition using reduced SVD:
If some of the i are zero,
put zero instead of 1/I .

26. Pseudo-inverse Pseudoinverse A+ exists for any matrix A.
Its properties:
If A is mn then A+ is nm
Acts a little like real inverse:

27. Solving least-squares systems
When A is not full-rank: ATA is singular
There are multiple solutions to the normal equations:
Thus, there are multiple solutions to the least-squares.
The SVD approach still works! In this case it finds the minimal-norm solution:

28. PCA – the general idea
PCA finds an orthogonal basis that best represents given data set.
The sum of distances2 from the x’ axis is minimized. y y’ x’ x

29. PCA – the general idea y v2 v1 x y y x x
This line segment approximates the original data set
The projected data set approximates the original data set

30. PCA – the general idea
PCA finds an orthogonal basis that best represents given data set.
PCA finds a best approximating plane (again, in terms of distances2) z
3D point set in
standard basis y x

31. For approximation
In general dimension d, the eigenvalues are sorted in descending order:
1  2  …  d
The eigenvectors are sorted accordingly.
To get an approximation of dimension d’ < d, we take the d’ first eigenvectors and look at the subspace they span (d’ = 1 is a line, d’ = 2 is a plane…)

32. For approximation
To get an approximating set, we project the original data points onto the chosen subspace:
xi = m + 1v1 + 2v2 +…+ d’vd’ +…+dvd
Projection:
xi’ = m + 1v1 + 2v2 +…+ d’vd’ +0vd’+1+…+ 0 vd

33. Principal components
Eigenvectors that correspond to big eigenvalues are the directions in which the data has strong components (= large variance).
If the eigenvalues are more or less the same – there is no preferable direction.
Note: the eigenvalues are always non-negative. Think why…

34. Principal components S looks like this: S looks like this:
There’s no preferable direction
S looks like this:
Any vector is an eigenvector
There is a clear preferable direction
S looks like this:
 is close to zero, much smaller than .

35. Application: finding tight bounding box
An axis-aligned bounding box: agrees with the axes y
maxY
minX
maxX x
minY

36. Application: finding tight bounding box
Oriented bounding box: we find better axes! x’ y’

37. Application: finding tight bounding box
Oriented bounding box: we find better axes!

38. Scanned meshes

39. Point clouds Scanners give us raw point cloud data
How to compute normals to shade the surface?
normal

40. Point clouds
Local PCA, take the third vector

41. Eigenfaces
Same principal components analysis can be applied to images

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

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

44. 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’

45. SVD for animation compression
Chicken animation

46. 3D animations Each frame is a 3D model (mesh)
Connectivity – mesh faces

47. 3D animations Each frame is a 3D model (mesh)
Connectivity – mesh faces
Geometry – 3D coordinates of the vertices

48. 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

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

50. 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 =

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

52. 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

53. 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

54. LCD Defect Detection
Defect types of TFT-LCD : Point, Line, Scratch, Region

55. Pattern Elimination Using SVD

56. Pattern Elimination Using SVD
Cutting the test images

57. Pattern Elimination Using SVD
Singular value determinant for the image reconstruction

58. Pattern Elimination Using SVD
Defect detection using SVD