1. Computer Graphics
Recitation 5

2. Motivation – Shape Matching
What is the best transformation that aligns the dinosaur with the dog?

3. Motivation – Shape Matching
Regard the shapes as sets of points and try to “match” these sets
using a linear transformation
The above is not a good matching….

4. Motivation – Shape Matching
Regard the shapes as sets of points and try to “match” these sets
using a linear transformation
To find the best transformation we need to know SVD…

5. Finding underlying dimensionalityx
Given a set of points, find the best line that
approximates it – reduce the dimension of the data set…

6. Finding underlying dimensionalityx’ x
Given a set of points, find the best line that
approximates it – reduce the dimension of the data set…

7. Finding underlying dimensionalityx’ x
When we learn PCA (Principal Component Analysis),
we’ll know how to find these axes that minimize
the sum of distances2

8. PCA and SVD
PCA and SVD are important tools not only in graphics but also in statistics, computer vision and more.
To learn about them, we first need to get familiar with eigenvalue decomposition.
So, today we’ll start with linear algebra basics reminder.

9. The plan today Basic linear algebra review.
Eigenvalues and eigenvectors.

10. Vector space Informal definition: V   (a non-empty set of vectors)
v, w  V  v + w  V (closed under addition)
v  V,  is scalar  v  V (closed under multiplication by scalar)
Formal definition includes axioms about associativity and distributivity of the + and  operators.

11. Subspace - example Let l be a 2D line though the origin
L = {p – O | p  l} is a linear subspace of R2 y O x

12. Subspace - example Let  be a plane through the origin in 3D
V = {p – O | p  } is a linear subspace of R3 z O y x

13. Linear independence
The vectors {v1, v2, …, vk} are a linearly independent set if:
1v1 + 2v2 + … + kvk = 0  i = 0  i
It means that none of the vectors can be obtained as a linear combination of the others.

14. Linear independence - example
Parallel vectors are always dependent:
Orthogonal vectors are always independent.
v = 2.4 w  v + (2.4)w = 0 v w

15. V = {1v1 + 2v2 + … + nvn | i is scalar}
Basis
{v1, v2, …, vn} are linearly independent
{v1, v2, …, vn} span the whole vector space V:
V = {1v1 + 2v2 + … + nvn | i is scalar}
Any vector in V is a unique linear combination of the basis.
The number of basis vectors is called the dimension of V.

16. Basis - example
The standard basis of R3 – three unit orthogonal vectors x, y, z: (sometimes called i, j, k or e1, e2, e3) z y x

17. Matrix representation
Let {v1, v2, …, vn} be a basis of V
Every vV has a unique representation
v = 1v1 + 2v2 + … + nvn
Denote v by the column-vector:
The basis vectors are therefore denoted:

18. Linear operators A : V  W is called linear operator if:
A(v + w) = A(v) + A(w)
A(v) =  A(v)
In particular, A(0) = 0
Linear operators we know:
Scaling
Rotation, reflection
Translation is not linear – moves the origin

19. Linear operators - illustration
Rotation is a linear operator:
R(v+w)
v+w w w v v

20. Linear operators - illustration
Rotation is a linear operator:
R(v+w)
v+w
R(w) w w v v

21. Linear operators - illustration
Rotation is a linear operator:
R(v+w)
v+w
R(v)
R(w) w w v v

22. Linear operators - illustration
Rotation is a linear operator:
R(v+w)
R(v)+R(w)
v+w
R(v)
R(w) w w v v
R(v+w) = R(v) + R(w)

23. Matrix representation of linear operators
Look at A(v1),…, A(vn) where {v1, v2, …, vn} is a basis.
For all other vectors: v = 1v1 + 2v2 + … + nvn
A(v) = 1A(v1) + 2A(v2) + … + nA(vn)
So, knowing what A does to the basis is enough
The matrix representing A is:

24. Matrix representation of linear operators

25. Matrix operations
Addition, subtraction, scalar multiplication – simple…
Matrix by column vector:

26. Matrix operations
Transposition: make the rows columns
(AB)T = BTAT

27. Matrix operations
Inner product can in matrix form:

28. Matrix properties Matrix A (nn) is non-singular if B, AB = BA = I
B = A-1 is called the inverse of A
A is non-singular  det A  0
If A is non-singular then the equation Ax=b has one unique solution for each b.
A is non-singular  the rows of A are linearly independent set of vectors.

29. Vector product v  w is a new vector: In matrix form in 3D:
||v  w|| = ||v|| ||w|| |sin(v,w)|
v  w  v, v  w  w
v, w, v  w is a right-hand system
In matrix form in 3D:
vw v w

30. Orthogonal matrices Matrix A (nn) is orthogonal if A-1 = AT
Follows: AAT = ATA = I
The rows of A are orthonormal vectors!
Proof: v1
I = ATA = v2 v1 v2 vn
= viTvj = ij vn
 <vi, vi> = 1  ||vi|| = 1; <vi, vj> = 0

31. Orthogonal operators
A is orthogonal matrix  A represents a linear operator that preserves inner product (i.e., preserves lengths and angles):
Therefore, ||Av|| = ||v|| and (Av,Aw) = (v,w)

32. Orthogonal operators - example
Rotation by  around the z-axis in R3 :
In fact, any orthogonal 33 matrix represents a rotation around some axis and/or a reflection
detA = rotation only
detA = 1 with reflection

33. Eigenvectors and eigenvalues
Let A be a square nn matrix
v is eigenvector of A if:
Av = v ( is a scalar)
v  0
The scalar  is called eigenvalue
Av = v  A(v) = (v)  v is also eigenvector
Av = v, Aw = w  A(v+w) = (v+w)
Therefore, eigenvectors of the same  form a linear subspace.

34. Eigenvectors and eigenvalues
An eigenvector spans an axis that is invariant to A – it remains the same under the transformation.
Example – the axis of rotation:
Eigenvector of the
rotation transformation O

35. Finding eigenvalues
For which  is there a non-zero solution to Ax = x ?
Ax = x  Ax – x = 0  Ax – Ix = 0  (A- I)x = 0
So, non trivial solution exists  det(A – I) = 0
A() = det(A – I) is a polynomial of degree n.
It is called the characteristic polynomial of A.
The roots of A are the eigenvalues of A.
Therefore, there are always at least complex eigenvalues. If n is odd, there is at least one real eigenvalue.

36. Example of computing A
Cannot be factorized over R
Over C:

37. Computing eigenvectors
Solve the equation (A – I)x = 0
We’ll get a subspace of solutions.

38. Spectra and diagonalization
The set of all the eigenvalues of A is called
the spectrum of A.
A is diagonalizable if A has n independent eigenvectors. Then: A 1 2 v1 v2 = vn v1 v2 vn n
AV = VD

39. Spectra and diagonalization
Therefore, A = VDV1, where D is diagonal
A represents a scaling along the eigenvector axes! 1 A 1 2 = v1 v2 vn v1 v2 vn n
A = VDV1

40. Spectra and diagonalization
A is called normal if AAT = ATA.
Important example: symmetric matrices A = AT.
It can be proved that normal nn matrices have exactly n linearly independent eigenvectors (over C).
If A is symmetric, all eigenvalues of A are real, and it has n independent real orthonormal eigenvectors.

41. Spectra and diagonalization
rotation
reflection
other
orthonormal
basis
orthonormal
basis

42. Spectra and diagonalization

43. Spectra and diagonalization

44. Spectra and diagonalization

45. Spectra and diagonalization

46. Why SVD… Diagonalizable matrix is essentially a scaling
Most matrices are not diagonalizable – they do other things along with scaling (such as rotation)
So, to understand how general matrices behave, only eigenvalues are not enough
SVD tells us how general linear transformations behave, and other things…

47. See you next time