1. Math Review Towards Fourier Transform
Image Processing
Math Review Towards Fourier Transform
Linear Spaces
Change of Basis
Cosines and Sines
Complex Numbers

2. Vector Spaces and Basis Vectors
Part I:
Vector Spaces and Basis Vectors

3. Basis Vectors
A given vector value is represented with respect to a coordinate system.
The coordinate system is arbitrarily chosen.
A coordinate system is defined by a set of linearly independent vectors forming the system basis.
Any vector value is represented as a linear sum of the basis vectors.
a= 0.5 *v1+1*v2 (0.5 , 1)v
v1, v2 are basis vectors
The representation of a with respect to this basis is (0.5,1) a v1 v2

4. Orthonormal Basis Vectors
If the basis vectors are mutually orthogonal and are unit vectors, the vectors form an orthonormal basis.
Example: The standard basis is orthonormal:
V1=( …)
V2=( …)
V3=( …) ….
V2=(0 1) a
V1=(1 0)

5. Change of Basis v2 v1 a u2 u1
Question: Given a vector av, represented in an orthonormal basis {vi} , what is the representation of av in a different orthonormal basis {ui}?
Answer:

6. Change of Basis: Matrix Formv2 v1 a u2 u1
Defining the new basis as a collection of columns in a matrix form

7. Signal (image) Transforms
Basis Functions.
Method for finding the transform coefficients given a signal.
Method for finding the signal given the transform coefficients.

8. Standard Basis Functions - 1D
The Orthonormal Standard Basis - 1D
Wave Number 1 2 3 4 5 6 7 8 9
N = 16
Standard Basis Functions - 1D

9. The Orthonormal Hadamard Basis – 1D
Wave Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
N = 16

10. Hadamard Transform Standard Basis New Basis Grayscale Image
Transformed Image
spatial Coordinate
Transform Coordinate
Standard Basis:
[ ]standard=
2 [ ] + 1 [ ] + 6 [ ] + 1 [ ]
Hadamard Transform:
[ ]standard =
= [ ]/ [ ] /2 +
2 [ ] / [ ] /2
 [ ] Hadamard

11. Finding the transform coefficients
X = [ ] standard
Signal:
Hadamard Basis:
T0 = [ ] /2
T1 = [ ] /2
T2 = [ ] /2
T3 = [ ] /2
Hadamard Coefficients:
a0 = <X,T0 > = < [ ] , [ ] > /2 = 5
a1 = <X,T1 > = < [ ] , [ ] > /2 = -2
a2 = <X,T2 > = < [ ] , [ ] > /2 = 2
a3 = <X,T3 > = < [ ] , [ ] > /2 = -3
Signal:
[ ] Standard  [ ] Hadamard

12. Transforms: Change of Basis - 2D
Grayscale
Image
Transformed
Image
V Coordinates
Y Coordinate
X Coordinate
U Coordinates
The coefficients are arranged in a 2D array.

13. Hadamard Basis Functions
Black = +1 White = -1
size = 8x8

14. Transforms: Change of Basis - 2D
Standard Basis:
1 0
0 0 [ ]
0 1
+ 1
+ 6
= 2
Hadamard Transform:
1 1 [ ]
-1 -1
-1 1
1 -1 -3 +2 -2
= 5 /2 [ ] 
Hadamard

15. Finding the transform coefficients
6 1 [ ]
X =
Signal:
standard
New Basis: [
]/2
1 1
-1 -1 [
]/2
1 1
T11 =
T12 = [
-1 1
1 -1
]/2 [
]/2
-1 1
T21 =
T22 =
Signal:
X = a11T11 +a12T12 + a21T21 + a22T22
New Coefficients:
a11 = <X,T11 > = sum(sum(X.*T11)) = 5
a12 = <X,T12 > = sum(sum(X.*T12)) = -2
a21 = <X,T21 > = sum(sum(X.*T21)) = 2
a22 = <X,T22 > = sum(sum(X.*T22)) = -3 ] [
X 
new

16. [ ] [ ] [ ] [ ] ] [ [ ] [ ] Standard Basis: Hadamard Transform: 1 0
1 0
0 0 ]
0 1
0 0
0 1 [
6 1 ]
coefficients
Standard
Basis Elements
Hadamard Transform:
1 1 [ ] [ ]
1 1
-1 -1 ] [ /2 /2
-1 1
1 -1 [ ]
-1 1 [ ]
Hadamard
coefficients /2 /2
Hadamard
Basis Elements

17. Part II:
Sines and Cosines

18. Wavelength and Frequency of Sine/Cosin1
sin()  1
cos() 1 x
The wavelength of sin(x) is 2 .
The frequency is 1/(2) .

19. 1 x 1 x
2/2 1 x

20. The wavelength of sin(2x) is 1/ . The frequency is  .
Define K=2 1 x
The wavelength of sin(2x) is 1/ .
The frequency is  .

21. Changing Amplitude: A x
Changing Phase: x

22. } Sine vs Cosine sin(x) = cos(x) with a phase shift of p/2.

23. sin(x) + cos(x) = sin(x) scaled by with a phase shift of p/4.
Sine vs Cosine
sin(x) + cos(x) = sin(x) scaled by with
a phase shift of p/4.
3 sin(kx) + 4 cos(kx) = sin(kx) with amplitude
scaled by 5 and phase shift of 0.3p
3 sin(kx)
4 cos(kx)
5 sin(kx + 0.3p)

24. Combining Sine and Cosine
If we add a Sine wave to a Cosine wave with the same frequency we get a scaled and shifted (Co-) Sine wave with the same frequency:
What is the result if a=0?
What is the result if b=0?
(prove it!)
tan()

25. Part III:
Complex Numbers

26. Complex Numbers Two kind of representations for a point
Imaginary
(a,b) b R
The Complex Plane 
Real a
Two kind of representations for a point
(a,b) in the complex plane
The Cartesian representation:
The Polar representation:
(Complex exponential)
Conversions:
Polar to Cartesian:
Cartesian to Polar

27. Conjugate of Z is Z*: Cartesian rep. Polar rep. Imaginary b R  Real a- R -b

28. Algebraic operations:
addition/subtraction:
multiplication:
inner Product:
norm:

29. Number MultiplicationIm
A*B B A   Re

30. The (Co-) Sinusoid The (Co-)Sinusoid as complex exponential: Or
What about generalization?
S sin(kx) + C cos(kx) = ?

31. S C We saw that : S sin(kx) + C cos(kx) = R sin(kx + )
Using Complex exponentials:
Scaling and phase shifting can be represented as a multiplication with
and also æ C ö
where R = S 2 + C 2
and q =
tan - 1 ç ç è S ø
Example
6sin(kx) + 4cos(kx) = 1/2 [(4-6i) eikx + (4+6i) e-ikx ]

32. T H E E N D