1. Image Enhancement and Restoration
Digital Image
from sensor
Too dark, Too bright, or Noisy
Better quality
Image
Pixel (Spatial Domain)
Frequency Domain
i/p
Frequency Transform
Adjust freq. coefficient
Inverse Transform
o/p

2. Fourier Analysis Color lights DFT Signals with different frequencies
White Light
Signals with different frequencies
Signal
DFT .
Fourier Analysis

3. Frequency Components
Frequency Domain
Time Domain

4. Fourier Analysis & Transform
นักคณิตศาสตร์ชาวฝรั่งเศส ชื่อเต็มว่า
Jean Baptiste Joseph Fourier
ทฤษฎีของ Fourier ได้ถูกตีพิมพ์ในหนังสือของ Fourier ชื่อ
1822: “La The’orie Analitique de la Chaleur”
1878: ถูกนำมาแปลเป็นภาษาอังกฤษโดย Freeman
“ฟังก์ชันใดๆ ที่มีคุณสมบัติการวนซ้ำค่า (repeat periodically) สามารถแสดงในรูปของผลรวมของสัญญาณ sine และ cosine ของความถี่ที่แตกต่างกันได้ ซึ่ง sine และ cosine แต่ละความถี่จะมีขนาดหรือถูกคูณด้วยค่าสัมประสิทธิ์ที่มีค่าแตกต่างกันไป”
Fourier Analysis & Transform

5. Fourier Theory Any functions or signals = Fourier Fourier Series
Fourier Transform
Periodic function
Aperiodic function
Continuous
Discrete
Continuous
Discrete
Fourier Theory

6. 1D Discrete Fourier Transform
Forward 1D DFT:
Inverse 1D DFT (IDFT):
1D Discrete Fourier Transform

7. Basis Set: Generalized Harmonics
The set of generalized harmonics -> an orthonormal basis set for functions: {ei2pst} where each harmonic has a different frequency s. Remember: ei2pst = cos(2pst) + i sin(2pst) The real part is a cosine of frequency s. The imaginary part is a sine of frequency s.`

8. How to Interpret the Weights F(u)
The weights F(u) are complex numbers:
Real part
Imaginary part
How much of a cosine of frequencys you need.
How much of a sine of frequencys you need.
Magnitude
Phase
How much of a sinusoid of frequencys you need.
What phase that sinusoid needs to be.

9. 4-point 1D DFT x 4 12 x 4
- 4 +

10. b0 = 0
a0 = 36
8-point DFT

11. 8-point DFT b1= -9.6569 a1 = - 4 b2 = -4 a2 = 4
[0.7071, 0, , -1, , 0, , 1]
[0.7071, 1, , 0, , -1, , 0]
b2 = -4
a2 = 4
8-point DFT
[0, -1, 0, 1, 0, -1, 0, 1]
[1, 0, -1, 0, 1, 0, -1, 0]

12. 8-point DFT a7 = -4 b7 = 9.6569 b0 = 0 a0 = 36 a1 = - 4 b1= -9.6569
[0.7071, 0, , -1, , 0, , 1]
[ , -1, , 0, , 1, , 0]
b0 = 0
a0 = 36
a1 = - 4
b1=
a2 = 4
b2 = -4 …
a7 = -4
b7 =
a0 – j b0 = 36
a1 – j b1 = j
a2 – j b2 = 4 + j 4
a7 – j b7 = - 4 – j
8-point DFT

13. 8-point 1D DFT x x x x x x x x x x x x x x x x

14. 2D Discrete Fourier Transform (DFT)
f(x,y) M N
F(u,v) M N
2D DFT
Forward 2D DFT:
Inverse 2D DFT (IDFT):
2D Discrete Fourier Transform (DFT)

15. Sine Wave Image (1) 1D sine wave v = 3; A = 127; v = 6; A = 127;

16. Sine Wave Image (2) 2D sine wave u = 4; v = 0; A = 127;

17. 2D DFT Basic Functions

18. Fourier Transform Properties
Shifting property

19. 2D DFT Cosine Periodic Property

20. 2D DFT Shifted Sine Periodic Property

21. Fourier Transform property (1)
1 = 4* 1 2
2 = 3* 3 4

22. Fourier Shifting

23. Fourier Transform property (2)
Unshifted
Magnitude Spectrum
Shifted
Magnitude Spectrum
Fourier Transform property (2)

24. 2D DFT Separable Property
1D DFT (row)
1D DFT (column)

25. Fast Fourier Transform (FFT)
1D DFT (x-direction)
2D image
1D DFT (y-direction)
1D DFT
1D Fast Fourier Transform (FFT)
Separate discrete data points into several groups
Calculate DFT in hierarchical manner

26. Fast Fourier Transform
If we let
WN = e-i2p /N
the Discrete Fourier Transform can be written
F(u) = f [x] WN
If N is a multiple of 2, N = 2M for some positive integer M, substituting 2M for N gives
F(u) = f [x] W2M 1 N
N -1 S
n = 0 sn 1 2M
2M -1 S
n = 0 ux

27. Fast Fourier Transform
Separating out the M even and M odd terms,
F(u) = f [2x] W2M f [2x+1] W2M
Notice that
W2M = e-i2p u (2x)/(2M) = e-i2p u x /M =WM
and
W2M = e-i2p u (2 x+1)/(2M) = e-i2p u x/M e-i2p u /2M = WM W2M
So,
F(u) = f [2x] WM f [2x+1] WM W2M
{ } 1 2 1 M
M -1 S
n = 0 1 M
M -1 S
n = 0
u(2x)
u(2x+1)
u(2x) ux
u(2x+1) ux u
{ }
M -1 S
n = 0 1 M 2 ux u

28. Fast Fourier Transform
{ }
M -1 S
n = 0 1 M 2 ux u
F(u) = f [2x] WM f [2x+1] WM W2M
Can be written as
F(u) = {Feven(u) + Fodd(u)W2M}
Simplifying further, the first M terms of the Fourier transform of 2M items can be computed by
F(u) = {Feven(u) + Fodd(u)W2M}
and the last M terms can be computed by
F(u) = {Feven(u) - Fodd(u)W2M} 1 2 u 1 2 u 1 2 u

29. Fast Fourier Transform
If M is itself a multiple of 2, do it again!
If N is a power of 2, keep recursively subdividing until you have one element, which is its own Fourier Transform.
FourierTransform FFT(Signal f) {
if (length(f) == 1) return f;
evenpart = FFT(EvenTerms(f));
oddpart = FFT( OddTerms(f));
for (s = 0; s < length(f) / 2; s++) {
result[s ] = evenpart[s] + W_2M ^ s * oddpart[s];
result[s+M] = evenpart[s] – W_2M ^ s * oddpart[s]; }

30. DIT-FFT (Scalable) (1) 2-point FFT butterfly 2-point a b a+b a-b a b
weight
2-point
FFT - 1
f(0)
f(2)
f(1)
f(3)
F(0)
F(1)
F(2)
F(3)
butterfly
DIT-FFT (Scalable) (1)

31. DIT-FFT (Scalable) (2) 8-point FFT weight butterfly 4-point 4-point8 W 1 2 3 - ) ( F 4 5 6 7
f(4)
f(2)
f(6)
f(1)
4-point
FFT
f(5)
f(3)
f(7)
DIT-FFT (Scalable) (2)

32. Fast Fourier Transform (DIT-FFT)
Direct calculation = N2
FFT = 2log2N
8-point FFT
Fast Fourier Transform (DIT-FFT)

33. Fast Fourier Transform
Computational Complexity: Remember: The Fast Fourier Transform is just a faster algorithm for computing the Discrete Fourier Transform — it does not produce a different result.
Discrete Fourier Transform 
O(N2)
Fast Fourier Transform
O(2 log2 N)

34. Processing Time (DFT vs FFT)

35. DFT Magnitude response |F(u,v)|
Unit step response

36. DFT Magnitude response |F(u,v)|
Small vertical line response
-45 degree rotated line response
Linear Combination response