1. The Fourier Transform
Jean Baptiste Joseph Fourier

2. A sum of sines and cosines=
3 sin(x) A
sin(x) A
+ 1 sin(3x) B
A+B
+ 0.8 sin(5x) C
A+B+C
Accept without proof that every function is a sum of
sines/cosines
As frequency increases – more details are added
Low frequency – main details
Hight frequency – fine details
Coef decreases with the frequency
+ 0.4 sin(7x) D
A+B+C+D

3. Higher frequencies due to sharp image variations
(e.g., edges, noise, etc.)

4. The Continuous Fourier Transform

5. Complex Numbers
Imaginary
Z=(a,b) b
|Z| 
Real a

6. The 1D Basis Functions x The wavelength is 1/u . The frequency is u .

7. The Continuous Fourier Transform
1D Continuous Fourier Transform:
The Inverse Fourier Transform
The Fourier Transform
2D Continuous Fourier Transform:
The Inverse Transform
The Transform

8. The 2D Basis Functions V U The wavelength is . The direction is u/v .

9. Discrete Functions f(x) f(n) = f(x0 + nDx) The discrete function f:
f(x0+2Dx)
f(x0+3Dx)
f(x0+Dx)
f(x0) x0
x0+Dx
x0+2Dx
x0+3Dx
N-1
The discrete function f:
{ f(0), f(1), f(2), … , f(N-1) }

10. The Discrete Fourier Transform
1D Discrete Fourier Transform:
(u = 0,..., N-1)
(x = 0,..., N-1)
2D Discrete Fourier Transform:
(x = 0,..., N-1; y = 0,…,M-1)
(u = 0,..., N-1; v = 0,…,M-1)

11. The Fourier Image Image f Fourier spectrum |F(u,v)|
Fourier spectrum log(1 + |F(u,v)|)

12. Frequency Bands Image Fourier Spectrum
Percentage of image power enclosed in circles (small to large) :
90%, 95%, 98%, 99%, 99.5%, 99.9%

13. Low pass Filtering
90%
95%
98%
99%
99.5%
99.9%

14. Noise Removal
Noisy image
Noise-cleaned image
Fourier Spectrum

15. High Pass Filtering
Original
High Pass Filtered

16. High Frequency Emphasis+
Original
High Pass Filtered

17. High Frequency Emphasis
Original
High Frequency Emphasis
High Frequency
Emphasis
Original

18. High Frequency Emphasis
Original
High pass Filter
High Frequency
Emphasis
High Frequency Emphasis +
Histogram Equalization

19. Properties of the Fourier Transform –
Developed on the board… (e.g., separability of the 2D transform, linearity, scaling/shrinking, derivative, rotation, shift  phase-change, periodicity of the discrete transform, etc.) We also developed the Fourier Transform of various commonly used functions, and discussed applications which are not contained in the slides (motion, etc.)

20. 2D Image
2D Image - Rotated
Fourier Spectrum
Fourier Spectrum

21. Fourier Transform -- Examples
Image
Domain
Frequency
Domain

22. Fourier Transform -- Examples
Image
Domain
Frequency
Domain

23. Fourier Transform -- Examples
Image
Domain
Frequency
Domain

24. Fourier Transform -- Examples
Image
Domain
Frequency
Domain

25. Fourier Transform -- Examples
Image
Fourier spectrum

26. Fourier Transform -- Examples
Image
Fourier spectrum

27. Fourier Transform -- Examples
Image
Fourier spectrum

28. Fourier Transform -- Examples
Image
Fourier spectrum