1. Jean Baptiste Joseph Fourier
Fourier Transform
Jean Baptiste Joseph Fourier
(1768 – 1830)

2. Fourier Transform
Fourier: Every cyclic function f(x) can be decomposed into a set of sin() and cos() functions of different frequencies, given by
The original function can be reconstructed from its Fourier set without any loss of data!
Reminder: Euler’s Formula

3. Example

4. Continuous FT
Fourier Transform
Inverse Fourier Transform

5. Discrete Fourier Transform (DFT)
Inverse DFT:
Note: f(x) is treated as a cyclic function!

6. Discrete Fourier Transform (DFT)
Spatial domain
(Standard basis)
Frequency domain
(Furier basis)
The Discrete Fourier Transform is more intuitive basis transform.
Image processing is performed over a discrete world

7. DFT Basis Change Matrix
Where

8. Fourier Spectrum Fourier decomposition: Fourier spectrum:
Fourier Phase:
Polar decomposition:

9. FT Properties
1. Linearity:
2. Periodicity:

10. FT Properties
3. Symmetry*:
2. Scaling: if
then

11. Examples
Fourier
Fourier
Fourier

12. Examples (Cont.)
Fourier
Fourier

13. Discrete Fourier Transform (2D)
2D Fourier Transform:
2D Inverse Fourier Transform:

14. 2D DFT – Simple Examples
Low frequency
Medium frequency
High frequency

15. 2D DFT – Simple Examples
2D rect
Gaussian
2D comb

16. Real Example

17. Fourier Decompsition 1D Fourier is sufficient to do 2D Fourier
– Do 1D Fourier on each column.
– On result: Do 1D Fourier on each row.
– (Multiply by N – application dependant)
• 1D Fourier Transform is enough to do Fourier for ANY dimension

18. Fourier Decompsition

19. Decomposition Example

20. Image Translation

21. Image Rotation Polar Coordinates: x = r ∙ cos Ө , y = r ∙ sin Ө
u = ω ∙ cos φ , v = ω ∙ sin φ
Linearity

22. Low vs. High Frequencies
Fuorier Transform supplies a global representation of the image in the frequency domain.
features in the image can not be assigned specific frequencies.
In general:
Low frequencies represent the coarse structure of the image (large homogenous parts like walls, sky, etc.)
High frequencies represent the fine details in the image (fine texture, wrinkles, noise, etc.)

23. Image Derivatives

24. Image derivatives To compute the x derivative of f (up to a constant):
Compute the Fourier Transform F
Multiply each Fourier coefficient F(u,v) by u
Compute the inverse Fourier Transform
To compute the y derivative of f (up to a constant):
Multiply each Fourier coefficient F(u,v) by v

25. Image derivatives – Intuition
Image derivative is the inverse Fourier Transform of the weighted frequency domain.
High frequencies contribute more to the image derivative.

26. How to use the Fourier Transform?
Finding a more accurate derivative
Understanding some image features
What else?

27. Multiply the Fourier transform by the filter:
Notch Filter Example
Reminder:
Multiply the Fourier transform by the filter:

28. Filtering Scheme Input image f(x) Fourier Transform – F(u,v)
Filter function – H(u,v) ∙ F(u,v)
Inverse Fourier Transform
Output filtered image g(x)

29. Ideal Low-pass Filters
Input
Output
Cutoff = 10

30. Ideal Low-pass Filters
Input
Output
Cutoff = 20

31. Cutoff = 40
Input
Output
Cutoff = 40

32. Ideal High-pass Filters
Input
Output
Cutoff = 10

33. Ideal High-pass Filter
Input
Output
Cutoff = 10

34. Ideal High-pass Filter
Input
Output
Cutoff = 40

35. Ideal Band-pass Filter
Input
Output
Cutoff = [20,30]

36. Gaussian Filter
Input
Output
Width = 10

37. Gaussian Filter
Input
Output
Width = 40

38. Resemblance to Convolution
Fourier Filter
Low-pass filter
High-pass filter
Convolution Filter

39. The Convolution Theorem:
Reminder:
The Convolution Theorem:

40. Convolution Vs. Fourier
Convolution by Fourier:
Complexity (using the FFT algorithm): O(N∙logN)
Different Fourier transform phenomena can be explained via convolution, and vice versa.
In practice: Fourier transform intuition is used to design convolution filters.

41. Why do we get the rings?
1D Simplification:
Fourier

42. In the image domain (1D):
What happens to an edge?
In the image domain (1D):

43. Cross-Correlation Theorem
Definition:
Cross Correlation is a measure of similarity of two signals as a function of a time-lag applied to one of them.
Commonly used filter due to similar propagation direction (e.g. pattern matching in an image)
The Cross-Correlation Theorem: