1. Lecture 4 Image Enhancement in Frequency Domain
Machine Vision
R. Ebrahimpour, 2008

2. Objective
Basic understanding of the frequency domain, Fourier series, and Fourier transform.
Image enhancement in the frequency domain.

3. Summary of FT, DTFT/DSFT, DFT and FFT
Fourier Transform
(FT):
(continuous)
(continuous)
Discrete Time/Space Fourier Transform
(DTFT/DSFT):
(discrete)
(continuous,
periodic)

4. Summary of FT, DTFT/DSFT, DFT and FFT
Discrete Fourier Transform (DFT):
(discrete,
finite)
(discrete,
finite)
Fast Fourier Transform (FFT): Fast algorithm for computing DFT

5. Two-Dimensional Discrete Fourier Transform (2D-DFT)

6. 2D DFT and Inverse DFT (IDFT)
f(x, y)
F(u, v)
often used short notation:
M, N: image size
x, y: image pixel position
u, v: spatial frequency

7. The Meaning of DFT and Spatial Frequencies
Important Concept
Any signal can be represented as a linear combination of a set of basic components
Fourier components: sinusoidal patterns
Fourier coefficients: weighting factors assigned to the Fourier components
Spatial frequency: The frequency of Fourier component
Not to confused with electromagnetic frequencies (e.g., the frequencies associated with light colors)

8. Real Part, Imaginary Part, Magnitude, Phase, Spectrum
Magnitude-phase representation:
Magnitude
(spectrum):
Phase
(spectrum):
Power
Spectrum:

9. 2D DFT Properties Mean of image/ DC component:
Highest frequency component:
“Half-shifted”
Image:
Conjugate Symmetry:
Magnitude Symmetry:

10. 2D DFT Properties Spatial domain differentiation:
Frequency domain differentiation:
Distribution law:
Laplacian:
Spatial domain
Periodicity:
Frequency domain periodicity:

11. Computation of 2D-DFT
To compute the 1D-DFT of a 1D signal x (as a vector):
To compute the inverse 1D-DFT:
To compute the 2D-DFT of an image X (as a matrix):
To compute the inverse 2D-DFT:

12. Computation of 2D-DFT Fourier transform matrices: rememberW N - 1
(N-1) 2
2(N-1) W N - 1
(N-1) 2
2(N-1)
relationship:
relationship:
In particular, for N = 4:
In particular, for N = 4:

13. Computation of 2D-DFT: Example
A 4x4 image
Compute its 2D-DFT:
MATLAB function: fft2

14. Computation of 2D-DFT: Example
Real part:
Imaginary part:
Magnitude:
Phase:

15. Computation of 2D-DFT: Example
MATLAB function: ifft2

16. Spectrum of images

17. Centered Representation
MATLAB function: fftshift
Example:

18. Log-Magnitude Visualization
2D-DFT
centered

19. 2D-DFT  centered  log intensity transformation
Apply to Images
2D-DFT  centered  log intensity transformation

20. 2D-DFT (Frequency) Domain Filtering

21. Convolution Theorem f (x,y) h(x,y) g(x,y) G(u,v) = F(u,v) H(u,v)
input image
impulse response
(filter)
output image
DFT
IDFT
DFT
IDFT
DFT
IDFT
G(u,v) =
F(u,v)
H(u,v)

22. Frequency Domain Filtering
Filter design: design H(u,v)

23. 2D-DFT Domain Filter Design
Ideal lowpass, bandpass and highpass

24. 2D-DFT Domain Filter Design- Ideal LP
Project # 5
Salt and pepper denoising with Median and ILP
2D-DFT Domain Filter Design- Ideal LP
Ideal lowpass, bandpass and highpass
Cutoff frequency

25. 2D-DFT Domain Filter Design- Ideal LP
Ideal lowpass filtering with cutoff frequencies set at radii values of 5, 15, 30, 80, and 230, respectively
Blurring &Ringing Effect

26. 2D-DFT Domain Filter Design- Ideal LP

28. 2D-DFT Domain Filter Design- Ideal LP

29. 2D-DFT Domain Filter Design- Ideal LP

30. Cutoff frequency

31. No Visible Ringing Effect
Blurring &
No Visible Ringing Effect

32. 2D-DFT Domain Filter Design- Butterworth LP

33. 2D-DFT Domain Filter Design- Butterworth LP

34. 2D-DFT Domain Filter Design- Gaussian LP
Gaussian lowpass

35. 2D-DFT Domain Filter Design- Gaussian LP
Effect of Gaussian lowpass filter

36. 2D-DFT Domain Filter Design- Gaussian LP
Effect of Gaussian lowpass filter
Blurring &
No Ringing Effect

37. 2D-DFT Domain Filter Design- Gaussian LP

38. 2D-DFT Domain Filter Design- Gaussian LP
Effect of Gaussian lowpass filter

40. 2D-DFT Domain Filter Design
Gaussian lowpass filtering
Gaussian highpass filtering

41. 2D-DFT Domain Filter Design
Choices of highpass filters
Ideal
Butterworth
Gaussian

42. 2D-DFT Domain Filter Design

43. 2D-DFT Domain Filter Design- Ideal HP

44. 2D-DFT Domain Filter Design- Butterworth HP

45. 2D-DFT Domain Filter Design- Gaussian HP

46. 2D-DFT Domain Filter Design
Ideal
Butterworth
Gaussian

47. 2D-DFT Domain Filter Design
Gaussian filter with different width

48. 2D-DFT Domain Filter Design- Laplacian HP

49. 2D-DFT Domain Filter Design- Laplacian HP

50. 2D-DFT Domain Filter Design- Laplacian HP

51. 2D-DFT Domain Filter Design- Laplacian HP

53. 2D-DFT Domain Filter Design
Orientation selective filters

54. 2D-DFT Domain Filter Design
Narrowband Filtering
by combining radial and orientation selection *

55. Homomorphic filtering

56. Homomorphic filtering

57. Homomorphic filtering
C controls the sharpness of the slope

58. Homomorphic filtering