1. Frequency Domain Filtering (Chapter 4)
CS474/674 - Prof. Bebis

2. Frequency Domain Methods
Spatial Domain

3. Major filter categories
Typically, filters are classified by examining their properties in the frequency domain:
(1) Low-pass
(2) High-pass
(3) Band-pass
(4) Band-stop

4. Example Original signal Low-pass filtered High-pass filtered
Band-pass filtered
Band-stop filtered

5. Low-pass filters (i.e., smoothing filters)
Preserve low frequencies - useful for noise suppression
frequency domain
time domain
Example:

6. High-pass filters (i.e., sharpening filters)
Preserves high frequencies - useful for edge detection
frequency
domain
time
domain
Example:

7. Band-pass filters Example: Preserves frequencies within a certain band
frequency
domain
time
domain
Example:

8. Band-stop filters
How do they look like?
Band-pass
Band-stop

9. Frequency Domain Methods
Case 1: H(u,v) is specified in
the frequency domain.
Case 2: h(x,y) is specified in
the spatial domain.

10. Frequency domain filtering: steps
F(u,v) = R(u,v) + jI(u,v)

11. Frequency domain filtering: steps (cont’d)
(case 1)
G(u,v)= F(u,v)H(u,v) = H(u,v) R(u,v) + jH(u,v)I(u,v)

12. Example f(x,y) fp(x,y) fp(x,y)(-1)x+y F(u,v) H(u,v) - centered
G(u,v)=F(u,v)H(u,v)
g(x,y)
gp(x,y)

13. h(x,y) specified in spatial domain: how to generate H(u,v) from h(x,y)?
If h(x,y) is given in the spatial domain (case 2),
we can generate H(u,v) as follows:
Form hp(x,y) by padding with zeroes.
2. Multiply by (-1)x+y to center its spectrum.
3. Compute its DFT to obtain H(u,v) 13

14. Example: h(x,y) is specified in the spatial domain
Important: need to preserve
odd symmetry (i.e., H(u,v)
should be imaginary)
(read details on page 268)
In this example, we start with a spatial mask and show how to generate its corresponding filter in the frequency domain. Then, we compare the filtering results obtained using frequency domain and spatial techniques. We use the 3x3 Sobel vertical edge detector. The left one is a 600x600 pixel image, and its spectrum is shown on the right.
Sobel 14

15. Results of Filtering in the Spatial and Frequency Domains
spatial domain
filtering
frequency domain
filtering
In this example, we start with a spatial mask and show how to generate its corresponding filter in the frequency domain. Then, we compare the filtering results obtained using frequency domain and spatial techniques. We use the 3x3 Sobel vertical edge detector. The left one is a 600x600 pixel image, and its spectrum is shown on the right. 15

16. Low-pass (LP) filtering
Preserves low frequencies, attenuates high frequencies.
ideal
in practice
D0: cut-off frequency

17. Lowpass (LP) filtering (cont’d)
In 2D, the cutoff frequencies lie on a circle.

18. Specifying a 2D low-pass filter
Specify cutoff frequencies by specifying the radius of a circle centered at point (N/2, N/2) in the frequency domain.
The radius is chosen by specifying the percentage of total power enclosed by the circle.

19. Specifying a 2D low-pass filter (cont’d)
Typically, most frequencies are concentrated around the center of the spectrum.
r=8 (90% power)
r=18 (93% power)
original
r: radius
r=43 (95%)
r=78 (99%)
r=152 (99.5%)

20. How does D0 control smoothing?
Reminder: multiplication in the frequency domain implies convolution in the time domain
time domain
freq. domain * =

21. How does D0 control smoothing? (cont’d)
D0 controls the amount of blurring
r=78 (99%)
r=8 (90%)

22. Ringing Effect
Sharp cutoff frequencies produce an overshoot of image features whose frequency is close to the cutoff frequencies (ringing effect).
h=f*g

23. Low Pass (LP) Filters Ideal low-pass filter (ILPF)
Butterworth low-pass filter (BLPF)
Gaussian low-pass filter (GLPF)

24. Butterworth LP filter (BLPF)
In practice, we use filters that attenuate high frequencies smoothly (e.g., Butterworth LP filter)  less ringing effect
n=1
n=4
n=16

25. Spatial Representation of BLPFs
n= n= n= n=20

26. Comparison: Ideal LP and BLPF
ILPF
BLPF
D0=10, 30,
60, 160,
460
D0=10, 30,
60, 160,
460
n=2

27. Gaussian LP filter (GLPF)27

28. Gaussian: Frequency – Spatial Domains

29. Example: smoothing by GLPF (1)

30. Examples of smoothing by GLPF (2)
D0=100
D0=80
4/14/2017

31. High-Pass filtering
A high-pass filter can be obtained from a low-pass filter using:
= 1 - D0

32. High-pass filtering (cont’d)
Preserves high frequencies, attenuates low frequencies.

33. High Pass (LP) Filters Ideal high-pass filter (IHPF)
Butterworth high-pass filter (BHPF)
Gaussian high-pass filter (GHPF)
Difference of Gaussians
Unsharp Masking and High Boost filtering

34. Butterworth high pass filter (BHPF)
In practice, we use filters that attenuate low frequencies smoothly (e.g., Butterworth HP filter)  less ringing effect

35. Spatial Representation of High-pass Filters
IHPF
BHPF
GHPF

36. Comparison: IHPF and BHPF

37. Gaussian HP filter
GHPF
BHPF

38. Comparison: BHPF and GHPF

39. Example: High-pass Filtering and Thresholding for Fingerprint Image Enhancement
BHPF
(order 4 with a cutoff frequency 50)

40. Difference of Gaussians: Frequency – Spatial Domains
This is a high-pass filter!

41. Difference of Gaussians: Frequency – Spatial Domains (cont’d)
High-pass filter!

42. Frequency Domain Analysis of Unsharp Masking and Highboost Filtering
(alternative definition)
previous definition:
Frequency
domain:

43. Revisit: Unsharp Masking and Highboost Filtering

44. Highboost and High-Frequency-Emphasis Filters1
1+k k1
k1+k2
Highboost
High-emphasis

45. Example D0=40 High-Frequency Emphasis filtering Using Gaussian filter
GHPF
D0=40
High-emphasis
High-emphasis
and hist. equal.
High-Frequency
Emphasis filtering
Using Gaussian filter
k1=0.5, k2=0.75

46. Homomorphic filtering
Many times, we want to remove shading effects from an image (i.e., due to uneven illumination)
Enhance high frequencies
Attenuate low frequencies but preserve fine detail.

47. Homomorphic Filtering (cont’d)
Consider the following model of image formation:
In general, the illumination component i(x,y) varies slowly and affects low frequencies mostly.
In general, the reflection component r(x,y) varies faster and affects high frequencies mostly.
i(x,y): illumination
r(x,y): reflection
IDEA: separate low frequencies due to i(x,y)
from high frequencies due to r(x,y)

48. How are frequencies mixed together?
Low and high frequencies from i(x,y) and r(x,y)
are mixed together.
When applying filtering, it is difficult to handle
low/high frequencies separately.

49. Can we separate them?
Idea:
Take the ln( ) of

50. Steps of Homomorphic Filtering
(1) Take
(2) Apply FT: or
(3) Apply H(u,v)

51. Steps of Homomorphic Filtering (cont’d)
(4) Take Inverse FT: or
(5) Take exp( ) or

52. Example using high-frequency emphasis
Attenuate the contribution made by illumination and amplify the contribution made by reflectance
Attenuate the contribution made by illumination and amplify the contribution made by reflectance

53. Homomorphic Filtering: Example

54. Homomorphic Filtering: Example