1. Image Enhancement in the Frequency Domain Part III
Image Processing
Image Enhancement in the Frequency Domain
Part III

2. Some Basic Filters
Low frequencies in the Fourier transform are responsible for the general gray-level appearance of an image over smooth areas.
High frequencies are responsible for detail such as edges and noise.
Lowpass Filter : Attenuates high frequency while “passing” low frequency.
Less sharpen the details because the high frequencies are attenuated
highpass Filter : Attenuates low frequency while “passing” high frequency.
Less gray level variation in smooth areas and emphasized transitional.

3. Some Basic Filters

4. Correspondence between Filtering in the spatial and Frequency domains
Relationship
It is more computationally efficient to do the filtering in the frequency domain
Filtering is more intuitive in the frequency domain
It makes more sense to filter in the spatial domain using small filter mask
filter in the frequency domain
filter in the spatial domain
DFT
IDFT
guide for
small filter mask

5. Gaussian Filters Example( Filters based on Gaussian functions )
both functions are real
two functions behave reciprocally with respect to one another

6. Gaussian Filters

7. Ideal low pass
The transfer function for the ideal low pass filter can be given as:
where D0 is a specified nonnegative quantity, and D(u,v) is the distance from point (u,v) to the origin of the frequency rectangle.

8. Ideal Low pass Filters
The center of frequency rectangle is (u,v) = (M/2,N/2)
In this case, the distance from any point (u,v) to the center (origin) D(u,v) of the Fourier transform is given by
The lowpass filters considered here are radially symmetric about the origin

9. Ideal Low pass Filters
The transition point is called the cutoff frequency
Standard cutoff frequency for comparing filters
α percent of the power

10. Ideal Low pass Filters
The severe blurring in (b) is a clear indication that most of the sharp detail information in the picture is contained in the 8% power removed by the filter

11. Ideal Low pass Filters (c)~(e) “ringing” behavior
The ringing behavior is a characteristic of ideal filters

12. Ideal Low pass Filters

13. Butterworth Low pass Filters
Butterworth low pass filter (BLPF) of order n, and with cutoff frequency at a distance D0 from the origin
A cutoff frequency is defined at points for which H(u,v) is down to a certain fraction of its maximum value
In this case, H(u,v) = 0.5 when D(u,v) = D0

14. Butterworth Low pass Filters
Example No
ringing effect

15. Gaussian Low pass Filters
Gaussian lowpass filters (GLPFs) of two dimensions are given by
σ is a measure of the spread of the Gaussian curve
By letting σ = D0
where D0 is the cutoff frequency
When D(u,v) = D0, the filter is down to of its maximum value

16. Sharpening Freq. Domain Filters
Image can be blurred by attenuating the high-frequency components of its Fourier transform
Edges and other abrupt changes in gray levels are associated with high-frequency components
The transfer function of the highpass filters can be obtained using the relation
Where Hlp(u, v) is the corresponding lowpass filter

17. Sharpening Freq. Domain Filters
ideal highpass filter
Butterworth highpass filter
Gaussian highpass filter

18. Ideal Highpass Filters
:cutoff frequency

19. Butterworth highpass filter

20. Gaussian Highpass Filters
Gaussian highpass filter (GHPF)
The results obtained are smoother than with the previous two Filters.
Smaller objects and thin bars is cleaner with the GHPF

21. The Laplacian in Freq. Domain
It can be shown that
It follow that
Which result

22. The Laplacian in Freq. Domain
Laplacian can be implemented in the frequency domain by using the filter
The center of the filter function also needs to be shifted

23. The Laplacian in Freq. Domain
2-D image of Laplacian in the frequency domain
Laplacian in the frequency domain
Inverse DFT of Laplacian in the frequency domain
Zoomed section of the image on the left compared to spatial filter

24. The Laplacian in Freq. Domain

25. Assignments
Read the following sections from ch.4
1, 2, 3 ,4