1. IMAGE PROCESSING FREQUENCY DOMAIN PROCESSING
Editor by
DR. FERDA ERNAWAN
Faculty of Computer Systems & Software Engineering

2. Today’s Lesson Frequency Domain Processing
Basic steps in frequency domain
Low pass Filter (smoothing filters)
Ideal lowpass, Butterworth lowpass, Gaussian lowpass
High pass Filter (sharpening filters)
Ideal highpass, Butterworth highpass , Gaussian highpass
Learning Outcomes:
To understand frequency filter operation.

3. Basic steps for filtering
Images taken from Gonzalez and Woods, 2016
Frequency domain filtering

4. The Basics of Filtering
1. Fourier Spectrum
Images taken from Gonzalez and Woods, 2016
Scanning electron microscopy of a damaged circuit (b) Fourier Spectrum

5. The Basics of Filtering
2. Frequency Domain Filtering Fundamentals
Result of filtering the image

6. DFT

7. DFT
The results of DFT can be visualised by showing the spectrum signals as depicted in below:
Images taken from Gonzalez and Woods, 2016
DFT

8. Scanning electron microscope image
DFT
DFT
Images taken from Gonzalez and Woods, 2016
Scanning electron microscope image
Fourier spectrum

9. Fourier Transform Spectrum
FT Spectrum
Original Image
Fourier Spectrum
Centered Spectrum
Centered Spectrum after log transformation
Images taken from Gonzalez and Woods, 2016
Centered
FT Spectrum
FT Spectrum (centered + log)

10. Fourier Transform Spectrum
Vertical rectangle image
The corresponding spectrum of vertical rectangle image
Diagonal rectangle image
The corresponding spectrum of diagonal rectangle image
Images taken from Gonzalez and Woods, 2016
Spectrum is insensitive to translation

11. Filter Function Low Pass Filter High Pass Filter
Images taken from Gonzalez and Woods, 2016
Spectrum is insensitive to translation
High Pass Filter

12. The Basic of Filtering 3. Steps for Filtering:
Spectrum is insensitive to translation

13. The Importance of Zero Padding
Images taken from Gonzalez and Woods, 2016
Zero padding is used to produce smoother spectrum without further interpolation.
Zero padding before FFT is computationally efficient method of interpolating (parabolic) a large number of points..

14. The Importance of Zero Padding
(a) An image, (b) Result of blurring without padding (c) Result of blurring with padding

15. Smoothing
Smoothing can be done by reducing high frequency values. An image filtering can be given as:
G(u,v) = H(u,v)F(u,v)
where F(u,v) denotes as DFT function and H(u,v) represents filter function.

16. ILF
Ideal lowpass filter (ILF) is defined by:
where D0 >0 and D(u,v) denotes the distance between a point (u,v) and the frequency center. D(u,v) can be defined as:

17. ILF
H(u,v)=1
(a) an ideal low-pass filter (b) Filter (c) Filter radial

18. ILF Example: Sample Image Fourier spectrum
Images taken from Gonzalez and Woods, 2016
Sample Image
Fourier spectrum

19. ILF
(b)
(a)
(a) Representation of an ideal lowpass, radius 5 (b) Intensity profile

20. BLF
Butterworth lowpass filter (BLF) of order n with cut-off frequency at distance D0 is defined as:
Images taken from Gonzalez and Woods, 2016
(a)Perspective plot of a BLF (b) Filter (c) Filter radial of order 1 through 4

21. BLF
Images taken from Gonzalez and Woods, 2016
Spatial representation of Butterworth lowpass filter of order 1, 2, 5, and 20 and the corresponding intensity

22. BLF
Example:

23. BLF Original image BLPF n=2, D0=5 BLPF n=2, D0=15 BLPF n=2, D0=30
Less ringing than ILPF due to smoother transition
BLPF n=2, D0=230

24. GLF Gaussian lowpass filter (GLPF) can be defined as:
(a) GLPF function (b) filter function (c) Filter radial

25. GLF Original image Gaussian D0=5 Gaussian D0=15 Gaussian D0=30
Less ringing than BLPF but also less smoothing
Gaussian D0=230

26. Lowpass Filters Comparison
BLPF n=2,
D0=15
ILPF D0=15
Gaussian D0=15

27. Lowpass Filtering Examples
Gaussian lowpass filter can be used to connect broken text from scanning image.
Blurring can help reading broken characters

28. Lowpass Filtering Examples
Lowpass filtering can be used for publishing industry and printing. and. “cosmetic” processing
Image
GLPF with Do=100
GLPF with
Do=80

29. Sharpening (Highpass filter)
High frequency coefficients contribute significantly in edges and fine detail of images. Highpass filter is a reverse of lowpass filter:
HPF(u, v) = 1 – LPF(u, v)
D0 represents a cut-off frequency and n denotes the order of Butterworth filter.

30. Ideal Highpass Filters
An ideal highpass filter is defined by:
D0 denotes a cut off distance.

31. Image Sharpening (Highpass filter)
Ideal Highpass Filters
Image Sharpening (Highpass filter)
IHPF D0 = 30
IHPF D0 = 80
IHPF D0 = 15

32. Butterworth Highpass Filters
Butterworth highpass filter is defined as:
n represents the order and D0 denotes a cut off distance.

33. Image Sharpening (Highpass filter)
Butterworth Highpass Filters
Image Sharpening (Highpass filter)
BHPF n=2,
D0 =15
BHPF n=2,
D0 =30
BHPF n=2,
D0 =80

34. Gaussian Highpass Filters
Gaussian highpass filter is defined as:
D0 represents a cut off distance.

35. Image Sharpening (Highpass filter)
Gaussian High-pass Filters
Image Sharpening (Highpass filter)
Gaussian HPF n=2, D0 =15
Gaussian HPF n=2, D0 =30
Gaussian HPF n=2, D0 =80

36. Image Sharpening (Highpass filter)
Gaussian High-pass Filters
Image Sharpening (Highpass filter)
Gaussian HPF n=2, D0 =15
BHPF n=2,
D0 =15
Gaussian HPF n=2, D0 =15

37. Image Sharpening (Highpass filter)
Highpass filtering Examples
Example: using high pass filtering and thresholding for image enhancement
Highpass filtering
Thresholding
Image Sharpening (Highpass filter)
Thumb print (b) Result obtained from highpass filter
Thresholding result
Images taken from US National Institute of standards

38. References
R.C. Gonzalez and R.E. Woods, Digital Image Processing, Pearson Education India; Third edition.
A.K. Jain, Fundamentals of Digital Image Processing, Pearson Education India; First edition.
R.C. Gonzalez, R.E. Woods and S.L. Eddins, Digital Image Processing Using MATLAB. McGraw Hill Education; 2 edition.
S. Jayaraman, T. Veerakumar, S. Esakkirajan, 2017.Digital Image Processing, McGraw Hill Education; 1 edition.