1. IMAGE PROCESSING IN FREQUENCY SPACE 19.4.2015Erkki Rämö1

2. 19.4.2015Lauri Toivio2

3. Images frequency domain  2D spatial domain image can be altered into frequency domain by applying Fourier transformation  Frequency image has the same dimensions as the original, but the components are complex numbers  Frequency image is a map of image frequencies in the spatial image 19.4.20153

4. Images frequency domain  Components of frequency image are complex numbers  Consists of magnitude and phase components  Frequency image is visualized by showing its magnitude components  Calculated from spatial images first by rows then by columns 19.4.20154

5. Example of frequency images 5  Low frequencies are near origin  Frequency is symmetrical in relation to the coordinate axis

6. Numeral scope of frequency image  Complex number consists of magnitude and phase components  Magnitude components differencies of samples are so big that a logarithmic scaling is needed to visualize the frequency image 19.4.20156

7. Visualization of frequency image 7 Original Magnitude component Logarithmic scaling

8. Directional dependency of frequency image 19.4.20158

9. Lauri Toivio 9

10. Directional dependency – application  Straightening of scanned text 19.4.201510 Threshold FFT

11. Some hardcore mathematics 19.4.201511

12. Fourier-transform Fourier –transform in one dimension: 19.4.201512 Fourier –counter transform:

13. Fourier-transform  If using angular frequen instead of oscillation frequency, the formulas are: 19.4.201513

14. Discrete Fourier trasform X(k) and its counter transform x(n): 19.4.201514

15. 2D Fourier-transform 19.4.201515 = =

16. DFT - 2D 19.4.201516

17. Euler formula Lauri Toivio17  Example: for (i=0;i<m;i++) { x2[i] = 0; y2[i] = 0; arg = - dir * 2.0 * 3.141592654 * (double)i / (double)m; for (k=0;k<m;k++) { cosarg = cos(k * arg); sinarg = sin(k * arg); x2[i] += (x1[k] * cosarg - y1[k] * sinarg); y2[i] += (x1[k] * sinarg + y1[k] * cosarg); }

18. Fast Fourier Transform - FFT  Speed up calculation by decreasing values to be calculated 19.4.201518 where

19. Single-frequency images frequency domain  In image, only one vertical frequency  Shows as a dot in frequency image 19.4.201519

20. 19.4.2015Lauri Toivio 20

21. 19.4.2015Lauri Toivio 21

22. Threshold 200 2 pixel wide vertical lines FFT

23. Frequency filtering  Chosen frequencies are masked off of frequency image 19.4.201523

24. FFT-filtering Low-pass filtering High-pass filtering

25. 19.4.2015Lauri Toivio 25

26. 19.4.201526

27. 19.4.2015Lauri Toivio27

28. 19.4.201528

29. 19.4.201529

30. Image restoration by Photoshop 19.4.2015Lauri Toivio 30

31. Group discussion Discuss application areas for frequency based image processing 19.4.2015Lauri Toivio31

32. Fourier-transform in Matlab >> load trees >> I=ind2gray(X,map); >> FI=fft2(I); >> SFI=fftshift(FI); >> abs(SFI); >> max(max(abs(SFI))) ans = 3.7987e+004 >> m=3.7987e+004 >> imshow(abs(SFI)/m,64) 19.4.201532

33. More information:  http://www.dai.ed.ac.uk/HIPR2/fourier.htm http://www.dai.ed.ac.uk/HIPR2/fourier.htm 19.4.201533