Tutorials 12,13 discrete signals and systems

1. Tutorials 12,13discrete signals and systems Technion, CS department, SIPC 236327 Spring 2014

2. Discrete LSI system • Linear • Space invariant

3. Discrete LSI system • Linear • Space invariant

4. Example • For compression, a rule to predict the pixel value is used:Is the system linear? Space invariant?

5. Discrete LSI system • System is defined with its impulse response

6. Cyclic convolution Convolution • Infinite support • DTFT • Finite support • DFT • Efficient implementation

7. Exercise Q: How can we use this system to calculate a linear convolution? A: Zero padding, and truncation of the result. H Q: If both signals are of length N, how many zeros will we add? A: N-1 zeros

8. Exercise Q: How can we use this system to calculate a cyclic convolution? A: Duplicate one signal, and truncation of the result. H Q: If both signals are of length N, how much should we duplicate A: N-1 cells

9. Discrete Fourier Transform (DFT, FFT) Infinite support Infinite support Continiuous Continuous Finite support Finite support Discrete Discrete

10. DFT • התמרות הDFT וDFT-1מתבצעות בדרך הרגילה • המקדמים מחזוריים: • לכן במקום להתייחס לתחום [0,N-1] בד"כ מסתכלים על התחום [-N/2,N/2-1].

11. הפעלת DFT

12. דוגמאות DFT

13. Summary – Fourier Transforms • Fourier transform • Time domain – non-periodic infinite signals • Continuous time (t) • Continuous frequency (f) • Formulas

14. Summary – Fourier Transforms • DTFT: Discrete Time Fourier Transform • Time domain – non-periodic infinite signals • Discrete time (n) • Continuous frequency (f) • Formulas לא נלמד בקורס

15. Summary – Fourier Transforms • Fourier series • Time domain – periodic infinite signals • Continuous time (t) • Discrete frequency (f) • Formulas

16. Summary – Fourier Transforms • DFTor Discrete Time Fourier Series • Time domain – periodic infinite signals • Discrete time (n) • Discrete frequency (f) • Formulas

17. DFT ומערכת LSI

18. Exercise • We have an N-length filter with impulse response h[n].We create a new filter as follows: Express F[k] with H[k], where H[k]=DFT{h[n]},F[k]=DFT{f[n]} • Instructions: calculate

19. Example – discrete frequency filtration • Noisy image of size 256X256 Im_out[m,n]=Im_in[m,n]+noise[m,n] • Harmonic noise: • f = 1/(8 pixels) • Amplitude A and phase φ are random and independent for each line.

20. Example – added noise in line 100

21. Example – discrete frequency filtration

22. Example – discrete frequency filtration - smoothing

23. Example – discrete frequency filtration– smoothing vs median (8 pixels) No noisebut image is blurred

24. Example – discrete frequency filtration • DFT of the noise in line i

25. Example – discrete frequency filtration • Design an LSI filter • Such filter multiplies each frequency with a complex number • Can handle each frequency separately • In this example, we want to handle frequencies 32 and -32. • Notch filter – attenuates specific frequency

26. Example – discrete frequency filtration Original signal in frequency domain Filtered signal in frequency domain

27. Example – discrete frequency filtration • Noise removed completely • Original image not fully restored • We cannot restore the attenuated frequencies

28. Example – discrete frequency filtration Smoothing filter of 8 pixels Notch filter

29. Example –frequency filtration - implementation Notch filter in freq. domain • Filter in freq. domain: Filter=ones(1,256); Filter(32+1)=0; Filter(224+1)=0; • Filtration: For k=1:size(I,1), Y=fft(I(k,:)).*Filter; I(k,:)=ifft(Y); end

30. Tutorials 12,13discrete signals and systemsPart II: 2D Technion, CS department, SIPC 236327 Spring 2014

31. 2D - definitions 2Dconvolution:

32. 2D - definitions • Cyclic 2D-convolution: • 2D DFT:

33. 2D - notes • DFT is linear, we have an operation matrix: • 2D-DFT can be implemented as: • If the input is separable:

34. Example • Noisy image 512X512 • The noise:Add 100 gray levels for all 16i lines

35. Example Noisy image Average filter

36. Example Noisy image Average filter

37. Example • How does the noise look like in the frequency domain?

38. Example After freq. filtration • Filter implementation in the freq. domain: H=ones(512,512); for n=1:32:512 H(n,1) = H(1,n) = 0; end H(1,1) = 1; • Image filtration: out = ifft( fft(img).*H );

39. לפני סינון תדר

40. לפני סינון תדר (הגדלה של מרכז)

41. אחרי סינון תדר (הגדלה של מרכז)

42. Image filtration

43. Edge detection of Image A • Roberts • Prewitt • Sobel

44. Edge detection of Image A Original Roberts Prewitt Sobel

45. Unsharp masking – edge enhancement