DREAM

1. DREAM IDEA PLAN IMPLEMENTATION

3. Introduction to Image Processing Present to:Amirkabir University of Technology (Tehran Polytechnic) & Semnan University Dr. Kourosh Kiani Email: kkiani2004@yahoo.com Email: Kourosh.kiani@aut.ac.ir Email: Kourosh.kiani@semnan.ac.ir Web: www.kouroshkiani.com

4. Lecture 10 2D Discrete Fourier Transform (DFT)

5. The two-dimensional Fourier transform and its inverse • Fourier transform (discrete case) DFt • Inverse Fourier transform: • u, v : the transform or frequency variables • x, y : the spatial or image variables

6. DFT

7. |F|=abs(F)= log10 |F|=log10(abs(F)) =

8. angleF = radToDegF=

10. Inverse Fourier transform

11. Inverse Fourier transform

12. Magnitude and Phase of DFT • What is more important? magnitude phase

13. Magnitude and Phase of DFT Reconstructed image using magnitude only Reconstructed image using phase only

14. Magnitude and Phase of DFT amplitude phase original

15. Example: DFT of 2D rectangle function Input function Fourier spectrum Spectrum displayed as an intensity function

16. Extending DFT to 2D 2D cos/sin functions

17. 2D - DFT Base-functions are waves v u

23. Why is DFT Useful? • Easier to remove undesirable frequencies. • Faster perform certain operations in the frequency domain than in the spatial domain.

24. Properties in the frequency domain • Fourier transform works globally • No direct relationship between a specific components in an image and frequencies • Intuition about frequency • Frequency content • Rate of change of gray levels in an image

25. Example: Removing undesirable frequencies frequencies noisy signal To remove certain frequencies, set their corresponding F(u) coefficients to zero! remove high frequencies reconstructed signal

26. How do frequencies show up in an Signal? • Low frequencies correspond to slowly varying information • High frequencies correspond to quickly varying information

27. How do frequencies show up in an image? • Low frequencies correspond to slowly varying information (e.g., continuous surface). • High frequencies correspond to quickly varying information (e.g., edges)

28. How do frequencies show up in an image?

29. How do frequencies show up in an image?

30. How do frequencies show up in an image?

31. How do frequencies show up in an image?

32. How do frequencies show up in an image?

33. How do frequencies show up in an image?

34. How do frequencies show up in an image?

35. The 2D DFT and its inverse • Centered spectrum for display

36. 2-D Fourier transform • Frequency axis 0 x u u F shift y v v

46. Low and high frequencies Frequencies of the 2D DFT High Low Low Low Low Low High High Low Low Low

47. Periodicity of 2-D DFT f(x,y) 2-D DFT: -M For an image of size NxM pixels, its 2-D DFT repeats itself every N points in x-direction and every M points in y-direction. 0 M We display only in this range 2M -N 0 N 2N

48. Conventional Display for 2-D DFT F(u ,v) has low frequency areas at corners of the image while high frequency areas are at the center of the image which is inconvenient to interpret. High frequency area Low frequency area

49. 2-D FFT Shift : Better Display of 2-D DFT 2-D FFT Shift is a MATLAB function: Shift the zero frequency of F(u,v) to the center of an image. 2D FFTSHIFT High frequency area Low frequency area

50. 2-D FFT Shift : How it works -M 0 Display of 2D DFT After FFT Shift M Original display of 2D DFT 2M -N 0 N 2N