1. Reminder Fourier Basis: t  [0,1] nZnZ Fourier Series: Fourier Coefficient:

2. Example - Sinc rect(t)

3. Sinc - Pictures

4. Discrete Fourier Transform Fourier Transform ( notations: f(x) = s(x/N), F(u) =  a u+Nk ) Inverse Fourier Transform Complexity: O(N 2 ) (10 6 10 12 ) FFT: O(N logN) (10 6 10 7 )

5. Fourier of Delta

6. 2D Discrete Fourier Fourier Transform Inverse Fourier Transform

7. Display Fourier Spectrum as Picture 1. Compute 2. Scale to full range Original f0124100 Scaled to 10000010 Log (1+f)00.691.011.614.62 Scaled to 10012410 Example for range 0..10: 3. Move (0,0) to center of image (Shift by N/2)

8. Fourier Displays

9. Decomposition

10. Decomposition (II) 1-D Fourier is sufficient to do 2-D Fourier –Do 1-D Fourier on each column. On result: –Do 1-D Fourier on each row –(Multiply by N?) 1-D Fourier Transform is enough to do Fourier for ANY dimension

11. Decomposition Example

12. Translation

13. Periodicity & Symmetry (Only for real images)

14. Rotation

15. Linearity

16. Derivatives I Inverse Fourier Transform

17. Derivatives II To compute the x derivative of f (up to a constant) : –Computer the Fourier Transform F –Multiply each Fourier coefficient F(u,v) by u –Compute the Inverse Fourier Transform To compute the y derivative of f (up to a constant) : –Computer the Fourier Transform F –Multiply each Fourier coefficient F(u,v) by v –Compute the Inverse Fourier Transform

18. Convolution Theorem Convolution by Fourier: Complexity of Convolution: O(N logN)

19. Filtering in the Frequency Domain Low-Pass Filtering Band-Pass Filtering High-Pass Filtering Picture FourierFilter Filtered Picture Filtered Fourier

20. (0 0 1 1 0 0)  Sinc (0 0 1 1 0 0) * (0 0 1 1 0 0 ) = (0 1 2 1 0 0)  Sinc 2 (0 1 4 6 4 1 0) = (0 0 1 1 0 0 ) 4  Sinc 4 Fourier (Gaussian)  Gaussian Low Pass: Frequency & Image

21. Continuous Sampling · = T * = 1/T ·=Image: * = 1/T Fourier: