1. Digital Image Processing Homework II Fast Fourier Transform 2012/03/28 Chih-Hung Lu ( 呂志宏 ) Visual Communications Laboratory Department of Communication Engineering National Central University ChungLi, Taiwan

2. The Goal of This Homework Implement Fast Fourier Transform. Implement Inverse Fast Fourier Transform. Implement Notch Filter.

3. Work Chart 3 FFT Filtering (Notch Filter) IFFT

4. To compute a discrete Fourier transform: N: number of pixel f(x): value of pixel x: pixel position F(u): value of frequency u: frequency Rewritten as: where Fast Fourier Transform(1/3) 4

5. Fast Fourier Transform(2/3) Assume N= 2 n, Let N=2M. 5

6. Fast Fourier Transform(3/3) 6 Because (e jx = cos(x) + j sin(x))

7. Fast Fourier Transform(8 points) 7

8. Inverse Fast Fourier Transform 8 To compute a Inverse Fourier transform: N: number of pixel f(x): value of pixel x: pixel position F(u): value of frequency u: frequency where

9. Notch Filter 9 Ideal Notch Reject Filter: H: Filter Set F(0,0) to zero & leave all other frequency components of the Fourier transform untouched.

10. Display of Frequency Spectrum 10 Modulation in space domain: Moving (0,0) to the central in image. Log transformation Example:

11. Grading FFT (3 points) Filtering (3 points) IFFT (2 points) Report (2 points) Computation complexity (Bonus 1 points) Other properties (Bonus 1 points)

12. Demo Schedule Thursday night, 19 April, 2012. - Begin around at 19:00 at R3-307.(VCLab) NOTE:  We will use another image to test your code.  The details will be announced on our course website. ( http://140.115.156.251/vclab/html/course/DIP2012.html )

13. References Paul Heckbert, Fourier Transforms and the Fast Fourier Transform (FFT) Algorithm www.cs.cmu.edu/afs/andrew/.../fourier/fouri er.pdf www.cs.cmu.edu/afs/andrew/.../fourier/fouri er.pdf J.W. Cooley and J.W. Tukey, “An algorithm for the machine calculation of complex Fourier series”, Math. Computation, 19:297-301, 1965.