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Digital Image Processing

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. The Goal of This Homework. Implement Fast Fourier Transform.

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Digital Image Processing

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  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 FFT Filtering (Notch Filter) IFFT

  4. Fast Fourier Transform(1/3) • 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

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

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

  7. Fast Fourier Transform(8 points)

  8. Inverse Fast Fourier Transform 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 • 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 • 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/fourier.pdf • J.W. Cooley and J.W. Tukey, “An algorithm for the machine calculation of complex Fourier series”, Math. Computation, 19:297-301, 1965.

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