1. Fast Fourier Transform & Assignment 2
Yong-Fong Lin
Visual Communications Lab Department of Communication Engineering National Central University Chungli, Taiwan
Oct. 4, 2007

2. Outline Assignment Description Discrete Fourier Transform Review
Introduction to Fast Fourier Transform(FFT)
Example：4 Point FFT & FFT Butterfly
Experimental Result
Notice
Reference

3. Assignment Description
Part 1 Transform an gray image from spatial domain into frequency domain using Fast Fourier Transform(FFT). And show the spectrum of the image.
Part 2 Rotate the gray image by , and show the spectrum of the rotated image.

4. Discrete Fourier Transform(DFT) Review
One-Dimensional DFT for u = 0,1,2, …. ,M-1

5. Fast Fourier Transform (1/5)
for u = 0, 1, 2, … ,M-1 If (n N) then M can be expressed as M=2K (K N)
(Keep this in mind !!)

6. Fast Fourier Transform (2/5)
(1) for u = 0,1, … , K-1

7. Fast Fourier Transform (3/5)
By (1)

8. Fast Fourier Transform (4/5)
(2) for u = K, K+1, … , 2K-1

9. Fast Fourier Transform (5/5)
Conclusion：
We can perform the DFT by using FFT as follow steps for the first K points (u = 0 ~ K-1) for the rest K points (u = K ~ 2K-1) the rest K points doesn’t need extra computation, it can just be obtained by the result of first K points.

10. Example：4 Point FFT & FFT Butterfly
Consider a sequence：f(x) for x= 0 ~ 3 need to be transformed.
The transformed result is F(u) for u = 0 ~ 3.
F(0)
F(2)
F(1)
F(3)
F(0)
F(2)
F(1)
F(3)
F(0)
F(1)
F(2)
F(3) + -

11. Experimental Result (1/2)
Original image and the corresponding spectrum

12. Experimental Result (2/2)
Rotated image and the corresponding spectrum

13. Notice
Don’t forget to multiply
According to the property of “ Separability ” , we can perform two-dimensional DFT by using one-dimensional DFT. (p197)
is just
To deal with complex number , we must have 2 buffer.
One for the real part
The other for the imaginary part

14. Reference
Rafael C. Gonzalez , Richard E. Woods , “Digital Image Processing, ” second edition , pp