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

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

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.

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

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 !!)

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

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

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

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.

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) + -

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

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

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

Reference Rafael C. Gonzalez , Richard E. Woods , “Digital Image Processing, ” second edition , pp.208-213