1. EE 4780
2D Fourier Transform

2. Fourier Transform What is ahead?
1D Fourier Transform of continuous signals
2D Fourier Transform of continuous signals
2D Fourier Transform of discrete signals
2D Discrete Fourier Transform (DFT)
Bahadir K. Gunturk

3. Fourier Transform: Concept
A signal can be represented as a weighted sum of sinusoids.
Fourier Transform is a change of basis, where the basis functions consist of sines and cosines (complex exponentials).
Bahadir K. Gunturk

4. Fourier Transform
Cosine/sine signals are easy to define and interpret.
However, it turns out that the analysis and manipulation of sinusoidal signals is greatly simplified by dealing with related signals called complex exponential signals.
A complex number has real and imaginary parts: z = x + j*y
A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a)
Bahadir K. Gunturk

5. Fourier Transform: 1D Cont. Signals
Fourier Transform of a 1D continuous signal
“Euler’s formula”
Inverse Fourier Transform
Bahadir K. Gunturk

6. Fourier Transform: 2D Cont. Signals
Fourier Transform of a 2D continuous signal
Inverse Fourier Transform
F and f are two different representations of the same signal.
Bahadir K. Gunturk

7. Fourier Transform: Properties
Remember the impulse function (Dirac delta function) definition
Fourier Transform of the impulse function
Bahadir K. Gunturk

8. Fourier Transform: Properties
Fourier Transform of 1
Take the inverse Fourier Transform of the impulse function
Bahadir K. Gunturk

9. Fourier Transform: Properties
Fourier Transform of cosine
Bahadir K. Gunturk

10. Examples
Magnitudes are shown
Bahadir K. Gunturk

11. Examples
Bahadir K. Gunturk

12. Fourier Transform: Properties
Linearity
Shifting
Modulation
Convolution
Multiplication
Separable functions
Bahadir K. Gunturk

13. Fourier Transform: Properties
Separability
2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.
Bahadir K. Gunturk

14. Fourier Transform: Properties
Energy conservation
Bahadir K. Gunturk

15. Fourier Transform: 2D Discrete Signals
Fourier Transform of a 2D discrete signal is defined as
where
Inverse Fourier Transform
Bahadir K. Gunturk

16. Fourier Transform: Properties
Periodicity: Fourier Transform of a discrete signal is periodic with period 1. 1 1
Arbitrary integers
Bahadir K. Gunturk

17. Fourier Transform: Properties
Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.
Bahadir K. Gunturk

18. Fourier Transform: Properties
Linearity
Shifting
Modulation
Convolution
Multiplication
Separable functions
Energy conservation
Bahadir K. Gunturk

19. Fourier Transform: Properties
Define Kronecker delta function
Fourier Transform of the Kronecker delta function
Bahadir K. Gunturk

20. Fourier Transform: Properties
Fourier Transform of 1
To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1.
Bahadir K. Gunturk

21. Impulse Train Define a comb function (impulse train) as follows
where M and N are integers
Bahadir K. Gunturk

22. Impulse Train
Fourier Transform of an impulse train is also an impulse train:
Bahadir K. Gunturk

23. Impulse Train
Bahadir K. Gunturk

24. Impulse Train
In the case of continuous signals:
Bahadir K. Gunturk

25. Impulse Train
Bahadir K. Gunturk

26. Sampling
Bahadir K. Gunturk

27. Sampling
No aliasing if
Bahadir K. Gunturk

28. Sampling
If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.
Bahadir K. Gunturk

29. Sampling
Aliased
Bahadir K. Gunturk

30. Sampling
Anti-aliasing filter
Bahadir K. Gunturk

31. Sampling Without anti-aliasing filter: With anti-aliasing filter:
Bahadir K. Gunturk

32. Anti-Aliasing
a=imread(‘barbara.tif’);
Bahadir K. Gunturk

33. Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25);
c=imresize(b,4);
Bahadir K. Gunturk

34. Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25);
c=imresize(b,4);
H=zeros(512,512);
H(256-64:256+64, :256+64)=1;
Da=fft2(a);
Da=fftshift(Da);
Dd=Da.*H;
Dd=fftshift(Dd);
d=real(ifft2(Dd));
Bahadir K. Gunturk

35. Sampling
Bahadir K. Gunturk

36. Sampling
No aliasing if and
Bahadir K. Gunturk

37. Interpolation
Ideal reconstruction filter:
Bahadir K. Gunturk

38. Ideal Reconstruction Filter
Bahadir K. Gunturk