1. EE 7730 2D Fourier Transform

2. Bahadir K. Gunturk EE 7730 - Image Analysis I 2 Summary of Lecture 2 We talked about the digital image properties, including spatial resolution and grayscale resolution. We reviewed linear systems and related concepts, including shift invariance, causality, convolution, etc.

3. Bahadir K. Gunturk EE 7730 - Image Analysis I 3 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)

4. Bahadir K. Gunturk EE 7730 - Image Analysis I 4 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.

5. Bahadir K. Gunturk EE 7730 - Image Analysis I 5 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: z = x + j*y A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a)

6. Bahadir K. Gunturk EE 7730 - Image Analysis I 6 Fourier Transform: 1D Cont. Signals ■ Fourier Transform of a 1D continuous signal ■ Inverse Fourier Transform “Euler’s formula”

7. Bahadir K. Gunturk EE 7730 - Image Analysis I 7 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.

8. Bahadir K. Gunturk EE 7730 - Image Analysis I 8 Examples Magnitude: “how much” of each component Phase: “where” the frequency component in the image

9. Bahadir K. Gunturk EE 7730 - Image Analysis I 9 Examples

10. Bahadir K. Gunturk EE 7730 - Image Analysis I 10 Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions

11. Bahadir K. Gunturk EE 7730 - Image Analysis I 11 Fourier Transform: Properties ■ Separability 2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.

12. Bahadir K. Gunturk EE 7730 - Image Analysis I 12 Fourier Transform: Properties ■ Energy conservation

13. Bahadir K. Gunturk EE 7730 - Image Analysis I 13 Fourier Transform: Properties ■ Remember the impulse function (Dirac delta function) definition ■ Fourier Transform of the impulse function

14. Bahadir K. Gunturk EE 7730 - Image Analysis I 14 Fourier Transform: Properties ■ Fourier Transform of 1 Take the inverse Fourier Transform of the impulse function

15. Bahadir K. Gunturk EE 7730 - Image Analysis I 15 Fourier Transform: 2D Discrete Signals ■ Fourier Transform of a 2D discrete signal is defined as where ■ Inverse Fourier Transform

16. Bahadir K. Gunturk EE 7730 - Image Analysis I 16 Fourier Transform: Properties ■ Periodicity: Fourier Transform of a discrete signal is periodic with period 1. 1 1 Arbitrary integers

17. Bahadir K. Gunturk EE 7730 - Image Analysis I 17 Fourier Transform: Properties ■ Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.

18. Bahadir K. Gunturk EE 7730 - Image Analysis I 18 Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions ■ Energy conservation

19. Bahadir K. Gunturk EE 7730 - Image Analysis I 19 Fourier Transform: Properties ■ Define Kronecker delta function ■ Fourier Transform of the Kronecker delta function

20. Bahadir K. Gunturk EE 7730 - Image Analysis I 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.

21. Bahadir K. Gunturk EE 7730 - Image Analysis I 21 Impulse Train ■ Define a comb function (impulse train) as follows where M and N are integers

22. Bahadir K. Gunturk EE 7730 - Image Analysis I 22 Impulse Train Fourier Transform of an impulse train is also an impulse train:

23. Bahadir K. Gunturk EE 7730 - Image Analysis I 23 Impulse Train

24. Bahadir K. Gunturk EE 7730 - Image Analysis I 24 Impulse Train In the case of continuous signals:

25. Bahadir K. Gunturk EE 7730 - Image Analysis I 25 Impulse Train

26. Bahadir K. Gunturk EE 7730 - Image Analysis I 26 Sampling

27. Bahadir K. Gunturk EE 7730 - Image Analysis I 27 Sampling No aliasing if

28. Bahadir K. Gunturk EE 7730 - Image Analysis I 28 Sampling If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.

29. Bahadir K. Gunturk EE 7730 - Image Analysis I 29 Sampling Aliased

30. Bahadir K. Gunturk EE 7730 - Image Analysis I 30 Sampling Anti-aliasing filter

31. Bahadir K. Gunturk EE 7730 - Image Analysis I 31 Sampling ■ Without anti-aliasing filter: ■ With anti-aliasing filter:

32. Bahadir K. Gunturk EE 7730 - Image Analysis I 32 Anti-Aliasing a=imread(‘barbara.tif’);

33. Bahadir K. Gunturk EE 7730 - Image Analysis I 33 Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25); c=imresize(b,4);

34. Bahadir K. Gunturk EE 7730 - Image Analysis I 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, 256-64:256+64)=1; Da=fft2(a); Da=fftshift(Da); Dd=Da.*H; Dd=fftshift(Dd); d=real(ifft2(Dd));

35. Bahadir K. Gunturk EE 7730 - Image Analysis I 35 Sampling

36. Bahadir K. Gunturk EE 7730 - Image Analysis I 36 Sampling No aliasing if and

37. Bahadir K. Gunturk EE 7730 - Image Analysis I 37 Interpolation Ideal reconstruction filter:

38. Bahadir K. Gunturk EE 7730 - Image Analysis I 38 Ideal Reconstruction Filter