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EE 7730

EE 7730. 2D Fourier Transform. 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. Fourier Transform.

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EE 7730

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  1. EE 7730 2D Fourier Transform

  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. 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) 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. 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) EE 7730 - Image Analysis I

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

  7. Fourier Transform: 2D Cont. Signals • Fourier Transform of a 2D continuous signal • Inverse Fourier Transform • Fandfare two different representations of the same signal. EE 7730 - Image Analysis I

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

  9. Examples EE 7730 - Image Analysis I

  10. Fourier Transform: Properties • Linearity • Shifting • Modulation • Convolution • Multiplication • Separable functions EE 7730 - Image Analysis I

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

  12. Fourier Transform: Properties • Energy conservation EE 7730 - Image Analysis I

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

  14. Fourier Transform: Properties • Fourier Transform of 1 Take the inverse Fourier Transform of the impulse function 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 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 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. EE 7730 - Image Analysis I

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

  19. Fourier Transform: Properties • Define Kronecker delta function • Fourier Transform of the Kronecker delta function 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. EE 7730 - Image Analysis I

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

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

  23. Impulse Train EE 7730 - Image Analysis I

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

  25. Impulse Train EE 7730 - Image Analysis I

  26. Sampling EE 7730 - Image Analysis I

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

  29. Sampling Aliased EE 7730 - Image Analysis I

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

  31. Sampling • Without anti-aliasing filter: • With anti-aliasing filter: EE 7730 - Image Analysis I

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

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

  35. Sampling EE 7730 - Image Analysis I

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

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

  38. Ideal Reconstruction Filter EE 7730 - Image Analysis I

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