EE 4780. 2D Fourier Transform. 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). Fourier Transform: Concept.
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EE 4780
2D Fourier Transform
“Euler’s formula”
Take the inverse Fourier Transform of the impulse function
Magnitudes are shown
2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.
where
1
1
Arbitrary integers
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.
where M and N are integers
No aliasing if
If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.
Aliased
Anti-aliasing filter
a=imread(‘barbara.tif’);
a=imread(‘barbara.tif’);
b=imresize(a,0.25);
c=imresize(b,4);
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));
No aliasing if and
Ideal reconstruction filter: