Fourier Transform of Gaussian Kernel in Matlab

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Gobert
Gobert 2022-4-12
编辑： Gobert 2022-4-13
Hi everyone,
I need your help!
I was reading a document on Discrete Fourier Transform and found the following example:
Here's the mentioned gaussian kernel:
g_k = (1/256)*[1 4 6 4 1;...
4 16 24 16 4;...
6 24 36 24 6;...
4 16 24 16 4;...
1 4 6 4 1];
As can be seen, the size of g_k or g(x,y) is 5 x 5 - while the size of G(u,v) is around 380 x 450
Can you please show me the Matlab code that can generate the above G(u,v) image or result?

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Matt J
Matt J 2022-4-12
编辑：Matt J 2022-4-12
Ran in:
If you download gaussfitn from
https://www.mathworks.com/matlabcentral/fileexchange/69116-gaussfitn
then you can do,
g0=[1 4 6 4 1]';
params=gaussfitn((-2:2)',g0,[],{0,[],0},{0,[],0});
Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance.
[A,mu,sig2]=deal(params{2:4});
sig=1/2/pi/sqrt(sig2);
fun=@(x) exp(-(x/sig).^2/2);
Gu=fun(linspace(-6*sig,+6*sig,450));
Guv=imresize(Gu'*Gu,[380,450]);
imshow(Guv);

更多回答（1 个）

Matt J
Matt J 2022-4-12
Ran in:
One could also do as below. This gives only an approximately Gaussian spectrum, however,
g_k = (1/256)*[1 4 6 4 1;...
4 16 24 16 4;...
6 24 36 24 6;...
4 16 24 16 4;...
1 4 6 4 1];
Guv=fftshift( abs(fft2(g_k,380,450)) );
imshow(Guv)
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Matt J
Matt J 2022-4-12
编辑：Matt J 2022-4-12
@Paul Yes, that is why. If you don't do it, you will see a 2-pixel shift in the output.
Gobert
Gobert 2022-4-13
编辑：Gobert 2022-4-13
@Paul - These simple examples can give an idea of what "circshift" does to that "zero-padded matrix": See "ans"
% When m~=n
A = [16 2 3 13; 5 11 10 8; 9 7 6 12]
[n,m]=size(A)
A(2*n,2*m)=0
circshift(A,[-2,-2])
A =
16 2 3 13
5 11 10 8
9 7 6 12
n =
3
m =
4
A =
16 2 3 13 0 0 0 0
5 11 10 8 0 0 0 0
9 7 6 12 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
ans =
6 12 0 0 0 0 9 7
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
3 13 0 0 0 0 16 2
10 8 0 0 0 0 5 11
% When m=n
B = [16 2 3 13; 5 11 10 8; 9 7 6 12;4 14 15 1]
[n,m]=size(B)
B(2*n,2*m)=0
circshift(B,[-2,-2])
B =
16 2 3 13
5 11 10 8
9 7 6 12
4 14 15 1
n =
4
m =
4
B =
16 2 3 13 0 0 0 0
5 11 10 8 0 0 0 0
9 7 6 12 0 0 0 0
4 14 15 1 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
ans =
6 12 0 0 0 0 9 7
15 1 0 0 0 0 4 14
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
3 13 0 0 0 0 16 2
10 8 0 0 0 0 5 11

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