Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself?

Question

I am trying to understand why 2D FFT is done on a Gaussian process in a particular code. From my understanding from these posts:

1. https://www.researchgate.net/post/Why_would_we_want_to_do_Fourier_transform_of_a_Gaussian_signal

2. Why use a gaussian low pass filter, when we can convolve with gaussian kernel

3. Zero padding effect on a FFT of gaussian noise

It appears that it is used to

• reduce noise
• reduce the sharpness on the edges

I am working with MATLAB, and the code snippet is as follows. Please note the peak has been moved to the top-left in the gaussian_shaped_label function, with wrap-around:

%window size, taking padding into account
window_sz = floor(target_sz * (1 + padding));

%   we could choose a size that is a power of two, for better FFT
%   performance. in practice it is slower, due to the larger window size.
%   window_sz = 2 .^ nextpow2(window_sz);


%create regression labels, gaussian shaped, with a bandwidth proportional to target size
output_sigma = sqrt(prod(target_sz)) * output_sigma_factor / cell_size;
yf = fft2(gaussian_shaped_labels(output_sigma, floor(window_sz / cell_size)));

I was trying to plot the two results.

1. The Gaussian (left) zoomed on the top-left edge(right):

Plotting the code:

gaussian_shaped_labels(output_sigma, floor(window_sz / cell_size)));

2. The absolute value of the 2D FFT of the Gaussian mentioned in point 1 (left) zoomed on the top-left(right): Plotting the code-

fft2(gaussian_shaped_labels(output_sigma, floor(window_sz / cell_size)));

The plot of 2D FFT seems sharper than the Gaussian. Shouldn't 2D FFT actually blur the edges and make it smoother or something? I think I am going wrong somewhere and any help will be appreciated.

P. S.
If you can help me understand the mathematically too, it would extra helpful but not a hard requirement here.

Answer 1

Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself?

Have a look at the Fourier Transfrom of a Gaussian Signal.

$$ \mathcal{F}_{x} \left\{ {e}^{-a {x}^{2} } \right\} \left( \omega \right) = \sqrt{\frac{\pi}{a}} {e}^{- {\pi}^{2} \frac{ {\omega}^{2} }{a} } $$

First, Gaussian Signal stays Gaussian under Fourier Transform.

As you can see, the parameter which multiplies the variable is inverted.
Let's say $ a = 5 $, then it means that in time we will have very sharp and thin Gaussian while in frequency we will have very smooth and wide Gaussian.

This is related to a property of Fourier Transform.
In simple words, what's thin on Time / Spatial Domain is wide on Frequency Domain and vise versa.
This is basically the Uncertainty Principle of Fourier Transform.