In the context of the 1D Fourier transform, for a real function $f(x)$, the Fourier transform has a property: $F(\omega) = F^*(-\omega)$. This implies that the real part of the Fourier transform is an even function, while the imaginary part is an odd function.
I am attempting to generalize this property to 2D functions but am encountering difficulties. To illustrate, consider the following matrix as an example: $$ \begin{bmatrix} 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix} $$ Using NumPy (in python) for the Fast Fourier Transform, I printed out the imaginary part of the result:
$$ \begin{bmatrix} 0.0 & 2.0 & 0.0 & -2.0 \\ 0.0 & -2.0 & 2.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -2.0 & 2.0 \\ \end{bmatrix} $$
Contrary to my expectations, the output was not symmetrical. I am seeking guidance to understand the properties of the Fourier transform of a real function in 2D space.
I found it. The property is as follow: $F(i, j) = F^*(-i, -j)$. In my example the $(0, 0)$ of Fourier transformed is set to be the zero value at $(3, 2)$.