I have a 2D function $f:R\times R \rightarrow R$ that represents periodical axis-aligned spatial bumps at specific spatial periods (frequencies), like $f=\sin^2(2\pi \nu_1 x) \sin^2(2 \pi \nu_2 y)$. I expect the bump frequencies to be clearly visible as maximums on 2D Fourier transform of $F(kx,kx)= Fourier(f)$ at points $kx=2\pi\nu_1, ky=2\pi\nu_2$.
How does $F$ responds to the rotation of the $f$ around the origin? Would I see the maximums of the spatial frequencies on their correct places? Is there any other transformation that is invariant against the rotation and can show the spatially-periodic background of the function $f$?