Functions used to compute the discretised direct Fourier transform (DFT) for an ideal unpolarised interferometer. The DFT for an ideal interferometer is defined as
V(u,v,w) = \int I(l,m) e^{-2\pi i
\left( ul + vm + w(n-1)\right)}
\frac{dl dm}{n}
where u,v,w are data (visibility V) space coordinates and l,m,n are signal (image I) space coordinates. We adopt the convention where we absorb the fixed coordinate n in the denominator into the image. Note that the data space coordinates have an implicit dependence on frequency and time and that the image has an implicit dependence on frequency. The discretised form of the DFT can be written as
V(u,v,w) = \sum_s e^{-2 \pi i
(u l_s + v m_s + w (n_s - 1))} \cdot I_s
where s labels the source (or pixel) location. This can be cast into a matrix equation as follows
V = R I
where R is the operator that maps an image to visibility space. This mapping is implemented by the :func:`~africanus.dft.im_to_vis` function. An imaging algorithm also requires the adjoint denoted R^\dagger which is simply the complex conjugate transpose of R. The dirty image is obtained by applying the adjoint operator to the visibilities
I^D = R^\dagger V
This is implemented by the :func:`~africanus.dft.vis_to_im` function. Note that an imaging algorithm using these operators will actually reconstruct \frac{I}{n} but that it is trivial to obtain I since n is known at each location in the image.
.. currentmodule:: africanus.dft
.. autosummary::
im_to_vis
vis_to_im
.. autofunction:: im_to_vis
.. autofunction:: vis_to_im
.. currentmodule:: africanus.dft.dask
.. autosummary::
im_to_vis
vis_to_im
.. autofunction:: im_to_vis
.. autofunction:: vis_to_im