Direct Fourier Transform
Functions used to compute the discretised direct Fourier transform (DFT) for an ideal interferometer. The DFT for an ideal interferometer is defined as
where \(u,v,w\) are data space coordinates and where visibilities \(V\) have been obtained. The \(l,m,n\) are signal space coordinates at which we wish to reconstruct the signal \(B\). Note that the signal correspondes to the brightness matrix and not the Stokes parameters. 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
where \(s\) labels the source (or pixel) location. If only a single correlation is present \(B = I\), this can be cast into a matrix equation as follows
where \(R\) is the operator that maps an
image to visibility space. This mapping is
implemented by the im_to_vis()
function. If multiple correlations are present then
each one is mapped to its corresponding visibility.
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
This is implemented by the
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.
Numpy

Computes the discrete image to visibility mapping of an ideal interferometer: 

Computes visibility to image mapping of an ideal interferometer: 
 africanus.dft.im_to_vis(image, uvw, lm, frequency, convention='fourier', dtype=None)[source]
Computes the discrete image to visibility mapping of an ideal interferometer:
\[{\Large \sum_s e^{2 \pi i (u l_s + v m_s + w (n_s  1))} \cdot I_s }\] Parameters:
 image
numpy.ndarray
image of shape
(source, chan, corr)
The brighness matrix in each pixel (flatten 2D array per channel and corr). Note not Stokes terms uvw
numpy.ndarray
uvw coordinates of shape
(row, 3)
with u, v and w components in the last dimension. lm
numpy.ndarray
lm coordinates of shape
(source, 2)
with l and m components in the last dimension. frequency
numpy.ndarray
frequencies of shape
(chan,)
 convention{‘fourier’, ‘casa’}
Uses the \(e^{2 \pi \mathit{i}}\) sign convention if
fourier
and \(e^{2 \pi \mathit{i}}\) ifcasa
. dtypenp.dtype, optional
Datatype of result. Should be either np.complex64 or np.complex128. If
None
,numpy.result_type()
is used to infer the data type from the inputs.
 image
 Returns:
 visibilties
numpy.ndarray
complex of shape
(row, chan, corr)
 visibilties
 africanus.dft.vis_to_im(vis, uvw, lm, frequency, flags, convention='fourier', dtype=None)[source]
Computes visibility to image mapping of an ideal interferometer:
\[{\Large \sum_k e^{ 2 \pi i (u_k l + v_k m + w_k (n  1))} \cdot V_k}\] Parameters:
 vis
numpy.ndarray
visibilities of shape
(row, chan, corr)
Visibilities corresponding to brightness terms. Note the dirty images produced do not necessarily correspond to Stokes terms and need to be converted. uvw
numpy.ndarray
uvw coordinates of shape
(row, 3)
with u, v and w components in the last dimension. lm
numpy.ndarray
lm coordinates of shape
(source, 2)
with l and m components in the last dimension. frequency
numpy.ndarray
frequencies of shape
(chan,)
 flags
numpy.ndarray
Boolean array of shape
(row, chan, corr)
Note that if one correlation is flagged we discard all of them otherwise we end up irretrievably mixing Stokes terms. convention{‘fourier’, ‘casa’}
Uses the \(e^{2 \pi \mathit{i}}\) sign convention if
fourier
and \(e^{2 \pi \mathit{i}}\) ifcasa
. dtypenp.dtype, optional
Datatype of result. Should be either np.float32 or np.float64. If
None
,numpy.result_type()
is used to infer the data type from the inputs.
 vis
 Returns:
 image
numpy.ndarray
float of shape
(source, chan, corr)
 image
Dask

Computes the discrete image to visibility mapping of an ideal interferometer: 

Computes visibility to image mapping of an ideal interferometer: 
 africanus.dft.dask.im_to_vis(image, uvw, lm, frequency, convention='fourier', dtype=numpy.complex128)[source]
Computes the discrete image to visibility mapping of an ideal interferometer:
\[{\Large \sum_s e^{2 \pi i (u l_s + v m_s + w (n_s  1))} \cdot I_s }\] Parameters:
 image
dask.array.Array
image of shape
(source, chan, corr)
The brighness matrix in each pixel (flatten 2D array per channel and corr). Note not Stokes terms uvw
dask.array.Array
uvw coordinates of shape
(row, 3)
with u, v and w components in the last dimension. lm
dask.array.Array
lm coordinates of shape
(source, 2)
with l and m components in the last dimension. frequency
dask.array.Array
frequencies of shape
(chan,)
 convention{‘fourier’, ‘casa’}
Uses the \(e^{2 \pi \mathit{i}}\) sign convention if
fourier
and \(e^{2 \pi \mathit{i}}\) ifcasa
. dtypenp.dtype, optional
Datatype of result. Should be either np.complex64 or np.complex128. If
None
,numpy.result_type()
is used to infer the data type from the inputs.
 image
 Returns:
 visibilties
dask.array.Array
complex of shape
(row, chan, corr)
 visibilties
 africanus.dft.dask.vis_to_im(vis, uvw, lm, frequency, flags, convention='fourier', dtype=numpy.float64)[source]
Computes visibility to image mapping of an ideal interferometer:
\[{\Large \sum_k e^{ 2 \pi i (u_k l + v_k m + w_k (n  1))} \cdot V_k}\] Parameters:
 vis
dask.array.Array
visibilities of shape
(row, chan, corr)
Visibilities corresponding to brightness terms. Note the dirty images produced do not necessarily correspond to Stokes terms and need to be converted. uvw
dask.array.Array
uvw coordinates of shape
(row, 3)
with u, v and w components in the last dimension. lm
dask.array.Array
lm coordinates of shape
(source, 2)
with l and m components in the last dimension. frequency
dask.array.Array
frequencies of shape
(chan,)
 flags
dask.array.Array
Boolean array of shape
(row, chan, corr)
Note that if one correlation is flagged we discard all of them otherwise we end up irretrievably mixing Stokes terms. convention{‘fourier’, ‘casa’}
Uses the \(e^{2 \pi \mathit{i}}\) sign convention if
fourier
and \(e^{2 \pi \mathit{i}}\) ifcasa
. dtypenp.dtype, optional
Datatype of result. Should be either np.float32 or np.float64. If
None
,numpy.result_type()
is used to infer the data type from the inputs.
 vis
 Returns:
 image
dask.array.Array
float of shape
(source, chan, corr)
 image