Direct Fourier Transform¶
Functions used to compute the discretised direct Fourier transform (DFT) for an ideal unpolarised interferometer. The DFT for an ideal interferometer is defined as
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
where \(s\) labels the source (or pixel) location. 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.
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¶
im_to_vis (image, uvw, lm, frequency[, dtype]) 
Computes the discrete image to visibility mapping of an ideal unpolarised interferometer : 
vis_to_im (vis, uvw, lm, frequency[, dtype]) 
Computes visibility to image mapping of an ideal unpolarised interferometer: 

africanus.dft.
im_to_vis
(image, uvw, lm, frequency, dtype=None)[source]¶ Computes the discrete image to visibility mapping of an ideal unpolarised 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)
The Stokes I intensity in each pixel (flatten 2D array per channel). 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,)
 dtype : np.dtype, optional
Datatype of result. Should be either np.complex64 or np.complex128. Defaults to np.complex128
Returns:  visibilties :
numpy.ndarray
complex of shape
(row, chan)
 image :

africanus.dft.
vis_to_im
(vis, uvw, lm, frequency, dtype=None)[source]¶ Computes visibility to image mapping of an ideal unpolarised 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)
The Stokes I visibilities of which to compute a dirty image 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,)
 dtype : np.dtype, optional
Datatype of result. Should be either np.float32 or np.float64. Defaults to np.float64
Returns:  image :
numpy.ndarray
float of shape
(source, chan)
 vis :
Dask¶
im_to_vis (image, uvw, lm, frequency[, dtype]) 
Computes the discrete image to visibility mapping of an ideal unpolarised interferometer : 
vis_to_im (vis, uvw, lm, frequency[, dtype]) 
Computes visibility to image mapping of an ideal unpolarised interferometer: 

africanus.dft.dask.
im_to_vis
(image, uvw, lm, frequency, dtype=<type 'numpy.complex128'>)[source]¶ Computes the discrete image to visibility mapping of an ideal unpolarised 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)
The Stokes I intensity in each pixel (flatten 2D array per channel). 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,)
 dtype : np.dtype, optional
Datatype of result. Should be either np.complex64 or np.complex128. Defaults to np.complex128
Returns:  visibilties :
dask.array.Array
complex of shape
(row, chan)
 image :

africanus.dft.dask.
vis_to_im
(vis, uvw, lm, frequency, dtype=<type 'numpy.float64'>)[source]¶ Computes visibility to image mapping of an ideal unpolarised 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)
The Stokes I visibilities of which to compute a dirty image 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,)
 dtype : np.dtype, optional
Datatype of result. Should be either np.float32 or np.float64. Defaults to np.float64
Returns:  image :
dask.array.Array
float of shape
(source, chan)
 vis :