Image reconstruction in radio interferometry is the process of solving the linear system of equations
represents visibilities calibrated for direction independent effects,
is a list of parameters that model the sky brightness distribution (for example, a image of pixels),
represents a 2D Fourier transform, and
represents a spatial frequency sampling function that can include direction-dependent instrumental effects. For a practical interferometer with a finite number of array elements, the spatial frequency plane sampling is incomplete and
is singular with no unique inverse. Therefore, this system of equations must be solved iteratively, applying constraints via various choices of image parameterizations and instrumental models.
Image reconstruction in CASA comprises an outer loop of major cycles
and an inner loop of minor cycles
. Together, they implement an iterative weighted
minimization process that solves the measurement equation. The major cycle
implements transforms between the data and image space. The data to image transform is called the imaging
step in which a pseudo inverse of
is computed and applied to the visibilities. Operationally, weighted visibilities are convolutionally resampled onto a grid of spatial-frequency cells, inverse Fourier transformed, and normalized. This step is equivalent to calculating the normal equations as part of a least squares solution. The image to data transform is called the prediction
step and it evaluates the measurement equation to convert a model of the sky brightness into a list of model visibilities that can be compared with the data. For both transforms, direction dependent instrumental effects can be accounted for via carefully constructed convolution functions. Iterations begin with an initial guess for the image model. Each major cycle consists of the prediction of model visibilities, the calculation of residual visibilities and the construction of a residual image. This residual image contains the effect of incomplete sampling of the spatial-frequency plane but is otherwise normalized to the correct sky flux units. In its simplest form, it can be written as a convolution of the true sky image with a point spread function. The job of the minor cycle
is to iteratively build up a model of the true sky by separating it from the point spread function. This step is also called deconvolutio
n and is equivalent to the process of solving the normal equations as part of a least squares solution.
Image Reconstruction Steps
Data : Calibrated visibilities
Input : UV-sampling function, stopping threshold, loop gain
Output : Model Image, Restored Image, Residual Image
Initialize the model image
Compute the point spread function
Compute the initial residual image
While ( not reached global stopping criterion ) /* Major Cycle */
While ( not reached minor-cycle stopping criterion ) /* Minor Cycle */
Find the parameters of a new flux component
Update the model and residual images
Use current model image to predict model visibilities
Calculate residual visibilities (data - model)
Compute a new residual image from residual visibilities
Smooth the final model image and add to the residual image
Algorithms and Options
Within the CASA implementation, numerous choices are provided to enable the end user to fine-tune the details of their image reconstruction. Images can be constructed as spectral cubes with multiple frequency channels or single-plane wideband continuum images. One or more sub images may be defined to cover a wide field of view without incurring the computational expense of very large images. The iterative framework described above is based on the Cotton-Schwab Clean algorithm [CSCLEAN], but variants like Hogbom Clean [HOGBOMCLEAN] and Clark Clean [CLARKCLEAN] are available as subsets of this framework. The major cycle allows controls over different data weighting schemes [LECTURES1] and convolution functions that account for wide-field direction-dependent effects during imaging and prediction [WPROJ,APROJ,WBAWP]. Deconvolution options include the use of point source vs multi-scale image models [MSCLEAN], narrow-band or wide-band models [MTMFS], controls on iteration step size and stopping criteria, and external constraints such as interactive and non-interactive image masks. Mosaics may be made with data from multiple pointings, either with each pointing imaged and deconvolved separately before being combined in a final step, or via a joint imaging and deconvolution [MOSAIC]. Options to combine single dish and interferometer data during imaging also exist [SDIMAGE]. More details about these algorithms can be obtained from [LECTURES1,LECTURES2,ANOTE184,OVERVIEW].
List of References
[HOGBOMCLEAN] : J. A. Hogbom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astron. and Astrophys. Suppl. Ser., vol. 15, pp. 417426, 1974.
[CLARKCLEAN] : B. G. Clark, "An efficient implementation of the algorithm 'clean'," Astron. and Astrophys., vol. 89, p. 377, Sept. 1980.
[CSCLEAN] : F. R. Schwab, "Relaxing the isoplanatism assumption in self-calibration; applications to low-frequency radio interferometry," Astron. J., vol. 89, pp. 10761081, July 1984.
[MSCLEAN] : T. J. Cornwell, "Multi-Scale CLEAN deconvolution of radio synthesis images," IEEE Journal of Selected Topics in Sig. Proc., vol. 2, pp. 793801, Oct 2008.
[MTMFS] : U.Rau, T.J.Cornwell, ''A multi-scale multi-frequency deconvolution algorithm for synthesis imaging in radio-interferometry'', Astronomy and Astrophysics, Volume 532, August 2011.
[WPROJ] : T. J. Cornwell, K. Golap, and S. Bhatnagar, "The non-coplanar base- lines effect in radio interferometry: The w-projection algorithm," IEEE Journal of Selected Topics in Sig. Proc., vol. 2, pp. 647657, Oct 2008.
[APROJ] : S. Bhatnagar, T. J. Cornwell, K. Golap, and J. M. Uson, "Correcting direction-dependent gains in the deconvolution of radio interferometric images," Astron. and Astrophys., vol. 487, pp. 419429, Aug. 2008.
[WBAWP] : S.Bhatnagar, U.Rau, K.Golap, ''Wide-field wide-band interferometric imaging: The WB-A-Projection and hybrid algorithms'', Astrophysical Journal, volume 770, issue 2, id 91, pp.9, June 2013
[MOSAIC] : T. J. Cornwell, "Radio-interferometric imaging of very large objects," Astron. and Astrophys., vol. 202, pp. 316321, Aug. 1988.
[SDIMAGE] : ?
[NRAOLECTURES] Taylor,G.B., Carilli,C.L., Perley, R.A.,''Synthesis Imaging in Radio Astronomy II'', Astron. Soc. Pac. Conf. Ser. 180, 1999
[LECTURES1] Briggs D.S., Schwab F.R., Sramek,R.A.,''Imaging'', Astron. Soc. Pac. Conf. Ser. Vol 180 : Synthesis Imaging in Radio Astronomy II, p127, 1999
[LECTURES2] Cornwell, T.J., Braun, R., Briggs, D.S.,''Deconvolution'', Astron. Soc. Pac. Conf. Ser. Vol 180 : Synthesis Imaging in Radio Astronomy II, p151, 1999
[ANOTE183] : Cornwell, T.J., "The Generic Interferometer: I Overfiew of Calibration and Imaging'', Aips++ note 183. Aug 1995
[ANOTE184] : Cornwell, T.J., "The Generic Interferometer: II Image Solvers'', Aips++ note 184. Aug 1995
[AIPS++] Cornwell, T.J., Wieringa,M.,''The Design and Implementation of Synthesis Calibration and Imaging in AIPS++'', Astron. Soc. Pac. Conf. Ser. 125: Astronomical Data Analysis Software and Systems VI, p10-17, 1997
[OVERVIEW] : U.Rau, S.Bhatnagar, M.A.Voronkov,T.J.Cornwell, ''Advances in Calibration and Imaging Techniques in Radio Interferometry'', Proceedings of the IEEE, Vol.97, No.8, p-1472, August 2009