National Radio Astronomy Observatory (NRAO), Socorro, NM - 87801

Emails: sbhatnag@aoc.nrao.edu, tcornwel@aoc.nrao.edu

Aperture synthesis radio telescopes measure the Fourier transform of
the sky brightness distribution. However the Point Spread Function
(PSF) of such telescopes has significant and widespread side-lobes,
which needs to be deconvolved from the images. Existing deconvolution
algorithms can be thought of as decomposing the image into a set of
delta functions (scale less basis). This uses more degrees of freedom
than necessary and is not optimal for extended emission. In this
paper we present an iterative scale sensitive deconvolution algorithm
for radio interferometric imaging, which attempts to minimize the
degrees of freedom used to represent the signal (spatially correlated
pixels).

A radio interferometric telescope incompletely measures the visibility
function, at discrete points. The Fourier transform of the visibility
function, called the Dirty Image (), is the convolution of the
true image () with the telescope Point Spread Function (PSF)

The goal of deconvolution algorithms is to estimate a sky image model
, such that the model visibility
fits the observed visibility to the extent allowed by the
noise. A generalized model image can be expressed as a linear
sum of Pixel models

The current popular image deconvolution algorithms (Karovska, 2002), like CLEAN (and its variants (Clark, 1980; Cornwell et al., 1990) and MEM (and its variants (Cornwell&Evans, 1985)) model in a scale-less basis (delta functions). Such algorithms also require regularizes to avoid over-fitting (which results into spurious compact sources in the image). Usually, this regularization is done via a user defined maximum number of components or/and global estimate of the noise in the image. Extended emission, which is at a very different scale than a compact component, is broken up into delta functions and later smoothed to suppress the high frequency errors made in such a representation. However since delta functions are at a scale smaller than even the resolution element, this results into the well known breaking-up of extended emission problem. In this paper we describe an algorithm which decomposes the sky image into parameterized Adaptive Scale Pixel (Asp) model. The parameters of the Aspen are determined using non-linear minimization. The algorithm is sensitive to the local spatial scale as well as the local signal-to-noise ratio.

The functional form for the Asp used in this paper is a symmetric two dimensional Gaussian. The algorithm searches for the locally best fit Asp to the Dirty Image, by estimating the location (), amplitude () and the size () of the Asp.

The dirty image is smoothed to a few scales ranging from the smallest to the largest expected scale. A global set of Aspen, is maintained and a new Asp added to this list at each iteration. The model image is computed using this set and Eq. 2. The image decomposition into Aspen basis then proceeds as follows:

- At each iteration, compute the model and residual
visibilities as
, where is the observed
visibilities. Compute the residual image
and smoothed versions at a few scales.
- Locate the peak among all the smoothed versions of . Use the location,
amplitude and the scale at which the peak was found as the initial
guess for the new Asp . Set
.
- Make a sub-set of Aspen which will
maximally affect the convergence.
- Simultaneously solve for the parameters of this sub-set such that
is minimized.
- Goto Step 1 till is noise like.

The set of active Aspen is determined by applying a threshold on the length of the vector of the first derivatives of the with respect to all the parameters () of each Aspen . is computed for each at the start of each iteration and all Aspen with are dropped from the problem at that iteration. Since this is done at the beginning of each cycle, mistakes in this estimate for the active set of Aspen are corrected in later iterations. This however implicitly assumes that the surface has the same curvature along all axes. Ideally, the active set should be determined by thresholding the covariance matrix, which is computationally expensive. Since the value of the off-diagonal elements and the structure of the covariance matrix is strongly dependent on the side-lobe levels of the PSF, the active set of Aspen can be estimated by using an Aspen decomposition of the PSF and its significant side-lobes. Such a PSF decomposition (Bhatnagar&Cornwell, 2003) appears to model the distant but significant side-lobes of a typical PSF well. Use of such an approximation for the PSF to compute approximate covariance matrix is being currently investigated.

Several tests were done using simulated model images (not shown here) which were convolved with PSF for typical VLA observations and then deconvolved using the above algorithm. The algorithm was found to be sensitive to local scales and even overlapping components with varied scales were separately detected. Consequently, the model image was represented with many fewer degrees of freedom compared to other scale-less or multi-scale algorithms (Holdaway&Cornwell, 2001).

A VLA observation of M31 was used as a more realistic test case. The best available deconvolved image was used as the ``ideal'' model image. The model image was then convolved with a PSF corresponding to a typical VLA C-array observation. The resulting dirty image was then deconvolved using the above algorithm. The original model image, and the model image from this algorithm are shown in Fig 1. The Asp model in the right panel is composed of Asp. For similar dynamic range, the multi-scale (MS) Clean used components. The scale-less Clean algorithm required about a factor 10 more components than even MS-Clean, and did not do as well as Asp or MS-Clean in terms of residual noise and fidelity.

Image deconvolution can be treated as a search for a model image, which when convolved with the PSF, minimizes an objective function (e.g. the ). The model image is represented as a parameterized function, which is estimated by the deconvolution algorithms. Since the residual image provides the update direction in an iterative scheme, a parameterization which fundamentally separates noise from the signal (the sky emission) will always produce better results in terms of dynamic range and fidelity. Correlation length (or equivalently the scale of emission) is one such parameter which strongly separates the noise (which is fundamentally scale-less) from the signal. The algorithm presented here uses scale as one of the parameters and given the PSF, attempts to separate noise and signal using the Adaptive Scale Pixel (Asp) model. It is sensitive to the local scale of emission and SNR and is shown to perform better even for complex images with a range of scales. The reconstruction is optimal at all scales using minimum degrees of freedom compared to other algorithms. Heuristics used to eliminate insignificant Aspen, which adaptively changes the dimensionality of the problem at each iteration, are shown to be effective. Other, possibly more effective methods for this using an estimate of the covariance matrix are being explored. Also, a pixel model with a tighter support constraint, with also convenient mathematical properties is also likely to improve speed of convergence. Work is in progress to incorporate these improvements in the algorithm.

Bhatnagar, S. & Cornwell, T.J., 2003, Proc. of the Annual SPIE Meeting, In press

Clark, B.G. 1980, A&A, 89, 377

Cornwell, T.J., Evans, K.F. 1985, A&A, 143, 77-83

Cornwell, T.J., Braun, R., & Briggs, D.S. 1990, ASP

Holdaway, M.H., Cornwell, T.J. 2001, in preparation

Karovska, M., 2002 BAAS, 201, 6302

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