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:
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.
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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