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Deconvolution


Bob Sault


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An example of deconvolution


But the aperture is NEVER completely filled Limited Limited Various Min and


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Deconvolution and Multi frequency synthesis


Basics (again!) Multi-frequency synthesis Characteristics of the dirty beam Linear deconvolution Constraints CLEAN Maximum entropy Restoration Multi-frequency deconvolution


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Filling the Fourier plane
Use many antennas (6 antennas or more) Use Earth rotation (12 h observations) Physically move antennas

observing time number of antennas interruptions to the observation max baselines
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Multi-frequency synthesis
As (u,v) coordinate is measured in wavelengths, another way of filling the Fourier plane is to observed at multiple wavelengths simultaneously.


Basic imaging relationship

Using a "direct Fourier transform" we produce the dirty image

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Convolution relationship
Fourier theory tells us that so where


Deviations from convolution relationship
Wide-field effects are usually neglected. These include
­ Time and bandwidth smearing ­ Primary beam effects ­ So-called non-coplanar baseline effects

Convolution relationship strictly applies only for continuous functions (not a sampled grid of pixels). "Aliasing" in the imaging process is also not accounted for. Finite extent assumption.


Jargon: The point-spread function is usually called the "beam".
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Dirty beams

Dirty beam characteristics
Differing "holes" in the Fourier plane lead to a wide variety of sidelobe structure

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Linear deconvolution


Linear deconvolution ...
Noise properties are well understood Generally non-iterative and computationally cheap


Inverse filter


then

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Wiener filters It does a very poor job Rarely used in practical radio interferometry


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Non-linear deconvolution


Prior Information or Assumptions
Bounded support ("CLEAN boxes"). Positivity The sky is mostly empty Use a goodness measure to pick "reasonable" solutions.


Linear deconvolution is fundamentally unable to extrapolated unmeasured spatial frequencies.


A function which is non-zero only in the unsampled part of the Fourier plane is called an invisible distribution. A good non-linear deconvolution algorithm is one that picks plausible invisible distributions to fill in the Fourier plane.

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CLEAN Algorithm (HЖgbom, 1974)
Assumes that the sky can be modelled as a collection of point sources. Iteratively decomposes the sky into a collection of point sources. In principle, CLEAN is guaranteed to converge, although in practice it can become unstable if pushed too far. Generally it is quite a robust algorithm.
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CLEAN algorithm
Search for the largest peak in the residual image Assume this is a result of a point source ­ a component! Subtract off some fraction ("damping factor" or "loop gain") of the point source. Add that fraction of the point source to a component list. Iterate

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Iteration stops when the residual is below some cut-off, when a negative component is encountered, or when a fixed number of components are found.
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CLEAN implementations


Strengths/weaknesses
CLEAN is good for fields of sources which are unresolved or just resolved. Generally quite robust in the face of many defects. CLEAN is very poor for very extended objects:


There are different implementations of the algorithms (with their individual strengths and weaknesses):


­ HЖgbom algorithm ­ the classical one ­ Clark algorithm ­ faster for large images with ­ Cotton-Schwab ("MX") algorithm ­ works pa visibility domain. Able to cope with extra arte ­ Steer Dewdney Ito algorithm ­ works best for objects.

many point sources. rtially in the facts. Can be slow. very extended

­ Slow! ­ Corrugation instability. ­ CLEAN poorly estimates broad structure (short spacings). The result is the so-called "negative bowl" effect.

CLEAN's procedural definition makes it difficult to analyse.


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Examples of CLEANed images


Bayesian Statistics and Maximum Entropy
Two basic views of probability theory: ­ Views probability distribution function as a measure of the relative frequency of an outcome. ­ Views probability distribution function as a reflection of our uncertainty. Principle of maximum entropy: Of all the possible probability distributions which are consistent with the available information, the one that has the maximum entropy is most likely the correct one. Maximum entropy image deconvolution: Of all the possible images consistent with the observed data, the one that has the maximum entropy is most likely to be the correct one.


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Maximum entropy


Maximum entropy
The solution is generally constrained so that a 2 measure is consistent: i.e. the 2 measure is consistent with the expected noise level. Integrated flux constraint can be included. "CLEAN box" constraint is readily added. The default image, M, can be chosen to be a uniform value, or can be set to some prior expectation of the source. Solution image must be positive-valued.


Of all the possible images, pick that one which maximises some goodness measure called "entropy".


The most popular choice is the entropy function

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Strengths/weaknesses
Fourier extrapolation tends to be more conservative than CLEAN. Tends to work better for images with a large amount of extended emission. Tends to be faster for large images ( > 1024x1024 pixels). Susceptible to analysis. Depends more critically on its control parameters (e.g. noise variance and integrated flux). More likely to blow up on poorl y calibrated data, or data that violates the convolution relationship in some way. Poorly deconvolves point sources.

CLEAN vs MEM
The answer is image dependent: "High quality" data, extended emission, large images Maximum entropy


"Poor quality" data, confused fields, point sources CLEAN


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"Restoration" Step


Why include the residuals?
The residuals give an easy wa y of seeing how believable the features in an image are. The residuals still contain emission from sources that have not been CLEANed out.


CLEAN and MEM "super-resolve", and the high spatial frequencies can be of poor quality (particularly CLEAN).


Solution: Downweight the high spatial frequencies by convolving with a gaussian "CLEAN beam".

The CLEAN beam usually has the same FWHM as the main lobe of the dirty beam.

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Multi-frequency deconvolution


Multi-frequency deconvolution algorithm
The algorithm models the spectral variation at each pixel as a constant and a linearly varying component with frequency. The response to the constant part of this variation is just the normal dirty beam. The response to the linearly-varying component can be represented by a second response function. The dirty image is the sum of the responses to the constant and varying components. A joint deconvolution, simultaneously solving for the two components can be performed. In Miriad, this is the so-called mfclean algorithm.


Multi-frequency synthesis uses observations at many frequencies to prove the Fourier plane coverage. Problem: Source structure is a function of frequency. For modest spread in fractional bandwidth (< 15%), and modest dynamic range (< 500), the errors caused by source structure varying with frequency can be ignored. When this is not the case, a multi-frequency deconvolution algorithm can be used to eliminate the resultant errors.
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