[SciPy-User] deconvolution of 1-D signals

Anne Archibald aarchiba@physics.mcgill...
Mon Aug 1 16:07:48 CDT 2011


On 1 August 2011 10:14, Charles R Harris <charlesr.harris@gmail.com> wrote:
>
>
> On Sun, Jul 31, 2011 at 11:20 PM, Anne Archibald
> <aarchiba@physics.mcgill.ca> wrote:
>>
>> I realize this discussion has gone rather far afield from efficient 1D
>> deconvolution, but we do a funny thing in radio interferometry, and
>> I'm curious whether this is normal for other kinds of deconvolution as
>> well.
>>
>> In radio interferometry we obtain our images convolved with the
>> so-called "dirty beam", a convolution kernel that has a nice narrow
>> peak but usually a chaos of monstrous sidelobes often only marginally
>> smaller than the main lobe. We use a different regularization
>> condition to do our deconvolution: we treat the underlying image as a
>> modest collection of point sources. (One can see why this appeals to
>> astronomers.) Through an iterative process (the "CLEAN" algorithm and
>> its many descendants) we obtain an estimate of this underlying image.
>> But we very rarely actually work with this image directly. We normally
>> convolve it with a sort of idealized version of our kernel without all
>> the sidelobes. This then gives an image one might have obtained from a
>> normal telescope the size of the interferometer array. (Apart from all
>> the CLEAN artifacts.)
>>
>> What I'm wondering is, is this final step of convolving with an
>> idealized version of the kernel standard practice elsewhere?
>>
>
> That's interesting. It sounds like fitting a parametric model, which yields
> points, followed by a smoothing that in some sense represents the error. Are
> there frequency aliasing problems associated with the deconvolution?

It's very like fitting a parametric model, yes, except that we don't
care much about the model parameters. In fact we often end up with
models that have clusters of "point sources" with positive and
negative emissions trying to match up with what is in reality a single
point source. This can be due to inadequacies of the dirty beam model
(though usually we have a decent estimate) or simply noise. In any
case smoothing with an idealized main lobe makes us much less
sensitive to this kind of junk. Plus if you're going to do this
anyway, it can make life much easier to constrain your point sources
to a grid.

(As an aside, this trick - of fitting a parametric model but then
extracting "observational" parameters for comparison to reduce
model-sensitivity - came up with some X-ray spectral data I was
looking at: you need to use a model to pull out the instrumental
effects, but if you report (say) the model luminosity in a band your
instrument can detect, then it doesn't much matter whether your model
thinks the photons are thermal or power-law. In principle you can even
do this trick with published model parameters, but you run into the
problem that people don't give full covariance matrices for the fitted
parameters so you get spurious uncertainties.)

As far as frequency aliasing, there's not so much coming from the
deconvolution, since our beam is so irregular. The actual observation
samples image spatial frequencies rather badly; it's the price we pay
for not having a filled aperture. So we're often simply missing
information on spatial frequencies, most often the lowest ones
(because there's a limit on how close you can put tracking dishes
together without shadowing). But I don't think this is a deconvolution
issue; in fact in situations where people are really pushing the
limits of interferometry, like the millimeter-wave interferometric
observations of the black hole at the center of our galaxy, you often
give up on producing an image at all and fit (say) an emission model
including the event horizon to the observed spatial frequencies
directly.

Anne


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