[alma-config] BOUNCE alma-config at majordomo.cv.nrao.edu: Non-member submission from [Stephane Guilloteau <sguillot at eso.org>]
Min Yun
myun at aoc.nrao.edu
Thu Feb 10 14:08:28 EST 2000
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Date: Wed, 09 Feb 2000 11:52:56 +0100
From: Stephane Guilloteau <sguillot at eso.org>
Organization: ALMA
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Subject: Complete UV coverage and Tapers
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Hi everybody,
I have been in contact with several persons from the Signal Processing
community,
either medical imaging (one of my former student is now in charge of a medical
imaging
research department) or other fields like image recognition and "blind"
deconvolution
(when you don't even know the transfer function).
There are a number of "well" known problems which are relevant to our
discussion.
Imaging can be seen as a linear operator on the data space, with give some
output
O = K I
where K is the operator kernel. In finite space (discrete representation), K can
be
represented by a matrix. Note that the operator K not only depends on the dirty
beam,
but also on additional constraints like the support of the image, its positivity
(if any),
etc...
a) Because our images are support-limited, their Fourier transform are NOT
support limited.
So to recover them, some extrapolation is required...
b) In all cases, one cannot expect to recover the "true" brightness
distribution, but
only a "regularized" (see below) one (i.e. you will NEVER know whether you
apparently
smooth image is in fact composed by a large number of point sources until
you observe
with adequate resolution).
c) Some regularisation must be applied. "Regularisation" stands for any
method
applying some sort of smoothness constraint. MEM (and all its variant) is an
example of
(Bayesian) a priori regularisation. CLEAN is an example of a posteriori
regularisation,
because of the convolution with the Clean beam.
Positivity is NOT a regularisation principle: it is a constraint on the
data space.
It is this requirement for a regularisation method which drives the need for
TAPERED
distribution.
d) Most deconvolution techniques are guaranteed to converge if the
convolution kernel is
positive-definite, i.e. in our language, if the UV coverage is complete...
Alas, that
includes the short spacing also... Getting complete UV coverage will
minimize the number
of zeroes in K.
e) In all cases, one cannot expect to recover the "true" brightness
distribution, but
only the "regularized" one (i.e. you will NEVER know whether you apparently
smooth image
is in fact composed by a large number of point sources until you observe
with adequate
resolution).
f) If the kernel is not positive-definite, some of the Fourier components
will be
ill-constrained, and thereby, poorly recovered in the deconvolution.
However, here, positivity (and support information as well) does help a
lot.
The key point is wether the ratio of
the highest eigen-value to the lowest (non-zero) eigen-value in the operator
K stays
reasonable or not. Any mode (i.e. structure in the image plane)
corresponding to a very
small eigenvalue will be poorly recovered. One can actually compute the
effective noise
level on any mode from the initial noise distribution and the eigenvalue
analysis.
Some regularisation methods actually limit the reconstruction by
neglecting all the
small eigen values (and hence ignoring the corresponding modes). This is
similar to a
Singular Value Decomposition (although it uses very different methods,
because the
matrix K is huge...).
It is even possible to see which mode are actually "uncertain". What
happens is that
most of the poorly constrained modes are highly unphysical (the most simple
example is
the stripes which CLEAN produces sometimes). Hence, they limit the dynamic
range and
image fidelity, but not the physical interpretation...
g) Note that even a "complete" (i.e. no holes) UV coverage may have a wide
range of weights
for all UV cells. The argument of Ed Fomalont applies here: we should
measure the
uniformity not by comparing 0 to any number, but all numbers between them.
I think the measure of the ratio between highest and smallest non-zero
eigenvalue is
a fair measure of the quality of the imaging. Alas, this ratio depends not
only on the
dirty beam, but also (and not surprisingly) on the support of the image...
h) Tapered distributions (seen as a few extra points beyond the uniform UV
coverage)
most likely give better eigenvalues than purely uniform UV coverage. I
can't prove it,
but that looks intuitive: a few constraints on how the extrapolation must
go will give
a better defined operator than no constraints at all. The result will be a
less sensitive
observation before deconvolution, for sure, but also a FAR LOWER noise
amplification
factor in the deconvolution. At the end, the deconvolved image may be less
noisy with
tapered distribution...
Now a couple of comments on medical imaging
i) Most of the medical imaging involves "filled" aperture, or tomography
which is
somewhat different from Fourier synthesis.
j) Most the medical imaging is actually very poor. I guess the image
fidelity hardly ever
exceed 3 to 5, but this is sufficient for their purpose which is typically
to distinguish a
good tissue from a bad one (a binary operation in some way...).
I don't know whether this information is helpful or not. From my own
experience with WIPE,
which can produce an upper bound on the error map, I found that with current mm
arrays, the
error maps is discouragingly large, unless heavy taper is used. But I found the
display of
highest error modes very useful to pinpoint possible artefacts in the
reconstruction.
I have not experienced WIPE on ALMA-like UV coverages, because of computation
limits. I
can get in touch with the experts in Toulouse: perhaps they could work from the
gridded
UV data, which would be faster.
Stephane
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