[alma-config] [Fwd: Re: Memo 538 draft]

Wed Oct 5 22:46:20 EDT 2005

Hi John,

> Hi,
> 
>  I was planning to write a detailed reply to all of
> Frederic's
> comments tomorrow (when I have a gap from teaching),
> but since debate
> on the Nyquist issue has started again I have
> decided to reply on this point
> straight away
> 
> Several years ago we debated endlessley the issue of
> the appropriate
> value of the edge taper 10db  20db etc and the
> related issue of the minimum
> Nyquist sampling to allow at that edge. There was no
> universal agreement.
> At the Socorro delta-PDR meeting I suggested that we
> could circumvent this
> problem by considering as a practical matter what
> the impact on performace
> atually is. This argument I summarise in point 1
> below. Once we establish that we
> are talking about small factors, we can relax and
> hopefully then not get
> hot under the collar when we discuss what the actual
> answer is, which I
> discuss in 2)
> 
> 1) Frederic points out that my 3.5km array has
> points seperated by
> 4*nyquist at the edge  

This is actually 6*Nyquist at the edge. Indeed, the
config size for Nyquist sampling and gaussian tapering
at 15dB is 0.6 km and this size is proportional to the
sampling wanted (alpha parameter).
So, for a size of 3.5 km you have a sampling of
3.5/0.6=6*Nyquist at the edge. 

Now if the density of samples follows a gaussian
function with a 15dB cutoff at 3.5km, the density is
multiplied by a factor of 4 (the sampling interval
divided by a factor of 2 i.e. equal to 3*Nyquist) at
2.7 km. 

In other words 40% of the uv-plane is sampled with
spacing >3*Nyquist.

Similarly it can be shown that 64% of the uv-plane is
sampled at >2*Nyquist (corresponding to a radius of
2.1 km).

And all the uv plane is at sub-Nyquist!
(another way to verify this is that 15dB corresponds
to 3.16%. But to reach Nyquist sampling the density
should be multiplied by 36 and 3.16*36>100%.)

>when an array with 1*nyqusit
> could be build instead
> but this would better fit the design goals. Now I
> don't have the actual numbers handy (I'm
> writing this between meetings) but the overall
> density distribution in my
> design is close to  gauusian  with a cutoff  at the
> edge of say about 15dB
> and Frederick  suggests we now redesign it instead
> to have a cutoff of say
> 10db; because those sub-Nyquist  sampled points 
> cannot be used
> (especially  for Mosaicing). Whether or not  this is
> correct  is the
> subject of point 2 below. But let us assume he is 
> right, what happens?,
> we must then delete the uv points between 10dB and
> 15dB, but because
> a Gaussian fall off very fat beyond teh 10dB point
> there
> are  very few points in this region (!). 

According to the computation above all the uv-plane is
sub-Nyquist!
Tell me if I made a mistake somewhere or if the
density you measure for real is different.
Anyway I expect a significant part (more than the half
of the uv-plane) to be sub-Nyquist.

> I beleive
> the result is only an
> increase in the noise at the percent level and  a
> reduction in resolution
> of a  similar amount (If I get time tomorrow I  will
> look at the exact
> values).
> 
> Still even if the savings are small why not built
> the array to avoid these (assumed) useless
> points?. 

Indeed, this is the question.

> Such a design would argue for a less
> centrally condensed array, but there are costs
> involved in this both in terms of imaging and money.
> While all this attention is given to
> the outer uv edge and the exact value of its
> sampling we are in danger of forgetting
> what I think is a much more critical area for
> imaging quality, the short spacing coverage
> and specifically the shortest baseline for a given
> array resolution. A less centrally
> condensed  usually has a larger shortest baseline.

The shortest baseline is not necessarily larger for
less centrally condensed configs and the weight given
to the shortest baselines usually increases although
this is not the goal (this can not be avoided).
Eventually for a ring (this is an extreme case that we
don't want here but this illustrates well the
statement above is not right) we have two peaks: one
at the shortest and another one  at the largest
baselines.

> Likewise a less centrally  condensed array
> requires more pads to span a given range of
> resolutions. 

Answer in the main mail (I think I proved it is
possible to have configurations for gaussian with
higher cutoffs and the same number of pads as for the
spiral).

> The project is very short of money,
> one of my goals in the redesign going to 50 antennas
> was to reduce the number of pads as much as
> possible, the consequence was a highly concentrated
> array. If we have less of an edge
> taper we will need more pads, since each including
> roads each pad costs at least $150,000, the
> small amount of sensitivity we recover by having
> more pads and a better edge taper we may
> lose again because we cannot afford an antenna
> (which then loses us 2% in sensitivity).
> A final point is  that as designed the pad
> distribution has a 1/r^2 density distribution
> which then allows continuous zoom configuration with
>  all the operational
> and observational advantages that gives. With such a
> constraint on the pad density
> distribution the range  of occupied radii which
> gives close to gaussian uv density is
> constrained, and this value of occupied raddi leads
> to the particular edge
> taper we have. If we decide  to keep a 1/r^2 pad
> density distribution to
> allow zooming and then adjust the range of occupied
> radii in any  configuration to
> give say 10dB edge  taper, then as well as needing
> more pads the  uv
> density function will be  far from  gaussian and we
> will  lose sensitity
> from this. I can of course relax  the 1/r^2 pad
> desity requirement but
> then I lose the zoom capability,

Answer about the continuous reconfiguration in the
main mail.

> 
> 2) Now we can consider the problem of whether these
> uv point at the edge of the
> uv coverage which are sub-Nyquist sampled really are
> useless or not. First I try
> to characterise the nature of the problem then give
> my views.
> 
> Consider a thought experiment where I sample at
> greater than Nyquist
> within a boundary in the uv plane. I am now given a
> few extra points to place beyond
> this boundary. Clearly  their sampling will be
> highly sub-Nyquist.
> If I am an optimist I see the glass as half full -
> and argue that at least
> these points give me SOME information about the
> region beyond the boundary
> to constrain extrapolation and that 'some
> information is always better than none'.
> If I am a pessimist I say the glass is half-empty. I
> admit that there is extra
> information in these points but I argue that there
> is no way of making use of this
> information. 
> If  I do try to make use of these
> points the disaster results, with huge
> increases in noise because I am trying to
> interpolate between samples which are too
> far apart.
> 

If I tell you I am convinced we can decide the
positions of the antennas by throwing coins on the map
of Chajnantor and we will still be able to do useful
images, do you classify me as a pessimist or optimist?
And is this in contradiction with the fact that when
trying to optimize an array my goal is to avoid
half-empty or half-full?

To be more accurate: YES the points outside the well
sampled area contain useful information about the
source. It is even possible that if the source is
compact enough so that the sampling required can be
sufficiently relaxed, the spacing between those point
will be OK. In this case they can be used in the
image.
If their spacing is too large compared to the one
required by the size of the source and interpolation
is impossible i.e. the estimation of the visibility
between the points will be erronated (whatever the
algorithm, e.g. CLEAN...), adding the Fourier
components corresponding to these samples in the image
will not improve the quality of the image. Because in
that case, by hypothesis (i.e. no interpolation
possible), the value obtained from the interpolation
will be far from reality. Adding those points implies
adding the interpolated values too and this will
result in uncontroled errors on the corresponding
Fourier components. 

If the information contained in those points can be
used in an other way that is great but if the goal is
to design configurations for imaging of extended
sources they should be avoided.

> To decide which of the two points is correct I can
> first
> look at linear methods of interpolation (and I
> define a linear
> operation F rigoursly such that F(u1 + u2) = F(u1) +
> F(u2), while for non-linear operations I have an
> inequality,  where u1 and u2 are subsets  of the uv 
> data). One can look at an infinite sequence
> of interleaved points with mean seperations Nyquist
> but with half the seperations larger
> than Nyquist and half smaller and show that the
> noise increases significantly when I
> have gaps which are much larger than Nyquist.
> Frederic has shown something
> similar but more general  in one of his  papers. 

Yes but not for an infinite sequence: only 8 samples.

> I
> am not sure how
> applicable these ideas are to the problem at hand
> however, where
> we have rapid decrease in density to the uv edge
> rather than an infinite sequence and where
> although  we will get larger errors towards the uv
> edge they will have less impact because
> of the restoration with a CLEAN beam.

I agree with this point. This is not only true for
CLEAN but for all the deconvolution methods: as long
as a decreasing function is wanted for the clean beam
less weight is given to the higher spatial frequencies
measured by the instrument. 

But here again we are in the situation of designing an
array: if we think it is important to have
configurations up to a given radius it means we think
those samples corresponding to the largest baselines
are important. Why would we design configurations up
to this radius if this is to throw these samples away
most of the time (if imaging extended sources is our
goal most of the time)?

> 
> To try to see whether highly sub-Nyquist samples
> caused a serious problem or not
> for linear deconvolution schemes I once simulated
> doing a pseudo-inverse restoration of a 1D
> uv coverage where the samples at the edge were
> highly sub-Nyquist, see
> 
>http://www.oso.chalmers.se/~jconway/ALMA/SIMULATIONS/SIM12/
> 
> I got good reconstructions
> and no large amplifications of the noise. 

I can understand that. As you said the weight of those
data is low so overall they have little influence on
the noise of the reconstructed image, but this does
not answer to the question: are they useful to the
image?

> Of course
> in a sense a pseudo-inverse method is
> a sophisicated version of throwing away data, but in
> this case these points at the edge of
> the uv coverage make some contributions to the
> eigenvectors which are eventually included in
> the final image, wheras if one just deletes points
> beyond a given radius they have no impact
> at all.

For me this is a contradiction: assuming some points
are too far away from each other for the deconvolution
method we want to use to be able to interpolate is
equivalent to assume we have no constraint on the
value of the visibility function between the points
and adding them will also add uncontroled noise (even
if this noise is relatively low compared to the global
noise because of low weights).
So if adding these points result in an improvement of
the image there is a contradiction: it implies the
method was able to use the points to interpolate the
visibility function.

> 
> In practice we will almost certainly continue to use
> non-linear deconolution methods
> to fit our visibility data, and we can ask whether
> these can make use of the sub-Nyquist
> sampled points.  Frederik in his paper characterises
> the impact of such algorithms as
> improving on Nyquist by some fixed factor, so they
> would allow the use of 2x Nyquist
> spacing points instead of Nyquist. 

This is not right, in the paper I use the parameter
alpha to characterize the sampling required in units
of Nyquist interval. 
I never said that for non-linear methods alpha=2. I
don't make any asumption about how much it is and I
keep this parameter free along the paper.

One thing I am sure about is that the closer the
sampling to Nyquist (here in the sense 1/source_size
not the primary beam) the better the image. I admit
some algorithms can do a better job in interpolating
and support lower sample densities that is the reason
why I introduced this alpha parameter. If you find
that alpha=10 for a given method I have no problem.

> I am sure that he
> would be first to admit (and I think its
> said in his papers) that this is just a crude
> approximation. In particular
> performance for non-linear
> methods will depend on the source structure. If on
> the longest baselines all the extended emission
> is resolved out except for one bright unresolved
> point the low denisty uv samples certainly
> are useful for constraining the position and flux
> density of this point [I anticipate
> an objection here from Frederic that I am yet 
> again mixing up imaging 
> and model-fitting, 

Yes indeed I think you are mixing model-fitting and
imaging, even though for me too (for everybody I
guess) imaging can be seen as a model fitting (e.g.
the least square fit approach in the appendix of my
paper). So instead of using the usual terminology we
should probably say model fitting with a small number
of free parameters (instead of just "model fitting")
and fitting with the maximum of free parameters (for
"imaging"). Then, using this terminology, of course
fitting models with a small number of parameters
requires much less sampling than fitting models with
the maximum of free parameters. Fitting point sources
is in the first category, but should this be the
driver for the design?

> I will
> explain my position more fully  on this issue 
> in a subsequent email, 
> briefly I consider imaging
> just as model fitting using a model  
> representation which is a bed of 
> delta functions and the
> model  parameters to be adjusted the fluxes of each 
> delta function]. In 
> the above case on the long
> baselines the size of the source which is
influencing 
> the visibilities 
> is much smaller than the
> primary beam. The effective size of the 
> source theta is much less than 
> the primary beam so the
> 'bandwidth' of the uv data is smaller 
> than the primary beam, and the 
> Nyquist sampling interval
> in the uv plane correspondingly larger.Hence 
> low density uv points can 
> become in effect more than
> Nyquist sampled and I can readily use  their
> information. 

Yes of course, I fully agree with that.
Samples separated by any distance can be used for
point source subtraction.

> Non-linear algorithms like CLEAN
> (and more advanced versions like Multi-resolution 
> or wavelet CLEAN) I 
> believe can identify regions
> of signal above the noise and then adjust the value
of the pixles  
> within these regions to fit the data. So  even 
> if there was not one but 
> several disconnected compact regions 
> contributing to the uv data on the longest 
> baseline outlier uv points
> even a handful of such outlier uv
> points would be  sifficient to 
> constrain
> the structures of the compact regions.

Here again I agree with you: for compact sources the
sampling required is relaxed and samples that could
not be used for extended sources because of poor
sampling may be used for compact sources.

> In conclusion I believe (A) Robust 
> deconvolution schemes exist and can 
> be built for which
> can deal with data with sub-Nyquist 
> sampled uv points, such that they 
> don't in any way casue disaster
> and high noise in the final  image

I think I answered above: if the weights of the points
is low of course they wil have little influence, also
the smaller the region concerned in the uv-plane the
less impact this will have (e.g. if this is really
only the border of the uv-plane this can probably be
ignored). 
The question is, are these points useful for imaging?

> (this is clearly true since CLEAN 
> and MEM daily work with VLA
> data with sub-Nyquist  sampling and 
> get final map noises very close to 
> the noise in Jy/beam
> for  the dirty map). 

The discussion has shifted: it started with the
problem of what to do with data which are so far away
from each other that no interpolation with a
reasonalbe error is possible. And now you mention an
example for which interpolation is performed every
day!
I guess there is a confusion coming from the fact I
use the Nyquist interval as a unit for the measure of
the sampling interval (what other unit would you
propose?), but I never said (or please tell me when)
that sub-Nyquist data can not be used for imaging.
Again it is clear that the closer the sampling to
Nyquist the better (and again Nyquist in the sense
1/source_size) but this is not necessarily required.

> (B) Algorithms exist  where in all cases the
> outlier points contribute -some- useful information
to the final image 
> without dramtically
> increasing the noise. 

Maybe your point is that we are never in the situation
where the points are so far away that we cannot
interpolate with a reasonable uncertainty? For me this
is impossible to admit. With noise the information is
local in the uv-plane so that there is nessarily a
limit in the sampling interval. When this limit is
crossed the error on the estimation of the visibility
function between the points is much bigger than the
instrumental error. So for me the discussion is not
whether this limit exists but what is the value of
this maximum spacing (in units of Nyquist interval for
example). Up to now all the interferometers have been
designed to optimize the sampling (except the recent
CARMA and ATA). With ALMA we have the luxury to
consider tapering the distributions because the
sampling can be close to Nyquist. This is great but
this does not imply sampling can be forgotten, this
close to Nyquist sampling is not garanteed for all
configurations.

> (C) the degree  to which outlier points are 
> useful for imaging is very source structure
dependant. 
> In the case that the 
> longest baselines only sees
> structure above the noise from a small faction of 
> the primary beam they 
> can very useful. 

I fully agree.

> Non-linear
> algorithms can be effective in 
> seperating signal from noise in the 
> image domain and concentrating
> on adjusting the parameters of the pixel
> delta functions to the uv data.

> I am sure there will be a lot of disagreement  with
what is written 
> above!

Altogether it seems to me there is a lot of agreement
(sorry to be in disagreement on this point ;-).

Frederic

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