Deconvolution

1
Lecture 15

• Deconvolution
• CLEAN
• Why does CLEAN work?
• Parallel CLEAN
• MEM

2
Deconvolution.

• How can we go from
• this
• to this?
• Deconvolution in the sky plane implies
  interpolation or reconstruction of missing values
  in the UV plane.
• But we, the human observer, can look at the top
  image and just know that it is 2 point sources
  on a blank field.

3
CLEAN

• The following naive algorithm gives
  surprisingly good results
• Find the brightest pixel xb,yb in the dirty
  image.
• Measure its brightness I(xb,yb).
• Subtract ?I(xb,yb)B(x-xb,y-yb) from the image,
  where ? is a number in the approximate range 0.01
  to 0.2.
• Repeat until satisfied.
• This is known as the CLEAN algorithm. The
  successive numbers ?I are called clean components.

4
Problems with CLEAN

• Mathematicians dont like CLEAN. They say it
  ought not to work. There are lots of papers out
  there proving it doesnt.
• But it does work, good enough for rough-and-ready
  astronomers, anyway.
• This is because the real sky obeys strong
  constraints
• Nearly always there are just a few smallish
  bright patches on a blank background
• Negative flux values dont occur in the real sky.
• CLEAN doesnt work really well on extended
  sources can get clean stripes or bowl
  artifacts.
• It is difficult to know when to stop CLEANing.
  Too soon, and you are missing flux. Too late, and
  you are just cleaning the noise in your image.

5
More problems with CLEAN

• Sources which vary in flux over the duration of
  the observation.
• Solution cut the observation into shorter
  chunks, clean separately, then recombine.
• Clunky, loses sensitivity.
• Parallel cleaning works well though.
• (Only relevant for wide-band case) different
  sources have different shapes of spectrum.
• Parallel cleaning is also good for this even
  when sources vary both in frequency and time!
• Sources which arent located at the centre of a
  pixel. Fixes
• Re-centre on each source, then CLEAN them away.
• You guessed it parallel cleaning can also help.

6
Parallel CLEAN

• First developed to cope with wide-band imaging.
  The fundamental paper is
• The basic idea is to construct a number of dirty
  beams, the jth beam (starting at j0) by setting
  Vj(?) to (?/?0)j/j! then FTing. Eg V01, V1?/?0,
  V2(?/?0)2/2, etc. Since the Vjs form a Taylor
  series, any spectrum can be approximated by a sum
  of Vjs thus any source by a sum of Bjs.

Sault R J Wieringa M H AA Suppl. Ser. 108,
585 (1994)
7
Description of the problem an example.
If both point sources have identical spectra,
there is no problem.
S
?
8
Description of the problem an example.
More realistic different spectra
S
?
S
?
This will not clean away.
9
The Sault-Wieringa algorithm
0th order
S

Taylor expansion
1st order

A source spectrum
2nd order
etc
10
Taylor-term beams
max 1.0
max 0.02
max 0.01
max 0.004
0th order
1st order
2nd order
3rd order
11
A simulation to test this
19 point sources from 0.001 to 1 Jy
Spectra cubics, with random coefficients.
eg
? (GHz)
12
Alternate cleaning(i) 1000 cycles of standard
clean
not good.
13
Alternate cleaning(ii) each spectral channel
cleaned, then co-added.
pretty good, but do we lose faint sources?
14
S-W clean to various orders
(All 1000 cycles with gain (?) 0.1)
0th order (equivalent to standard clean)
15
S-W clean to various orders
1st order
16
S-W clean to various orders
2nd order
17
S-W clean to various orders
3rd order
Not much left but numerical noise.
18
Time-varying sources
Source constant in time
Source flux varying with time
19
Time-varying sources
I M Stewart et al paper in preparation!
20
MEM the Maximum Entropy Method.

• (Content for this slide pretty much copied from
  T. Cornwell, chapter 7, NRAO 1985 Synthesis
  Imaging Summer School. I havent studied ME
  myself.)
• What does entropy mean in this context?
• Something which, when maximized, produces a
  positive image with a compressed range of pixel
  values.
• An example maximize
• I guess we would need to read Narayan and
  Nityananda 1984 to figure out what e is.

I is the image we end up with
M is our best guess starting image.