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Lecture 18. Deconvolution CLEAN Why does CLEAN work? Parallel CLEAN MEM. ‘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.

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Lecture 18
Lecture 18

  • Deconvolution

    • CLEAN

      • Why does CLEAN work?

      • Parallel CLEAN

    • MEM


Deconvolution
‘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.


The clean algorithm
The CLEAN algorithm.

  • Find the brightest pixel rb in the dirty image.

  • Measure its brightness D(rb).

  • Subtract λD(rb)B(r-rb) from the image, where λ is a number in the approximate range 0.01 to 0.2.

  • Repeat until satisfied.

  • The successive numbers λD(rb) are called clean components.

  • Högbom J A, Astron. Astrophys. Suppl. 15, 417 (1974)

    Further refinements have been added by Clark and later by Cotton and Schwab, but the essential nature of the algorithm remains the same.

    25/43


    Cleaning
    CLEANing

    Dirty image

    Cleaned image

    26/43


    Cleaning1
    CLEANing

    Dirty image

    Cleaned image

    27/43


    Cleaning2
    CLEANing

    Dirty image

    Cleaned image

    28/43


    Cleaning3
    CLEANing

    Dirty image

    Cleaned image

    29/43


    Problems with clean
    Problems with CLEAN:

    • Mathematicians don’t like CLEAN. They say it ought not to work. There are lots of papers out there proving it doesn’t.

      • 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 don’t occur in the real sky.

    • CLEAN doesn’t 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.


    More problems with clean
    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 aren’t 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.


    Wide band issues one can vastly increase the number of samples by extending the bandwidth
    Wide-band issues:one can vastly increase the number of samples by extending the bandwidth.

    Simulated (e)MERLIN UV samples, δ=+35°

    Δf/f = 0.0005

    Δf/f = 0.3

    30/43


    Description of the problem an example
    Description of the problem: an example.

    S

    f

    If both point sources have identical spectra, there is no problem.

    S

    f

    31/43


    Description of the problem an example1
    Description of the problem: an example.

    More realistic: different spectra:

    S

    f

    S

    f

    This will not clean away.

    32/43


    Parallel clean for wide band observations
    Parallel CLEAN for wide-band observations.

    Conway J E, Cornwell T J & Wilkinson P N, MNRAS 246, 490 (1990)

    Decomposed the dirty image as a sum of convolutions.

    Sault R J & Wieringa M H, A&A Suppl. Ser. 108, 585 (1994)

    Described the parallel CLEAN algorithm.

    I0

    I1

    I2

    *

    *

    *

    =

    +

    +

    B0

    B1

    B2

    Dirty image D

    33/43


    The sault wieringa algorithm
    The Sault-Wieringa algorithm:

    νref

    νref

    νref

    νref

    ν

    ν

    ν

    ν

    • The basic idea is to construct a number of dirty beams from components of a Taylor expansion of the spectrum.

    0th order

    S

    +

    Taylor expansion

    1st order

    +

    A source spectrum:

    2nd order

    etc…


    Taylor term beams
    Taylor-term beams

    max = 1.0

    max = 0.02

    max = 0.01

    max = 0.004

    0th order

    1st order

    2nd order

    3rd order


    A simulation to test this
    A simulation to test this:

    19 point sources from 0.001 to 1 Jy

    Spectra: cubics, with random coefficients.

    eg

    ν (GHz)


    Alternate cleaning i 1000 cycles of standard clean
    Alternate cleaning:(i) 1000 cycles of standard clean

    …not good.


    Alternate cleaning ii each spectral channel cleaned then co added
    Alternate cleaning:(ii) each spectral channel cleaned, then co-added.

    …pretty good, but do we lose faint sources?


    S w clean to various orders
    S-W clean to various orders

    (All 1000 cycles with gain (λ) = 0.1)

    0th order (equivalent to standard clean)




    S w clean to various orders3
    S-W clean to various orders

    3rd order

    Not much left but numerical noise.


    Time varying sources
    Time-varying sources:

    Source constant in time

    Source flux varying with time


    Time varying sources1
    Time-varying sources:

    Stewart I M, Fenech D M & Muxlow T W B, in preparation.

    5 sources, all ~1 Jy, varying in both flux and spectral index by ~x10.

    Input to the simulation (average flux; convolved with restoring beam)

    Dirty image

    Högbom clean: 2000 cycles, λ=0.1

    Parallel clean: 2000 cycles, λ=0.1, Nf=3, Nt=6

    All brightness scales -0.02 to +0.02 Jy


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