Hugh Garsden, Observatoire de Paris-Meudon

Generating Radio Images Using Compressive Sampling as an Alternative to CLEAN

1. Genera&ng  Radio  Images  Using   Compressive  Sampling  as  an   Alterna&ve  to  CLEAN   Hugh  Garsden  (CEA,France)   Jean-­‐Luc  Starck   Stéphane  Corbel   Cyril  Tasse  (Observatoire  de  Paris-­‐Meudon,  France)   LOFAR  TKP,  Oxford,  Jun  14th  2012  
2. Outline   •  Introduc&on  to  Compressive  Sampling   •  Radio  interferometry   •  Radio  image  examples   •  Compressive  sampling  behaviour   •  Real  world  
3. Compressive  Sampling  Example   •  Use  this  128x128  image   –  Dynamic  Range  (DR)  100   –  added  noise   •  Take  the  Fourier  Transform  and  select  the   following  885  points  (5%)   from  the  2-­‐D  FFT   •  That’s  the  sampling  step,   we  have  a  compressed  image  
4. Compressive  Sampling  Example   •  Can  the  image  be  reconstructed?  Try  inverse  FFT   using  the  5%  Fourier  coefficients:   •  Apply  CS  method:   original reconstruction DR = 376
5. Compressive  Sampling  Example   •  What's  good  about  it?   – Compression   – Clean  images   – Less  noise  (see  later)   – Image  in  a  func&on  representa&on  (below)   •  How  is  it  done?   •  The  reconstruc&on  is  made  in  the  wavelet   domain  (this  talk)   Fourier    >>>>    Method  >>>>  Wavelet  
6. Compressive  Sampling  Theory   •  Compressive  sampling  was  discovered   –  Candes,  Donoho,  Tao  ~  2004   •  Sample  few,  randomly,  in  sampling  domain   •  Sparse  reconstruc&on  (few  coefficients)  of  image   in  another  domain   •  Sampling  and  reconstruc&on  domains  must  be   incoherent  (~  uncorrelated,  orthogonal)   •  Minimiza&on/regulariza&on  procedure   •  Theory  describes  constraints,  condi&ons,   expecta&ons,  probable  outcomes   Introduc.on  to  Compressive  Sampling,  Candes  (2008),  IEEE  Signal  Processing  
7. Compressive  Sampling  Theory   •  Mathema&cal  form    sampling                  y  =  φ  I                   signal  representa&on         I  =  ψ  x        y  =  φ  ψ  x            y  =  A  x      y  =  A  x  +  ε   •  Given  y  and  A,  search  for  x.  If  x  is  sparse:      find      |y  -­‐  A  x|l2  <  ε                  min|x|l1                works  very  well  (see  theory)  due  to  l1     •  Implemented  by  A.  Woiselle  and  J-­‐L.  Starck     A is not invertible, square etc. but is subject to certain desirabilities
8. Radio  Interferometry   •  Radio  interferometer  records  Fourier  transform   of  sky     –  visibili&es  (u,v)   •  Represent  this  by      V  =  M  F  I  +  ε   •  Matches  compression     sampling  equa&on        y  =  A  x  + ε •  Solve for the image I McKean et al. (2010) LOFAR: Early imaging results from commissioning for Cygnus A. 10th European VLBI Network Symposium and EVN Users Meeting: VLBI and the new generation of radio arrays.
9. Radio  Image  Reconstruc&on  in  Words    Given  the  visibili&es,  find  an  image  (represented   by  a  few  wavelets)  that  has  a  Fourier  transform   that  matches  the  visibili&es  (where  the  visibili&es   exist).    In  the  rest  of  the  UV  plane,  there  are  interpolated   Fourier  coefficients  -­‐  inpain.ng    Compressive  sampling  can  do  this  well.    
10. •  Image -> noise -> FFT -> mask -> CS No dirty map, psf More  realis&c  image  simula&ons   • Heald  et  al.  (2000)  Progress  with  the     LOFAR  Imaging  Pipeline:   •  DR  1000   •  McKean  (previous)  has  3340   Cygnus A LOFAR observation Fourier(PSF). 0.8% coverage. -> find      |y  -­‐  A  x|l2  <  ε                  min|x|l1  
11. Compression  Sampling  Tests   •  Test  image,  Cygnus  A   Original (noise invisible) Reconstruction 200 search iterations reconstruction. Flux of residual = 1.4% DR of reconstruction = 2415 Residual
12. Compression  Sta&s&cs   •  “Residual”  does  not  have  the  same  meaning  as   in  CLEAN   -­‐  Here,  residual  =  original  minus  reconstruc&on   -­‐  No  model  as  in  CLEAN,  the  reconstruc&on  is   the  answer   •  Dynamic  Range  =  Peak/(Off  source  RMS)   -­‐    What  is  off  source  RMS  in  wavelet    reconstruc&on?   -­‐  It  isn’t  noise,  it’s  wavelet  firng  and   smoothing  of  noise   -­‐  “DR,  noise”  used  loosely  here  
13. Compression  Sampling  Tests   •  Test  image,  HII  region  in  M31   Original (noise invisible) Reconstruction 200 iterations reconstruction Flux of residual = 0.16% DR of reconstruction = 2592 Residual
14. What  about  CLEAN?   Using casapy ‘deconvolve’ Can match CS with CLEAN. DR = 800 Hoog F. 2011, The application of compressive sampling to radio astronomy. I. Deconvolution. Astronomy and Astrophysics 528, A31
15. Compressive  sampling  behaviour   A single run, iterations Could stop 20-40? Well-behaved Sim only
16. Compression  Sampling  and  LOFAR   •  Reduce  the  number  of  samples  in  the  mask   •  Doesn’t  maser  above  0.6?   •  Shape  of  the  mask  
17. Other  CS  Behaviour   •  Many  algorithms  and  new  ones  coming  out   Sparsity  Averaging  Reweighted  Analysis  (SARA):  a  novel  algorithm  for  radio-­‐ interferometric  imaging  (2012),  Carrillo,  McEwen,  Wiaux,  arXiv:1205.3123   •  Algorithms  all  have  their  own  parameters  to   tweak   -­‐  Thresholds   -­‐  Constraints   •  Which  wavelets?     -­‐  Here  used  isotropic  undecimated   •  Not  wavelets?  
18. •  Visibili&es  are  3-­‐D  (u,v,w)    -­‐  Compression  sampling  can  be  3-­‐D  (MRI)   •  Visibili&es  are  non-­‐equispaced  sampled  Fourier   coefficients  with  a  huge  range  (-­‐30000-­‐30000)   -­‐  Convolu&ons,  gridding,  W  projec&on,  beam    correc&ons  …  required,  as  in  AWImager     -­‐  Correlated  noise   •  Incorporate  CS  in  AWImager  (Cyril  Tasse)   •  Generate  images  during  the  observa&on  and   search  for  transients  during  the  observa&on   (Cyril)    Real  World