Difference imaging and multi-channel Clean

Transcription

1 Difference imaging and multi-channel Clean Olaf Wucknitz Algorithms 2008, Oxford, 1 3 Dezember 2008

2 Finding gravitational lenses through variability [ Kochanek et al. (2006) ] small field in SDSS: one candidate, no lens [ Lacki et al. (2008) ] titlepage introduction summary contents back forward previous next fullscreen 1

3 Optical difference imaging difference original image convolution kernel reference image [ Alard & Lupton (1998) ] titlepage introduction summary contents back forward previous next fullscreen 2

4 Radio difference imaging UV epoch 1 CLEAN convolve crap UV epoch 2 CLEAN convolve UV epoch 1 CLEAN convolve crap UV epoch 2 UV epoch 1 difference + convolve mean UV epoch 2 CLEAN convolve difference titlepage introduction summary contents back forward previous next fullscreen 3

5 Related problem: combining different arrays UV epoch 1 CLEAN convolve + crap UV epoch 2 CLEAN convolve UV epoch 1 + CLEAN convolve crap UV epoch 2 UV epoch 1 difference + convolve mean UV epoch 2 CLEAN convolve difference titlepage introduction summary contents back forward previous next fullscreen 4

6 Notation sky brightness distribution I(l, m) I visibilities Ĩ(u, v) Ĩ perfect measurement Fourier transform vectors Ĩ µ = j A µ j I j A µ j = e 2πi(l,m) j (u,v) µ Ĩ = AI residuals R 2 = (AI Ĩ) W(AI Ĩ) natural weighting W = diag(σ 2 j ) R 2 = χ 2 titlepage introduction summary contents back forward previous next fullscreen 5

7 Lazy interferometrists do it in image space expand R 2 = Ĩ WĨ + I A WAI 2I A WĨ define (using w = Tr W) dirty beam dirty map B = A WA w I D = A WĨ w residuals derived in image space R 2 = const + w(i BI 2I I D ) minimum BI = I D } {{ } R 2 titlepage introduction summary contents back forward previous next fullscreen 6

8 CLEAN as maximum likelihood fitting add components to the model I minimize residuals in each step empty model plus component I j optimal flux R 2 = I 2 j 2I j I D j I j = I D j residuals for optimal flux R 2 = I D 2 j optimal position: peak in dirty map subtract shifted beam from dirty map, start over titlepage introduction summary contents back forward previous next fullscreen 7

9 Simultaneous CLEANing two epochs Ĩ 1 and Ĩ2 two models/maps I 1 and I 2 combined residuals R 2 = const + w 1 (I 1 B 1 I 1 2I 1 I D1 ) + w 2 (I 2 B 2 I 2 2I 2 I D2 ) next component at same position in 1 and 2 fluxes independent optimal fluxes I D1 and I D2 optimal position maximum of w 1 I D w 2 I D 2 2 titlepage introduction summary contents back forward previous next fullscreen 8

10 Need for difference-clean disadvantages of simultaneous CLEANing deconvolution errors still independent no control over mean and difference alternative approach two channels I + and I I + = 1 2 (I 1 + I 2 ) I = 1 2 (I 1 I 2 ) dirty maps I D+ = w 1 I D1 + w 2 I D2 w 1 + w 2 I D = w 1 I D1 w 2 I D2 w 1 + w 2 titlepage introduction summary contents back forward previous next fullscreen 9

11 D-CLEAN procedure next component in either I + or I { 2 ID residuals R 2 = const (w 1 + w 2 ) + for + for I D 2 subtract according to ( ID+ I D ) = ( B B )( I+ B B I ) I + and I not independent dirty beams B = w 1 B 1 + w 2 B 2 w 1 + w 2 B = w 1 B 1 w 2 B 2 w 1 + w 2 titlepage introduction summary contents back forward previous next fullscreen 10

12 An experiment: VLA-like uv coverage, scale 1:4 input convolved CLEAN 1 2 titlepage introduction summary contents back forward previous next fullscreen 11

13 Alternative methods CLEAN simultaneous D-CLEAN + titlepage introduction summary contents back forward previous next fullscreen 12

14 Alternative methods: residual errors CLEAN simultaneous D-CLEAN + titlepage introduction summary contents back forward previous next fullscreen 13

15 Combining different arrays: input input convolved CLEAN 1 2 titlepage introduction summary contents back forward previous next fullscreen 14

16 Combining different arrays: output combined simultaneous D-CLEAN + + titlepage introduction summary contents back forward previous next fullscreen 15

17 Two-channel CLEANing of real data target lens B two bright images Einstein ring two VLA-A observations with Pie Town use 1 IF titlepage introduction summary contents back forward previous next fullscreen 16

18 Differencing images of B CLEAN simultaneous D-CLEAN titlepage introduction summary contents back forward previous next fullscreen 17

19 Combined images of B CLEAN combined simultaneous D-CLEAN titlepage introduction summary contents back forward previous next fullscreen 18

20 Test with simulated VLBA data uv coverage and model from real observations of Virgo A (thanks to Yuri Kovalev) [ details in Kovalev et al. (2007) ] following slides: simulation without/with noise, two epochs, slightly different uv coverage titlepage introduction summary contents back forward previous next fullscreen 19

21 Results for simulated data: no noise difference of Clean maps difference-clean mean Clean map titlepage introduction summary contents back forward previous next fullscreen 20

22 Results for simulated data: with noise difference of Clean maps difference-clean mean Clean map titlepage introduction summary contents back forward previous next fullscreen 21

23 Alternative bases so far: two epochs constant part (sum) variable part (difference) many epochs constant part one additional part for each epoch continuous observation (or ν instead of t) constant part linear slope higher derivatives or general orthogonal polynomials, or... titlepage introduction summary contents back forward previous next fullscreen 22

24 Constant + linear in frequency dirty maps I D (n) = ν w(ν)ν n I D (ν) ν w(ν) dirty beams B (n) = ν w(ν)ν n B(ν) ν w(ν) convolution equation up to linear order ( ID (0) I D (1) ) = ( ) (I B (0) B (1) ) (0) B (1) B (2) I (1) previous MFS methods only use I D (0) = B (0) I (0) + B (1) I (1) [ Conway et al. (1990), Sault & Wieringa (1994) ] titlepage introduction summary contents back forward previous next fullscreen 23

25 ATCA observations of Pictor-A 4800 / 4928 MHz [ data from Emil Lenc ] titlepage introduction summary contents back forward previous next fullscreen 24

26 Pictor-A: difference MHz difference of Clean maps difference channel of D-Clean titlepage introduction summary contents back forward previous next fullscreen 25

27 Pictor-A: correct for variability with AIPS (4800 MHz) five epochs with ATCA (different configurations) common model + variable parts titlepage introduction summary contents back forward previous next fullscreen 26

28 Summary etc these tasks difference imaging, variability combine different arrays / epochs wide-band imaging with spectral indices multi-resolution / multi-scale Clean have in common that multi-channel output is needed channels do not always correspond to parts of the data want Clean regularisation for output channels need to deconvolve everything simultaneously developing multi-channel Clean first tests encouraging work in progress titlepage introduction summary contents back forward previous next fullscreen 27

29 Contents 1 Finding gravitational lenses through variability 2 Optical difference imaging 3 Radio difference imaging 4 Related problem: combining different arrays 5 Notation 6 Lazy interferometrists do it in image space 7 CLEAN as maximum likelihood fitting 8 Simultaneous CLEANing 9 Need for difference-clean 10 D-CLEAN procedure 11 An experiment: VLA-like uv coverage, scale 1:4 12 Alternative methods 13 Alternative methods: residual errors 14 Combining different arrays: input 15 Combining different arrays: output 16 Two-channel CLEANing of real data 17 Differencing images of B titlepage introduction summary contents back forward previous next fullscreen 28

30 18 Combined images of B Test with simulated VLBA data 20 Results for simulated data: no noise 21 Results for simulated data: with noise 22 Alternative bases 23 Constant + linear in frequency 24 ATCA observations of Pictor-A 4800 / 4928 MHz 25 Pictor-A: difference MHz 26 Pictor-A: correct for variability with AIPS (4800 MHz) 27 Summary etc 28 Contents titlepage introduction summary contents back forward previous next fullscreen 29