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Difference imaging and multi-channel Clean
Olaf Wucknitz
Algorithms 2008, Oxford, 1–3 Dezember 2008
Finding gravitational lenses through variability
[ Kochanek et al. (2006) ]
• small field in SDSS: one candidate, no lens [ Lacki et al. (2008) ]
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Optical difference imaging
original image reference image
difference convolution kernel
[A
lard
&Lupto
n(1
998)
]
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Radio difference imaging
UV epoch 1 CLEAN convolve
convolve
crap
CLEAN
UV epoch 1
UV epoch 2
UV epoch 2
UV epoch 1
UV epoch 2
CLEAN convolve
difference
CLEAN
+
−
convolve
convolve
crap
mean
difference
−
−
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Related problem: combining different arrays
UV epoch 1 CLEAN convolve
convolve
crap
CLEAN
UV epoch 1
UV epoch 2
UV epoch 2
UV epoch 1
UV epoch 2
CLEAN convolve
difference
CLEAN
+
−
convolve
convolve
crap
mean
difference
+
+
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Notation
• sky brightness distribution I(l,m) I
• visibilities I(u,v) I
• perfect measurement
? Fourier transform Iµ = ∑j
Aµ jI j
Aµ j = e2π i(l,m) j·(u,v)µ
? vectors I = AI
• residuals R2 = (AI− I)† W(AI− I)
• natural weighting W = diag(σ−2j ) R2 = χ2
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Lazy interferometrists do it in image space
• expand R2 = I†WI+I†A†WAI−2I†A†WI
• define (using w = Tr W)
? dirty beam B =A†WA
w
? dirty map ID =A†WI
w
• residuals derived in image space
R2 = const+w(I†BI−2I†ID) minimum BI = ID︸ ︷︷ ︸R′2
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CLEAN as maximum likelihood fitting
• add components to the model I
• minimize residuals in each step
• empty model plus component I j R′2 = I2j −2I jID j
• optimal flux I j = ID j
• residuals for optimal flux R′2 =−ID2j
optimal position: peak in dirty map
• subtract shifted beam from dirty map, start over
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Simultaneous CLEANing
• two epochs I1 and I2
• two models/maps I1 and I2
• combined residuals
R2 = const+w1 (I1†B1I1−2I1
†ID1)+w2 (I2†B2I2−2I2
†ID2)
• next component at same position in 1 and 2
fluxes independent
optimal fluxes ID1 and ID2
• optimal position maximum of w1 ID21 +w2 ID
22
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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(I1 +I2)
I− = 12(I1−I2)
• dirty maps
ID+ =w1 ID1 +w2 ID2
w1 +w2
ID− =w1 ID1−w2 ID2
w1 +w2
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D-CLEAN procedure
• next component in either I+ or I−
• residuals R2 = const− (w1 +w2)×
{ID
2+ for +
ID2− for −
• subtract according to
(ID+
ID−
)=(
B‖ B×B× B‖
)(I+
I−
)
• I+ and I− not independent
• dirty beams
B‖ =w1 B1 +w2 B2
w1 +w2
B× =w1 B1−w2 B2
w1 +w2
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An experiment: VLA-like uv coverage, scale 1:4
1
2
input convolved CLEAN
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Alternative methods
+
–
CLEAN simultaneous D-CLEAN
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Alternative methods: residual errors
+
–
CLEAN simultaneous D-CLEAN
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Combining different arrays: input
1
2
input convolved CLEAN
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Combining different arrays: output
+
+
combined simultaneous D-CLEAN
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Two-channel CLEANing of real data
• target lens B0218+357
? two bright images
? Einstein ring
• two VLA-A observations
? 1992
? 2003 with Pie Town
• use 1 IF
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Differencing images of B0218+357
CLEAN simultaneous D-CLEAN
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Combined images of B0218+357
CLEAN combined simultaneous D-CLEAN
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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
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Results for simulated data: no noise
difference of Clean maps
difference-Clean
mean Clean map
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Results for simulated data: with noise
difference of Clean maps
difference-Clean
mean Clean map
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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 . . .
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Constant + linear in frequency
• dirty maps ID(n) = ∑ν w(ν)νn ID(ν)
∑ν w(ν)
• dirty beams B(n) = ∑ν w(ν)νn B(ν)∑ν w(ν)
• convolution equation
up to linear order
(ID
(0)
ID(1)
)=
(B(0) B(1)
B(1) B(2)
)(I(0)
I(1)
)
• previous MFS methods only use ID(0) = B(0)I(0) +B(1)I(1)
[ Conway et al. (1990), Sault & Wieringa (1994) ]
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ATCA observations of Pictor-A 4800 / 4928 MHz
[ data from Emil Lenc ]
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Pictor-A: difference 4800 – 4928 MHz
difference of Clean maps difference channel of D-Clean
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Pictor-A: correct for variability with AIPS (4800 MHz)
• five epochs with ATCA (different configurations)
• common model + variable parts
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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
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Contents
1 Finding gravitational lenses through variability2 Optical difference imaging3 Radio difference imaging4 Related problem: combining different arrays5 Notation6 Lazy interferometrists do it in image space7 CLEAN as maximum likelihood fitting8 Simultaneous CLEANing9 Need for difference-CLEAN
10 D-CLEAN procedure11 An experiment: VLA-like uv coverage, scale 1:412 Alternative methods13 Alternative methods: residual errors14 Combining different arrays: input15 Combining different arrays: output16 Two-channel CLEANing of real data17 Differencing images of B0218+357
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18 Combined images of B0218+35719 Test with simulated VLBA data20 Results for simulated data: no noise21 Results for simulated data: with noise22 Alternative bases23 Constant + linear in frequency24 ATCA observations of Pictor-A 4800 / 4928 MHz25 Pictor-A: difference 4800 – 4928 MHz26 Pictor-A: correct for variability with AIPS (4800 MHz)27 Summary etc28 Contents
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