interferometric imaging - utu · interferometric imaging lecture 6.10.2015 41 . deconvolution with...
TRANSCRIPT
Interferometric imaging Tuomas Savolainen Aalto University Metsähovi Radio Observatory Max-Planck-Institut f. Radioastronomie
Image credit: NRAO
Outline
• Visibility function and sky brightness distribution
• Aperture synthesis concepts
• Imaging process in practice, deconvolution
• Image quality and error recognition
Interferometric imaging lecture 6.10.2015
2
Partly adapted from the NRAO Synthesis Imaging School
lectures by David Wilner and Gregory Taylor, and from the
European Radio Interferometry School lectures by Robert Laing
Visibility function ℱ sky brightness distribution
The relationship between the visibility function V(u,v) and the sky
brightness distribution I(l,m) is a 2-D Fourier transform (assuming a
small field of view and neglecting the w-coordinate):
𝑉 𝑢, 𝑣 = 𝐼(𝑙,𝑚) 𝑒−𝑖2𝜋(𝑢𝑙+𝑣𝑚)𝑑𝑙𝑑𝑚
u and v are E-W and N-S spatial frequencies (unit: wavelength)
l and m correspond to E-W and N-S angles on the sky
Interferometric imaging lecture 6.10.2015
3
Some properties of Fourier transforms
Fourier transform:
𝑭(𝒖) = ℱ 𝒇(𝒙) ≡ 𝒇(𝒙)𝒆−𝒊𝟐𝝅𝒖𝒙𝒅𝒙∞
−∞
Inverse Fourier transform:
𝒇(𝒙) = ℱ−𝟏 𝑭 𝒖 ≡ 𝑭(𝒖)𝒆𝒊𝟐𝝅𝒖𝒙𝒅𝒖∞
−∞
Linearity:
ℱ 𝒇 + 𝒈 = ℱ 𝒇 + ℱ(𝒈)
Shifting:
ℱ(𝒇 𝒙 − 𝒙𝟎 ) = 𝑭(𝒖)𝒆𝒊𝟐𝝅𝒖𝒙𝟎
Scaling:
ℱ 𝒇 𝒂𝒙 =𝟏
𝒂𝑭𝒖
𝒂
Convolution:
𝒇 𝒙 ∗ 𝒈 𝒙 ≡ 𝒇 𝒙′ 𝒈 𝒙′ − 𝒙 𝒅𝒙′∞
−∞
ℱ 𝒇 ∗ 𝒈 = ℱ(𝒇)ℱ(𝒈)
Interferometric imaging lecture 6.10.2015
4
About visibilities
Remember that
• Single visibility V(u,v) contains
information from the whole image
I(l,m) – not just some part of it
• Think V(u,v) as follows: 1) sky
brightness distribution is multiplied
by sine and cosine patterns of a
wavelength and orientation
determined by (u,v), and then 2)
these products are summed over.
Interferometric imaging lecture 6.10.2015
5
Image credit: Rick Perley
About visibilities
• Visibility is a complex quantity expressed as (real, imaginary) or
(amplitude, phase)
• V(0,0) is integral over I(l,m)dldm, i.e., the total flux density
• Since I(l,m) is real valued, V(-u,-v)=V*(u,v), i.e., V is Hermitian and
one gets two visibilities for one measurement
Interferometric imaging lecture 6.10.2015
6
ℱ
⇋
amplitude phase
1-D examples of visibility functions
Visibility amplitude V and phase 𝝓 as a function of baseline length
Interferometric imaging lecture 6.10.2015
7
V
V
𝝓
𝝓 d
d d
d
1
1
0
0
Point source
on axis
Point source
off axis
1-D examples of visibility functions
Visibility amplitude V and phase 𝝓 as a function of baseline length
Interferometric imaging lecture 6.10.2015
8
V
V
𝝓
𝝓 d
d d
d
0
0
Two point
sources
Source of size
𝛥𝜑 on axis
1 𝑑 =𝜆
𝛥𝜑
𝑉 → 0 when 𝑑 →𝜆
𝛥𝜑
More 1-D examples
Interferometric imaging lecture 6.10.2015
9
ℱ
⇋
ℱ
⇋
Image credit: Robert Laing
1) Sharp edges in the image
cause ripples in the visibilities
2) FT of a Gaussian is Gaussian.
Small feature in the image gives a
wide feature in visibility space and
vice versa (scaling theorem!).
2-D examples of visibility functions
Interferometric imaging lecture 6.10.2015
10
Image credit: David Wilner
I(l,m) I(l,m)
You have your calibrated visibility data. Now what?
1. Fit simple brightness distribution
models to the visibility data.
Pros:
- Works also with poorly sampled and
noisy data
- Visibilities have well-defined noise
properties
- Resolution better than Rayleigh limit
achievable for high SNR data
Cons:
- Works only with simple source
structures
Interferometric imaging lecture 6.10.2015
11
Image credit: Rick Perley
You have your calibrated visibility data. Now what? 2. Recover an image by using inverse
Fourier transform: 𝑰 𝒍,𝒎 = ℱ−𝟏 𝑽 𝒖, 𝒗
≡ 𝑽 𝒖, 𝒗 𝒆𝒊𝟐𝝅 𝒖𝒍+𝒗𝒎 𝒅𝒖𝒅𝒗∞
−∞
Pros:
- Complex structures can be studied
- No need to assume certain brightness
distribution
Cons:
- Requires well-sampled visibilities
Interferometric imaging lecture 6.10.2015
12
Image credit: Rick Perley
Aperture synthesis concepts
Interferometric imaging lecture 6.10.2015
13
Aperture synthesis
• In principle, inverting 𝑉 𝑢, 𝑣 = 𝐼(𝑙,𝑚) 𝑒−𝑖2𝜋(𝑢𝑙+𝑣𝑚)𝑑𝑙𝑑𝑚 gives
the sky brightness distribution. This however requires measuring
𝑉 𝑢, 𝑣 everywhere in the (u,v) plane. Not possible!
• In reality, we aim to sample 𝑉 𝑢, 𝑣 sufficiently well in order to
constrain 𝐼(𝑙,𝑚). What is sufficiently well? Well, that is a
complicated question… In any case “(u,v) coverage” is one of the
main decisive factors between a high quality image and rubbish.
• To do well, we want:
• Many telescopes, since the number of instantaneous (u,v) samples is
N(N-1), where N is the number of telescopes
• Long synthesis time for changing baseline projections as Earth
rotates. However, be careful if the source is variable!
Interferometric imaging lecture 6.10.2015
14
Examples of (u,v) plane sampling
Interferometric imaging lecture 6.10.2015
15
VLA Visibility sampling for a VLA
snapshot
Examples of (u,v) plane sampling
Interferometric imaging lecture 6.10.2015
16
Image credit: Rick Perley
Examples of (u,v) plane sampling
Interferometric imaging lecture 6.10.2015
17
Image credit: Rick Perley
Sources at
different
declinations
What does (u,v) coverage mean to your image?
• Outer boundary limits the
angular resolution
• Inner boundary limits the
sensitivity to large-scale
emission structure
• Imperfect sampling in-between
limits the image fidelity – there
is information missing!
Interferometric imaging lecture 6.10.2015
18
VLBA – 11h track
Formal description of a discrete sampling of the (u,v) plane Visibility plane is sampled at discrete points given by sampling function:
𝐒 𝒖, 𝒗 = 𝜹(𝒖 − 𝒖𝒌)𝜹(𝒗 − 𝒗𝒌)
𝒌
If we take an inverse FT of the sampled visibility function, we get a “dirty”
image:
𝑰𝑫 𝒍,𝒎 = ℱ−𝟏(𝑺 𝒖, 𝒗 𝑽 𝒖, 𝒗 )
Convolution theorem says:
𝑰𝑫 𝒍,𝒎 = 𝒃(𝒍,𝒎) ∗ 𝑰 𝒍,𝒎
So, 𝑰𝑫 𝒍,𝒎 is a convolution of the true sky brightness distribution and the
interferometer beam:
𝒃 𝒍,𝒎 = ℱ−𝟏(𝑺 𝒖, 𝒗 )
Interferometric imaging lecture 6.10.2015
19
Interferometer beam
Interferometric imaging lecture 6.10.2015
20
ℱ
⇋
(u,v) plane sampling Interferometer beam
Dirty image
Interferometric imaging lecture 6.10.2015
21
= ∗
interferometer beam dirty image source structure
Example: Beam shape with increasing number of (u,v) samples
Interferometric imaging lecture 6.10.2015
22
(u,v) coverage
2 stations
25 min
Dirty image
Example: Beam shape with increasing number of (u,v) samples
Interferometric imaging lecture 6.10.2015
23
(u,v) coverage
4 stations
25 min
Dirty image
Example: Beam shape with increasing number of (u,v) samples
Interferometric imaging lecture 6.10.2015
24
(u,v) coverage
6 stations
25 min
Dirty image
Example: Beam shape with increasing number of (u,v) samples
Interferometric imaging lecture 6.10.2015
25
(u,v) coverage
8 stations
25 min
Dirty image
Example: Beam shape with increasing number of (u,v) samples
Interferometric imaging lecture 6.10.2015
26
(u,v) coverage
10 stations
25 min
Dirty image
Example: Beam shape with increasing number of (u,v) samples
Interferometric imaging lecture 6.10.2015
27
(u,v) coverage
10 stations
5 hours
Dirty image
Example: Beam shape with increasing number of (u,v) samples
Interferometric imaging lecture 6.10.2015
28
(u,v) coverage
10 stations
11 hours
Dirty image
Imaging process in practice
Interferometric imaging lecture 6.10.2015
29
A note about practical Fourier transformation • Fast Fourier Transform (FFT) is
typically used to invert the data,
since it is much faster than direct
FT (𝒪(𝑁2 log2𝑁) vs. 𝒪(𝑁4) ) for
an image of N × 𝑁 pixels and
~𝑁2data points
• FFT requires data points on a
rectangular grid V(u,v) needs to
be interpolated and resampled for
FFT
Interferometric imaging lecture 6.10.2015
30
Decisions about image and pixel size
Pixel size in the image
• Has to satisfy the Nyquist
sampling theorem: ∆𝑙 ≤ 1
2𝑢𝑚𝑎𝑥
and ∆𝑚 ≤ 1
2𝑣𝑚𝑎𝑥
• In practice ~5 pixels across the
beam is a good choice
Image size
• 𝑁∆𝑙 > 1
𝑢𝑚𝑖𝑛 and 𝑁∆𝑚 >
1
𝑣𝑚𝑖𝑛
• For compact arrays, often one
chooses the primary beam size
(beware of aliasing, though!)
• For VLBI, a much smaller image
size is typically chosen
Interferometric imaging lecture 6.10.2015
31
Weighting of the visibility data
Modify the sampling function by a
weighting function W(u,v)
• S(u,v) W(u,v)S(u,v)
• Modifies the interferometer beam
Natural weighting
• 𝑊 𝑢, 𝑣 = 1/𝜎𝑢,𝑣2 in occupied cells,
0 elsewhere
• Maximizes point source sensitivity
• Typically weights more the short
baselines loses in angular
resolution
Interferometric imaging lecture 6.10.2015
32
Weighting of the visibility data
Uniform weighting
• 𝑊 𝑢, 𝑣 = 1/𝜌(𝑢, 𝑣) where 𝜌(𝑢, 𝑣) is
the local density of visibilities in the
(u,v) plane. Depends on selected
“box size”.
• Weights more the long baselines,
enhancing angular resolution
• Degrades point source sensitivity
• Be careful, if sampling is sparse
Other weighting schemes
• Briggs’ robust weighting
Interferometric imaging lecture 6.10.2015
33
Tapering the visibility data
Use a Gaussian function to taper
the sampling function
• 𝑊 𝑢, 𝑣 = 𝑒−(𝑢2+𝑣2
𝑎2)
• Smoothing in the image plane
• Degrades angular resolution
• Can reveal extended emission
Interferometric imaging lecture 6.10.2015
34
Weighting and tapering – VLA snapshot beam examples
Interferometric imaging lecture 6.10.2015
35
Briggs et al. (1999)
Remember this? Despite 11 hours with 10 antennas, the dirty image is useless!
Interferometric imaging lecture 6.10.2015
36
(u,v) coverage
10 stations
11h track
Dirty image
Going beyond the dirty image – deconvolution
• There exists an infinite number of solutions 𝐼𝑠 𝑙, 𝑚 that satisfy 𝐼𝐷 𝑙, 𝑚 =𝑏(𝑙, 𝑚) ∗ 𝐼 𝑙, 𝑚 . This is because there exist functions 𝑍 with 𝑍 ∗ 𝐵 = 0. Therefore, if 𝐼𝑠 𝑙, 𝑚 is a solution, so is 𝐼𝑠 𝑙, 𝑚 + α𝑍(𝑙,𝑚), if no extra
constraints exist. Traditional linear deconvolution methods do not work!
• Typically one uses non-linear deconvolution algorithms to interpolate and
extrapolate the part of the visibility function that was not measured.
• These methods require some form of regularization. This means that we
need some a priori assumptions about the source structure in order to
recover it. Luckily, quite simple assumptions suffice: 1) finite source size,
2) positivity of the true brightness distribution, 3) smoothness of the true
brightness distribution.
Interferometric imaging lecture 6.10.2015
37
Deconvolution with CLEAN algorithm CLEAN is the most widely used algorithm (implementations in
CASA, AIPS, Difmap …)
• Fits and subtracts the interferometer beam iteratively
• Original version by Högbom (1974), several improvements later
• Assumes that source structure can be presented as a sum of a finite
number of point sources
• User can supply a priori information by restricting the area at which
CLEAN is allowed to work (“CLEAN windows”)
• Has problems with diffuse emission (creates “spotty” structures)
• Instabilities: striping around extended sources is a common artefact
Interferometric imaging lecture 6.10.2015
38
Deconvolution with CLEAN algorithm Basic algorithm:
Initialize: residual map = dirty map and list of δ-
components = empty
1. Find the peak in the residual map, identify it
as a point source
Interferometric imaging lecture 6.10.2015
39
Deconvolution with CLEAN algorithm Basic algorithm:
Initialize: residual map = dirty map and list of δ-
components = empty
1. Find the peak in the residual map, identify it
as a point source
2. Subtract this point source, scaled by
loop_gain and convolved with the
interferometer beam, from the residual
image
Interferometric imaging lecture 6.10.2015
40
Deconvolution with CLEAN algorithm Basic algorithm:
Initialize: residual map = dirty map and list of δ-
components = empty
1. Find the peak in the residual map, identify it
as a point source
2. Subtract this point source, scaled by
loop_gain and convolved with the
interferometer beam, from the residual
image
3. Save the position and subtracted flux to the
list of δ-components
Interferometric imaging lecture 6.10.2015
41
Deconvolution with CLEAN algorithm Basic algorithm:
Initialize: residual map = dirty map and list of δ-
components = empty
1. Find the peak in the residual map, identify it
as a point source
2. Subtract this point source, scaled by
loop_gain and convolved with the
interferometer beam, from the residual
image
3. Save the position and subtracted flux to the
list of δ-components
4. If stopping criteria are not met, go to step 1
Interferometric imaging lecture 6.10.2015
42
Deconvolution with CLEAN algorithm • Stopping criteria? Target noise level reached, target SNR reached, or
some maximum number of iterations reached.
• Final step – make “restored” image:
• Make a model image from the final list of δ-components
• Convolve the model image with a “CLEAN beam”, which is typically
a Gaussian fitted to the central peak of the interferometer beam
• Add the last residual map to present the noise
• The resulting image is an estimate of 𝐼 𝑙,𝑚 .
• The units are typically Jy / clean_beam_area.
Interferometric imaging lecture 6.10.2015
43
CLEAN example
Interferometric imaging lecture 6.10.2015
44
CLEAN iterations
= 0
Residual image CLEAN image (log) Dirty image
CLEAN example
Interferometric imaging lecture 6.10.2015
45
CLEAN iterations
= 100
Residual image CLEAN image (log) Dirty image
CLEAN example
Interferometric imaging lecture 6.10.2015
46
CLEAN iterations
= 500
Residual image CLEAN image (log) Dirty image
CLEAN example
Interferometric imaging lecture 6.10.2015
47
CLEAN iterations
= 1500
Residual image CLEAN image (log) Dirty image
Other deconvolution methods
Maximum entropy method
(MEM)
• Assumes that 𝐼 𝑙, 𝑚 is as smooth
as possible
• Minimizes the pixel variance, while
keeping 𝜒2of the fit acceptable
• Works better than CLEAN in
extended and diffuse sources
• Fails to remove sidelobes, if there is
a point source on top of an
extended source
Multi-scale CLEAN
• Promising results for extended
emission
Non-negative least squares
• Directly solves for the (point source)
model parameters assuming
positivity
• If the source is small, a unique
solution may exist
Compressed sensing methods
• Based on sparsity of the data,
implementations minimize 𝐿1-norm
Interferometric imaging lecture 6.10.2015
48
Effect of the antenna reception pattern
The antenna reception pattern
A(l,m) is not uniform
• One needs to correct for the
direction-dependent sensitivity
• Luckily, it is usually simple:
dividing I(l,m) by A(l,m) in the
image plane is enough
Interferometric imaging lecture 6.10.2015
49
Image quality and error recognition
Interferometric imaging lecture 6.10.2015
50
Is my image ok?
The final image depends on…
• Imaging parameters (image and pixel size, visibility weighting, gridding)
• Deconvolution (used algorithm, its parameters, stopping criteria)
• Any errors in calibration and/or editing of the visibilities, i.e. existence of bad data
• Noise
Interferometric imaging lecture 6.10.2015
51
Yes!
Go to do
science.
No!
Find the cause and
redo some of the
calibration /
imaging steps.
How can I tell? I. Identifying bad data in the (u,v) plane
Look at data in the (u,v) plane first:
• Easier to identify outliers in (u,v) plane
– their effect is spread throughout
image plane
• Plot visibilities vs. baseline length or
time – variations should be smooth
• Fraction f of slightly bad data gives
errors at the level of f in the image –
look for gross outliers
• Plot weights – look for large discrepant
values
• Beware of RFI – check spectral plots
Interferometric imaging lecture 6.10.2015
52
One antenna has amplitudes down by 50%.
Increases image noise by a factor of 100!
How can I tell? II. Imaging artefacts
Persistent errors sometimes easier to
find in the image plane:
• For example, a 5% antenna gain calibration
error is difficult to see in the data, but
causes artefacts in the image at 1% level
• Look for unnatural structures in the image:
• Stripes or rings around bright features
• Negative bowls around extended structure
• Spotty on-source structure or short-
wavelength ripples
• Features resembling the interferometer
beam
Interferometric imaging lecture 6.10.2015
53
How can I tell? II. Imaging artefacts Persistent errors sometimes easier to
find in the image plane:
• Are these artefacts additive (constant
over the field) or multiplicative (brighter
around bright sources)?
• Are they symmetric or antisymmetric
around bright sources?
Interferometric imaging lecture 6.10.2015
54
How can I tell? III. Noise in the image • Calculate the expected thermal
noise level in the final image from
the sensitivity of your interferometer
and integration time
• Measure off-source rms noise by
e.g., making a histogram of pixel
fluxes and fitting a Gaussian. Is the
distribution Gaussian?
• Compare expected and measured
rms noise. If you do not reach the
thermal noise level, find out why.
Interferometric imaging lecture 6.10.2015
55
Flux distribution of the image from the
previous slide. The expected rms noise was
0.0001 Jy/beam!
Identifying bad data in the image plane - short burst of bad data at all antennas
No errors (rms 0.11 mJy/beam) 10% amplitude error for all antennas at
one time (rms 2.0 mJy/beam)
Interferometric imaging lecture 6.10.2015
56
Results for a point source using VLA. 13 x 5min observation over 10 hr. Images
shown after editing, calibration and deconvolution.
VLA
beam
pattern
Image credit:
Greg Taylor
Identifying bad data in the image plane - short burst of bad data at one antenna
10 deg phase error for one antenna
at one time (rms 0.49 mJy/beam)
20% amplitude error for one antenna at
one time (rms 0.56 mJy/beam)
Interferometric imaging lecture 6.10.2015
57
anti-symmetric
ridges symmetric ridges
Image credit:
Greg Taylor
Identifying bad data in the image plane - persistent bad data 10 deg phase error for one antenna
all times (rms 2.0 mJy/beam)
20% amp error for one antenna all
times (rms 2.3 mJy/beam)
Interferometric imaging lecture 6.10.2015
58
rings – odd
symmetry rings – even symmetry
Image credit:
Greg Taylor
Other causes of problems: I. Missing short spacings
• If short (u,v) spacings are
missing from the data,
there is no information
about structures larger
than ~ 𝜆 2𝐵𝑚𝑖𝑛
• Negative bowl around an
extended source is often a
sign of unmeasured
power at short (u,v)
spacings
Interferometric imaging lecture 6.10.2015
59
Image credit: Robert Laing
Other causes of problems: II. Deconvolution errors
• Wrongly selected CLEAN
windows
• Too shallow or too deep CLEAN
• Poor choice of weighting
• Poor choice of deconvolution
algorithm for the particular
source structure
Interferometric imaging lecture 6.10.2015
60
Image credit: Robert Laing
Effect of CLEAN windows
Select tight enough CLEAN boxes to avoid
CLEANing noise interacting with sidelobes.
Other causes of problems: III. Miscellaneous • Is the image large enough to
cover all the significant
emission? If not, aliasing may
become a problem.
• Is the pixel size small enough
to properly sample the beam?
• Did my source vary during the
observation? Violates the
basic assumption of aperture
synthesis!
• Problems in wide-field imaging
(not discussed in this lecture) • Time-average smearing
• Frequency-average smearing
• w-term
• Direction-dependent antenna
response
Interferometric imaging lecture 6.10.2015
61
Summary
• Interferometer samples Fourier components of the sky brightness
distribution
• Inverse Fourier transform of the measured visibilities gives an image
• Deconvolution of the image is required due to incomplete sampling of
the visibility function. In many cases self-calibration is essential, too
(see previous lecture on calibration).
• There are an infinite number of brightness distributions that can fit the
observed visibilities. Astronomers must exercise judgement while
imaging interferometric data!
• Still, most of the time things converge and great results can be
obtained!
Interferometric imaging lecture 6.10.2015
62
Reading (and watching) material
Interferometric imaging lecture 6.10.2015
63
• Thompson, Moran & Swenson: “Interferometry and Synthesis in
Radio Astronomy”, Wiley (2004)
• Taylor, G. B., Carilli, C. L. & Perley, R. A.: “Synthesis Imaging in
Radio Astronomy II” ASP Conference Series Vol. 180 (1999) • Contents available online, look in the NASA ADS
• J. A. Zensus, P. J. Diamond, and P. J. Napier: “Very Long Baseline
Interferometry and the VLBA” ASP Conference Series, Vol. 82,
(1995) • Book available online: http://www.cv.nrao.edu/vlbabook/
• NRAO Synthesis imaging school 2014 lectures are online
• https://science.nrao.edu/science/meetings/2014/14th-synthesis-imaging-workshop
Good luck with imaging!
Interferometric imaging lecture 6.10.2015
64
Radio galaxy Cyg A
Image credit: NRAO