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COMPRESSED SENSING AND RADIO INTERFEROMETRY
M. Jiang∗, J. N. Girard∗, J.-L. Starck∗, S. Corbel∗, C. Tasse†
∗Service d’Astrophysique, CEA Saclay,
Orme des Merisiers
91410 GIF-Sur-YVETTE, France
†GEPI
Observatoire de Paris-Meudon
5, rue place Jules Janssen
92190 Meudon, France
ABSTRACT
Radio interferometric imaging constitutes a strong ill-posed
inverse problem. In addition, the next generation radio tele-
scopes, such as the Low Frequency Array (LOFAR) and the
Square Kilometre Array (SKA), come with an additional
direction-dependent effects which impacts the image restora-
tion. In the compressed sensing framework, we used the
analysis and synthesis formulation of the problem and we
solved it using proximal algorithms. A simple version of our
method has been implemented within the LOFAR imager and
has been validated on simulated and real LOFAR data. It
demonstrated its capability to super-resolve radio sources, to
provide correct photometry of point sources in a large field of
view and image extended emissions with enhanced quality as
compared to classical deconvolution methods. One extension
of our method is to use the temporal information of the data
to build a 2D-1D sparse imager enabling the detection of
transient sources.
Index Terms—sparsity, compressed sensing, interferom-
etry, imaging, transients
1. INTRODUCTION
1.1. Previous work on 2D radio imaging
The application of interferometry in radio astronomy opened
a new window to observe, image and collect spatial infor-
mation about extended radio sources. A radio interferome-
ter gives a limited set of noisy Fourier samples (the visibili-
ties) of the sky [1] inside the field of view of the instrument.
The inverse Fourier Transform (FT) of those visibilities gives
an approximate of the sky under a small field approxima-
tion. In addition, giant interferometers such as LOFAR [2],
are no longer coplanar arrays but are defined in a 3D space
and consequently, the measurements are subject to “direction-
dependent” effects [3] such as the non-coplanarity impacting
wide-field imaging (addressed with W-projection [4]) or vari-
∗We acknowledge the financial support from the UnivEarthS Labex
program of Sorbonne Paris Cit´
e (ANR-10-LABX-0023 and ANR-11-IDEX-
0005-02) and the Physis project (H2020-LEIT-Space-COMPET)
ation of the beam impacting polarization (corrected with A-
projection [5]). Moreover, due to the limited number of base-
lines, not all Fourier regions are sampled and one requires in-
tegrated measurements (in time or frequency) to increase the
accuracy of the recovered sky. The problem of aperture syn-
thesis comes down to process this incomplete Fourier map
either by solving a deconvolution problem using the instru-
mental point spread function (PSF), such as CLEAN and its
derivates (e.g. [6–8]) or by solving the inpainting problem
by recovering missing information in the Fourier plane. In
that scope, many teams have addressed this issue within the
framework of the Compressed Sensing (CS) theory (e.g. [9]
and references therein). In [9], we developed a 2D sparse ra-
dio imager working inside a LOFAR imager which corrects
for A and W effects [5]. We demonstrated: i) better signal re-
construction and lower residuals compared to results obtained
with classical methods, ii) high-dynamic & wide-field imag-
ing capabilities by recovering accurate point-source flux den-
sities ranging from 1 to 104flux unit and iii) the capability to
image super resolved features on real LOFAR data (contain-
ing extended sources) which were consistent with true source
structures.
1.2. Transient radio imaging
Classical imagers work well with time and frequency inte-
gration by assuming a steady sky. However, another class of
radio sources exists, the radio transients, the information of
which are crucial for studies of “local” dynamic systems (e.g.
planetary radio emissions, exoplanets) and (extra)galactic
sources described by high-energy emission processes. Their
detection and imaging are active fields of research in radio
and a lot of work has been devoted to the development of
detection pipelines (e.g. the LOFAR TRAnsient Pipeline –
TRAP [10], based on fast iterative closed-loop performing
calibration / imaging / source detection / catalogue cross-
matching).
When the sky is steady (in the RA/DEC reference sys-
tem), we rely on the time/frequency integration to improve the
sampling of the visibility plane as well as the SNR. However,
being variable and mostly point-like, they will suffer from the
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imaging rate. On the one hand, when a set of successive snap-
shot images should theoretically enables temporal monitoring
of a transient, each snapshot provides poor visibility cover-
age, therefore, images with low signal-to-noise ratio (SNR)
due to large fraction of missing data. On the other hand, long
time integration ensures a good sampling, but it will destroy
the temporal information of the transient by mixing and dilut-
ing “ON” state periods with “OFF” state periods. As a result,
the transient can be detected but with a large uncertainty on
its time localization and temporal profile (or light curve).
Consequently, it is difficult to use classical imagers to de-
tect and image transient source when the temporal variability
of the transient source is unknown. Therefore, a new imager,
still based on CS but handling the temporality of the data, is
required.
2. IMAGE RECONSTRUCTION WITH SPARSITY
CONSTRAINT
2.1. Compressed sensing theory
Compressed Sensing (CS) [11], or compressive sensing is a
sampling and compression theory based on the sparsity of an
observed signal. According to the theory, we could go beyond
the Shannon limit to capture and represent compressible sig-
nals, which can be sparsely represented in a certain dictionary
Φ, at a rate significantly below the Nyquist rate.
The CS theory is a paradigm for finding a probably exact
reconstruction in the case of an undetermined problem such
as the interferometry imaging problem, as we have fewer ob-
servations than unknowns. To achieve the perfect reconstruc-
tion from few samples, the CS theory relies on two principles:
sparsity and incoherence.
- Sparsity: Generally, the CS theory exploits the fact the
signal can be sparse or compressive, i.e. it has an econom-
ical representation in some dictionary Φ. For instance, a
signal x(t)may be not sparse in its time domain, but in
some space, for example, the wavelet space, x(t)can be
decomposed as
x=Φα=
T
i=1
α[i]ϕi,(1)
with T relatively small so that αis a sparse representation
of x(t)in Φ.
- Incoherence: This extends the duality between time and
frequency in the sense that a sparse signal in Φmust be
spread out in the domain where it is acquired. It means
that the sensing vectors must be as different as possible
from the dictionary atoms.
2.2. Inverse problem formulation
The imaging problem constitutes an ill-posed inpainting prob-
lem which can be described in its simplified form as in Eq. (2)
and represented in Fig. 1:
V=MFx +N(2)
with V, the measured visibility vector, Mthe sampling mask
which accounts for the lack of information in the Fourier
space, Fthe FT operator, xthe sky, and Nthe noise. The sky
x, expressed in the direct space, is a real quantity while the
noise Nis complex as it alters both amplitude and phase of
the visibility measurements.
As the number of visibility measurements is limited, the
image reconstruction problem can be regarded as a visibil-
ity inpainting problem, which is an inverse problem. From
the CS perspective, to best reconstruct an image of the sky x
from its visibilities is to use sparse recovery by solving the
following minimization problem derived from Eq. (2):
min ||Φtx||1s.t.||V−MFx||2
2<, (3)
where Vis the measured visibility vector, Mthe mask matrix
to represent the limited measured visibilities, Fthe Fourier
transform operator. The objective function to minimize is of
form ||Φt·||
1where the l1-norm (the sum of coefficients ab-
solute values) is well known to reinforce the sparsity of the
solution.
In Lagrangian form, the convex minimization problem (3)
can be formulated as:
min
x||V−MFx||2
2+k||λΦtx||1(4)
in the analysis framework, where λis in vector form associ-
ated with the Lagrange multiplier which depends implicitly
on of the Eq (3), and the operator denotes the element-
by-element multiplication. However, the use of the l1-norm,
in the proximal framework, involves a soft thresholding step
which has a well-known drawback of giving biased solutions
[12]. This is particularly unsuitable for scientific data analy-
sis, especially for photometry. Thus, the reweighted l1min-
imization proposed in [13] is one way to handle this issue.
In addition, the wanted signal xis always positive, however,
Eq (4) does not ensure the positivity of the solution. There-
fore, the expression of these minimization problems is im-
proved by imposing a positivity constraint and introducing a
weighting vector W, such as:
min
x||V−MFx||2
2+k1||WλΦtx||1+k2i
C
+(x)(5)
for the analysis framework, where i
C
+denotes the indicator
function in the positive set
C
+.
2.3. 2D-1D sparse representation
The data model for the 2D-1D imaging at a given frequency
has two dimensions of spatial information and the remaining
third dimension of the temporal information (corresponding
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Visibilities
Sk
y
FOURIER
V
M
F
x
+
N
u
v
Fig. 1. Formulation of interferometric imaging as an inverse problem. xis the sky brightness and the signal to restore, Fis
the FT, Mis the mask accounting for the available information (represented below in the Fourier plane), Nis the noise which
impact the visibilities measurements and Vis the complex visibility vector measured by the interferometer.
to the sampling by the correlator). According to the CS the-
ory, the corresponding 2D-1D sparse representation Φin our
study is vital to the final reconstruction. In 2D spatial sparse
representation, it was shown that the starlet [12] or curvelet
[14] dictionary were adapted to astronomical sources. Indeed,
recent work of [9] has proved that the 2D reconstruction using
starlet dictionary gives a better angular resolution and pho-
tometry resolution, especially for the extended sources, com-
pared to the classical CLEAN methods. To extend to the 2D-
1D sparse representation with the temporal dimension added,
a direct inspiration would be to use a 3D starlet dictionary
where one dimension will represent the time. However, such
dictionary is not optimal as the temporal information is not
correlated to the spatial information. Thus, we would like
to separate 2D-spatial and 1D-temporal information. There-
fore, as described in [15], an ideal wavelet function would be
ψ(x, y, t)=ψ(xy)(x, y )ψ(t)(t)where the space (xy) and time
(t) are independent, and ψ(xy)is the spatial isotropic undec-
imated wavelet function (the starlet) and ψ(t)is a decimated
wavelet function (the Haar or biorthogonal CDF 9/7 wavelets,
depending on the form of the transient time profile).
2.4. Algorithms
In this section, we describe the algorithm for the reweighted
l1minimization problem (5). We present the reweighted
scheme before focusing on the complete algorithm.
In order to eliminate the false detected sources in our
study, we define the weight function such as
wi,j =f(|αi,j |)=kσj
|αi,j |if |αi,j |≥,
kσj
else ,(6)
to update the weights for each entity iat scale j, where is
a small value constant to avoid the division by zero, σjis
the noise standard deviation, accessible by reliable estimators
such as the MAD (median of the absolute deviation), expected
at scale j.kacts as a detection level in scale j. According
to [13], the reweighted scheme is performed as:
1. Set the iteration count n=0and initialize W(0) =1.
2. Solve the minimization problem (5) yielding a solution
x(n), and α(n)is obtained by α(n)=Φtx(n).
3. Update the weights by the weight function (6).
4. Terminate on convergence or when reaching the maxi-
mum number of iterations Nmax. Otherwise, increment
nand go to step 2.
As for the minimization problem (5) where the 1-norm
regularization term is not differentiable, a variety of proxi-
mal algorithms can be used, such as the Vu splitting method
(VSM) [16], a kind of primal-dual algorithms. The main
idea of the VSM is to convert the minimization problem (5) to
a monotone inclusion for a bounded and existed primal-dual
pair (x, u)which can be solved by a forward-backward algo-
rithm. The summarized algorithm using reweighted scheme is
presented in Algo 1, where the step τand ηare chosen under
the convergence condition of 1−τη||Φ||2>τ||MF||2/2.
More details can be found in [16]. The parameter μin the
Algo 1 is a relaxation parameter used to accelerate the algo-
rithm. If μ=1, we are in the unrelaxed case.
3. EXPERIMENTS
We generated radio data observations by building (32×32×64)
datacubes which respectively represent the two Fourier spa-
tial frequencies and the time. We performed our signal re-
construction in the spatial domain (2D) and in the temporal
domain (1D), to enable an unambiguous detection and local-
ization of a transient source. The mask operator, which also
depends on time, was generated by using a uniform random
antenna distribution of 20 antennas observing at the zenith,
for ∼2 hours (1 slice is therefore ∼2 min). Our time depen-
dent sky model is constituted of a control steady source at
the center of the field and a transient source with a gaussian
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1D wavelet transform
T
each pixel/scale N' 1D temporal scales
Starlet transform
each 2D slice N 2D spatial scales
2D-1D
UV data
Primal/dual splitting
algorithm [Vu, 2013]
Reconstruction
of approximate signal
Computation
of (u,v) residuals
Step 2
WT1D
Step 1
iFFT
+
WT2D
Step 3Step 4
Fig. 2. Organization of the different steps of the reconstruction: i) Each 2D slice of the masked data cube undergoes inverse
FT(giving a dirty image) and a 2D starlet transform giving N scales. ii) the time series of each pixel of each scales undergoes
a 1D temporal wavelet transform. iii) wavelets coefficients are processed as in the Vu algorithm iv) the signal is transformed
back and the residuals are computed with the data.
Algorithm 1: Analysis reconstruction using VSM
Data: Visibility V; Mask M
Result: Reconstructed image x
Initialize
(x(0),u(0) ),W(0) =1,τ >0,η > 0,μ∈]0,1];
for n=0to Nmax −1do
p(n+1) =
Proj +(x(n)−τΦu(n)+τ(MF)∗(V−MFx(n)));
q(n+1) =
(Id −STλW)(u(n)+ηΦT(2p(n+1) −x(n)));
(x(n+1),u(n+1) )=
μ(p(n+1),q(n+1) )+(1−μ)(xn,un);
α(n)=Φtx(n);
Update Wby w(n+1)
i,j =f(|α(n)
i,j |);
end
return x(Nmax)
light curve (FWHM = 10 min located at time slice T=24).
Both sources have the same flux density of 10 arbitrary unit.
We took the FT of the sky and applied the mask cube on the
visibility cube after adding white gaussian noise with vari-
ous magnitude (σ=0.0,0.5,1.0,1.5flux unit). We show
on Fig. 3 (left), characteristic dirty cube slices during the
transient “OFF” state (first row) and its “ON” state (second
row). When the noise level is high, the transient source is
indiscernible from background artifacts.
Figure 3 (right) illustrates the profile of the transient
source from the sky model cube (dash line), the dirty cube
(red line) and the reconstructed cube obtained by the Vu algo-
rithm (green) described in Sect. 2.4. With no additional noise
(but with the sampling noise due to missing data), the CS
reconstruction shows very small bias in flux density (∼10−5
Flux unit relative error) as compared to the dirty profile. As
the gaussian noise level increases, the flux density degrades
in flux while keeping an approximate accurate light curve.
For σ=1.5, the reconstructed flux density has a similar bias
as that of the dirty image but the shape of the light curve
is still accurate in time, allowing the transient localization
and further data processing around this date. The flux of the
steady source (not shown) was also affected by the increasing
level of noise.
4. DISCUSSION & CONCLUSION
CS offers a sound framework for solving the imaging prob-
lem in radio interferometry. In previous studies, such as [9],
We have shown that simple algorithms and implementation
can outperform classical tools used for deconvolution. In
this work, we present an extension of our imager, solving
the inpainting problem on data containing radio transients. It
handles the third dimension being the temporal information
in the data. We implemented the Vu algorithm and used a
combination of 2D and 1D wavelets dictionaries to perform
the reconstruction. We presented preliminary results based
on simulated data cubes containing both steady and transient
sources. The reconstructed transient light curve is accurate
and is relatively robust to an increasing level on noise injected
in the data. A full comparison of the imager performance with
classical deconvolution methods is underway. A “time-agile”
imager, which enables the detection of radio transients, may
have a strong impact in radio astronomy for transient studies
which will be addressed with next generation of interferome-
ters, such as LOFAR and SKA.
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Flux (AU)
Fig. 3. (left) Dirty images from the benchmark data cube containing two point sources: a central steady source (x=15,y=15)
and a transient source (x=24,y=6). First row corresponds to the OFF state (T=10) and second row to the ON state (T=25) at
the maximum of the transient. Each column corresponds to various level of additive gaussian noise with σ=0,0.5,1.0,1.5
arbitrary flux units. (right) Time profiles at the spatial location of the transient source from the sky model (dash line), the dirty
cube (red) and the reconstructed cube (green), for various levels of additive gaussian noise.
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