# Scalable precision wide-field imaging in radio interferometry:
II. AIRI validated on ASKAP data

A. G. Wilber,1 A. Dabbech,1 M. Terris,1 A. Jackson,2 Y. Wiaux1
1Institute of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh
EH14 4AS, UK
2EPCC, The University of Edinburgh, Edinburgh EH8 9BT, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Accompanying Part I, this sequel delineates a validation of the recently
proposed AI for Regularisation in radio-interferometric Imaging (AIRI)
algorithm on observations from the Australian Square Kilometre Array
Pathfinder (ASKAP). The monochromatic AIRI-ASKAP images showcased in this work
are formed using the same parallelised and automated imaging framework
described in Part I: “uSARA validated on ASKAP data”. Using a Plug-and-Play
approach, AIRI differs from uSARA by substituting a trained denoising deep
neural network (DNN) for the proximal operator in the regularisation step of
the forward-backward algorithm during deconvolution. We build a trained shelf
of DNN denoisers which target the estimated image-dynamic-ranges of our
selected data. Furthermore, we quantify variations of AIRI reconstructions
when selecting the nearest DNN on the shelf versus using a universal DNN with
the highest dynamic range, opening the door to a more complete framework that
not only delivers image estimation but also quantifies epistemic model
uncertainty. We continue our comparative analysis of source structure, diffuse
flux measurements, and spectral index maps of selected target sources as
imaged by AIRI and the algorithms in Part I – uSARA and WSClean. Overall we
see an improvement over uSARA and WSClean in the reconstruction of diffuse
components in AIRI images. The scientific potential delivered by AIRI is
evident in further imaging precision, more accurate spectral index maps, and a
significant acceleration in deconvolution time, whereby AIRI is four times
faster than its sub-iterative sparsity-based counterpart uSARA.

###### keywords:

techniques: interferometric – techniques: image processing – radio continuum:
galaxies – galaxies: clusters: intracluster medium

††pubyear: 2023††pagerange: Scalable precision wide-field imaging in radio
interferometry:  II. AIRI validated on ASKAP data–Scalable precision wide-
field imaging in radio interferometry:  II. AIRI validated on ASKAP data

##  1 Introduction

The superior detection capabilities of modern and upcoming radio arrays –
namely, the Square Kilometre Array (SKA) – necessitate new and improved radio-
interferometric (RI) imaging algorithms that are precise, robust, and scalable
to larger quantities of data, wider fields-of-view, and broader frequency
bandwidths. The “Scalable precision wide-field imaging in radio
interferometry” series aims to showcase a novel imaging framework in action,
as applied to real radio observations from ASKAP. The proposed imaging
framework builds from compressed sensing techniques to reconstruct true signal
from incomplete, noisy data and operates at the interface of optimisation
theory and deep learning. Implemented in MATLAB, the framework is automated,
highly parallelised, and capable of producing wide-field, high-dynamic range,
super-resolved monochromatic intensity images. Two interchangeable image
regularisation denoisers can be “plugged” in as the ‘backward’ step of the
underlying iterative forward-backward (FB) deconvolution structure (Terris et
al., 2022; Dabbech et al., 2022) of the imaging framework, alternating with a
gradient descent ‘forward’ step promoting data fidelity.

The first algorithm, unconstrained Sparsity Averaging Reweighted Analysis
(uSARA), is purely optimisation-based. It leverages the proximal operator of a
state-of-the-art handcrafted sparsity-promoting regularisation function
(Carrillo et al., 2012; Terris et al., 2022) as regularisation denoiser. In
Part I: “uSARA validated on ASKAP data”, uSARA was validated against the
widely-used CLEAN-based imager for RI WSClean (Offringa et al., 2014; Offringa
& Smirnov, 2017). Our experiments with uSARA showed that we were able to take
wide-field, imperfectly calibrated data and create images with exceptional
resolution and enhanced sensitivity. The findings of Part I establish uSARA as
an advanced RI imaging algorithm capable of surpassing the state-of-the-art in
precision and robustness when applied to large-scale, real, and imperfect
data. A remaining caveat with uSARA lies in its computational cost due to the
iterative nature of the proximal operator underlying its image model.

In this sequel, we showcase AIRI (Terris et al., 2022) – the second algorithm
encapsulated in our automated, parallelised imaging framework – which combines
optimisation theory with the power of AI. Building on the recent success of
Plug-and-Play (PnP) approaches in various applications such as image
restoration (Zhang et al., 2021) and magnetic resonance imaging (Ahmad et al.,
2020), AIRI relies on the same FB iterative scheme as uSARA, with the proximal
operator enforcing the image prior model replaced by a learned DNN denoiser.
The learned denoiser, deployed on graphic processing units (GPUs), enables a
significant acceleration of the backward step of the iterative FB algorithm
when compared to the computationally heavy iterative proximal operator
powering uSARA. The speed combined with the learning power of the DNNs –
trained herein as denoisers for high dynamic-range images – make for a
scalable and robust tool in image reconstruction. Although the training of the
DNNs requires important computational time and resources, AIRI denoisers are
pre-trained independently of the data under scrutiny. They can generalise to
any RI data with a simple scaling procedure.

This paper specifically addresses testing AIRI on the same real and
imperfectly calibrated data from ASKAP, used in Part I. To summarise from Part
I, we selected three fields-of-view (FoVs) from ASKAP Early Science and Pilot
Survey observations hosting radio sources of primary interest that exhibit
emission with both complex diffuse and compact filamentary morphology. Our
targets of interest include the merging galaxy cluster system Abell 3391-95
(hosting a candidate radio phoenix; Brüggen et al., 2021), the merging galaxy
cluster SPT-CL J2023-5535 (hosting a radio halo and radio relic; HyeongHan et
al., 2020), the X-shaped radio galaxy PKS 2014-558 (e.g. Cotton et al., 2020),
and the “dancing ghosts,” known collectively as PKS 2130–538 (e.g. Norris et
al., 2021). We refer the reader to Table 1 of Part I for full details on the
ASKAP observations selected for imaging. Further details of the observations
selected, calibration and processing of the data, and the steps involved to
prepare for imaging are elaborated upon in Section 3 of Part I.

The remainder of this article is structured as follows. In Section 2, we
recall the parallelised, automated imaging framework from Part I and expand
upon the application of the AIRI algorithm as well as our approach to
selecting DNN denoisers for imaging. In Section 3, we provide details of the
ASKAP data used in this work and the imaging settings applied for the AIRI
algorithm. Reconstruction results of our selected fields are presented and
compared to the results of Part I in Section 4. The computational performance
of AIRI is studied in 5. Finally, conclusions are made in Section 6.

##  2 Methods

In this section, we briefly recall the RI data model in the context of wide-
field imaging and provide a summary of AIRI, building from the underlying
theory (Terris et al., 2022) and its first application to real RI data
(Dabbech et al., 2022). We also outline the encompassing framework for wide-
field imaging, focusing on the parallelisation of the AI denoiser and the
automated selection of associated parameters. A description of the framework’s
underpinning wide-field parallel measurement operator can be found in Section
2 of Part I.

###  2.1 RI data model

We recall the discrete RI data model in the context of wide-field imaging,
detailed in Part I, whereby the measured visibilities
$\bm{y}\in\mathbb{C}^{M}$ are modelled from a discrete representation of the
sought radio image $\bm{x}\in\mathbb{R}_{+}^{N}$ as follows

$\bm{y}=\bm{\Phi}\bm{x}+\bm{n},$ (1)

where $\bm{n}\in\mathbb{C}^{M}$ is a realisation of random Gaussian noise with
mean zero and standard deviation $\tau>0$, and
$\bm{\Phi}\in\mathbb{C}^{M\times N}$ is the measurement operator encompassing
the Fourier sampling and the so-called $w$-effect, a chirp-like phase
modulation emanating from the non-coplanarity of the array (Cornwell et al.,
2008). Often, a noise-whitening operation is applied to the measured
visibilities to ensure constant standard deviation of the noise (see Appendix
A of Terris et al., 2022, for more details). The operation can be applied in
combination with a weighting scheme compensating for the highly non-uniform
Fourier sampling (_e.g._ Briggs weighting; Briggs, 1995) to enhance the
effective resolution of the observation. Naturally, any transform applied to
the data is injected in the model of the measurement operator. Our imaging
framework is shipped with a parallel and memory-efficient measurement
operator, ensuring scalability to large data sizes (see Dabbech et al., 2022,
for a comprehensive summary).

###  2.2 AIRI algorithm

The recent PnP scheme established that proximal optimisation algorithms, such
as FB, enable not only the use of proximal operators of handcrafted
regularisation operators, but also the injection of learned DNN denoisers,
which define regularisation implicitly (Venkatakrishnan et al., 2013; Romano
et al., 2017). In order to preserve the convergence of the algorithm, and the
interpretability of its solution, the PnP denoiser must typically satisfy a
“firm non-expansiveness” constraint, ensuring that it contracts distances
(Pesquet et al., 2021; Hurault et al., 2022). Learning denoisers from rich
databases (as opposed to handcrafting proximal operators) opens the door to
more powerful regularisation. The speed of DNNs on GPU also offers a
significant acceleration over iterative proximal operators. In this context,
the AIRI imaging algorithm (Terris et al., 2022) is underpinned by the same FB
structure as the uSARA imaging algorithm (see Section 2 in Part I) with the
image update at each iteration alternating between a ‘forward’ step enforcing
data fidelity with respect to (1), and a ‘backward’ denoising step for image
regularisation:

$(\forall k\in\mathbb{N})\qquad\bm{x}^{(k+1)}={{\rm
D}}\left(\bm{x}^{(k)}-\gamma\nabla f(\bm{x}^{(k)})\right).$ (2)

The operator ${{\rm D}}$ denotes a learned DNN denoiser. Considering the
standard data-fidelity function
$f(\bm{x};\bm{y})=1/2\|\bm{y}-\bm{\Phi}\bm{x}\|^{2}_{2}$, where $\|.\|_{2}$
denotes the $\ell_{2}$-norm of its vector argument, the operator $\nabla f$
stands for its the gradient and reads $\nabla
f(\bm{x})=\text{Re}\\{\bm{\Phi}^{\dagger}\bm{\Phi}\\}\bm{x}-\text{Re}\\{\bm{\Phi}^{\dagger}\bm{y}\\}$.
The parameter $\gamma>0$ is a sufficiently small step size.

###  2.3 DNN training and noise level

Following Terris et al. (2022), the AIRI denoisers used in this work were
trained in a supervised approach to remove random Gaussian noise with zero-
mean and standard deviation $\widehat{\sigma}>0$ from noisy input images. The
denoisers rely on a simple denoising convolutional neural network (DnCNN)
architecture and are trained using a rich high-dynamic range database
synthesised from optical astronomy images, with groundtruth images normalised
to have a peak value equal to 1. Importantly, the training loss function is
regularised with an appropriate non-expansiveness term on the denoiser ${{\rm
D}}$.

Given the normalisation of the groundtruth images from the training database,
the DNN’s noise level can be interpreted as the inverse of a target dynamic
range, which can intuitively be adjusted to match the signal-to-noise ratio in
the observed data. Considering $L$ as the spectral norm of
$\text{Re}\\{\bm{\Phi}^{\dagger}\bm{\Phi}\\}$, the standard deviation of the
measurement noise in the image domain can be estimated as
$\sigma=\eta\tau/\sqrt{2L}$ (Thouvenin et al., 2022; Wilber et al., 2022a)
where $\eta>0$ is derived from the data-weighting operator when considered in
imaging and is set to 1 otherwise. Hence, the target dynamic range of the
sought image is given by $\operatorname{max}_{j}\\{{x}_{j}\\}/\sigma$.
However, the peak value of the true image of the sky is not accessible in
practice. We therefore resort to the dirty image defined as
$\overline{\bm{x}}^{\textrm{dirty}}=\beta\text{Re}\\{\bm{\Phi}^{\dagger}\bm{y}\\}\in\mathbb{R}^{N}$,
where $\beta>0$ is a normalisation factor111The factor $\beta$ corresponds to
the peak value of the non-normalised point spread function given by
$\text{Re}\\{\bm{\Phi}^{\dagger}\bm{\Phi}\\}\bm{\delta}$, where
$\bm{\delta}\in\mathbb{R}^{N}$ is the image with one at its centre and zero
otherwise., and approximate the peak value of the sought image by the peak
value of the dirty image
$\kappa=\operatorname{max}_{j}\\{{\overline{x}}^{\textrm{dirty}}_{j}\\}>0$.
The value of $\kappa$ constitutes an upper bound on the estimated peak
value222In our experiments, we observed that $\kappa$ is within one order of
magnitude from the peak value of the reconstructed image.. Consequently,
$\kappa/\sigma$ provides an estimate (in fact, an upper bound) on the target
dynamic range.

In this context, the RI inverse problem (1) is normalised by $\kappa$ to
ensure that the sought image $\bm{x}/\kappa$ satisfies the same normalisation
constraints of the training images, such that pixel values are below 1. The
DNN denoiser is trained for the removal of a zero-mean random Gaussian noise
with standard deviation

$\widehat{\sigma}=\sigma/{\kappa}\textrm{,
where~{}}\sigma=\eta\tau/\sqrt{2L}.$ (3)

The AIRI image estimate is later re-scaled back via multiplication by
$\kappa$.

The apparent dependency of the training noise level on the statistics of the
noise corrupting the RI data and the associated measurement operator raises a
generalisability concern for the AIRI denoisers. However, this can be
circumvented via further scaling of the inverse problem to bring the target
dynamic range of the reconstruction to the inverse of the noise level of an
already available DNN denoiser.

###  2.4 Denoiser selection

The selection of the appropriate denoiser can be conducted via two approaches.
The first approach relies on a pre-trained _shelf of denoisers_ , from which
the appropriate denoiser is selected depending on the target dynamic range of
the reconstruction. A set of denoisers are thus trained, with noise levels
sampled within a wide range of values, reflecting a whole range of dynamic
ranges of interest in modern RI imaging. For each RI dataset, AIRI’s denoiser
is selected from the shelf as the DNN with the nearest noise level
${\sigma}_{s}$ below the inverse of the target dynamic range
$\widehat{\sigma}$. This implies considering a slightly looser image peak
upper bound $\kappa\widehat{\sigma}/{\sigma}_{s}$ for the re-scaling of the
inverse problem (1), thus leading to a heuristic value ${\sigma}_{s}$ for the
training noise level in (3).

The second approach leverages a pre-trained _single (universal) denoiser_ to
be applied for the image formation of any RI dataset. The denoiser is trained
with a very low noise level ${\sigma}_{u}$, tailored for the highest target
dynamic ranges of interest for modern RI imaging. For any RI dataset of
interest, this amounts to considering a possibly much looser image peak upper
bound $\kappa\widehat{\sigma}/{\sigma}_{u}$ for the re-scaling of the inverse
problem (1), systematically leading to a heuristic value ${\sigma}_{u}$ for
the training noise level in (3). This second approach was already shown to be
efficient in the formation of high-quality radio maps when applied to
observations from the MeerKAT telescope (Dabbech et al., 2022).

###  2.5 Denoiser faceting

Owing to their convolutional nature and narrow receptive fields, AIRI’s
denoisers can be applied to facets of the image without causing any faceting-
related artefacts, provided that appropriate facet overlaps are considered.
This feature enables the scalability of AIRI denoisers to large image
dimensions through their parallel application to image facets, as well as
circumventing the memory limitation of the GPUs during inference.

Figure 1: The positioning of the inverse of the target dynamic range,
$\widehat{\sigma}$, associated with the different SPWs of the selected fields
(dashed lines), with respect to the noise level, $\sigma_{s}$, of the pre-
trained shelf of DNN denoisers (black solid lines). From left to right: plots
for the fields SB8275-15, SB9351-12, and SB9442-35, respectively. Specifically
to SB9442-35, we also add the inverse of the target dynamic range associated
with the full-band imaging experiment.

##  3 Data, imaging, and analysis

In validating the AIRI algorithm, we consider the same RI datasets imaged in
Part I. These data consist of three individual beam observations from ASKAP
Early Science and Evolutionary Map of the Universe Pilot survey (EMU-PS Norris
et al., 2021) scheduling blocks (SBs): SB8275-15, SB9351-12, and SB9442-35.
The measurement sets containing calibrated visibilities of these selected
observations were produced by ASKAPsoft (Hotan et al., 2021) and obtained from
the CSIRO ASKAP Science Data Archive (CASDA; Chapman et al., 2017. For our
imaging purposes, each of these observations, with bandwidths of 288 MHz
(ranging between [800,1158] MHz), was split into eight spectral windows
(SPWs). For the first two fields (SB8275-15 and SB9351-12), the resulting sub-
band data were imaged separately to form monochromatic images with dimensions
of $5500\times 5500$ pixels and a cell-size of 2.2 arcsec. For the third field
(SB9442-35), sub-band images and a single full-band image with dimensions of
$4096\times 4096$ pixels and cell-sizes of 2.2 arcsec were formed. The full-
band data of SB9442-35 were reconstructed into a monochromatic full-band image
with the aim to increase both the dimensionality and the sensitivity of the
data. Full details of the data and the imaging settings can be found in Tables
1 & 2 of Part I.

Specific to AIRI, in each imaging experiment we enabled the image-faceting
functionality of the denoiser by splitting the image into four facets of equal
dimensions. Faceting was found to be necessary for satisfying memory
requirements. In the selection of the appropriate denoiser, we first
determined the values of the inverse of the target dynamic range,
$\widehat{\sigma}$, of all formed sub-band and full-band images, following
(3). We primarily opted for the pre-trained shelf strategy for all AIRI
reconstructions, where the considered noise levels of the pre-trained
denoisers are $\sigma_{s}=[2,4,8]\times 10^{-5}$. We also investigated the
pre-trained universal denoiser strategy for the field SB9351-12. In this case,
the universal denoiser is chosen from the pre-trained shelf as the denoiser
with the lowest noise level (equivalently, the highest dynamic range), that is
$\sigma_{u}=2\times 10^{-5}$.

We note that training under the firm non-expansiveness constraint is highly
challenging. While Terris et al. (2022) demonstrated in simulation that the
AIRI training approach leads to a robust way to ensure convergence of the PnP
algorithms, we acknowledge that, when used for real data and at large image
sizes and dynamic ranges such as those of interest here, some denoisers lead
to algorithm instability, ultimately requiring further training. This
phenomenon was also witnessed by Dabbech et al. (2022).

AIRI experiments were run on Cirrus333http://www.cirrus.ac.uk, a UK Tier2
high-performance computing (HPC) service, and utilised its GPU compute nodes.
A Cirrus GPU node is comprised of 4 GPUs and 40 CPU cores with 384 GB of
shared memory. Each imaging experiment of AIRI is launched on one to five GPU
nodes, depending on the memory requirements, whereby CPU cores, allocated
dynamically, were utilised for the forward step, more precisely the
application of the measurement operator, and GPUs were exploited in the
parallel application of the denoising DNN on facets of the image.

For a quantitative assessment of AIRI’s performance, we focus on the same
primary sources of interest as presented in Part I and analyse their
associated flux measurements and spectral index maps. Results are compared to
those obtained with uSARA and WSClean for all spectral windows of each imaged
field. The AIRI flux measurements were computed from hand-drawn regions
(generated using the visualisation software SAOImageDS9; Joye & Mandel, 2003)
slightly different from those considered in the uSARA and WSClean images, to
better match the morphology of recovered emission. Spectral index maps
inferred from the AIRI-ASKAP sub-band images were also obtained following the
same procedure described in Part I. Likewise, only the first six sub-band
images were used to generate the spectral index maps due to the limited
diffuse signal recovered in the final two sub-bands. Similarly to uSARA,
blurring with a circular Gaussian beam of 5 arcsec was applied to AIRI-ASKAP
sub-band images to smooth the source structure before flux measurements were
fitted to the spectral curve: $S_{\nu}\propto\nu^{-\alpha}$, where $S_{\nu}$
is the flux density for a given beam area at a given frequency $\nu$ and
$\alpha>0$ is the spectral index. In presenting our AIRI images, we also make
use of optical images from the first data release of the Dark Energy Survey
(DES; Abbott et al., 2018).

We performed an additional assessment towards the quantification of epistemic
model uncertainty by comparing variations of AIRI (_i.e._ using different DNN
denoisers) when imaging each spectral window of the field SB9351-12. More
details on this analysis can be found in Section 4.4.

Figure 2:  SB8275-15 – AIRI: Full FoV image covering the merging cluster system Abell 3391-95, at the first sub-band (SPW:1, centred at 887 MHz). For visual comparison with WSClean and uSARA, refer to their respective Figures 1 and 2 from Part I. This monochromatic image is an AIRI model with a pixel resolution of $2.2\times 2.2$ arcsec. Panel (a) centred on the FR I radio galaxy in A3391; panel (b) centred on cluster member FR II radio galaxy; (c) panels centred on FR I and diffuse source in A3395. Middle (c) panel: r-band optical image from DES overlaid with AIRI model image, demarcated by blue contours at levels $\\{2^{n}\\}_{0\leq n\leq 10}~{}\mu$Jy pixel-1. Rightmost (c) panel: spectral index map obtained with the first six sub-band images of AIRI after smoothing with a common circular Gaussian beam of 5 arcsec. In Wilber et al. (2022b) are provided all sub-band images combined into the GIF ‘SB8275-15_AIRI’, and the spectral index map of Abell 3395 obtained with AIRI, uSARA, and WSClean in the GIF ‘SpectralIndexMap_Abell_$3395$’, together with a colour blind-friendly version in the GIF ‘SpectralIndexMap_Abell_$3395$_colorblind_friendly’.  A3395 Phoenix | $S_{887}$ | $S_{923}$ | $S_{959}$ | $S_{995}$ | $S_{1031}$ | $S_{1067}$ | $S_{1103}$ | $S_{1139}$
---|---|---|---|---|---|---|---|---
AIRI model | 25.3 | 27.0 | 17.9 | 8.5 | 11.3 | 12.1 | 9.7 | 1.6
Table 1: Integrated flux density values in [mJy] of the diffuse phoenix source
in Abell 3395 for each SPW imaged with AIRI. Central frequency of each SPW is
listed in MHz. See Table 3 in Part I for uSARA and WSClean flux measurements
of the diffuse phoenix source.

##  4 Results

In this section, we showcase high-resolution high-fidelity images of our three
selected fields produced by the AIRI algorithm encapsulated in our
parallelised and automated imaging framework and investigate two approaches
for the selection of the DNN denoiser, as described in Section 2.3. Figure 1
illustrates the positioning of the inverse of the target dynamic range
$\widehat{\sigma}$ of each imaged spectral window for each field with respect
to the noise levels of the learned DNNs.

Select AIRI-ASKAP images are displayed in Figures 2–4, in a format identical
to the uSARA-ASKAP and WSClean images presented in Part I. The AIRI-ASKAP
figures consist of full FoV images with zoomed-in views focused on complex
radio emission of interest, and their associated optical images and spectral
index maps. In Wilber et al. (2022b), we provide all sub-band AIRI-ASKAP
images of the three selected fields (as FITS files) and combine them into GIF
files to better show how emission and source morphology changes over the full
frequency band. Throughout this section, we refer the reader to Part I for
specific comparisons of our imaging results with the uSARA and WSClean
reconstructions from Part I.

Upon visual inspection, our monochromatic AIRI-ASKAP images of all three
fields capture more extended structure than seen in the pure-optimisation
counterpart uSARA-ASKAP images. In particular, the faintest structures of our
diffuse targets of interest appear more pronounced and defined than they did
in the uSARA-ASKAP images. However, the intensity of the faintest point
sources, as reconstructed by AIRI, seems to be diminished. In the following
subsections, we focus on each ASKAP field and address these differences by
examining specific sources of interest. We present detailed comparisons of
source morphology, flux density measurements, and spectral index maps between
the three imaging algorithms. In an experiment toward uncertainty
quantification of AIRI denoisers, we also showcase AIRI reconstructions made
via the universal denoiser approach.

###  4.1 First field: SB8275-15

This field contains the massive, merging galaxy clusters Abell 3391 (in the
north) and Abell 3395 (in the south). The cluster pair is connected by a warm
gas bridge, recently discovered in eROSITA X-ray observations (Reiprich et
al., 2021). In Figure 2, we present our AIRI image of this full imaged FoV
(3.36∘) of the first spectral window (SPW:1). The figure includes zoomed-in
views of the FR-I in Abell 3391 (a: top right panel), a FR-II cluster member
in the east (b: middle right panel), and multiple sources in Abell 3395 (c:
bottom panels). The bent-tail FR I radio galaxies at the centres of Abell 3391
and Abell 3395 (see Table 1 in Part I for source names) are reconstructed with
similar brightness and resolution when compared to our uSARA image from Part I
(the peak pixel flux of the FRI in Abell 3391 is 20 mJy in both the AIRI and
uSARA images). The highly resolved detail of the ‘braiding’ in these FRI jets,
not resolved in WSClean images, is present in both the AIRI and uSARA images.
However, the radio galaxies as captured by AIRI exhibit more blended edges
than seen in the uSARA image. In addition, the ring-like artefacts emanating
from these bright FRI sources take on a much more extended structure in our
AIRI image – their appearance is dimmed and they propagate further out from
their point of origin. Mainly, there is a noticeable difference in the diffuse
structure recovered in the candidate phoenix of Abell 3395 and the background
FR-II radio galaxy (c and b panels, respectively, in Figure 2).

Figure 3: SB9351-12 – AIRI: Full FoV image covering the merging cluster SPT2023 and the X-shaped radio galaxy PKS 2014-55, at the first sub-band (SPW:1, centred at 817 MHz). For visual comparison with WSClean and uSARA, refer to their respective Figures 3 and 4 from Part I. This monochromatic image is an AIRI model with a pixel resolution of $2.2\times 2.2$ arcsec. Panel (a) centred on the merging galaxy cluster SPT2023; panel (b) centred on a field containing compact and point sources; (c) panels centred on the X-shaped radio galaxy PKS 2014-55. Middle (c) panel: r-band optical image from DES overlaid with the AIRI model image, demarcated by blue contours at the levels $\\{2^{n}\\}_{0\leq n\leq 10}~{}\mu$Jy pixel-1. Rightmost (c) panel: spectral index map obtained with the first six sub-band images of AIRI after smoothing with a common circular Gaussian beam of 5 arcsec. In Wilber et al. (2022b) are provided all sub-band images combined into the GIF ‘SB9351-12_AIRI’, and the spectral index map of the X-shaped radio galaxy obtained with AIRI, uSARA, and WSClean in the GIF ‘SpectralIndexMap_PKS_2014_55’, together with a colour blind-friendly version in the GIF ‘SpectralIndexMap_PKS_2014_55_colorblind_friendly’.  X-Shaped RG | $S_{817~{}{\rm MHz}}$ | $S_{853~{}{\rm MHz}}$ | $S_{889~{}{\rm MHz}}$ | $S_{925~{}{\rm MHz}}$ | $S_{961~{}{\rm MHz}}$ | $S_{997~{}{\rm MHz}}$ | $S_{1033~{}{\rm MHz}}$ | $S_{1069~{}{\rm MHz}}$
---|---|---|---|---|---|---|---|---
AIRI model | 678.7 | 580.0 | 485.0 | 427.7 | 436.2 | 352.7 | 302.3 | 190.5
Table 2: Integrated flux density values in [mJy] of the X-shaped radio galaxy PKS 2014-55 for each SPW imaged with AIRI. The listed flux densities are totals from summing the flux densities measured in regions mapping the east wing, the west wing, and the core. See Table 4 in Part I for uSARA and WSClean flux measurements of the X-shaped radio galaxy. SPT2023 Relic | $S_{817~{}{\rm MHz}}$ | $S_{853~{}{\rm MHz}}$ | $S_{889~{}{\rm MHz}}$ | $S_{925~{}{\rm MHz}}$ | $S_{961~{}{\rm MHz}}$ | $S_{997~{}{\rm MHz}}$ | $S_{1033~{}{\rm MHz}}$ | $S_{1069~{}{\rm MHz}}$
---|---|---|---|---|---|---|---|---
AIRI model | 4.3 | 3.2 | 1.8 | 2.0 | 2.8 | 2.3 | 1.7 | 0.5
Table 3: Integrated flux density values in [mJy] of the radio relic in SPT2023
for each SPW imaged with AIRI. See Table 5 in Part I for uSARA and WSClean
flux measurements.

####  4.1.1 Abell 3395

In the middle panel (c) of Figure 2, contours mapping the AIRI recovered
emission above $1~{}\mu$Jy pixel-1 are overlaid on an r-band optical image
from DES. In comparison with uSARA, the recovered emission of the candidate
phoenix by AIRI seems to grow in size. Although appearing slightly fainter and
smoother, more extended structure is revealed in the AIRI reconstruction of
the Abell 3395 phoenix, as the north-west arm now bridges the dim core and the
compact sources at the north-west edge of the cluster. The structure of the
recovered emission also changes from one spectral window to the next, with
noticeable fading as the frequency increases (see associated GIF in Wilber et
al., 2022b). Flux density measurements of the phoenix from the sub-band AIRI-
ASKAP images are provided in Table 1. In comparing the flux density
measurements to those taken from uSARA and WSClean images, we see some slight
variations across the spectral windows with the exception of the highest
frequency where there is clearly less flux recovered in the AIRI image.

The most exciting result is perhaps the improvement of the spectral index map
of the phoenix obtained with AIRI when compared to uSARA (see rightmost panel
(c) of Figure 2). Since more diffuse structure is recovered by AIRI overall,
even at the higher frequencies, the spectral index map has more coverage over
the full source morphology. This coverage aids in source classification,
enabling the identification of a trend in the spectral index as the emission
shifts from the dim core to the north-west and south-west arms. There is
clearly more steepness of the spectra as compared to the uSARA map shown in
Part I. The dim core recovered by AIRI has a steeper index ($2.1\leq\alpha\leq
2.8$), and the north-west arm shows a sharp, rather than gradual, drop-off
from the core. There still exists the ring of flatter emission around the
core, matching the results from uSARA and WSClean. Our AIRI results are in
line with the hypothesis that this source is no longer receiving a fresh
injection from an active nucleus and that the surrounding emission may be
undergoing some gentle re-energisation, which in turn is causing brightening
and flattening of old and faded AGN emission. Interestingly, both the FR-I to
the east and the compact source to the north-west exhibit consistent spectral
behaviour between uSARA and AIRI.

###  4.2 Second field: SB9351-12

The second selected field covers the merging galaxy cluster SPT-CL J2023-5535
(hereafter SPT2023) and the X-shaped radio galaxy PKS 2014-55. As stated in
Part I, two recent studies have been separately published for these sources of
interest: HyeongHan et al. (2020) confirmed the detection of a radio halo and
radio relic in SPT2023 with the same data used in this work and Cotton et al.
(2020) used MeerKAT observations to generate total intensity, polarisation,
B-field maps, and spectral index maps of the X-shaped radio galaxy. In Figure
3, we present our AIRI image of the full FoV (3.36∘) of the first spectral
window (SPW:1) of SB89351-12. The figure includes zoomed-in views on the
merging cluster SPT2023 (a: upper right panel), a field of compact and point-
like sources (b: middle right panel), and the X-shaped radio galaxy PKS
2014-55 (c: bottom panels).

In comparison with the uSARA image (Figure 3 of Part I), there is an
undeniable improvement in the recovery of faint emission within the zoomed-in
views seen in the AIRI image. The diffuse emission stretching east-to-west in
the WSClean image of SPT2023, which was not recovered by uSARA, clearly
emerges in the AIRI image. Similarly, faint point sources seen in the WSClean
image but not in the uSARA image, are captured by AIRI, though appearing to be
somewhat fainter and smoother (see the panel (b) of Figure 3). Finally, the
calibration artefacts are still noticeable at the southern edge of the
pointing, taking the form of ring-type artefacts emanating from the bright
quasar RX J2024.3-5723 and propagating radially up to 1 deg. Compared to
uSARA, these artefacts are generally fainter but extend further.

Figure 4: SB9442-35 – AIRI: Full FoV image covering PKS 2130-538, formed using the full-band data (centred at 943 MHz). For visual comparison with WSClean and uSARA, refer to their respective Figures 5 and 6 from Part I. This monochromatic image is an AIRI model with a pixel resolution of $2.2\times 2.2$ arcsec. Panel (a) centred on a field containing extended and point-like radio galaxies; panel (c) centred on the star-forming galaxy NGC 7090; (b) panels centred on “the dancing ghosts” (PKS 2130-538). Middle (b) panel: image made with only the first sub-band of data (SPW:1, centered at 817), shown for a comparison of sensitivity. Rightmost (b) panel: spectral index map made with the first six sub-band images of AIRI after smoothing with a common circular Gaussian beam of 5 arcsec. In Wilber et al. (2022b) are provided all sub-band images combined into the GIF ‘SB9442-35_AIRI’, and the spectral index map of “the dancing ghosts” obtained with AIRI, uSARA, and WSClean in the GIF ‘SpectralIndexMap_PKS_2130_538’, together with a colour blind-friendly version in the GIF ‘SpectralIndexMap_PKS_2130_538_colorblind_friendly’.  Dancing Ghosts | $S_{\rm fullband-943~{}{\rm MHz}}$ | $S_{817~{}{\rm MHz}}$ | $S_{853~{}{\rm MHz}}$ | $S_{889~{}{\rm MHz}}$ | $S_{925~{}{\rm MHz}}$ | $S_{961~{}{\rm MHz}}$ | $S_{997~{}{\rm MHz}}$ | $S_{1033~{}{\rm MHz}}$ | $S_{1069~{}{\rm MHz}}$
---|---|---|---|---|---|---|---|---|---
AIRI model | 116.7 | 128.3 | 123.8 | 118.1 | 113.2 | 109.2 | 105.7 | 102.6 | 97.7
Table 4: Integrated flux density values of “the dancing ghosts” PKS 2130-53
for each SPW imaged with AIRI. See Table 6 in Part I for uSARA and WSClean
flux measurements of “the dancing ghosts”.

####  4.2.1 X-shaped Radio Galaxy

In the middle panel (c) of Figure 3, we overlay AIRI-ASKAP emission of the
X-shaped radio galaxy as contours on an r-band optical map from DES. Compared
to uSARA, the X-shape radio galaxy as reconstructed by AIRI appears to have a
greater extent of diffuse emission, though with a noticeable loss in the
resolution of the compact structure within the lobes. This behaviour is
consistent across all the sub-band images of AIRI, where smoother and more
diffuse edges are observed. Table 2 reports the measured flux densities per
spectral window for the X-shaped radio galaxy. AIRI flux measurements are
consistently lower than the uSARA flux measurements for SPW:1 through SPW:5,
then increase for SPW:6 and SPW:7, and drop lower again in spectral window 8.

A spectral index map of the X-shaped radio galaxy is shown in the rightmost
(c) panel of Figure 3. There is an incredible improvement in the coverage of
the AIRI spectral index map compared to uSARA, thanks to the diffuse flux
consistently recovered at the edges of the lobes, even at the higher
frequencies, with AIRI. The – most likely – artificial steepening at the edges
of the lobes seen in the uSARA spectral index map is now corrected in the AIRI
map. Nonetheless, AIRI recovers slightly steeper spectra on the borders of the
lobes when compared to the WSClean spectral index map. Owing to the superb
resolution achieved by AIRI, turbulent activity can be traced where plasma in
the lobes exhibits a flatter spectral index. The south-east leg of the east
wing shows a spectral index of $1.4\leq\alpha\leq 2.1$, which is much flatter
than the emission in the west wing (with an average spectral index of about
$\alpha=3$. The furthest north-west portion of the west wing exhibits an
ultra-steep spectrum $3.5\leq\alpha\leq 5$. Wide-band deconvolution is
necessary to confirm these ultra-steep values.

####  4.2.2 SPT-CL J2023-5535

In Part I, our monochromatic WSClean image shows the radio halo in SPT2023 as
an increase in noise at the cluster centre, but our uSARA image did not show
any diffuse structure resembling a radio halo. In Figure 3, the panel (a)
focuses on the diffuse emission present in SPT2023. Here, our AIRI image does
in fact recover the diffuse structure of the radio halo – elongating from the
western relic towards the east. Across the sub-bands the radio halo is most
clearly detected in AIRI images SPW:1 and SPW:2. It is quite remarkable that
the radio halo is detected in these narrow, sub-band AIRI images since the
full 288 MHz bandwidth (in the form of a wideband WSClean image) was used to
detect and measure the radio halo’s signal in HyeongHan et al. (2020). We also
find that the SPT2023 radio relic has a smoother, wider, and fainter
morphology when compared to uSARA. AIRI flux measurements of the relic for
SPW:6 and 7 are slightly higher than uSARA flux measurements, but lower for
all other spectral windows, as reported in Table 3.

###  4.3 Third field: SB9442-35

This final selected field is centred on the complex radio source PKS 2130-538,
nicknamed “the dancing ghosts,” owing to its peculiar and mirrored ghost-like
shape. Two radio lobes, bridged by arching jets from the primary AGN host,
extend southwards and blend into each other. A secondary AGN in the south-east
produces a similar arched jet with a bent-tail that curls back around to the
eastern primary lobe. With the original images generated from the ASKAP
Evolutionary Map of the Universe Survey (EMU; Norris et al., 2011), Norris et
al. (2021) mention that the interaction between the primary and secondary AGN
is unclear. With our super-resolved uSARA images, presented in Part I, we have
been able to distinguish a clear physical separation between the secondary
curling jet and the primary western lobe. Nonetheless, this strange source
offers an interesting case study of turbulent dynamics in bent-tail radio
galaxies.

For this field, we produced eight sub-band images as well as a monochromatic
full-band image with AIRI. In Figure 4, we present the AIRI image of a $\sim
2.5^{\circ}$ FoV formed using the full-band (288 MHz) data of SB89442-35. This
figure includes zoomed-in views on a field containing an extended radio galaxy
and points sources (a: upper right panel), the star-forming galaxy NGC 7090
(c: middle right panel), and the “dancing ghosts” (b: bottom panels). The
bottom panels include a view of PKS 2130-538 from the full-band image
(leftmost) and the first sub-band image SPW:1, covering 36 MHz (middle) for a
visual comparison of the sensitivity in both imaging settings. Also included
in the bottom rightmost panel is a spectral index map of PKS 2130-538,
generated with the first six sub-band images.

SPW:2 | SPW:3 | SPW:4 | SPW:5 | SPW:6 | SPW:7 | SPW:8
---|---|---|---|---|---|---
99.6% | 99.7% | 99.7% | 99.3% | 99.5% | 99.2% | 98.9%
Table 5:  The percentage of the pixels in the error maps between
$\sigma_{u}$-AIRI and $\sigma_{s}$-AIRI images with values below the estimated
standard deviation of the noise in the image domain, $\sigma$, for each sub-
band image of the field SB9351-12.  Figure 5: Comparison of uSARA and AIRI
reconstructions of the seventh sub-band data (SPW:7) of the field SB9351-12,
focusing on the X-shaped radio galaxy (top) and the galaxy cluster SPT2023
(bottom). From left to right: uSARA reconstruction, $\sigma_{s}$-AIRI
reconstruction (using the shelf-appropriate DNN denoiser), $\sigma_{u}$-AIRI
reconstruction (using a universal DNN denoiser), and the error map between
$\sigma_{s}$-AIRI and $\sigma_{u}$-AIRI, normalised by the estimated standard
deviation of the noise in the image domain, $\sigma$.

####  4.3.1 The Dancing Ghosts

The separation between the curling secondary jet and the eastern lobe of the
primary AGN in PKS 2130-538 is less distinct in our AIRI image when compared
to the uSARA image from Part I. However, there is a more drastic difference in
the improvement of resolution when moving to the full-band with AIRI. Our AIRI
sub-band image shows a much smoother structure than the full-band image,
particularly noticeable when focusing on the sharpness of the jet bridge
linking the two lobes from the primary AGN. Faint point sources emerge more
clearly in the AIRI full-band image, with a slight improvement over its uSARA
counterpart.

The filamentary emission extending from the eastern lobe (possibly a
synchrotron thread similar to those discovered in Ramatsoku et al., 2020)
appears slightly fainter and more diffuse in the AIRI image, with overall
steeper spectra ($3.5\leq\alpha\leq 6$) than seen in the uSARA maps. It is
interesting that this eastern extending filament has such an ultra-steep
spectrum in the AIRI map since the spectral index over the rest of the source
morphology remains similar between the uSARA and AIRI maps. When comparing the
flux density measurements in Table 4 to the corresponding measurements in Part
I, there appears to be a clear consistency between uSARA and AIRI and WSlean,
with AIRI recovering slightly less flux at the higher frequencies. This source
arguably has the most consistency in flux density measurements across the
three different imaging methods, perhaps due to the overall flatter spectral
index of the source and the lack of strong calibration artefacts in this
field.

###  4.4 Universal Denoiser and model uncertainty

In an experiment toward modelling epistemic uncertainty, we measure
differences between AIRI reconstructions of the field SB9351-12 produced by
the two denoiser selection strategies proposed in Section 2.4, namely the
denoiser shelf and universal denoiser strategies. The two approaches differ by
the denoiser instance used. The AIRI reconstructions leveraging a pre-trained
shelf of denoisers were presented in Section 4.2. Here we also present the
reconstruction results when utilising the universal denoiser approach, and
study the robustness of AIRI reconstructions to denoiser (_i.e._ model)
variations. We recall that the considered universal denoiser corresponds to
the lowest training noise level on the shelf, $\sigma_{u}=2\times 10^{-5}$.
Under this consideration, we note that the first sub-band (SPW:1) is not
included in this analysis since its shelf-appropriate denoiser corresponds to
the universal denoiser (_i.e._ $\sigma_{s}=\sigma_{u}$); therefore, only
spectral windows 2–8 were re-imaged.

Focusing on our target sources of interest in this field – namely, the
X-shaped galaxy and the merging galaxy cluster SPT2023 – reconstruction
results of SPW:7, obtained using uSARA and the two AIRI denoiser selection
strategies (the nearest shelf-appropriate DNN with $\sigma_{s}=8\times
10^{-5}$ and the universal DNN denoiser with $\sigma_{u}=2\times 10^{-5}$) are
showcased in Figure 5. The seventh spectral window is chosen for a visual
comparison due to its high signal and lower dynamic range when compared to
other spectral windows. All other spectral windows imaged via the AIRI
universal denoiser strategy are provided as FITS files and combined into a GIF
for easier viewing in Wilber et al. (2022b). In what follows, we refer to the
AIRI reconstructions generated via their associated denoiser strategy as
$\sigma_{u}$-AIRI (universal approach) or $\sigma_{s}$-AIRI (shelf approach).

The most evident visual difference in the AIRI reconstructions – particularly
noticeable for the X-shaped galaxy – is in the smoothness of the emission
recovered in $\sigma_{s}$-AIRI and the arguably more detailed emission
recovered in $\sigma_{u}$-AIRI which targets higher dynamic ranges. Both AIRI
images recover significantly more diffuse emission than uSARA, yet with a
slight compromise in resolution, as can be seen in the radio galaxies of the
merging galaxy cluster SPT2023.

We examine the absolute difference between the $\sigma_{u}$-AIRI and
$\sigma_{s}$-AIRI images. The resulting error map444A hard thresholding
operation is applied to the error map, keeping only values above $10^{-8}$. of
SPW:7 is displayed in Figure 5, following a normalisation by the associated
noise estimate in the image domain $\sigma$ (see Eq. 3). From the full-FoV
error maps associated with the sub-band images, we conduct a numerical
analysis of AIRI reconstructions on the basis of the percentage of the pixels
with values above $\sigma$. As shown in Table 5, for each sub-band error map,
we find that $0.8-1.1\%$ of the pixels are of values higher than $\sigma$.
This very small percentage corresponds mainly to pixel intensities within the
brightest point-like sources. Because the integrated flux densities of these
bright sources recovered by both AIRI reconstructions are very close, the
discrepancy most likely arises from differences in individual pixel
intensities which are slightly spatially offset from each other.

When comparing the uSARA reconstruction to each of the AIRI reconstructions,
we find that the percentage of the pixels with absolute difference above
$\sigma$ is slightly more significant, at $2.5\%$ for SPW:7. These findings
suggest that uSARA and AIRI reconstructions are very similar with respect to
each other and that AIRI is highly robust to variations of the denoiser
instance used (highlighting a small epistemic uncertainty of the learned
denoiser approach). This also validates the simpler high dynamic-range
universal denoiser strategy for AIRI, as opposed to the denoiser shelf
approach.

##  5 Computational performance

For all AIRI imaging experiments performed in this work, the decomposition of
the measurement operator and consequently the number of CPU cores allocated to
enforce data fidelity are identical to uSARA (see Tables 7–9 in Part I for
further details). While uSARA was deployed on the CPU nodes of Cirrus, AIRI
was run on its GPU nodes comprising both CPU cores and GPUs (see Section 3 for
details of the compute nodes). In this setting, the computing time of the
forward step in AIRI was found to be up to 1.2 faster than its counterpart in
uSARA, which we attribute to the newer processors used in these GPU nodes.

The faceting functionality of AIRI was enabled, whereby the image is
decomposed into $F=4$ facets. Hence, four GPUs were deployed for the parallel
application of the DNN denoiser on each image facet. As such, the learned
denoiser brought a drastic reduction of the computing time of AIRI’s backward
step by a factor 10 to 30 (depending on the image dimensions) compared to its
pure optimisation counterpart uSARA. For each imaging experiment of each
field, AIRI’s total compute time and computational cost in CPU core hour are
reported in Table 6. In light of its extremely fast denoiser, AIRI’s
computational cost, solely driven by its forward step, is on average four
times lower than uSARA (see Tables 7–9 in Part I) and five times higher than
WSClean (see Table 10 in Part I).

Interestingly, preliminary experiments further leveraging GPUs to perform the
Fourier Transforms involved in the forward step have shown a reduction of
AIRI’s total compute time, and consequently its computational cost by nearly a
factor of two. However, a similar consideration in uSARA or WSClean would not
necessarily bring significant acceleration. The computational cost of uSARA
would still be driven by its sub-iterative denoiser. Similarly, the Fourier
transforms are typically not the dominating the computational cost of WSClean.

The speed and learning power of the DNN denoisers, pre-trained independently
of the RI data under scrutiny, are significantly narrowing the gap between
optimisation-based imaging algorithms and the standard CLEAN-based imager,
thus highlighting their prospective potential for scalability and
computational efficiency when handling extreme data and image dimensions.

|  | SB8275 |  |  | SB9351 |  |  | SB9442 |
---|---|---|---|---|---|---|---|---|---
| $F$ | CImage | TImage | $F$ | CImage | TImage | $F$ | CImage | TImage
|  | [CPUh] | [h] |  | [CPUh] | [h] |  | [CPUh] | [h]
Full-band | – | – | – | – | – | – | 4 | 203 | 1.1
SPW:1 | 4 | 48 | 1 | 4 | 95 | 2.1 | 4 | 45 | 1.4
SPW:2 | 4 | 63 | 1.1 | 4 | 95 | 2.2 | 4 | 54 | 1.5
SPW:3 | 4 | 104 | 1.4 | 4 | 101 | 2.6 | 4 | 51 | 1.6
SPW:4 | 4 | 104 | 1.6 | 4 | 105 | 2.7 | 4 | 50 | 1.6
SPW:5 | 4 | 119 | 1.7 | 4 | 99 | 2.3 | 4 | 52 | 1.5
SPW:6 | 4 | 129 | 1.7 | 4 | 104 | 2.5 | 4 | 62 | 1.6
SPW:7 | 4 | 103 | 1.5 | 4 | 125 | 2.9 | 4 | 66 | 1.6
SPW:8 | 4 | 144 | 2 | 4 | 91 | 2 | 4 | 69 | 1.6
Table 6: AIRI computational costs for all imaging experiments: $F$ refers to
the number of image facets, each deployed on one GPU core; CImage [CPUh] is
the computational cost of the deconvolution in CPU core hours and TImage is
the time in hours of the deconvolution. Memory and computational costs
specific to the measurement operator are identical to those considered in
uSARA (see Tables 7–9 in Part I for more details).

##  6 Conclusions

The results of this work show that the PnP-RI image reconstruction algorithm
AIRI is on par with the precision-capability of its pure optimisation
counterpart uSARA and surpasses the precision and robustness capabilities of
WSClean. A main and consistent feature of AIRI reconstructions is their
sensitivity to the diffuse components of faint emission. This gives AIRI a
distinct advantage over uSARA when the scientific goal is to detect and fully
reconstruct low-surface-brightness diffuse emission at or near the noise
level.

Building a shelf of suitable denoisers covering a range of potential target
dynamic ranges has proven to be a solid approach for implementing AIRI. When
resorting to a single universal high dynamic-range denoiser, high-fidelity
reconstruction is also achieved, with comparable results to nearest-on-the-
shelf reconstructions. In fact, we find that AIRI realisations reconstructed
from denoisers with different training noise levels have about a 1%
discrepancy in terms of the percentage of pixel intensities above the
estimated noise level of the imaged data.

In comparing the flux density values of the scrutinised sources reported in
Parts I and II, we observe varying levels of consistency between uSARA and
AIRI. A strong agreement is found in the case of the source “ dancing ghosts”,
likely due to its relatively flat spectra and absence of strong calibration
errors at its vicinity. However, when at least one of these conditions is not
met, the flux density values of the other sources compare differently, yet
with consistent variations across sub-bands. This is expected as AIRI has been
shown to capture more faint and diffuse emission with slightly different
morphology than uSARA. Concerning WSClean, its flux measurements are generally
higher than both AIRI and uSARA, particularly for the faint sources whose
brightness is near or at the noise level, indicating possible over-estimation
of the measurements taken from the restored images. Since each sub-and was
imaged separately, one cannot assert that the measurements of one method are
more reliable than the other, particularly since these early ASKAP
observations have not been validated against standard flux catalogues. Wide-
band variants of uSARA and AIRI are expected to provide more accurate
measurements in the future.

For the analysed ASKAP FoVs, AIRI has demonstrated a four-fold acceleration on
average over its pure optimisation counterpart, uSARA. The higher
computational efficiency achieved by AIRI is attributed to its substantially
faster denoiser, and is a clear indication of its scalability power to large
image dimensions. An in-depth investigation of practical scalability to
extreme data dimension is warranted. The study will appear as part of on-going
work towards a professional parallel C++ implementation of AIRI.

## Acknowledgements

The first two authors contributed equally to this work. This work was
supported by the UK Research and Innovation under the EPSRC grants
EP/T028270/1 and EP/T028351/1, and the STFC grant ST/W000970/1. The research
used Cirrus, a UK National Tier-2 HPC Service at EPCC funded by the University
of Edinburgh and EPSRC (EP/P020267/1). ASKAP, from which the data under
scrutiny originate, is part of the Australia Telescope National Facility
managed by CSIRO. This project used public archival data from the Dark Energy
Survey (DES).

## Data Availability

The ASKAP data underlying this article (calibrated visibilities and mosaic
images of Scheduling Blocks) are made publicly available for viewing and
download on the CSIRO ASKAP Science Data Archive (CASDA; Chapman et al.,
2017), and can be accessed with the unique Project Identifiers AS034 and
AS101. The reconstructed images in FITS format as well as the GIF files
showing the imaged fields over the spectral windows are made available in
Wilber et al. (2022b). The uSARA and AIRI code will become available in a
later release of the Puri-Psi library for RI imaging.

## References

  * Abbott et al. (2018) Abbott T. M. C., et al., 2018, ApJS, 239, 18
  * Ahmad et al. (2020) Ahmad R., Bouman C. A., Buzzard G. T., Chan S., Liu S., Reehorst E. T., Schniter P., 2020, IEEE Signal Process. Mag., 37, 105
  * Briggs (1995) Briggs D. S., 1995, in American Astronomical Society Meeting Abstracts. p. 112.02
  * Brüggen et al. (2021) Brüggen M., et al., 2021, A&A, 647, A3
  * Carrillo et al. (2012) Carrillo R. E., McEwen J. D., Wiaux Y., 2012, MNRAS, 426, 1223
  * Chapman et al. (2017) Chapman J. M., Dempsey J., Miller D., Heywood I., Pritchard J., Sangster E., Whiting M., Dart M., 2017, in ADASS XXV. p. 73
  * Cornwell et al. (2008) Cornwell T. J., Golap K., Bhatnagar S., 2008, IEEE J. Sel. Top. Signal Process., 2, 647
  * Cotton et al. (2020) Cotton W. D., et al., 2020, MNRAS, 495, 1271
  * Dabbech et al. (2022) Dabbech A., Terris M., Jackson A., Ramatsoku M., Smirnov O. M., Wiaux Y., 2022, ApJ, 939, L4
  * Hotan et al. (2021) Hotan A. W., et al., 2021, Publ. Astron. Soc. Australia, 38, e009
  * Hurault et al. (2022) Hurault S., Leclaire A., Papadakis N., 2022, arXiv preprint arXiv:2201.13256
  * HyeongHan et al. (2020) HyeongHan K., et al., 2020, ApJ, 900, 127
  * Joye & Mandel (2003) Joye W. A., Mandel E., 2003, in ADASS XII. p. 489
  * Norris et al. (2011) Norris R. P., et al., 2011, Publ. Astron. Soc. Australia, 28, 215
  * Norris et al. (2021) Norris R. P., et al., 2021, Publ. Astron. Soc. Australia, 38, e046
  * Offringa & Smirnov (2017) Offringa A. R., Smirnov O., 2017, MNRAS, 471, 301
  * Offringa et al. (2014) Offringa A. R., et al., 2014, MNRAS, 444, 606
  * Pesquet et al. (2021) Pesquet J.-C., Repetti A., Terris M., Wiaux Y., 2021, SIAM J. on Imaging Sci., 14, 1206
  * Ramatsoku et al. (2020) Ramatsoku M., et al., 2020, A&A, 636, L1
  * Reiprich et al. (2021) Reiprich T. H., et al., 2021, A&A, 647, A2
  * Romano et al. (2017) Romano Y., Elad M., Milanfar P., 2017, SIAM J. on Imaging Sci., 10, 1804
  * Terris et al. (2022) Terris M., Dabbech A., Tang C., Wiaux Y., 2022, MNRAS
  * Thouvenin et al. (2022) Thouvenin P.-A., Abdulaziz A., Dabbech A., Repetti A., Wiaux Y., 2022, MNRAS
  * Venkatakrishnan et al. (2013) Venkatakrishnan S. V., Bouman C. A., Wohlberg B., 2013, in 2013 IEEE Global Conf. on Signal and Inf. Processing. pp 945–948
  * Wilber et al. (2022a) Wilber A., Dabbech A., Jackson A., Wiaux Y., 2022a, in prep. HWU research portal ID: 52551784
  * Wilber et al. (2022b) Wilber A., Dabbech A., Terris M., Jackson A., Wiaux Y., 2022b, uSARA and AIRI reconstructions of selected ASKAP observations, doi:10.17861/5a60f25b-d43b-4807-ba02-a69bc460db03
  * Zhang et al. (2021) Zhang K., Li Y., Zuo W., Zhang L., Van Gool L., Timofte R., 2021, IEEE PAMI, 44, 6360