# Improving Log-Cumulant Based Estimation of Roughness Information in SAR
imagery

Jeova Farias Sales Rocha Neto, and Francisco Alixandre Avila Rodrigues J. F.
S. Rocha Neto is with the Department of Computer Science, Haverford College,
Haverford, PA, 19041 USA e-mail<EMAIL_ADDRESS>A. A. Rodrigues is
with the Centro de Ciências e Tecnologia, Federal University of Cariri,
Juazeiro do Norte, Brazil e-mail<EMAIL_ADDRESS>
###### Abstract

Synthetic Aperture Radar (SAR) image understanding is crucial in remote
sensing applications, but it is hindered by its intrinsic noise contamination,
called speckle. Sophisticated statistical models, such as the
$\mathcal{G}^{0}$ family of distributions, have been employed to SAR data and
many of the current advancements in processing this imagery have been
accomplished through extracting information from these models. In this paper,
we propose improvements to parameter estimation in $\mathcal{G}^{0}$
distributions using the Method of Log-Cumulants. First, using Bayesian
modeling, we construct that regularly produce reliable roughness estimates
under both $\mathcal{G}^{0}_{A}$ and $\mathcal{G}^{0}_{I}$ models. Second, we
make use of an approximation of the Trigamma function to compute the estimated
roughness in constant time, making it considerably faster than the existing
method for this task. Finally, we show how we can use this method to achieve
fast and reliable SAR image understanding based on roughness information.

###### Index Terms:

Synthetic Aperture Radar images, Bayesian Estimation, Image Analysis,
Statistical Modeling.

##  I Introduction

Synthetic Aperture Radar (SAR) imaging is a powerful tool for global earth
monitoring due to its advantages in operating under day or night and all-
weather conditions. In has found many applications, such as change detection
[1], image segmentation [2, 3] and target detection [4], to name a few.
Unfortunately, processing it is difficult due to the presence of an inherent
speckle noise, since it degrades image details that are relevant for image
analysis and interpretation. Therefore, statistical modeling of SAR data is a
crucial step in understanding it [5, 6]. Over the years, many models were
proposed to that end, with various degrees of success [6].

In particular, the $G^{0}$ family of distributions [7] arose to prominence in
the SAR scientific community due to its flexibility in describing rough and
extremely rough areas better than the $K$ distribution, besides not involving
Bessel functions in their density functions. One of the most important
features in these distributions is the interpretation of their $\alpha$
parameter, which varies according to the roughness of terrain being observed.
Recently, directly estimating it from real data has found success in improving
SAR imaging tasks [8, 9, 4].

In this work, we focus our attention on the method of Log-cumulants ($LCum$)
to estimate $\alpha$ because of its mathematical simplicity and its previous
success in preprocessing SAR imagery for segmentation [8]. On the other hand,
$LCum$ based estimators suffer from being computationally expensive and prone
to non-convergence, especially in small sample scenarios [9]. Here, we aim at
improving the estimation procedure in $LCum$ in two ways: (1) by making it
less susceptible against estimation failures and by (2) speeding it up. For
(1), we adapt the Bayesian approach pursued in [10] to the $G^{0}$ models and,
for (2), we employ an approximation to the Trigamma function to expedite the
inversion of a crucial equation present in $LCum$. Our final estimator is
shown to, besides being more robust and faster than the traditional one,
considerably reduce its estimation error.

This paper unfolds as follows. In Section II, we briefly review SAR data
modeling and the method of Log-cumulants in the context of the $G^{0}$
distributions. We then introduce our main methodology in Section III. In
Section IV, we present experimental results and briefly discuss them, and, in
Section VI, we summarize our findings and conclude this work.

##  II Background

###  II-A Statistical Models for SAR data

SAR imagery is characterized by a typical multiplicative model in which the
return in monopolarized SAR image is modeled by the product of the backscatter
($X$) and the noise ($Y$), both independent random variables. Thus, the SAR
data return ($Z$) can be obtained through the product $Z=XY$ [11].

Here, we adopt the $G^{0}$ family of distributions to model the return $Z$
[12], a flexible, tractable and descriptive model proposed by [7], that is
able to modeling homogeneous, heterogeneous and extremely heterogeneous
regions in SAR images [9]. In that sense, considering that the data arises
from intensity evaluations at the image level, the model is called
$\mathcal{G}^{0}_{I}$ distribution and has its probability density function
(pdf) defined by:

$f_{\mathcal{G}^{0}_{I}}(z,\theta)=\frac{L^{L}\Gamma(L-\alpha)}{\gamma^{\alpha}\Gamma(-\alpha)\Gamma(L)}z^{L-1}(\gamma+Lz)^{\alpha-L}.$
(1)

For amplitude SAR data, the model is called $\mathcal{G}^{0}_{A}$
distribution, whose probability density function is defined as:

$f_{\mathcal{G}^{0}_{A}}(z,\theta)=\frac{2L^{L}\Gamma(L-\alpha)}{\gamma^{\alpha}\Gamma(-\alpha)\Gamma(L)}z^{2L-1}(\gamma+Lz^{2})^{\alpha-L}$
(2)

For both distributions, $-\alpha,\gamma,L,z>0$ . The parameters $\alpha$ and
$\gamma$ correspond to the roughness and scale, respectively, $L$ the number
of looks and $\Gamma(.)$ is the gamma function. In this paper, we assume the
parameter $L$ is known.

From Eqs. (1) and (2), the $r$th order non-central moment for theses
distributions are, respectively:

$E_{\mathcal{G}^{0}_{I}}[Z^{r}]=\left(\frac{\gamma}{L}\right)^{r}\frac{\Gamma(-\alpha-r)\Gamma(L+r)}{\Gamma(-\alpha)\Gamma(L)},\quad\alpha<-r$
(3)

and

$E_{\mathcal{G}^{0}_{A}}[Z^{r}]=\left(\frac{\gamma}{L}\right)^{\frac{r}{2}}\frac{\Gamma(-\alpha-{\frac{r}{2}})\Gamma(L+{\frac{r}{2}})}{\Gamma(-\alpha)\Gamma(L)},\quad\alpha<-{\frac{r}{2}}.$
(4)

###  II-B The Log-Cumulants method (LCum) for $\mathcal{G}_{I}^{0}$ and
$\mathcal{G}_{A}^{0}$ parameter estimation

Let $Z$ be a continuous random variable with pdf $f_{Z}(z,\theta)$ defined
over $\mathbb{R}^{+}$, where $\theta$ can be either a real-valued parameter or
a vector. The construction of the $LCum$ is based on the Mellin transform of
$f_{Z}(z,\theta)$, which is defined by [13]:

$\phi_{Z}(s)\triangleq\int^{\infty}_{0}u^{s-1}f_{Z}(u,\theta)\mathrm{d}u=E[Z^{s-1}].$
(5)

Based on the natural logarithm of $\phi_{Z}(s)$, the log-moments (${m}_{v}$)
and log-cumulants (${k}_{v}$) of order $v\in\mathbb{N}^{*}$ can be obtained
by, respectively [13, 14]:

${m}_{v}=\frac{\mathrm{d}^{v}\phi_{z}(s)}{\mathrm{d}s^{v}}\bigg{|}_{s=1}$ (6)

and

${k}_{v}\triangleq\frac{\mathrm{d}^{v}\psi_{z}(s)}{\mathrm{d}s^{v}}\bigg{|}_{s=1},$
(7)

with $\psi_{z}(s):=\log(\phi_{z}(s))$.

For the pdf in Eq. (1) the function $\phi_{Z}(s)$ can be obtained using Eqs.
(5) and (3), which readily gives us expressions for log-cumulants of order 1
and 2 for the $\mathcal{G}_{I}^{0}$ distribution:

${k}_{1}=\log\left(\frac{\gamma}{L}\right)+\Psi^{0}(L)-\Psi^{0}(-\alpha),\\\ $
(8) ${k}_{2}=\Psi^{1}(L)+\Psi^{1}(-\alpha),$ (9)

where $\Psi^{0}(.)$ and $\Psi^{1}(.)$ are the digamma and trigamma functions,
respectively. A similar approach can be applied to the Eq. (4) in order to
calculate the first and second log-cumulants for the $\mathcal{G}_{A}^{0}$
distribution [8]:

$2{k}_{1}=\log\left(\frac{\gamma}{L}\right)+\Psi^{0}(L)-\Psi^{0}(-\alpha),\\\
$ (10) $4{k}_{2}=\Psi^{1}(L)+\Psi^{1}(-\alpha).$ (11)

Usually, systems involving Eqs. (8)-(9) and (10)-(11) are solved using the
relation between sample log-moments and log-cumulants, which lead us to the
following estimators for $k_{1}$ and $k_{2}$, respectively

$\widehat{{k}}_{1}=\frac{1}{n}\sum_{i=1}^{n}\log
z_{i},\quad\widehat{{k}}_{2}=\frac{1}{n}\sum_{i=1}^{n}(\log
z_{i}-\widehat{{k}}_{1})^{2},$ (12)

where $z_{i}$, $i\in\ \\{1,2,...,n\\}$, is a sample of $Z$.

##  III Estimation Methodology

Using Eqs. 9 and 11, we suggest a strategy to find an approximation of the
inverse of $\Psi^{1}(-\alpha)$ based on the identity:

$\Psi^{1}(-\widehat{\alpha})=\widehat{\eta},$ (13)

where $\widehat{\eta}=c_{\alpha}\widehat{{k}}_{2}-\Psi^{1}(L)$, where
$c_{\alpha}=1$ for the $\mathcal{G}_{I}^{0}$ and $c_{\alpha}=4$ in
$\mathcal{G}_{A}^{0}$. In order to estimate $\gamma$ as well, we can plug our
estimation for $\widehat{\alpha}$ into the derivations of ${k}_{1}$ for the
$\mathcal{G}_{I}^{0}$ and $\mathcal{G}_{A}^{0}$ models and find that:

$\widehat{\gamma}=L\exp\left(\sqrt{c_{\alpha}}{k}_{1}-\Psi^{0}(L)+\Psi^{0}(-\widehat{\alpha})\right).$
(14)

In our work, we concentrate ourselves on the estimation of $\alpha$, leaving
the analysis of the estimation of $\gamma$ to future work.

Estimating $\alpha$ using $LCum$ requires the inversion of Eq. 13, which is
usually done via out-of-the-shelf root finding algorithms [9]. This
methodology, however, as it is stated, suffers from two main problems: it
requires a considerable computational processing time and fails for most cases
when the estimator for $\widehat{\eta}$ returns a negative value. In other to
solve such problems, we propose the following methodological improvements

###  III-A Bayesian Approach to Reduce Failure Rate

Inspired by the bayesian approach presented in [10] and the fact that
$\Psi^{1}(x)>0,\forall x\in\mathbb{R}^{+}$ [15], we proposed an expression for
the posterior distribution of $\widehat{\eta}$ of the Eq. (13), described
below.

Let $\eta_{m}$ be a random variable that corresponds to the posterior mean
under this setting. Mathematically, the posterior PDF of $\eta_{m}$ given
$\widehat{\eta}$ can be calculated by applying the Bayes rule in the following
fashion:

$f(\eta_{m}|\widehat{\eta})=\frac{f(\widehat{\eta}|\eta_{m})f(\eta_{m})}{f(\widehat{\eta})}.$
(15)

Our prior knowledge on $\eta_{m}$ states that it should only assume positive
values. Saying so, a convenient way to express that is by imposing a
$\text{Uniform}(0,b)$ prior on it, i.e.,
$f(\eta_{m};b)=1/b\mathds{1}_{\eta_{m}\in(0,b)}$, where $\mathds{1}_{\cdot}$
is the indicator function. The likelihood function can be readily written as:

$f(\widehat{\eta}|\eta_{m})=\frac{1}{\sqrt{2\pi{\sigma}^{2}}}\exp\left(-\frac{(\hat{\eta}-\eta_{m})^{2}}{2{\sigma}^{2}}\right),$
(16)

where $\sigma^{2}$ is the variance of $\hat{\eta}$, whose estimation is
described at the end of this section. From Eq. 15, the posterior distribution
becomes:

$f(\eta_{m}|\widehat{\eta};b)=\frac{\exp\left(-\frac{(\hat{\eta}-\eta_{m})^{2}}{2\sigma^{2}}\right)}{\sqrt{2\pi\sigma^{2}}\left(\Phi\left(\frac{\hat{\eta}}{\sigma}\right)-\Phi\left(\frac{\hat{\eta}-b}{\sigma}\right)\right)}$
(17)

where $\Phi(\cdot)$ is the Cumulative Distribution Function of a Standard
Normal. Again following the approach in [10], we compute the expected
posterior mean as our estimation $\widehat{\eta_{m}}$ in the limit where
$b\to\infty$:

$\displaystyle\widehat{\eta}_{m}$
$\displaystyle=\lim_{b\to\infty}\mathbb{E}[\eta_{m}|\widehat{\eta};b]$
$\displaystyle=\lim_{b\to\infty}\int_{0}^{b}\frac{\eta_{m}\exp\left(-\frac{(\hat{\eta}-\eta_{m})^{2}}{2\sigma^{2}}\right)}{\sqrt{2\pi\sigma^{2}}\left(\Phi\left(\frac{\hat{\eta}}{\sigma}\right)-\Phi\left(\frac{\hat{\eta}-b}{\sigma}\right)\right)}\mathrm{d}\eta_{m}$
$\displaystyle=\hat{\eta}+\frac{\sigma}{\sqrt{2\pi}\Phi\left(\frac{\hat{\eta}}{\sigma}\right)}\exp\left(-\frac{\hat{\eta}^{2}}{2\sigma^{2}}\right).$
(18)

Notice that Eq. III-A proposes a correction on $\hat{\eta}$ in order to ensure
that the posterior mean is positive.

In order to estimate $\sigma$, for both $\mathcal{G}_{I}^{0}$ and
$\mathcal{G}_{A}^{0}$, we first notice that the randomness on $\hat{\eta}$ is
uniquely due to $\widehat{k}_{2}$. Using Eq. 12 and defining the random
variable $W=\log Z$, we can rewrite $\hat{\eta}$ as:

$\hat{\eta}=c_{\alpha}\frac{1}{n}\sum_{i=1}^{n}\left(w_{i}-\frac{1}{n}\sum_{j=1}^{n}w_{j}\right)^{2}-\Psi^{1}(L).$
(19)

Note that the first term in the above expression is the sample variance of
$W$, $S_{w}^{2}$. We can then rearrange the Eq. 19 as
$\hat{\eta}=c_{\alpha}S_{w}^{2}-\Psi^{1}(L)$. Then, the variance $\hat{\eta}$
is therefore given by the variance of $c_{\alpha}S_{w}^{2}$, which is computed
via [16]:

$\operatorname{Var}[\hat{\eta}]=\frac{c_{\alpha}^{2}}{n}\left(\mathbb{E}[W^{4}]-\frac{n-3}{n-1}\operatorname{Var}[W]^{2}\right).$
(20)

We estimate $\sigma$ in Eq. III-A as
$\sigma=\sqrt{\operatorname{Var}[\hat{\eta}]}$.

Figure 1: MSE values for the $G_{I}^{0}$ experiments. Figure 2: MSE values for
the $G_{A}^{0}$ experiments.

###  III-B Trigamma Approximation to Improve Estimation Speed

Now, back to Eq. 13, we can further improve our algorithm performance by
approximating the Trigamma function by an efficiently invertible one. The
authors in [17] propose strong bounds for $\Psi^{1}(\cdot)$ that lead us to
state the following approximation on $\Psi^{1}(x)$:

$\displaystyle\Psi^{1}(x)\approx\frac{1}{x}+$
$\displaystyle\frac{1}{2x^{2}}+\frac{1}{6x^{3}}-\frac{1}{30x^{5}}+\frac{1}{42x^{7}}-\frac{1}{30x^{9}}.$

Considering that in practice $x=-\alpha\in(1,15]$, we notice that the error
for such approximation, $O\left(1/x^{11}\right)$ [17] is negligible. Using our
estimate for $\widehat{\eta}_{m}$ on Eq. 13 and the above approximation, we
define the polynomial:

$P(\hat{\alpha})\triangleq
210\widehat{\eta}_{m}\hat{\alpha}^{7}+210\hat{\alpha}^{6}-105\hat{\alpha}^{5}+35\hat{\alpha}^{4}-7\hat{\alpha}^{2}+5,$
(21)

such that $P(\hat{\alpha})=0$ approximately corresponds to Eq. 13. Notice that
polynomial roots can be efficiently computed by the solution of an eigenvalue
problem on its companion matrix.

##  IV Numerical Experiments

###  IV-A Data and Algorithmic Setup

We demonstrate the performance of our proposed methodology using synthetic and
real SAR data. The synthetic experiments were conducted on both intensity and
amplitude data drawn from $G_{I}^{0}$ and $G_{A}^{0}$, respectively. To
generate synthetic amplitude data, $Z_{A}$, we used the inverse transform
method according to [6] , in which:

${Z}_{A}=\sqrt{-\frac{\gamma}{\alpha}\Upsilon^{-1}_{2L,-2\alpha}(U)},$ (22)

where $\Upsilon^{-1}_{2L,-2\alpha}$ is the cumulative distribution inverse
function of Snedecor’s $F$ law with $2L$ and $-2\alpha$ degrees of freedom,
and $U$ is an uniformly distributed random variable in $(0,1)$. And to
generate intensity images, $Z_{I}$, we used the relation between amplitude and
intensity data that states ${Z}_{A}^{2}={Z}_{I}$. Without any loss of
generality, for each chosen $\alpha$, we chose the scale parameter ($\gamma$)
such that $E_{G_{I}^{0}}[Z_{I}]=1$, i.e., $\gamma=-\alpha-1$ [9].

As a real experiment, we make use of a $600\times 450$ pixel SAR image with
$L=4$, in L-band and HH polarization, acquired by Airborne Synthetic Aperture
Radar (AIRSAR) over the San Francisco Bay [8]. Our estimation methodology for
this image is similar to what it is done in [8]: we sweeping it with a
$\ell\times\ell$ window centered on each pixel and estimate $\alpha$ for that
pixel using the data from the window.

For both synthetic and real experiments, we evaluate the performance of our
proposed methodology under two settings: one where we estimate $\alpha$
directly from Eq. 13 using our proposed approximation algorithm, as described
in Section III-B, and another one where we first correct our $\hat{\eta}$
estimate using Eq. III-A and then proceed with the approximation algorithm. We
compare these methods to the estimation approach employed in [8], here named
Traditional LCum, where the authors use MATLAB’s fsolve function to invert the
expression in Eq. 13. This function, in turn, utilizes a Trust-region
optimization technique [18] to solve this inversion problem. In this paper, we
focus on the performance of LCum for brevity, leaving the comparisons with
other estimation approaches such as Maximum Likelihood and Fractional Moments
to future work. Here we focus on assessing the estimation of $\alpha$, once
$\gamma$ can be simply estimated from it using Eq. 14.

These experiments were run on an Intel® Core™ i5-6200U CPU 2.30GHz with 8Gb of
memory and the software was entirely coded in MATLAB111The code
implementations from this paper can be found at
github.com/jeovafarias/Improved-Roughness-Estimation-SAR.

TABLE I: Failure rates (%) on synthetic data. Model | $L$ | Proposed Method |  Traditional LCum
---|---|---|---
Non-corrected | Corrected
$G_{I}^{0}$ | $1$ | 31.85 | 1.25 | 31.61
$3$ | 11.97 | 1.73 | 11.37
$8$ | 4.41 | 1.80 | 4.13
$G_{A}^{0}$ | $1$ | 30.97 | 1.40 | 32.11
$3$ | 11.62 | 2.00 | 12.22
$8$ | 4.31 | 1.27 | 4.01
TABLE II: Average estimation time (s) on synthetic data. Model | $n$ | Proposed Method |  Traditional LCum
---|---|---|---
Non-corrected | Corrected
$G_{I}^{0}$ | 9 | 0.56e-04 | 1.13e-04 | 2.28e-03
121 | 0.87e-04 | 1.14e-04 | 2.13e-03
1000 | 2.33e-04 | 3.43e-04 | 2.17e-03
$G_{I}^{0}$ | 9 | 0.59e-04 | 1.05e-04 | 2.71e-03
121 | 0.91e-04 | 1.37e-04 | 2.16e-03
1000 | 2.55e-04 | 3.41e-04 | 2.18e-03

###  IV-B Assessment Methodology

To analyze the robustness of the algorithms experimented here (i.e., how
likely they are to fail their estimation), we computed their failure rates
according to the criteria [9]:

  * •

For estimates using the technique in Sec. III-B:

    * –

$P(\hat{\alpha})$ presents more than one real root or none,

    * –

If the real root found is lesser than -15 or positive.

  * •

For estimates arising from the Traditional LCum:

    * –

The solver does not converge,

    * –

The solution found is lesser than -15 or positive.

The quantitative assessment of our methods’ estimation performance compared
the estimated $\hat{\alpha}_{i}$ with their underlying values $\alpha_{i}$ for
a sample set $i$ via Mean Square Error (MSE):

$\operatorname{MSE}\triangleq\frac{1}{N}\sum_{i=1}^{N}(\hat{\alpha}_{i}-\alpha_{i})^{2}$
(23)

where $N$ is the number of estimates compared, since we only consider those
that did not fail, according to our failure criteria.

Finally, we also consider the average runtime of our proposed methods for
speed assessment.

(a) Original

(b) Traditional LCum
(ET: $604.8s$, #Fail: 30242)

(c) Proposed (non-corrected)
(ET: $11.2s$, #Fail: 30304)

(d) Proposed (corrected)
(ET: $39.18.3s$, #Fail: 35)

Figure 3: Qualitative results for $\alpha$ estimation in a real SAR image
using an $11\times 11$ window. The color bar on the right is shared by all
roughness maps. ET stands for each algorithm’s elapsed runtime and #Fail is
the absolute number of failures encountered when computing each map.

###  IV-C Results on Synthetic Data

We start by evaluating our estimation methods on synthetic samples of size
$n\in\\{9,25,49,121,1000\\}$, generated using roughness
$\alpha\in\\{-1.5,-3.0,-5.0\\}$ and number of looks $L\in\\{1,3,8\\}$. For
each setting, we performed a Monte Carlo experiment with 1000 intensity and
amplitude data vectors. We depict their average MSE in Figures 1 and 2. We
note that, while our proposed non-corrected estimator performs generally as
well as the traditional one, our corrected estimator consistently over
performs both for most scenarios.

In Table I, we use this same experimental data to compute the failure rates
per value of $L$, averaging our the other parameters and averaging over
experiments. Here, we again note the resemblance between our non-corrected
estimator results and the ones from Traditional LCum. In both cases, we also
see how increasing the number of looks helps the estimation by making the
failure rate lower, a fact also observed by [9]. The table also shows that our
corrected estimator keeps a consistent low failure rate, a clear evidence of
the effectiveness of our proposed Bayesian correction approach.

Finally, we depict in Table II the average runtime of each method again using
the same Monte Carlo data, but only considering the samples of sizes
$n\in\\{9,121,1000\\}$. The non-corrected and corrected versions of our
estimation algorithm are 6 and 40 faster than the traditional estimator. Upon
inspection, most of the computation spent in our methods is due to the
averaging when computing sample log-moments, which is evident when we note
that the runtime increases with the sample size. That phenomenon is not
explicit in the traditional LCum method, as it spends most of its time in
optimizing.

###  IV-D Results on Real Data

Figure 3 depicts the roughness maps generated by each method on our sample
real SAR image using a $11\times 11$ estimation window. It also presents the
runtime and the absolute number of failures when computing them. For each
estimation map, we set the values of $\hat{\alpha}$ to zero on locations where
a failure occurred.

Note that the estimated $\alpha$’s in all maps follow their usual
understanding: higher values (greater than $-3$) are related to highly
textured, urban areas; moderate values (in $[-6,-3]$) suggest forest/park
zones and low values (below $-6$) correspond to textureless regions, such as
the ocean [9]. This qualitatively demonstrate the effectiveness of all
presented methods in this experiment. Also note the similarity between the
results attained by both the traditional method and our proposed non-corrected
one, in terms of both estimation quality and number of failure cases. Despite
this similarity, our method is around 50$\times$ faster. Furthermore, although
being slower than its non-corrected counterpart, these results show that our
corrected estimation encountered a negligible number of failure cases, making
it more suitable to further imaging applications, such as segmentation and
classification.

##  V Conclusion

In this paper, we improve the traditional Log Cumulant-based estimation of
parameters from both $G_{I}^{0}$ and $G_{A}^{0}$ in two ways. We first propose
a correction to their roughness estimates using Bayesian methodology. Second,
we make use of an approximation of the Trigamma function to quicken the
estimation process. Our synthetic experiments demonstrate that these
improvements give rise to fast and performant parameter estimation algorithms
that overcome the drawbacks from its traditional approach. In fact, the
approximation alone is shown to be able to speed up estimation by over 50
times, despite keeping the same estimation performance and failure rate from
the traditional method. When accompanied by our Bayesian correction, the final
algorithm is demonstrated to have low failure rate and higher estimation
efficacy, while still being 10 times faster than its traditional counterpart.
We finally also demonstrate that our proposed method is able to quickly and
efficiently extract roughness maps from a real SAR image that can used for
further understanding it.

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# Functionalized Tellurene; a candidate large-gap 2D Topological Insulator

Raghottam M. Sattigeri  Prafulla K. Jha<EMAIL_ADDRESS>1Department of
Physics, Faculty of Science, The Maharaja Sayajirao University of Baroda,
Vadodara-390002, Gujarat, India

###### Abstract

The discovery of group IV and V elemental Xene’s which exhibit topologically
non-trivial characters natively in their honeycomb lattice structure (HLS) has
led to extensive efforts in realising analogous behaviour in group VI
elemental monolayers. Although; it was theoretically concluded that group VI
elemental monolayers cannot exist as HLS but recent experimental evidence
suggests otherwise. In this letter we report that, HLS of group VI elemental
monolayer (such as, Tellurene) can be realised to be dynamically stable when
functionzalised with Oxygen. The functionalization leads to, peculiar orbital
filtering effects (OFE) and broken spatial inversion symmetry which gives rise
to the non-trivial topological character. The exotic quantum behaviour of this
system is characterized by, spin-orbit coupling induced large-gap $\approx$
0.36 eV with isolated Dirac cone along the edges indicating perspective room
temperature spin-transport applications. Further investigations of spin Hall
conductivity and the Berry curvatures unravel high conductivity as compared to
previously explored Xene’s. The non-trivial topological character is
quantified in terms of the $\mathbb{Z}_{2}$ invaraint as $\nu=$ 1 and Chern
number $\mathit{C}=$ 1\. Also, for practical purposes, we report that,
hBN/TeO/hBN quantum-wells can be strain engineered to realize a sizable non-
trivial gap ($\approx$ 0.11 eV). We finally conclude that, functionalization
of group VI elemental monolayer with Oxygen gives rise to, exotic quantum
properties which are robust against surface oxidation and degradations while
providing viable electronic degrees of freedom for spintronic applications.

Topological Insulators (TI) are exotic quantum phase of matter characterized
by the D dimensional bulk gap and the D$-$1 dimensional conducting states
which are robust against weak deformations and external perturbations Kane and
Mele (2005a, b); Bernevig and Zhang (2006); Hasan and Kane (2010); Han, Kang,
and Jeon (2020). In two-dimensional (2D) systems this effect gives rise to,
gapless spin-polarized helical edge states with insulating non-magnetic bulk
gap Chen  _et al._ (2018). Since such systems are governed by the time-
reversal symmetry, the dissipationless spin currents are protected against
backscattering from non-magnetic impurities leading to quantized Hall
conductance in absence of magnetic field Hu  _et al._ (2018); Gusev  _et al._
(2020). This makes them highly desirable candidates for spintronic,
valleytronic, electronic and quantum computing applications.

Figure 1:  a) HLS of monolayer Te. b) PDC alongside PhDOS of monolayer Te
which exhibits imaginary modes indicating dynamically unstable structure. c)
HLS of monolayer TeO wherein, Te monolayer is functionalized with Oxygen at
alternative sublattice positions on either side of the monolayer. d) PDC
alongside PhDOS of TeO exhibits absence of imaginary modes indicating
dynamically stable structure. e) EBS alongside DOS indicates the semi-metallic
nature of TeO (inset zoomed version DOS and the 2D BZ with highlighted edge
BZ). f) EBS without and with SOC indicating non-trivial large-gap alongside
the Fermi energy scan ($\Delta\mu$) of SHC
($\sigma^{\textit{spinz}}_{\textit{xy}}(\omega)$) in the units of
$(\hbar/e)\Omega^{-1}cm^{-1}$ (red line indicates the Fermi energy and the
green lines indicate the SOC induced band gap).

Although, with the discovery of graphene several 2D TI based on group IV and V
elements were theoretically predicted Yao  _et al._ (2007); Konschuh, Gmitra,
and Fabian (2010); Deng  _et al._ (2018); Liu, Feng, and Yao (2011); Xu  _et
al._ (2013); Liu  _et al._ (2015); Yang  _et al._ (2017); Wang  _et al._
(2016a, b); Zhao, Zhang, and Li (2015); Song  _et al._ (2014); Xian  _et al._
(2017) but, the phenomena was only experimentally realised and limited to
HgTe/CdTe and InAs/GaSb quantum wells (QW) at ultra-low temperatures and
ultra-high vacuum by the virtue of weak spin-orbit coupling (SOC) effects
Zhang  _et al._ (2017). However, this sparked several investigations based on
first-principles and theorical modelling methods which have driven the
experimental efforts over the years. One of the major challenges is, realising
large band gaps which would make room temperature applications of 2D TI
materials viable. This has been well addressed by the functionalization of
monolayers of group IV, V Xene’s and MXene’s etc., which leads to the -pz
orbital saturation and enhancement of the SOC by virtue of orbital filtering
effects (OFE) Hu  _et al._ (2018); Xu  _et al._ (2013); Zhou  _et al._ (2014);
Si  _et al._ (2014); Weng  _et al._ (2015); Wang  _et al._ (2017); Liang  _et
al._ (2017); Wu, Gu, and Li (2019); Hu, Wang, and Li (2020); Huang  _et al._
(2020); Liu  _et al._ (2021). For example, it was experimentally shown that
halogenation of Bi monolayers can host non-trivial TI states but, these states
are quite sensitive due to their strong hybridizations with the substrates
Miao  _et al._ (2015). However, the robustness of TI nature against surface
oxidation and degradation is not well addressed which might hinder their
practical applications.

Similar to group IV and V elemental monolayers, efforts were made to realise
2D TI nature in group VI Xene’s with honeycomb lattice structure (HLS) based
on Tellurium (Te) and Selenium (Se). But, the failure is attributed to
electronic configuration of group VI elements and their bulk structures which
inhibit fostering the hexagonal HLS rendering them to exist in ring, chain,
square and rectangular lattice structures Xian  _et al._ (2017); Huang  _et
al._ (2017); Zhu  _et al._ (2017); Qin  _et al._ (2017). It was eventually
concluded that, elemental monolayers of Te and Se cannot exist in HLS Zhao and
Wang (2020). However, recent, experimental evidence suggests otherwise, i.e.,
thickness-dependent transition from Au-Te surface alloy to HLS of Te (i.e., 3
$\times$ 3 honeycomb-like superstructures) can be realised by employing the
self-organization mechanism of Te atoms on Au(111) surface (when exposed to
high temperature vapor deposition of Te in ultra-high vacuum) at 0.5 monolayer
coverage Bouaziz  _et al._ (2020).

Inspired by this, in this letter we explore alternative approach to realise
dynamically stable HLS of Te and Se. For this purpose, we perform Oxygen and
Sulphur functionalization of Te and Se monolayers since; the pristine
monolayers are unstable which was found to be in agreement with previous
studies Zhao and Wang (2020). Off these, only Oxygen functionalised Te and Se
and Sulphur functionalised Se were found to be dynamically stable. However, by
the virtue of OFE and large SOC (due to high atomic number of Te) Oxygen
functionalised Te exhibits a large-gap 2D TI nature with viable room
temperature applications by exploiting the robust Dirac dispersion of spin-
polarized edge states. Hence, we restrict our discussions to non-trivial
nature of Oxygen functionalised Te (while the other systems are presented in
the supplementary material (SM)) which is also analysed in terms of the
$\mathbb{Z}_{2}$ invariant and the Chern number ($\mathit{C}$).

We implement first-principles based Density Functional Theory in Quantum
Espresso package Giannozzi  _et al._ (2009) to obtain the ground state energy
of the system (with a vacuum of c = 25.00 Å along the [001] crystal direction
to isolate the interactions between periodic images) under plane wave self
consistent formulation and calculate the electronic band structures (EBS),
density of states (DOS) and orbital/elemental projected density of states
(PDOS). We use, scalar-relativistic and norm conserving Martins-Troullier
pseudopotentials for calculations without SOC and fully-relativistic Projector
augemented wave pseudopotentials (for calculations with SOC) under generalized
gradient approximation with Perdew-Burke-Ernzerhof type of exchange-
correlation functional in our calculations Perdew, Burke, and Ernzerhof
(1996); Blöchl (1994). The converged value of kinetic energy cutoff was 80 Ry
and the corresponding uniform momentum (k) Monkhorst-Pack grid was of 8
$\times$ 8 $\times$ 1 Monkhorst and Pack (1976). The structures were fully
relaxed with force convergence criteria of $<$ 10-6 a.u. The dynamic stability
of the system is investigated in terms of Phonon dispersion curves (PDC) and
Phonon density of states (PhDOS) under the Density Functional Perturbation
Theory Baroni  _et al._ (2001) regime with a q-mesh of 5 $\times$ 5 $\times$
1\. We then project the DFT wave functions onto the maximally localized
Wannier functions (with convergence of the spread function) by using Wannier90
code Mostofi  _et al._ (2014) to compute the quantum transport properties and
create the tight-binding hamiltonian as an input for the WannierTools code Wu
_et al._ (2018). We compute the spin Hall conductivity (SHC)
($\sigma^{\textit{spinz}}_{\textit{xy}}(\omega)$) using the Kubo-Greenwood
formula Qiao  _et al._ (2018); Guo  _et al._ (2008) within the independent-
particle approximation, and the Berry curvature ($\Omega_{\textit{z}}(k)$) and
k-resolved spin Berry curvature ($\Omega^{\textit{spinz}}_{\textit{xy}}(k)$)
to understand the topological properties and the quantum transport phenomena.
For this purpose, a dense k-mesh of 100 $\times$ 100 $\times$ 1 is used; since
large contributions of spin Berry curvatures occur in minute regions of k
space which leads to slow convergence Sui  _et al._ (2017). We finally compute
the $\mathbb{Z}_{2}$ invariant, Chern number ($\mathit{C}$) and edge states
using the WannierTools code Wu  _et al._ (2018).

Figure 2:  PDOS of TeO (a) without and (b) with SOC indicating exchange of -p
orbitals across the Fermi highlighted by downward arrows. Orbital ($\pi$ and
$\sigma$) projected EBS of TeO (c) without and (d) with SOC. e) Berry
curvature ($\Omega_{\textit{z}}(k)$) mapped along the entire 2D BZ indicates
opposite signs. f) k-resolved spin Berry curvature
($\Omega^{\textit{spinz}}_{\textit{xy}}(k)$) mapped along the BZ valleys K’
and K.

Similar to graphene and several other group IV,V elemental monolayers we begin
with designing of elemental monolayers composed of Te (and Se) with HLS as
shown in Fig. 1(a) (and Fig. 1(b)(SM)). The optimized lattice constant of Te
(and Se; presented in SM Table I) with HLS is a = 4.94 Å, with a buckling
height of h = 0.72 Å and the angle between Te atoms $\approx$ 114.14∘.
However, we find these structures to be dynamically unstable due to imaginary
modes throughout the Brillouin Zone (BZ) as evident from Fig. 1(b) (and Fig.
1(a)(SM)), which is in agreement with previous studies Zhao and Wang (2020).
Such a HLS is not favoured due to, the electronic configuration of Te (and Se)
which does not allow the formation of typical -sp hybridizations and the
reactive surface due to the dangling bonds Wang  _et al._ (2016c). However, on
functionalization with Oxygen at alternative sublattice positions on both
sides of Te (and Se) monolayer (Fig. 1(c) and Fig. 1(c)(SM)) and Sulphur on Se
(Fig. 1(f)(SM)), the imaginary modes are eliminated rendering the system
(Tellurene Oxide-TeO, Selenene Oxide-SeO and Selenene Sulphide-SeS) to be
dynamically stable (Fig. 1(d) and Fig. 1(d,e)(SM)) in the much sought-after
HLS. This structure is governed by the D3d point group symmetry which is lower
than the D6h point group symmetry of graphene; leading to broken spatial
inversion symmetry in the system. The optimized lattice constant of TeO is, a
= 5.26 Å, with reduced buckling height of h = 0.57 Å (which indicates viable
structural design), layer thickness t $\approx$ 4.18 Å, $\angle$O-Te-Te $=$
100.74∘ and the angle between Te atoms as $\approx$ 116.61∘. Apart from
stability, the functionalization contributes by, saturating the out of plane
-pz orbitals (dangling bonds) of Te giving rise to the -sp3 hybridization and
strong SOC.

The 2D hexagonal BZ and the orthorhombic edge BZ are presented in inset of
Fig. 1(e). It is evident that, when the SOC is not considered TeO exhibits
graphene-like semi-metallic behaviour along the high symmetry point K in the
BZ hosting degenerate Dirac-cone states (Fig. 1(e)) (whereas, SeO and SeS are
semi-conducting with gaps of the order of $\approx$ 10-3 eV as evident from
band structure presented in Fig. 2(SM)). The electronic band structure of TeO
along the 2D hexagonal BZ path with and without SOC is presented in Fig. 1(f).
SOC lifts off the degeneracy giving rise to a global large-gap of $\approx$
0.365 eV due to strong SOC and OFE. The strong SOC is due to the domination of
$\sigma$ orbitals around the Fermi, rather than the $\pi$ orbitals with weaker
SOC as evident from the orbital projected band structures presented in Fig.
2(c-d) Si  _et al._ (2014) apart from the high atomic number of Te. Also, from
the projected density of states (Fig. 2) it is evident that the orbital band
inversion mechanism involves -p orbitals of Te and O. This sort of band
inversion along with large global gap indicates the potential non-trivial TI
character of TeO. The -p orbital near the vicinity of Fermi splits into two
groups -pz and -px, -py since the system is governed by D3d point group
symmetry and the D3 groups of the wave vector at Dirac points K and K’ in the
BZ. The flat bands (below the Fermi) made up of -px and -py orbitals and the
massive Dirac cones make TeO unique as compared to graphene which indicates
that TeO maybe a novel orbital analog of quantum anomalous Hall effect and
Wigner crystallization Song  _et al._ (2014).

We further analyse the $\mathbb{Z}_{2}$ invariant by using Wilson loops method
around Wannier Charge Centers Wu  _et al._ (2018); which turns out to be
$\nu=$ 1, confirming the non-trivial TI character of TeO and the corresponding
quantum spin Hall effect. The Berry curvature distribution in the entire BZ is
presented in Fig. 2(e) for the nth occupied band. By integrating the Berry
curvature throughout the BZ, we find the Chern number to be $\mathit{C}=$ 1
which further confirms the non-trivial TI character of the system. Also, non-
trivial TeO will host bulk-boundary correspondance which is a key ingredient
in realising a potential 2D TI. For this purpose we compute the edge states
along the high symmetry paths of orthorhombic edge BZ. From Fig. 3(a), it is
clearly evident that, 2D TeO is a TI hosting robust conducting edge states
along the high symmetry point S in the edge BZ. We further analyse the origin
and nature of SHC in this system.

Since the strong SOC effects split the band degeneracies along the high
symmetry point K in the hexagonal BZ with the Fermi level lying within the
gap, we can expect significantly large SHC Sui  _et al._ (2017). This is
because, the spin Berry curvature contributions (presented in Fig. 2(f)) from
the positive and negative spins around Fermi do not cancel out excluding one
sign of spin Berry curvature. The corresponding spin Berry curvature governed
topology of TeO gives rise to unconventionally large transverse SHC
($\sigma^{\textit{z}}_{\textit{xy}}$) of the order of $\approx$ 63.25
$(\hbar/e)\Omega^{-1}cm^{-1}$ (evident from Fig. 1(g) which shows the Fermi
energy scan $\delta\mu$ to study the SHC). This is superior as compared to
other systems Farzaneh and Rakheja (2020); Matusalem  _et al._ (2019) which
indicates higher charge-spin-current conversion efficiency. The absence of
quantization in the SHC can be attributd to the presence of buckling in the
system Farzaneh and Rakheja (2020).

Figure 3:  a) Edge states along high symmetry point S in the edge BZ
indicating towards bulk-boundary correspondence in TeO (temperature scale is
in arbitrary units). b) Schematic of the van der Waals hBN/TeO/hBN quantum
well heterostructure device subjected to uniform tensile strain. c) EBS of
hBN/TeO/hBN indicating the persistence of non-trivial gap along high symmetry
point K in the BZ.

From the Berry curvature plot presented in Fig. 2(e) we observe sharp peaks in
the valley region with opposite signs for K and K’ which implies the
possibility of valley Hall effects in the system. This implies that, when the
TeO monolayer is subjected to voltage bias, the electrons from different
valleys would experience opposite Lorentz forces leading to electronic motion
in opposite directions perpendicular to the drift current thus creating
charge-spin segregation making the system a potential quantum valley Hall
insulator Zhou  _et al._ (2018).

For practical purposes, apart from the free standing monolayers of TeO, we
explore their quantum well heterostructures with $2\times 2$ hexagonal-Boron
Nitride (hBN) presented in Fig. 3(b). We can sandwitch TeO between hBN
creating quantum wells of the form hBN/TeO/hBN with small lattice mismatch of
$\approx$ 4.79% and interlayer separation was varied from 3 Å to 6 Å. Upon
optimization of this system under van der Waals appromixation Grimme  _et al._
(2010), we observe that, the TeO and hBN retain their original structures.
When subjected to tensile strain as high as $\approx$ 14% the system hosts a
non-trivial gap of the order of $\approx$ 0.11 eV (evident from Fig. 3c) which
is considerably high as compared to room temperature thermal energy and
majorly arising from TeO states around the Fermi due to weak coupling with hBN
with interlayer separation of 6 Å. This would electrically insulate the layers
of TeO protecting the spin-polarised helical edge states. Such aggrangements
are known to increase the number of edge transport channels to support the
dissipationless charge-spin transport Wang  _et al._ (2017).

In summary, using first-principles based DFT calculations, we report a novel
large-gap 2D TI phase in HLS of TeO (with a possibility of exploring SeO and
SeS for their potential TI properties under modest strain/electric field).
This non-trivial phase is characterised in terms of the $\mathbb{Z}_{2}$
invariant, Chern number $\mathit{C}$ and the edge state spectrum. From
practical applications point of view, we propose, strained quantum well
heterostructure of the form hBN/TeO/hBN which can be used to realise charge-
spin transport phenomena at room temperature. We also discussed the origin of
unconventionally large SHC in this system and the possibility of valley Hall
insulating phenomena by analysing the Berry curvature and spin Berry
curvature. Our predictions may further inspire experimental efforts to realise
this novel phase of TeO with room temperature applications as spin Hall
insulators / valley Hall insulators.

See the supplementary material for additional information on the structural
parameters of Se, SeO, SeS and TeS along with their Phonon Dispersion Curves
and the electronic band structures of SeO and SeS.

## ACKNOWLEDGEMENT

The authors acknowledge, the Interdisciplinary Centre for Mathematical and
Computational Modelling UW (ICM UW), Warsaw, Poland for providing
computational facilities. RMS acknowledges Dr. Som Narayan, Warsaw University
of Technology, Poland and Dr. Trupti Gajaria, GSFC University, India for
fruitful discussions and sugesstions.on

### CONFLICT OF INTEREST

The authors have no conflicts of interest to disclose.

## AUTHORS’ CONTRIBUTIONS

R.M.S. and P.K.J. contributed equally to this work.

## DATA AVAILABILITY

The data that support the findings of this work are available from the
corresponding author upon reasonable request.

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## SUPPLEMENTARY MATERIAL

### Structure, phonon dispersion curves and electronic band structure:

Figure 4: Honeycomb lattice structure alongside phonon dispersion curves of
(a, b) Se, (c, d) SeO, (e, f) SeS and (g, h) TeS.

Figure 4 presents the predicted honeycomb lattice structure and phonon
dispersion curves of Se (Fig. 4(a, b)), SeO (Fig. 4(c, d)), SeS (Fig. 4(e, f))
and TeS Fig. (4(g, h)) which clearly indicates that, SeO and SeS are
dynamically stable (since imaginary modes are absent in the entire Brillouin
Zone (BZ)) whereas Se and TeS are dynamically unstable which can be attributed
to the soft modes evident from the imaginary modes in the BZ. Also, from the
phonon dispersion curves in Fig. 4(d) and (e) we observe softening of the
phonon modes along the M $\rightarrow$ K $\rightarrow$ $\Gamma$ high symmetry
points in the BZ which indicates structural instability at higher
temperatures.

Table 1:  Lattice constants (a) and structural parameters of 2D systems of type A and AB where, A = Se, Te and B = O, S (with angles, buckling height (h), layer thickness (t) and distance between A and B (d)). AB (A = Se, Te; B = O, S) |  a (Å) |  $\angle$BAA |  $\angle$AAA |  h (Å) |  t (Å) |  d (Å)
---|---|---|---|---|---|---
Se | 4.36 | $-$ | 114.29∘ | 0.63 | $-$ | $-$
SeO | 4.79 | 103.87∘ | 114.44∘ | 0.68 | 4.00 | 1.66
SeS | 4.80 | 108.12∘ | 110.79∘ | 0.91 | 5.04 | 2.06
TeS | 5.36 | 104.68∘ | 113.80∘ | 0.81 | 5.28 | 2.23

The optimized lattice constants and structural parameters of the proposed
(stable and unstable) two-dimensional (2D) systems are tabulated in Table 1.
Since, Se and TeS are dynamically unstable, we present the electronic band
structure of SeO and SeS only. These systems exhibit semi-conducting nature
(as evident from the electronic band structure presented in Fig. 5) with gaps
of the order of $\approx$ 10-3 eV (SeO $\approx$ 1.8 meV, SeS $\approx$ 2.3
meV) which can be exploited to realise topologically non-trivial behaviour
under the influence of modest tensile/compressive strain or external electric
field or under influence of spin-orbit coupling effects.

Figure 5: Electronic band structure of (a) SeO and (b) SeS with inset
indicating the semi-conducting gap along the high symmetry point K in the BZ.


# A Radio Continuum Study of NGC 2082

Balzan J. C. F. 11affiliationmark:
<EMAIL_ADDRESS>Filipović M. D. 11affiliationmark:
<EMAIL_ADDRESS>Dai S. 11affiliationmark:
22affiliationmark:
<EMAIL_ADDRESS>Alsaberi R. Z. E. 11affiliationmark:
<EMAIL_ADDRESS>Barnes L. 11affiliationmark:
<EMAIL_ADDRESS>
###### Abstract

We present radio continuum observations of NGC 2082 using ASKAP, ATCA and
Parkes telescopes from 888 MHz to 9000 MHz. Some 20 arcsec from the centre of
this nearby spiral galaxy, we discovered a bright and compact radio source,
J054149.24–641813.7, of unknown origin. To constrain the nature of
J054149.24–641813.7, we searched for transient events with the Ultra-Wideband
Low Parkes receiver, and compare its luminosity and spectral index to various
nearby supernova remnants (SNRs), and fast radio burst (FRB) local
environments. Its radio spectral index is flat ($\alpha=0.02\pm 0.09$) which
is unlikely to be either an SNR or pulsar. No transient events were detected
with the Parkes telescope over three days of observations, and our
calculations show J054149.24–641813.7 is two orders of magnitude less luminous
than the persistent radio sources associated with FRB 121102 & 190520B. We
find that the probability of finding such a source behind NGC 2082 is
$P=1.2\%$, and conclude that the most likely origin for J054149.24–641813.7 is
a background quasar or radio galaxy.

00footnotetext: 1Western Sydney University, Locked Bag 1797, Penrith South DC,
NSW 1797, Australia00footnotetext: 2CSIRO Astronomy and Space Sciences,
Australia Telescope National Facility, PO Box 76, Epping, NSW 1710, Australia

Keywords Radio continuum: galaxies, Galaxies: spiral, Galaxies: ISM

##  1 Introduction

In the absence of an active galactic nucleus (AGN), a spiral galaxy’s radio
emission primarily derives from non-thermal synchrotron radiation from
supernova remnants (SNRs), and thermal bremsstrahlung from Hii regions
(Condon, 1992; Filipović and Tothill, 2021b, a). Thus, deep and wide radio
surveys from the new generation of radio telescopes such as the Australian
Square Kilometre Array Pathfinder (ASKAP) and MeerKAT can shed important light
on the processes by which star formation shapes the interstellar medium (ISM).

NGC 2082 is G-type spiral galaxy (of SB(r)b morphology) in the Dorado
constellation. It has an absolute B-band magnitude of $M_{B}=12.79$ (Lauberts
and Valentijn, 1989), a diameter of 10.16 kpc, and is located at a distance of
18.5 Mpc (Olivares E. et al., 2010) and redshift $z=0.00395$. Unlike some
other galaxies in the Dorado constellation (e.g. NGC 1566), NGC 2082 remains
poorly studied, with its most notable feature being a type II supernova,
SN1992ba (Evans and Phillips, 1992).

Here, we study the radio properties of NGC 2082 using ASKAP, Australia
Telescope Compact Array (ATCA) and Parkes radio telescope observations. We
will also draw on Hubble Space Telescope (HST) observations. Section 2
presents our observations and data analysis of NGC 2082. Section 3 gives our
results and discussion, and conclusions are presented in Section 4.

##  2 Observations & Data

NGC 2082 has been observed in the ASKAP–EMU 888 MHz radio continuum survey of
the Large Magellanic Cloud (LMC; Pennock et al., 2021; Filipović et al.,
2022), as well as in the ATCA 20 cm mosaic survey (Filipović et al., 2021). We
have also made new observations from Parkes radio telescope, and obtained new
and archival data from ATCA (pre-CABB) and the HST.

###  2.1 Australian Telescope Compact Array

####  2.1.1 CABB

We observed NGC 2082 on 2019 November 30th using the Australian Telescope
Compact Array (ATCA) (project code C3275 with 1.5C array configuration). The
observations were carried out in ‘snap-shot’ mode, with 1-hour of integration
over a 12 hour period as a minimum. We used the Compact Array Broadband
Backend (CABB) (with 2048 MHz bandwidth), centred at a) wavelengths of 3/6 cm
($\nu$ = 4500–6500 and 8000–10000 MHz), totalling 43.2 minutes of integration
and, b) 13 cm ($\nu$ = 2100 MHz), totalling 43.2 minutes of integration. The
primary calibrator (flux) was PKS B1934–638 and secondary calibrator (phase)
was PKS B0530–727.

The miriad111http://www.atnf.csiro.au/computing/software/miriad/ (Sault et
al., 1995) and karma222http://www.atnf.csiro.au/computing/software/karma/
(Gooch, 1995) software packages were used for reduction and analysis. Imaging
was completed using the multi-frequency synthesis invert task with natural
Briggs weighting (robust=0 for all images), and beam size of $4.5\times 4.1$
arcsec, $1.9\times 1.8$ arcsec, and $1.3\times 1.0$ arcsec for 2100, 5500, and
9000 MHz images, respectively. The mfclean and restor algorithms were used to
deconvolve the images, with primary beam correction applied using the linmos
task. We follow the same process with Stokes Q and U parameters to produce
polarisation maps, except with beam size of $5\times 5$ arcsec (see Section
3.2 below).

####  2.1.2 Pre-CABB

We analysed archival333Australia Telescope Online Archive (ATOA), hosted by
the Australia Telescope National Facility (ATNF): https://atoa.atnf.csiro.au
ATCA data (project code C466 with 6A array configuration) from 16th November
1995. The observations used pre-CABB receiver (with 128 MHz bandwidth),
centered at wavelengths of 3/6 cm (4800 and 8640 MHz) totalling of 48 minutes
of integration. The primary calibrator (flux) was PKS B1934–638 and the
secondary calibrator (phase) was PKS B0355-483.

###  2.2 Australian Square Kilometre Array Pathfinder

NGC 2082 was observed serendipitously in the ASKAP-EMU radio continuum survey
of the Large Magellanic Cloud (Pennock et al., 2021), at the edge of the 120
deg2 field. This survey was performed at 888 MHz with a 288 MHz bandwidth and
$13.9\times 12.1$ arcsec beam size.

###  2.3 Parkes Radio Telescope

We observed NGC 2082 on 2021 June 14th, 15th and 18th with the Parkes radio
telescope (project code PX075) using the Ultra-Wideband-Low (UWL) receiver
(Hobbs et al., 2020), which delivers radio frequency coverage from 704 MHz to
4032 MHz. All observations were pointed at J054149.24–641813.7 and executed
using the transient search mode where data are recorded with 2-bit sampling
every 64 $\mu$s in each of the 0.125 MHz wide frequency channels (26624
channels across the whole band).

The full UWL band was split into multiple 512 MHz subbands for the search of
bursts. The search was performed using the pulsar searching software package
PRESTO (Ransom, 2001). Radio-frequency interference (RFI) were identified and
marked using the PRESTO routine RFIFIND with a 1 s integration time. To
determine the optimal dispersion measure (DM) steps of the search, we used the
DDPLAN.PY routine of PRESTO for a DM range of 200 to 3000 cm-3 pc. Data were
then dedispersed at each of the trial DMs using the PREPDATA routine with RFI
removal based on the mask file produced by RFIFIND.

Single pulse candidates with a signal-to-noise ratio larger than seven were
identified using the SINGLE_PULSE_SEARCH.PY routine for each dedispersed time
series and for different boxcar filtering parameters (from 1 to 300 samples).
Burst candidates were manually examined.

###  2.4 Hubble Space Telescope

NGC 2082 was first imaged by the Hubble Space Telescope in 1997, following the
type II supernova SN1992ba (Evans and Phillips, 1992), revealing a bright,
face-on spiral galaxy (Fig. 1). Fig. 1 is a 3-colour image created with APLpy
(Robitaille and Bressert, 2012), using archival HST data444Based on
observations made with the NASA/ESA Hubble Space Telescope, and obtained from
the Hubble Legacy Archive, which is a collaboration between the Space
Telescope Science Institute (STScI/NASA), the Space Telescope European
Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre
(CADC/NRC/CSA). (Carollo et al., 2002), where the red channel uses I-band data
(F814W filter), the blue channel uses B-band data (F435W filter), and the
green channel is pseudo-green which has been constructed by stacking the red
and blue channels (B+I-band). No detection of SN1992ba is reported; it cannot
be seen in any of these images.

Fig. 1 : 3-colour HST image of NGC 2082 overlaid with ASKAP and ATCA contours.
The blue contours are ASKAP 888 MHz (0.3, 0.5, 0.7, 0.9, 1.3, 1.4, 2.1 mJy
beam-1), and the light-blue, orange and red contours are ATCA 2100 MHz (0.4,
1.0, 1.5 mJy beam-1), 5500 MHz (0.5, 1.0, 1.5 mJy beam-1), and 9000 MHz
(0.4346, 1.5198, 3.0, 3.4 mJy beam-1) respectively. The inset image in the
bottom-left provides a zoom-in of J054149.24–641813.7, showing the absence of
any optical counterpart. The subplot in the top-right provides a zoom-in of a
888 MHz flux density peak of 1.3 mJy beam-1, almost directly opposite
J054149.24–641813.7. The magenta triangle denotes the position of SN1992ba.

##  3 Results and Discussion

A striking feature in all our radio images of NGC 2082 is strong point radio
source (J054149.24–641813.7) positioned 20 arcsec from the galaxy centre, as
seen in the bottom-left image of Fig. 1 by the ATCA contours. We also note no
detection of SN1992ba in any of our images. The 9000 MHz ATCA observations,
with our highest resolution, shows an unresolved point radio source,
regardless of the parameters used in the data reduction. The top-right subplot
in Fig. 1 provides a better look at an 888 MHz emission peak flux density of
0.0013 Jy beam-1 opposing J054149.24–641813.7, and also away from the centre
of galaxy. It is unlikely that the two radio sources are related. Finally, we
note HST observations with F435W and F814W filters show no optical
counterparts to either source, and there are no counterparts at other
wavelengths.

###  3.1 Spectral index

In Table 1 we show the flux densities of J054149.24–641813.7, measured using
CARTA555https://cartavis.org/ and treated as a point source. We assume that
the flux density errors are $<$10 per cent. We estimate a flat radio spectral
index of $\alpha=+0.02\pm 0.09$ suggesting that the emission is predominantly
of thermal origin if the source is located in NGC 2082 (Fig. 2). Such a flat
spectral index would be very unusual among SNRs and radio pulsar sources
(Urošević, 2014; Bates et al., 2013; Dai et al., 2015) unless this source is
an unresolved pulsar wind nebulae (PWN). However, a background galaxy (quasar)
could explain this radio spectrum (see Section 3.5).

Table 1 : The radio flux densities of NGC 2082’s extended radio emission and radio point source J054149.24–641813.7 (point source). A ’—’ represents a non-detection.  Freq. & Telescope | SNGC 2082 | S${}_{\rm point\,source}$
---|---|---
(MHz) | (mJy) | (mJy)
888 & ASKAP | 13.4$\pm$1.3 | 3.5$\pm$0.4
2100 & ATCA CABB | 11.2$\pm$1.1 | 2.7$\pm$0.3
4800 & ATCA Pre-CABB | — | 4.0$\pm$0.4
5500 & ATCA CABB | — | 4.0$\pm$0.4
8640 & ATCA Pre-CABB | — | 3.0$\pm$0.3
9000 & ATCA CABB | — | 3.6$\pm$0.4
$\alpha\pm\Delta\alpha$ | –0.15$\pm$0.23 | +0.02$\pm$0.09
Fig. 2 : The spectral index of J054149.24–641813.7 ($\alpha=0.02\pm 0.09$)
with assumed 10% error bars, based on the measured flux densities from Table
1.

To measure the flux densities of NGC 2082’s entire extended emission, we use
the method described in Hurley-Walker et al. (2019) and Hurley-Walker et al.
(2019), which includes careful region selection that also excludes
J054149.24–641813.7. We measure reliable NGC 2082 flux densities at two
frequencies (888 and 2100 MHz; Table 1), which allows us to estimate the
spectral index of $\alpha=-0.15\pm 0.23$. This flat radio spectral index is
unusual for spiral galaxies Gioia et al. (1982), but consistent with thermal
emission from Hii regions across NGC 2082. As the nucleus of NGC 2082 does not
show any radio compact source or emission above 0.1 mJy beam-1, we suggest
this might account for the unusually flat radio spectral index.

###  3.2 Polarisation

We also investigate if any polarisation from J054149.24–641813.7 or NGC 2082
can be detected in our ATCA images. The fractional linear polarisation ($P$)
of NGC 2082 was calculated using the equation:

$P=\frac{\sqrt{S^{2}_{Q}+S^{2}_{U}}}{S_{I}}~{},$ (1)

where $P$ is the mean fractional linear polarisation, $S_{Q}$, $S_{U}$, and
$S_{I}$ are integrated intensities for the $Q$, $U$, and $I$ Stokes
parameters, respectively. We calculate $P_{5500\,\text{MHz}}=6\pm 2\,\%$ (see
Fig. 3a) and $P_{9000\,\text{MHz}}=8\pm 4\,\%$ (see Fig. 3c). Their associated
polarisation intensity maps are seen in Figures 3b and 3d respectively. This
weak polarisation associated with J054149.24–641813.7 is most likely explained
if the source is of background origin (see Section 3.5).

(a) Fractional polarisation map of J054149.24–641813.7 at 5500 MHz, where
average $P_{5500\,\text{MHz}}=6\pm 2\,\%$.

(b) Polarisation intensity map of J054149.24–641813.7 at 5500 MHz, with
contours of 0.0003 and 0.002 Jy beam-1.

(c) Fractional polarisation map of J054149.24–641813.7 at 9000 MHz, where
average $P_{9000\,\text{MHz}}=8\pm 4\,\%$.

(d) Polarisation intensity map of J054149.24–641813.7 at 9000 MHz, with
contours of 0.0002 and 0.002 Jy beam-1.

Fig. 3 : Left: Fractional polarisation maps of the radio point source
J054149.24–641813.7, where the circle on the lower left corner represents a
synthesised beam of $5\times 5$ arcsec and the blue line below he circle
represents 100 per cent polarisation. The bar on the right side represents the
gray-scale gradients for the ATCA image in Jy beam-1. Right: Polarisation
intensity maps of J054149.24–641813.7, where the bar on the right represents
the colour-scale gradients for the ATCA image in Jy beam-1.

###  3.3 Luminosity of J054149.24–641813.7

Assuming a concordance cosmology with parameters provided by Planck
Collaboration et al. (2016), the luminosity distance of NGC 2082 is
$D_{L}=17.5\,\text{Mpc}$. The radio luminosity of J054149.24–641813.7 at 888
MHz is then $L_{888\text{MHz}}=1.29\times 10^{20}$ W Hz-1 (with
$\alpha=0.02\pm 0.09$). For comparison, the luminosity of the LMC’s SN 1987A
SNR (SNR 1987A) at its peak flux density ($S_{1000\text{MHz}}=0.15$ Jy;
(Turtle et al., 1987)) is $L_{1000\text{MHz}}=4.74\times 10^{16}$ W Hz-1 (with
$\alpha=-0.74\pm 0.02$ between 72 MHz and 8640 MHz (Callingham et al., 2016))
— 4 orders of magnitude smaller than J054149.24–641813.7. In addition to the
very different radio spectral indexes of J054149.24–641813.7 and SNR 1987A,
which indicate different emission origins, J054149.24–641813.7 is probably too
bright to be an SNR originating from NGC 2082.

###  3.4 An FRB embedded in J054149.24–641813.7?

Fast radio bursts (FRBs) are extremely bright transient events of unknown
origin (Lorimer et al., 2007). While FRBs can be associated with a variety of
types of galaxies (e.g., Bhandari et al., 2020), two of the most active
repeating FRBs 121102 (Chatterjee et al., 2017) and 190520B (Niu et al., 2021)
are associated with compact persistent radio sources (PRSs). They also show
large and variable DM and rotation measure (Michilli et al., 2018; Dai et al.,
2022), indicating extreme magneto-ionic local environment. The compact nature
of J054149.24–641813.7 and its location at the outskirt of NGC 2082 are
reminiscent of those of FRBs 121102 and 190520B.

FRB 121102’s PRS (QRS 121102) has a flat spectrum, with a spectral index of
$\alpha=0.07\pm 0.03$ (Resmi et al., 2021), and its luminosity at 1600 MHz is
$L_{1600\text{MHz}}=2.8\times 10^{22}$ W Hz-1 (Law et al., 2021) — two orders
of magnitude greater than the luminosity of J054149.24–641813.7,
$L_{1600\text{MHz}}\approx L_{888\text{MHz}}=1.29\times 10^{20}$ W Hz-1. Chen
et al. (2022) suggested that QRS121102 is too luminous to be an SNR,
consistent with our findings for J054149.24–641813.7 in Section 3.3.

190520Bs PRS (henceforth ”QRS 190520B”) has a luminosity
$L_{3000\text{MHz}}=3\times 10^{22}$ W Hz-1 and a spectral index of
$\alpha=-0.41\pm 0.04$ (Niu et al., 2021). Similarly, this is 2 orders of
magnitude greater than the luminosity of J054149.24–641813.7,
$L_{3000\text{MHz}}\approx L_{888\text{MHz}}=1.29\times 10^{20}$ W Hz-1.
Unlike QRS 121102 and J054149.24–641813.7, QRS 190520Bs spectral index is
steep and negative — suggesting non-thermal emission as its origin.

Our Parkes observations (see Section 2.3) over 3 days detected no transient
events. Despite this, considering both FRB 121102 and 190520B show sporadic
outbursts (Rajwade et al., 2020; Dai et al., 2022), it is plausible
J054149.24–641813.7 could host a repeating FRB and we observed during a
quiescent period.

###  3.5 J054149.24–641813.7 as a background object?

If intrinsic sources associated with J054149.24–641813.7 are implausible, the
most likely remaining possibility is an extragalactic background source, such
as a quasar, radio galaxy or AGN. If so, we may expect to see some Hi
absorption; however, there is currently no high resolution Hi data for NGC
2082. The flat spectral index together with somewhat weak polarisation at 5500
and 9000 MHz images argue in favour of J054149.24–641813.7 background origin.

Our observations (Tab. 1) show that the flux density at 5500 MHz is $\sim$4.0
mJy. From Wall (1994), we find that for observations at 5500 MHz there are
$\sim$15 sources/deg2 at $\geq$4.0 mJy. The probability of finding a source of
such brightness behind NGC 2082 is then,

$P=A\times 15\,\text{sources/deg}^{2}$

where $A$ is NGC 2082s area on the sky in deg2. We calculate $P=1.2\%$, given
the radius of NGC 2082 is $r=0.016\,\text{deg}$ (de Vaucouleurs et al., 1991).

##  4 Conclusions

Nearby spiral galaxy NGC 2082 was found to contain an bright, compact radio
source, J054149.24–641813.7 (Fig. 1), which is most likely of background
origin. The flux densities reveal J054149.24–641813.7 has a flat spectral
index, indicating its source may be of thermal origin. We compare the
luminosity of J054149.24–641813.7 to SNR 1987A, QRS 121102, and QRS 190520B,
finding that J054149.24–641813.7 is likely too bright and flat to be a
supernova, and is probably not bright enough to be a persistent radio source
with an embedded FRB progenitor.

Acknowledgements

The Australia Telescope Compact Array (ATCA) and Australian SKA Pathﬁnder
(ASKAP) are part of the Australia Telescope National Facility which is managed
by CSIRO. Operation of the ASKAP is funded by the Australian Government with
support from the National Collaborative Research Infrastructure Strategy. The
ASKAP uses the resources of the Pawsey Supercomputing Centre. Establishment of
the ASKAP, the Murchison Radio-astronomy Observatory, and the Pawsey
Supercomputing Centre are initiatives of the Australian Government, with
support from the Government of Western Australia and the Science and Industry
Endowment Fund. We acknowledge the Wajarri Yamatji people as the traditional
owners of the Observatory site. SD is the recipient of an Australian Research
Council Discovery Early Career Award (DE210101738) funded by the Australian
Government. This research has made use of the NASA/IPAC Extragalactic Database
(NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. This research made use of APLpy, an open-source plotting
package for Python (Robitaille and Bressert, 2012). This research is based on
observations made with the NASA/ESA Hubble Space Telescope, and obtained from
the Hubble Legacy Archive, which is a collaboration between the Space
Telescope Science Institute (STScI/NASA), the Space Telescope European
Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre
(CADC/NRC/CSA). We thank the anonymous referee for a constructive report and
useful comments.

## Data Availability

All data are publicly available:

  * •

Parkes (project code PX075):
https://data.csiro.au/

  * •

ATCA CABB (project code C3275) and
Pre-CABB (project code C466):
https://atoa.atnf.csiro.au/query.jsp

  * •

ASKAP (project code AS101):
https://data.csiro.au/

  * •

HST (proposal ID 9395):
https://hla.stsci.edu/hlaview.html

## Author Contribution

Miroslav Filipović and Shi Dai contributed to the original discovery,
conception and design of this study. Parkes data collection and analysis were
performed by Joel Balzan and Shi Dai. The first draft of this manuscript was
written by Joel Balzan and all authors commented on previous versions of the
manuscript. Rami Alsaberi observed and reduced ATCA data. All authors read and
approved the final manuscript.

## Funding

No funding was acquired for this research.

## Declarations

## Conflict of interest

All authors declare that they have no conflicts of interest.

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