# On Odd Radio Circles as Supernova Remnants

Sumit K. Sarbadhicary,1,2,3 Todd A. Thompson1,2,3 Laura A. Lopez2,3 and Smita
Mathur2,3
1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
2Center for Cosmology & Astro-Particle Physics, The Ohio State University,
Columbus, Ohio 43210, USA
3Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The origin of arcmin-sized Odd Radio Circles (ORCs) found in modern all-sky
radio surveys remain uncertain, with explanations ranging from starburst/AGN-
driven shocks to supernova remnants (SNRs) in the low-density ambient medium.
Using well-calibrated radio light curve models, we assess the possibility that
ORCs are radio SNRs evolving in low ambient densities. Our models imply that
ORCs 1-5 and J0624–6948 (near the LMC) as SNRs must be within 200 kpc and 100
kpc from the Sun respectively, given their observed flux densities and angular
sizes. To be evolving in the circumgalactic medium of the Milky Way, our
models require ORCs 1-5 to be ejecta-dominated SNRs within 50 kpc, evolving in
ambient densities of $(0.2-1.2)\times 10^{-3}$ cm-3. However, this is
statistically unlikely because ORCs 1-5 would have ages $<640$ yrs, much
smaller than their expected lifetimes of $\gtrsim$105 yrs at these densities,
and because the low SN rate and steep profile of the stellar halo imply a
negligible number of ORC-like SNRs within 50 kpc. The circumgalactic medium
SNR scenario for J0624–6948 is more likely (though still low probability)
compared to ORCs 1-5, as our models allow J0624–6948 to be $\lesssim$3000 yrs.
On the other hand, the interpretation of J0624–6948 as a Sedov-Taylor SNR at
50 kpc (LMC) distance is possible for a wide range of ambient densities
($6\times 10^{-4}-0.5$ cm-3) and ages $\sim$$(0.2-2.6)\times 10^{4}$ yr, while
also being consistent with the local HI environment.

###### keywords:

ISM: supernova remnants – acceleration of particles – shock waves – radio
continuum: ISM – (galaxies:) intergalactic medium

††pubyear: 2015††pagerange: On Odd Radio Circles as Supernova Remnants–A

##  1 Introduction

Recently, a class of low radio surface-brightness, circular-shaped objects has
been discovered by the Australian Square Kilometer Pathfinder Array (ASKAP)
Evolutionary Map of the Universe (EMU) survey (Norris et al., 2021b, a;
Filipović et al., 2022), the Giant Meterwave Radio Telescope (GMRT, Koribalski
et al., 2021) and the Low Frequency Array (LOFAR, Omar, 2022b). These objects,
named ORCs 1-5111We restrict our sample to the ORCs detected at $\sim 1$ GHz,
so we leave out the LOFAR ORC (Omar, 2022b) from our analysis for consistency.
However, as Omar (2022b) mentioned, the object is an arcmin across and would
have a 1 GHz flux density of 4.6 mJy assuming a spectral index $=-1$, similar
to ORCs 1-5. are between 1-3 arcmins in size, spanning integrated radio flux
densities of 0.1-7 mJy at 1 GHz. Observations of ORCs thus far have not
revealed any corresponding detection in X-ray or optical emission lines,
preventing distance measurements and potentially confirming the origin of the
structures. Three of the ORCs are coincident with an galaxy at redshifts of
0.2-0.6 in the center, raising the possibility that they are produced by few-
hundred kpc-scale shocks driven by a starburst/AGN (Norris et al., 2022) or
galactic mergers (Dolag et al., 2022).

Since the morphology of ORCs closely resemble supernova remnants (SNRs), but
without any X-ray or optical emission, we are motivated to consider whether a
subset of the observed objects may be SNRs in the extended halo of the Galaxy.
The majority of SNRs in the Galaxy and the Magellanic Clouds are observable or
were discovered in the radio band (Green, 2019; Badenes et al., 2010; Bozzetto
et al., 2017). Norris et al. (2021b) found that the latitude distribution of
the ORCs are higher than that of known SNRs in the Galaxy from the Green
(2019) catalog. The Galactic SNR population, however, is a heterogenous
compilation of SNRs from various different surveys, the majority of which were
restricted to the Galactic plane.

Omar (2022a) recently argued that ORCs could be SNRs evolving in the low-
density intragroup medium. Filipović et al. (2022) found an object (called
J0624–6948) that bears similarity to ORCs and is located in the outskirts of
the LMC, about 3∘ from the eastern edge of LMC’s radio continuum region
(3${}^{\circ}\sim$ 2.6 kpc if the object is at the 50 kpc distance of the LMC:
Pietrzyński et al. 2019a). Given its lack of any central galaxy, steep
spectral index, and large angular size, Filipović et al. (2022) preferred to
interpret J0624–6948 as an intergalactic Type Ia SNR produced by a progenitor
in the old LMC stellar halo, possibly the first discovered object of its kind.
Therefore, like other phenomena in astrophysics, it is possible that ORCs are
composed of two diverse populations, with one being relatively nearby SNRs in
the extended Milky Way halo and another being at extragalactic distances.

Meanwhile, more sensitive, contemporary radio surveys have been increasingly
discovering new SNRs (e.g. Anderson et al., 2017; Hurley-Walker et al., 2019;
Heywood et al., 2022), highlighting the incompleteness in the Galactic SNR
population due to selection effects from different sensitivity limits, source
confusion, and surveyed region of the sky (Green, 2019). Analysis of the
luminosity function of SNRs in external galaxies, where distances,
interstellar medium (ISM), and stellar population environments around SNRs are
well characterized, have shown that a non-negligible fraction of SNRs are
missing, and many of them could be of Type Ia origin evolving in low-density
ISM (Chomiuk & Wilcots, 2009; Sarbadhicary et al., 2017). Several new SNRs
have been discovered in recent surveys covering higher galactic latitudes,
with properties consistent with evolution in the low-density halo of the Milky
Way (Fesen et al., 2020, 2021; Becker et al., 2021; Churazov et al., 2021,
2022; Araya et al., 2022). SN Ia progenitors are expected to be abundant in
the old stellar disk and halo that extend beyond 1 kpc height above the
midplane (Mateu & Vivas, 2018). In distant galaxies, Type Ia SNe have been
observed to explode at scale heights of $\gtrsim 1$ kpc (Hakobyan et al.,
2017), and a small fraction even appear to be going off in the
intergalactic/cluster medium (Gal-Yam et al., 2003; Maoz et al., 2005; Sand et
al., 2011; Graham et al., 2015). If ORCs are indeed a new variety of SNRs,
they can serve as novel probes of the circumgalactic medium (CGM) environment,
shock physics in hot tenuous media, and SN progenitors in remote, low-
metallicity stellar populations.

In this paper, we use the radio light curve models of Sarbadhicary et al.
(2017) (hereafter, S17) to assess the possibility that the measured flux
densities and sizes of ORCs 1-5 and J0624–6948 are consistent with SNRs
evolving in a low-density, halo medium. The high Galactic latitudes of the
ORCs (favoring heights well above the Galactic midplane), the projected
location of J0624–6948 away from the main LMC disk, and the lack of X-ray and
optical emission from any of these objects suggest a hot, low-density
environment. These make the ORCs distinct from the Galactic SNRs near the
plane, and therefore models of synchrotron emission from SNRs as a function of
ambient medium density are needed to properly characterize these objects. The
S17 models have been used previously to reproduce observations of extra-
galactic and Galactic SNRs in a diversity of environments (S17, Russell et
al., 2020) as well as young Type Ia SNRs evolving in the warm/hot interstellar
medium, such as G1.9+0.3 and SN 1885A (Sarbadhicary et al., 2019).

The paper is organized as follows. In Section 2, we summarize the the physics
of the S17 light curve model, and assumptions about model parameters. Section
3 describes the evolution of the light curves at different values of the
ambient density and electron acceleration efficiency. We then assess whether
these light curves can reasonably reproduce the observed properties of ORCs
1-5 and J0624–6948 .

##  2 Method

Radio detection of SNRs depends sensitively on the ambient density, explosion
energy, age, properties of the accelerated electrons producing synchrotron
radiation, and the SN rate, which in turn depends on the stellar mass, age,
and metallicity distributions. In S17, we developed a model of SNR radio light
curves to interpret SNR populations evolving in different environments with
different explosion properties and used it to understand the properties of
SNRs in M33. Here, we use this model to understand under what conditions and
for what parameters radio SNRs may explain ORCs.

###  2.1 Radio Light Curve Model of SNRs

The S17 model predicts the synchrotron radio luminosity of SNRs by combining
the synchrotron radio emission models of Chevalier (1998) with the self-
similar non-radiative shock models of Truelove & McKee (1999). The radio
emission model assumes that radio emission is synchrotron radiation produced
by electrons accelerated in the turbulent magnetic field amplified by the SN
shock (Fermi, 1949; Bell, 1978; Blandford & Ostriker, 1978). The S17 radio
luminosity models reproduce the radio luminosity function and sizes of SNRs in
M33 (Gordon et al., 1999; Chomiuk & Wilcots, 2009) and are consistent with the
luminosity-sizes of Galactic SNRs from Case & Bhattacharya (1998). The model
was also applied to individual SNRs, including SN1885A and G1.9+0.3
(Sarbadhicary et al., 2019) and other extragalactic populations of SNRs (e.g.
in M83, Russell et al., 2020). The analytical nature of the S17 model is ideal
for scanning the wide parameter space of SNR shocks with observations, and it
was verified by Truelove & McKee (1999) and later studies, such as Tang &
Chevalier (2017), that these smooth solutions connecting the ejecta-dominated
phase through Sedov-Taylor evolution are consistent with numerical results to
within a few percent.

Qualitatively, the S17 model predicts that the radio luminosity of a SNR
initially increases in the ejecta-dominated stage and then decreases in the
Sedov-Taylor stage (Figure 2 in S17). The initial increase is due to the
emitting volume increasing faster than the rate of deceleration (which sets
the magnetic field and cosmic ray electron energy densities). In the Sedov-
Taylor stage, the shock experiences stronger deceleration due to the swept-up
ISM, causing the deceleration effects to dominate and the radio luminosity to
decrease. Higher radio luminosities are expected for greater ISM densities and
ejecta energies, and larger remnants with longer lifetimes are expected for
lower ISM densities and higher ejecta energies. These results are consistent
with numerical investigations of the radio emission from SNRs (e.g. Berezhko &
Völk, 2004; Pavlović et al., 2018).

###  2.2 Model assumptions

We assume a standard 1.4 M⊙ ejecta mass, a 1051 erg Type Ia explosion with an
ejecta density profile of $\rho_{ej}(v)\sim v^{-n}$ (here $\rho_{ej}$ is the
ejecta density, $v$ is the ejecta velocity, and the power-law index $n=10$ as
assumed in S17 and Chomiuk et al., 2012), interacting with a uniform density
ambient medium. Type Ia SNe are associated with old stellar populations (Maoz
et al., 2014), and their remnants are generally more symmetric than core-
collapse SNRs in multi-wavelength images (Lopez et al., 2011; Peters et al.,
2013). The remote locations and symmetric morphology of the ORCs can therefore
be more easily explained with a Type Ia origin. In the case of J0624–6948 ,
Filipović et al. (2022) argued that a core-collapse origin is only possible
with a high-velocity progenitor, with required velocities of 80-400 km/s at
the distance of LMC to explain its current location. This scenario is
statistically rare and also in tension with the symmetric morphology of the
ORCs.

The light curve models depends on the assumed ambient density and temperature,
magnetic field strength, parameters describing the amplified magnetic field
and accelerated electron spectrum, observing frequency and the radio detection
limit. We explore a broad range of parameters to see whether or not SNRs of
the angular size, brightness, and frequency of ORCs can be produced in the
halo of the Galaxy.

####  2.2.1 Ambient Density and Temperature

We assume densities $n_{0}=10^{-1}-10^{-4}\ \mathrm{cm}^{-3}$ that are typical
of the warm-hot phase of the interstellar/circumgalactic medium (CGM) (Gupta
et al., 2012) where we believe such SNRs may be evolving. For example, the ORC
J0624–6948 had an ISM density constrained to $\sim 7\times 10^{-3}$ cm-3 based
on the size of the SNR, assuming a distance of 50 kpc and that the SNR is in
the Sedov-Taylor stage (Filipović et al., 2022). Such low densities are also
consistent with the lack of any X-ray or forbidden line emission tracing the
ORCs. Consistent with the symmetric morphology of the ORCs (Norris et al.,
2021b, 2022; Filipović et al., 2022) as well as Type Ia SNRs (Badenes et al.,
2007; Lopez et al., 2011), we assume that the ambient medium is homogeneous.

We set the temperature of the surrounding medium to be $10^{6}$ K, similar to
measurements of the Galactic CGM and the virial temperature of a Milky-Way-
sized halo (Mo et al., 2010; Das et al., 2019; Gupta et al., 2021; Martynenko,
2022). While the supersonic SNR shock evolution does not directly depend on
the temperature, the duration of the light curve will be affected by the
larger sound speed and lower Mach-number shocks (Tang & Wang, 2005). We
explore this caveat further in Appendix A, where we find that while the Tang &
Wang (2005) model predicts a deviation away from the conventional Sedov-to-
radiative transition in SNR shocks, the difference in the radio light curve
evolution and lifetime of the SNR is not significant. We therefore continue
using our default S17 model dynamics.

####  2.2.2 Electron acceleration physics

The radio luminosity depends on the physics of electron acceleration, which
remains an active topic of research (Caprioli, 2015; Marcowith et al., 2016).
Chevalier (1998) accounted for the energy in the magnetic field and
accelerated electrons by parameterizing the fraction of the shock energy going
into the amplified magnetic field as $\epsilon_{b}$, the fraction of shock
energy going into the accelerated electrons as $\epsilon_{e}$ and the spectral
index of the accelerated electron spectrum as $p$, i.e. $N(E)\propto E^{-p}$
for $E>m_{e}c^{2}$.

Similar to most SN and SNR studies using the Chevalier et al. models, we
assume these parameters to be constant, with the exception of $\epsilon_{b}$
where we take $\epsilon_{b}\propto\epsilon_{p}(v_{s}/c+1/M_{A})$ (S17), where
$\epsilon_{p}\approx 0.1\%$ is the cosmic-ray proton acceleration efficiency
(Lacki et al., 2010), $v_{s}$ is the shock velocity, and $M_{A}$ is the Alfvén
Mach number. The relation follows results from particle-in-cell simulations
from Caprioli & Spitkovsky (2014b, a) that show that magnetic fields are
amplified in the shock precursor by streaming cosmic-ray protons initially
driving non-resonant instabilities and later, resonant instabilities as the
shock decelerates (Bell, 1978, 2004).

The cosmic-ray spectrum has typically been observed to have $p\approx 2.2-2.4$
in young SNRs (Caprioli, 2012, and references therein), so we assume a median
value of $p=2.3$. More recent simulations by Diesing & Caprioli (2021) suggest
that the spectrum will likely steepen at higher shock velocities, which may
affect the shorter-duration ejecta-dominated phase in our light curves. We
plan to incorporate this in future iterations of the S17 model.

The $\epsilon_{e}$ parameter is typically the least constrained and
understood. Values vary between $\epsilon_{e}=0.1$ for SNe/GRBs (Chevalier &
Fransson, 2017, and references therein) to $10^{-2}-10^{-4}$ in young SNRs
(e.g. Morlino & Caprioli, 2012; Reynolds et al., 2021). In S17, we found that
$\epsilon_{e}\sim 10^{-3}-10^{-2}$ for the sample of SNRs in M33 that are
mostly in the Sedov-Taylor stage. This is also consistent with expectations
from the Galactic cosmic-ray spectrum where the fraction of shock energy in
cosmic-ray protons $\epsilon_{p}\sim 0.1$ (e.g. Lacki et al., 2010) and the
normalization ratio of the cosmic ray protons and electrons $K_{ep}\sim
10^{-2}$ (Park et al., 2015), so $\epsilon_{e}\sim K_{ep}\epsilon_{p}\sim
10^{-3}$. We assume for our purposes that $\epsilon_{e}=10^{-3}-10^{-2}$, and
we report results for variation in this range.

####  2.2.3 Ambient Magnetic field

In S17, the luminosity depends on the amplified magnetic field
$B\sim(8\pi\epsilon_{b}\rho v_{s}^{2})^{1/2}$, which is much larger than the
ambient magnetic field ($B_{0}$). This relation however means that
$B\rightarrow 0$ as $v_{s}\rightarrow 0$, so in S17 we set
$B=\mathrm{max}(B,4B_{0})$ to ensure the post-shock field in the non-radiative
stage is at least a simple compression of the ambient field, similar to the
models of Berezhko & Völk (2004) and observations of old SNRs (e.g. Loru et
al., 2021). This causes the synchrotron light curve to plateau in the late
Sedov-Taylor stage because the effects of expansion and deceleration of the
shock roughly cancel out (Fig 2 in S17; also seen in the numerical simulations
of Berezhko & Völk, 2004). This plateau luminosity in the late Sedov-Taylor
stage becomes dependent on the assumed value of $B_{0}$.

The value of the ambient magnetic field $B_{0}$ in the halo of the Milky Way
is uncertain. S17 assumed $B_{0}\sim(9\mu\mathrm{G})n_{0}^{0.47}$, based on
Crutcher (1999) who showed from Zeeman splitting measurements of molecular
clouds that $B_{0}\sim n_{0}^{0.47}$, and global measurements of $B_{0}\sim
5-9$ $\mu G$ from radio intensity and polarization maps of spiral galaxies
(Beck, 2015). While this dependence accounts for local variations in $B_{0}$
in the star-forming ISM, it implies that $B_{0}\ll 1\mu\mathrm{G}$ in the
ambient Galaxy halo where $n_{0}$ may be $\lesssim 10^{-3}$ cm-3. While star-
forming galaxies tend to have $1-10$ kpc-scale radio haloes where the
equipartition fields are of order $\mu G$ (e.g. Planck Collaboration et al.,
2016; Mora-Partiarroyo et al., 2019), the value of $B_{0}$, on 100 kpc scales
is not directly known. As the supernova remnant slows its expansion, the
shock-amplified field weakens, and eventually the magnetic field energy
density ($U_{B}$) will eventually becomes less than the energy density of the
background cosmic microwave background radiation field ($U_{cmb}$),where
cooling via inverse-Compton scattering will dominate over synchrotron cooling.
Thus, the value of $B_{0}$ on 100 kpc scales matters to the late-time radio
luminosity of remnants. For the purposes of this paper, we set
$U_{B}=U_{cmb}$, where $U_{B}=B^{2}/8\pi$ and $U_{cmb}=(4\sigma/c)T_{cmb}^{4}$
where $\sigma$ is the Stefan-Boltzmann constant, we get a lower limit on
$B_{0}=B_{low}\approx 2\mu\mathrm{G}(T_{cmb}/2.7\mathrm{K})^{2}$. We assume
$B_{low}=2\mu$G for $T_{cmb}=2.7$K, and therefore
$B_{0}=\mathrm{max}(B_{0},B_{low})$.

####  2.2.4 Detection Limit

We assume that the SNR will remain detectable as long as it is 3$\sigma$ above
the RMS noise of the survey. In this paper, we use the detection limits of the
ASKAP survey of Norris et al. (2021a), which reached an RMS = 30$\mu$Jy/bm, so
the detection limit is about 90$\mu$Jy for objects smaller than the
synthesized beam (or point-spread function) of 12′′. The limit for objects
larger than the beam size will scale as the number of beams spanned by the
object, i.e. $\sqrt{D^{2}/\theta^{2}}$, where $D$ is the angular size of the
object, and $\theta$ is the angular size of the beam.

We adopt two characteristic distances for the SNRs, as in Filipović et al.
(2022): 5 kpc, representing the Milky Way stellar halo, and 50 kpc,
representing the LMC distance for assessing J0624–6948 . At 5 kpc, the
detection limit corresponds to a 1-GHz luminosity limit of $2\times 10^{18}$
ergs/s/Hz and a beam physical size of 0.29 pc, while at 50 kpc (LMC), the
luminosity limit is $2.7\times 10^{20}$ ergs/s/Hz and the beam physical size
is 2.9 pc. At these distances, most SNRs will be resolved in radio surveys, so
their visibility will be surface-brightness-limited, as seen in Figure 1.

Figure 1: The 1 GHz SNR radio luminosity vs diameter predicted by the S17
model for $n_{0}$ = [$10^{-1},10^{-2}$, $10^{-3}$ , $10^{-4}$] cm-3 (from top-
left to bottom right) for different values of electron acceleration efficiency
$\epsilon_{e}$ = [$10^{-2}$, $10^{-3}$] (solid, dashed blue lines). Horizontal
scale with ticks represent predicted SNR ages for a given diameter. Blue
circles at the end of the light curves mark the time when $M_{s}=1$, i.e. the
shock merges with the ISM. Shaded regions denote 3$\sigma$-detection limit
(90$\mu$Jy/bm) of the EMU survey at 5 kpc (representative of the Milky Way
Halo) and 50 kpc (representative of LMC). Purple square with error-bars
represent J0624–6948 at a distance of 5 kpc and 50 kpc. The diagonal dashed
line with the vertical tick marks represent distances at which we assume ORCs
1-5 are located for the discussion in Section 3.2. The light curves indicate
dimmer, smaller and longer lived SNRs with decreasing densities, and they
would remain detectable above $3\sigma$ for the entirety of their lifetimes
only for $n_{0}>10^{-3}$ cm-3 and $\epsilon_{e}>10^{-3}$ at 5 kpc and
$\epsilon_{e}>10^{-2}$ at 50 kpc.

####  2.2.5 Observing Frequency

We predict light curves at 1 GHz, roughly the frequency at which the ORCs were
discovered (Norris et al., 2021b, a; Filipović et al., 2022, e.g.). Both older
SNRs (Dubner & Giacani, 2015) and the ORCs (Norris et al., 2021b; Filipović et
al., 2022) are bright at these low frequencies because they have steep spectra
($\alpha\sim 0.5-0.6$) that are characteristic of non-thermal emission. Most
modern synthesis arrays such as VLA, MeerKAT and ASKAP can observe in the
0.8-2 GHz frequency range. Historically, such low frequencies have been
challenging to observe because of radio-frequency interference and dynamic
range issues, but these have been substantially mitigated in modern surveys
with sophisticated algorithms (e.g. Bhatnagar et al., 2013; Tasse et al.,
2018), making it feasible to obtain sensitive images where objects can be
securely identified

####  2.2.6 When does radio emission terminate?

When exactly radio emission stops in the SNR lifetime is difficult to model
because of the poorly understood physics of electron acceleration. The SNR
phase ends when the shock velocity becomes of the order of the sound speed of
the ISM, but even before this stage, a cooling front will develop at the shock
due to radiative losses (Draine, 2011). Additionally, cosmic ray acceleration
becomes inefficient, and more cosmic rays escape from the post-shock region,
as the shock slows (Blasi et al., 2007; Xu et al., 2020). As a result, models
typically assume that radio emission stops at the onset of radiative phase
(e.g. S17, Berezhko & Völk, 2004). However, at such high ambient temperatures
(and possibly low metallicites if exploding in the halo), there is not a well-
defined radiative phase as the internal energy is not radiated away before the
SNR merges with the ISM (Tang & Wang, 2005; Karpov et al., 2020; Li et al.,
2020). We also note that radio emission from low-Mach number shocks
($M_{s}\sim 2$) has been observed in galaxy clusters and attributed to quasi-
perpendicular shocks that preferentially accelerate electrons (Guo et al.,
2014; Xu et al., 2020).

We therefore conservatively assume that radio emission continues until the
shock merges with the ambient medium (i.e. $M_{s}=1$). Here
$M_{s}=v_{s}/c_{s}$, where $v_{s}$ is the shock velocity, and $c_{s}=(0.12\
\mathrm{km/s})\mu^{-1/2}T^{1/2}$ is the isothermal sound speed of the ambient
medium. We assume $\mu=0.6$ and $T=10^{6}$ K for a hot, fully-ionized ambient
medium, making $c_{s}=155$ km/s.

##  3 Results and Discussion

###  3.1 Radio properties of SNRs at low ambient densities

We show our predicted 1-GHz luminosities and diameters as a function of SNR
age for different ISM densities from $10^{-1}$ (top-left panel) to $10^{-4}$
cm-3(bottom-right panel) for two different values of
$\epsilon_{e}=10^{-2},10^{-3}$ (solid, dashed) in Figure 1. As mentioned in
Section 2.1, the luminosity increases with age during the ejecta-dominated
phase and decreases during the Sedov-Taylor phase. The radius monotonically
increases with time, initially as $R_{s}\propto t^{0.7}$ for our assumed
$\rho_{ej}\propto v^{-10}$ profile, and then $R_{s}\propto t^{0.4}$ during the
Sedov-Taylor phase. The plateau portion in the late Sedov-Taylor phase is due
to $B\rightarrow 4B_{0}$, as explained in Section 2.2.3. Note that the plateau
luminosity is the same for all panels since we assume the same $B_{low}$
(defined in Section 2.2.3). Overall, the SNR evolves more slowly as the
ambient density decreases, as shown by the vertical age ticks shifting
rightwards in Figure 1. So for a given size, a SNR evolving in a lower density
ambient medium will be younger than a higher density one. The radio emission
is assumed to stop once the shock merges with the ISM, effectively ending the
SNR phase.

As seen in Figure 1, for the current detection limits of ASKAP (marked by the
gray shaded regions for 5 and 50 kpc), any SNR out to 5 kpc from the Sun will
be detectable for the entirety of its lifetime, provided it is evolving in
ambient densities $n_{0}>10^{-3}$ cm-3, with $\epsilon_{e}\geq 10^{-3}$.
Sources are brighter at a given radius when expanding into a higher density
medium (except during the plateau phase set by B0 in Section 2.2.3).
Similarly, the more efficient the electron acceleration, the brighter the SNR.
The size of the SNR is larger at lower densities for a given age
($R_{s}\propto n_{0}^{-1/5}$ during the Sedov-Taylor phase). The final size of
the modeled SNRs, set by when $M_{s}\rightarrow 1$, is larger at low densities
because $M_{s}\propto v_{s}\propto n_{0}^{-1/5}$ during the Sedov-Taylor
phase). With decreasing density, the SNR becomes fainter and can drop below
the detection limit of the survey. For example, at $n_{0}=10^{-4}$ cm-3, the
SNR will not be detectable at the distance of LMC, unless it has an
$\epsilon_{e}>10^{-3}$. For $n_{0}=10^{-2}$ cm-3, SNRs are detectable
throughout their lifetime at the distance of 5 kpc. At 50 kpc, SNRs are still
detectable, though for $\epsilon_{e}<10^{-2}$ they start to become surface
brightness limited during the late Sedov phase.

To summarize, SNRs in the Milky Way halo, assuming a fiducial distance of 5
kpc, are likely to be detectable at $>3\sigma$ throughout their lifetime in
surveys that can achieve EMU-like sensitivity, if they are evolving in ambient
densities $\gtrsim 10^{-3}$ cm-3 and $\epsilon_{e}>10^{-3}$. The same is true
for SNRs at the distance of LMC, but for $\epsilon_{e}\gtrsim 10^{-2}$. The
detectability and visibility time noticeably decreases for lower densities,
for larger distances, and for lower $\epsilon_{e}$ values.

###  3.2 Can ORCs 1-5 be SNRs?

Figure 2: Ambient density needed to reproduce the 1GHz luminosities and
diameters of ORCs 1-5 at a given distance from the Sun, assuming
$10^{-3}\leq\epsilon_{e}\leq 10^{-2}$. From 0-100 kpc, the required density
increases with distance as the ORC is in ejecta-dominated phase, while at
100-200 kpc, it is in the Sedov-Taylor phase, as explained in Section 3.2.1.
Grey colorscale represents the logarithmic age of the SNR. The red shaded
region represents the density profile of the Milky Way CGM used by Fang et al
(2013) based on the models of Maller & Bullock (2004), explained further in
Section 3.2.2.

We discuss here whether the ORCs 1-5 can be reproduced by our radio light
curve models of SNRs at low densities. Since their distances are
unconstrained, we calculate their luminosities and diameters at increasing
distances from the Sun and check if the values are reproducible by our light
curve models. Since ORCs 1-5 are similar in size and flux density, we use a
median projected size of 80′′and median flux density of 4 mJy for comparison
with models. We then estimate the range of ambient densities versus distance
within which the observed radio luminosities and diameters coincide with the
model light curves for values of $\epsilon_{e}=10^{-3}-10^{-2}$.

####  3.2.1 Required ambient density and age range

Our results for plausible densities at various distances, are summarized in
Figure 2, and we refer the reader to the light curves in Figure 1 for
reference. We also plot the age of the SNR, since older and larger SNRs are
more frequent in a SNR population than younger and smaller ones.

Figure 2 suggests an upper limit of $\sim$200 kpc, which coincidentally is
similar to the virial radius of the Milky Way (Dehnen et al., 2006). This can
be understood from Figure 1. As the distance increases, the ORCs are more
luminous and have larger diameters for a given flux density and projected
size. In our models, more luminous SNRs can be produced in a denser ambient
medium, but this also results in smaller SNRs with shorter lifetimes as they
decelerate and eventually merge more quickly with the ambient medium. In other
words, more distant SNRs move towards the top right of Figure 1, while model
light curves in denser ambient media move towards the top left. Therefore,
beyond a certain distance, the light curves and the observations cannot
coincide, yielding an upper limit to their distance. We note that Omar (2022a)
had similarly estimated that ORCs can be interpreted as intra-group medium
SNRs going off at $0.1-3$ Mpc distances assuming a 500 pc upper limit to the
SNR diameter, though we find that these are unlikely to be farther than the
virial radius of the Milky Way.

Within 80-200 kpc, the ORCs are consistent with a wide range of ambient
densities spanning $\sim$$10^{-3}-2$ cm-3, resulting in the parameter space
extending vertically in Figure 2. This is because at around 100 kpc, the ORCs
coincide with the Sedov-Taylor part of the radio light curves as seen in
Figure 1, where luminosity decreases with diameter. With increasing density,
the 100 kpc luminosity and diameter is still encompassed by the Sedov-Taylor
part of the light curve, but at progressively older ages. This trend continues
up to densities of $\sim 2$ cm-3, above which the predicted light curves are
over-luminous and larger compared to the ORCs.

At distances less than 80 kpc, the required ambient density range is less than
$6\times 10^{-3}$ cm-3, and this range decreases with decreasing distance as
seen in Figure 2. This can again be understood from Figure 1 where the SNR
luminosities and diameters for distances less than 80 kpc can only be
encompassed by the ejecta-dominated part of the light curves for a smaller
range of ambient densities than the Sedov-Taylor stage. The ages of the ORCs
also decrease with decreasing distance, as shown by the progression of the
gray colorscale with distance in Figure 2. For example, ORCs at 4-10 kpc can
only be reproduced by ambient densities $\sim(0.3-1)\times 10^{-3}$ cm-3 and
SNR ages of a few decades, and at 40 kpc, by an ambient density of $\sim
5\times 10^{-4}$ cm-3 and ages of a few hundred yrs. We note that the youngest
SNRs known in the Local Group are SN1885A and G1.9+0.3, both Type Ia SNRs with
ages of about 137 yrs and 100-140 yrs, respectively (Fesen et al., 2007;
Reynolds et al., 2008).

####  3.2.2 Comparison with Milky Way Circumgalactic Medium

Is the required density parameter space consistent with densities expected in
the Milky Way CGM? The CGM primarily consists of hot, low-density, ionized gas
(Spitzer, 1956; Tumlinson et al., 2017; Das et al., 2021), and a variety of
observations along multiple sightlines through the CGM indicate decreasing
densities with distance, with typical values of 10-3 cm-3 at few kpc from the
Sun, and 10-5 cm-3 out to the virial radius of the Milky Way (see discussions
in Das et al., 2019; Bhattacharyya et al., 2022). For this paper, we compare
our ORC density parameter space with the widely used CGM model of Maller &
Bullock (2004), which assumes the CGM as a hot, multi-phase adiabatic gas in
hydrostatic equilibrium with the dark matter halo. We use the form of the
model used by Fang et al. (2013), who found consistency between the model and
a variety of observations such as X-ray emission measures (Henley & Shelton,
2010; Gupta et al., 2012; Gupta et al., 2017, e.g.), pulsar dispersion
measures along the LMC/SMC (Crawford et al., 2001; Manchester et al., 2006),
and non-detection of neutral hydrogen in dwarf galaxies within 300 kpc of the
Milky Way (Grcevich & Putman, 2009), while also being able to account for the
missing baryonic mass of the Milky Way. We use the form of the density profile
defined in Eq (1) of Fang et al. (2013). The density range at a given distance
is based on the assumption in Fang et al. (2013) that the integrated hot gas
mass of the Milky Way CGM within the virial radius is in the plausible range
of $0.5-2.7\times 10^{11}$ M⊙.

We show our results in Figure 2. The density profile is somewhat anti-
correlated with the required density versus distance constrained for the ORCs.
Considering that most SNRs are found in their Sedov Taylor phase at ages of a
few $\times 10^{4}$ yr, ORCs 1-5 as SNRs would most likely be evolving in
ambient densities greater than 10-2 cm-3 at a distance of $\sim 100-200$ kpc,
where the parameter space in Figure 2 gets darker. However, these densities
are several orders of magnitude higher than the CGM density profile in Figure
2 at those distances. While turbulence and inhomogeneities can produce
overdense regions in the bulk CGM (Oppenheimer et al., 2016; Bhattacharyya et
al., 2022), these are not expected to be high enough to explain the required
densities for ORCs.

One can also consider ORCs 1-5 to be SNRs at distances less than 50 kpc since
their required density-distance parameter space overlaps with CGM densities in
Figure 2, but such a scenario conflicts with the low SN rate within this
volume and with the required ages of $\lesssim 640$ yr, as shown by the
greyscale in Figure 2. These ages are much smaller than their total lifetimes
of nearly $10^{5}$ yr (Figure 1), making it highly improbable that they will
be observed at such an age. Given the total stellar halo mass of $1.4\times
10^{9}$ M⊙ (Deason et al., 2019) with an average age of 10 Gyrs, the currently
measured form of the SN Ia delay-time distribution of $(4\times
10^{-13}\mathrm{yr^{-1}M_{\odot}^{-1}})(t/\mathrm{Gyr})^{-1}$ (Maoz &
Mannucci, 2012) implies a SN Ia rate of $5.6\times 10^{-5}$ yr-1. Assuming
these SNRs would be visible up to 105 yr based on Figure 1, we expect only 6
SNRs in the halo. The stellar halo density profile is roughly $n_{*}(r)\sim
r^{-3}$ (e.g. Slater et al., 2016; Iorio et al., 2018), which means the number
of stars at a given distance is
$N_{*}(r)\sim\int_{r_{0}}^{r_{1}}n_{*}(r)d^{3}r\sim\mathrm{ln}(r_{1}/r_{0})$.
Thus, assuming a uniform distance distribution of SNe between $r_{0}=1$kpc and
$r_{1}=260$kpc (roughly the virial radius), there is only a 19$\%$ probability
of stars exploding within 50 kpc, meaning only $1-2$ SNRs are likely to exist
within this distance today, and as mentioned earlier, these SNRs would most
likely be older and larger SNRs. Indeed, some of the recently discovered and
confirmed radio-faint halo SNRs by eROSITA are a few degrees across (Churazov
et al., 2021, 2022; Becker et al., 2021), much larger than the arcmin-sized
ORCs.

####  3.2.3 Effects of model assumptions

We checked if the results are robust to deviations from the fiducial values of
model parameters in Section 2.2. The biggest effect comes from the assumption
about $\epsilon_{e}$. A greater range of $\epsilon_{e}$ enlarges the allowed
density-distance parameter space in Figure 2 because a larger range of
luminosities are now allowed by the models in Figure 1. For example, changing
the $\epsilon_{e}=0.1$ leads to an upper limit of 350 kpc for ORCs 1-5,
although such efficient electron acceleration efficiencies may not be physical
in SNRs. The stopping condition, $M_{s}=1$ does not have an effect on the
distance upper limit, but affects the maximum density allowed by the models as
it leads to smaller final sizes for SNRs. E.g. if we use a stopping condition
of $M_{s}=5$, the maximum density in Figure 2 reduces from 1 cm-3to 0.1 cm-3.
Changing $B_{low}$ within a factor of 2 of its fiducial value, i.e. $0.5-4$
$\mu$G does not change the maximum allowed distance of 200 kpc, and produces a
minor shift, i.e. by few kpc, in the gray colorscale parameter space in Figure
2 to larger distances. This is because a larger $B_{low}$ increases the
luminosity of the plateau portion of the light curve, producing a more
luminous SNR at a given diameter.

####  3.2.4 Summary

In summary, our S17 model implies that if ORCs 1-5 are SNRs, they must be
going off within 200 kpc, and if they are interacting with the bulk CGM, they
are most likely going off within 50 kpc in densities of $(0.2-1.2)\times
10^{-3}$ cm-3. However, statistically their interpretation as SNRs is unlikely
given their very young required ages and the low SN rate at the above distance
range.

Figure 3: Same as Figure 2, but for ORC J0624–6948 . Vertical red line denotes
the distance to LMC of 50 kpc, while horizontal red line denotes the line of
sight HI density of 0.017 cm-3. See Section 3.3 for details.

###  3.3 Can ORC J0624–6948 be an SNR?

We apply the above analysis to J0624–6948 as well, and show the results in
Figure 3. Similar to ORCs 1-5, the distance to J0624–6948 is unconstrained,
but its close proximity to LMC also raises the possibility that it is located
at the distance of LMC (50 kpc, Pietrzyński et al., 2019b). We assume
J0624–6948 has a flux density of 11.16 mJy (based on scaling the ASKAP 888 MHz
flux density of 11.7 mJy to 1 GHz assuming a spectral index of $-0.4$ as
constrained by Filipović et al., 2022) and use the measured angular size of
196′′. We find a distance upper limit of about 100 kpc in Figure 3, about a
factor of 2 less than ORCs 1-5. The density-distance parameter space behaves
similar to the case of ORCs 1-5. The vertical region between $\sim 40-80$ kpc
is where the observed luminosity and diameter at a given distance is
reproduced by the Sedov Taylor part of the SNR light curves bounded by the
values for $\epsilon_{e}=[10^{-3},10^{-2}$], while for distances less than 40
kpc, the observations are reproduced by the ejecta-dominated portion of the
light curve (Figure 1).

We first consider the scenario where J0624–6948 is evolving in the Milky Way
CGM and happens to be projected along the LMC. Indeed there is some overlap
between the CGM density profile and the ambient densities required by our
model within a distance of 100 kpc, as shown in Figure 3. The maximum age that
an ORC can have in this overlap region is about 3000 yr (based on the gray
colorscale in Figure 3). Compared to ORCs 1-5 in Section 3.2.1, the allowed
age range for J0624–6948 in the CGM scenario is 3 times larger, but still much
smaller than their lifetimes of $\sim 10^{5}$ yr. The CGM scenario is
therefore more feasible for J0624–6948 compared to ORCs 1-5, but still
statistically unlikely due to the required young age.

On the other hand, the interpretation of ORC J0624–6948 as a Sedov-Taylor SNR
in the LMC outskirts appears to be far more feasible based on Figure 3. At 50
kpc, the observed properties of J0624–6948 can be reproduced by a wide range
of densities ($6\times 10^{-4}-0.5$ cm-3) and ages $\sim(0.2-2.6)\times
10^{4}$ yr. As discussed in Filipović et al. (2022), the best available
constraint in the literature on the local environment comes from the HI4PI
survey (HI4PI Collaboration et al., 2016), which showed the presence of a
diffuse HI cloud, about a degree across, along the line of sight of J0624–6948
. The cloud is detected in emission integrated within the velocity range of
190-320 km/s, similar to the velocity component of the LMC shown in their
Figure 9. With a column density of $6\times 10^{19}$ cm-2 after opacity
correction, and taking the lateral extent of the cloud of 0.9 kpc as its line
of sight depth, Filipović et al. (2022) got an average density of 0.017 cm-3,
which falls well within the vertical density-distance region in our Figure 3.
If the ORC is indeed evolving inside this cloud at 50 kpc, the ORC would
correspond to an age of about 7000 years.

To summarize, the interpretation of J0624–6948 as an SNR is more feasible
compared to ORCs 1-5. Its interpretation as an ejecta-dominated SNR evolving
in the CGM is more likely than ORCs 1-5 (though its required age is still much
less than its expected lifetime at CGM densities) and its interpretation as a
Sedov-Taylor SNR evolving in the diffuse HI environment in the LMC outskirts
is consistent with the plausible model parameter space in Figure 3. The total
stellar mass of the LMC is about an order an magnitude smaller than the Milky
Way (van der Marel et al., 2002; Harris & Zaritsky, 2009; Licquia & Newman,
2015), so we expect the stellar halo mass to be similarly small. Thus,
extending the SN rate calculation in Section 3.2.1 to the LMC stellar halo
implies $\lesssim$ 1 SNR, which is consistent with J0624–6948 being the only
ORC-like SNR discovered near the LMC in the EMU survey (Filipović et al.,
2022). It is possible therefore that some fraction of ORCs are SNRs like
J0624–6948 evolving in the low-density halo outside the star-forming disks of
galaxies. Deep multi-wavelength follow-ups of existing ORCs, and continued
deep radio-continuum surveys at high Galactic latitudes as well as the halo of
nearby galaxies is warranted to further characterizing the nature of ORCs.

## Acknowledgements

SKS acknowledges support from the CCAPP Postdoctoral Fellowship, and useful
discussions with Joy Bhattacharya, David Weinberg, and Sanskriti Das at the
weekly OSU Galaxy/ISM meetings. LAL acknowledges support from the Heising-
Simons Foundation. This research made use of Astropy, a community-developed
core Python package for Astronomy (Astropy Collaboration et al., 2018, 2013),
NumPy (Harris et al., 2020), matplotlib, a Python library for publication
quality graphics (Hunter, 2007), and NASA’s Astrophysics Data System. The
acknowledgements were compiled using the Astronomy Acknowledgement Generator.

## Data Availability

No new data were generated in support of this research. All analyzed data are
publicly available in the citations mentioned in the text. The light curve
model is hosted on GitHub.

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##  Appendix A Modified light curves in hot ISM

Figure 4: Sonic Mach number ($v_{s}/c_{s}$) and radio luminosity as a function
of age for SNR for our default S17 model (dashed) and the S17 model including
the deviation expected in the Sedov Taylor stage at high ambient temperatures
(solid), based on Tang & Wang (2005). Red circle and square denotes the age
when $M_{s}=2$ for the S17 light curve and modified light curve for hot ISM
respectively. Case here is for $n_{0}=10^{-2}$ cm-3, $T=10^{6}$ K.

S17 model uses equations based on SNR evolution in the cold ISM ($T\lesssim
10^{4}$ K), where the SNR develops a cooling shell as it transitions from the
Sedov to radiative phase. However, in the hot ($10^{6}$ K) ISM, the SNR never
reaches the radiative phase as cooling becomes inefficient. Tang & Wang (2005)
verified this with 1D simulations and found that at late times the shock
radius and velocity evolution deviates from the Sedov-Taylor solution, and the
shock velocity asymptotically reaches the sound speed of the ISM. The shock
velocity in such a scenario was parameterized by Tang & Wang (2005) as

$v_{s}(t)=c_{s}\left(\frac{t_{c}}{t}+1\right)^{3/5}$ (1)

which gives the shock radius as
$R_{s}(t)=\int_{0}^{t}v_{s}(t^{\prime})dt^{\prime}$. Here $t_{c}$ is a
characteristic time defined in Tang & Wang (2005) and corresponds to when the
thermal energy added to the SNR by the swept-up ISM becomes twice the
explosion energy. The expression goes as

$t_{c}=\left[\left(\frac{2\xi}{3}\right)^{5}\frac{E_{SN}}{\rho_{0}c_{s}^{5}}\right]^{1/3}$
(2)

where $\xi=1.14$, $E_{SN}$ is the SN explosion energy, $\rho_{0}=\mu
n_{0}m_{p}$ is the particle density in g cm-3, and $c_{s}$ is the sound
velocity of the plasma.

We check the difference introduced by these results in the radio light curves
in Figure 4. As shown in Tang & Wang (2005), the solution for the shock
velocity deviates from the Sedov solution and asymptotically reaches the sound
speed for the corresponding ISM temperature at $M_{s}=1$. The radio luminosity
evolution closely follows the Sedov-Taylor solution we had in our S17 model,
but starts to deviate around the same time the velocity solution deviates.
Ultimately, the light curve appears to increase rapidly, as the shock velocity
changes very slowly, while the shock emitting volume increases rapidly,
similar to the cause for increasing luminosity in the ejecta-dominated phase.
Both the visibility time and luminosity in our modified Sedov solution at
$M_{s}=2$ is within a factor of 2 of the corresponding values for our S17
model, which is a relatively small effect compared to the systematic
uncertainties introduced by other parameters in our model (e.g.
$\epsilon_{e}$) that affect the luminosity by order of magnitude.

While we could adopt this solution into our model for the present study, we
refrain from doing that until these have been tested properly with observed
SNRs, particularly since it predicts a rise in luminosity at late times.
Additionally, these results are borne of 1D simulations, which do not
necessarily capture physics at higher dimensions that can affect the shock
evolution, such as fluid instabilities that enhance mixing of the hot and cold
phases (Gentry et al., 2019).


[acronym]long-short acronymnohyperfirsttrue

# End-to-End Lidar-Camera Self-Calibration for Autonomous Vehicles
††thanks: We gratefully acknowledge that this work has been supported by the
Bavarian State Ministry of Economic Affairs, Regional Development, and Energy
(StMWi) under the EmPer project (grant no. DIK 0179/01).

Arya Rachman1 2 Jürgen Seiler1 André Kaup1
1Multimedia Communications and Signal Processing 2Product Segment Autonomous
Driving Friedrich-Alexander-Universität Erlangen-Nürnberg AVL Software and
Functions, GmbH Erlangen, Germany Regensburg, Germany Email:
<EMAIL_ADDRESS>
###### Abstract

Autonomous vehicles are equipped with a multi-modal sensor setup to enable the
car to drive safely. The initial calibration of such perception sensors is a
highly matured topic and is routinely done in an automated factory
environment. However, an intriguing question arises on how to maintain the
calibration quality throughout the vehicle’s operating duration. Another
challenge is to calibrate multiple sensors jointly to ensure no propagation of
systemic errors. In this paper, we propose Camera Lidar Calibration Network
(CaLiCaNet), an end-to-end deep self-calibration network which addresses the
automatic calibration problem for pinhole camera and Lidar. We jointly predict
the camera intrinsic parameters (focal length and distortion) as well as
Lidar-Camera extrinsic parameters (rotation and translation), by regressing
feature correlation between the camera image and the Lidar point cloud. The
network is arranged in a Siamese-twin structure to constrain the network
features learning to a mutually shared feature in both point cloud and camera
(Lidar-camera constraint). Evaluation using KITTI datasets shows that we
achieve 0.154$\degree$ and 0.059 $\mathrm{m}$ accuracy with a reprojection
error of 0.028 pixel with a single-pass inference. We also provide an ablative
study of how our end-to-end learning architecture offers lower terminal loss
(21% decrease in rotation loss) compared to isolated calibration.

###### Index Terms:

self-calibration, Lidar, camera, end-to-end learning, autonomous vehicle,
multi-sensor

##  I Introduction

Modern vehicles are equipped with multi-modal sensors, enabling perception
algorithms to better understand the environment. Compared to classical camera-
only perception, adding depth-capable sensors such as Radar and Lidar enables
unambiguous recognition in a 3D world. Therefore, a camera and Lidar
combination is the most used in autonomous vehicle setup, often with multiple
instances of the same modality (e.g., 2$\times$ Lidar plus 3$\times$ camera).

The use of such multi-sensor setups comes with the need for cascaded
calibration: first, each sensor needs to be both calibrated intrinsically
relative to its internals, and after that, it needs to be extrinsically
calibrated to other sensors. Both calibrations are prerequisites for sensor
fusion, a building block of vehicle perception. In order to perceive the
environment consistently, the vehicle has to use sensor measurements in a
common coordinate system.

The quality of intrinsic calibration generally impacts extrinsic calibration.
Minor errors in the intrinsic calibration may affect the extrinsic
calibration’s accuracy. To tackle this problem, jointly-optimized intrinsic
and extrinsic calibration is proposed in[1][2]. The approach offers better
performance compared to isolated calibration. However, it requires a static
target which is not applicable when a vehicle is in use. For this case, a
targetless calibration method is needed.

The classical model-based self-calibration based on Structure-from-Motion
(SfM) [3], optical flow[2], and odometry[4] also exist. However, in order to
accommodate a broad operating domain and avoid heuristically tuned algorithms,
deep-learning-based approaches have been increasingly used for self-
calibration (see [5, 6, 7]). Notwithstanding, the comparison between model-
based vs. deep-learning-based approaches remains difficult since there is
little-to-no back-to-back comparison, and each approach has a different
operating domain, not necessarily automotive. Finally, the deep learning
approach primarily relies on supervised networks: this means there is a need
for a sufficient amount of labelled, high quality data for the network to
properly generalize.

Our previous work[8] introduced a self-calibration pipeline to perform
intrinsic camera calibration using deep neural networks and a back-to-back
comparison against a target-based method. In this paper, we extend the
pipeline with a novel end-to-end learning network called CaLiCaNet, aiming to
calibrate a camera intrinsically from scratch and, simultaneously,
extrinsically to a Lidar. This is a distinction from the majority of existing
Lidar-Camera calibration methods that assume perfect camera intrinsic
parameters to be available.

To measure the impact of the proposed architecture, we use evaluation metrics
that are directly comparable to conventional pattern-based calibration and
benchmark our proposal to a dataset that reflects the target domain: KITTI
Datasets[9]. We also include an ablation study treating the intrinsic and
extrinsic networks as sub-networks. The aim is to understand how end-to-end
networks perform better than isolated networks for Lidar-Camera calibration.

To summarize, our contributions consist of the following:

  1. 1.

We propose end-to-end CaLiCaNet, which to the best of our knowledge is the
first application of end-to-end learning for the purpose of sensor
calibration.

  2. 2.

We evaluate our approach using KITTI Dataset with the specific metrics
directly comparable to the conventional calibration method. This facilitates
an informed decision to switch from the state-of-the-practice to the state-of-
the-art calibration approach.

  3. 3.

We propose a streamlined data collection and label generation pipeline for a
self-calibrating automotive Lidar-Camera system.

##  II Related Works

Lidar-Camera extrinsic calibration is a well-visited topic. Typically, as a
first step, a camera is intrinsically calibrated using a geometric pattern
like a checkerboard[10]. Then, its extrinsic calibration to Lidar is done by
matching 3D features from the Lidar point cloud to the camera equivalent
(i.e., in the image plane). The 3D features from the camera image can be
generated from the 3D checkerboard plane or 3D reconstruction method, such as
SfM, given the intrinsic parameters.

In practice, the camera intrinsic calibration and Lidar-Camera extrinsic
calibration are done (1) sequentially and (2) performed using a specialized
pattern before the sensor system is used for intended operation (i.e.,
offline). Calibrating the camera first, followed by the Lidar-Camera, poses
the risk of a systemic error propagating to extrinsic calibration when
intrinsic parameters are not sufficiently accurate. Offline calibration also
does not consider mechanical shifts occurring during the sensor system
operation. The shift is especially relevant for an automotive system,
requiring modern vehicles equipped with an array of perception sensors to be
periodically recalibrated.

To address the first challenge, we refer to the formulation of joint Lidar-
Camera calibration as proposed in [11, 12, 1]. The general idea is to reach a
more robust and globally optimal solution bounded by physical measurements of
multi-modal sensors. With regard to the second challenge, a global
optimization is ideally done without the need for a pattern-like target and
uses a natural environment to enable the possibility of online calibration.
The SfM-based approach[11] comes close to our needs: natural features are
detected using SfM and used as inputs to the Bundle Adjustment algorithm to
predict intrinsic parameters. Lastly, Iterative Closest Point (ICP) is used
for extrinsic calibration.

Notwithstanding, feature detection and matching using an explicit model
require a stable and feature-rich environment and, therefore, generally is not
very robust when applied to online calibration. Furthermore, in [11], the
evaluation using KITTI dataset is limited to road scenarios (e.g., stable
environment), similar limitation can also be found in [3], where it relies on
pillar-like structures for the calibration to work. The model-based feature
extractor, such as SfM or odometry, typically requires heuristics parameter
tuning that needs to be changed once the operating domain (e.g., camera type
or environment) is shifted.

Acknowledging these general difficulties, deep-learning-based perception
sensor calibration is proposed, as seen in [5, 6, 7]. The main idea is to
leverage well-validated and pre-trained image detectors (e.g., ImageNet,
Inception, ResNet), which are conventionally used for image classification and
object detection, as an environmental feature extractor.

For automotive purposes, DeepPTZ[7] is able to calibrate vehicle cameras with
the extension we provided in [8]. We also consider LCCNet[6] and CFNet[13],
which evaluate their method directly on KITTI Dataset for Lidar-Camera
calibration. Still, these deep-learning approaches have yet to address the
need to jointly calibrate camera intrinsic and camera extrinsic.

Unlike the classical model-based online calibration, we believe there is no
learning-based solution yet that performs Lidar-Camera calibration end-to-end,
the implicit assumption being that perfect camera intrinsic is always
available and always rigid notwithstanding with a vehicle dynamics (e.g.,
vibration) influencing sensor and outdoor environment (e.g., temperature)
affecting sensor mechanical setup.

Finally, to the best of our knowledge, end-to-end learnings were widely
investigated, albeit within a limited domain (e.g., self-driving[14] or text
recognition[15]). Therefore, the behaviour of end-to-end network design is
only little visited in the field of vehicle perception, especially sensor
calibration.

Figure 1: End-to-end Lidar-Camera Self-Calibration Pipeline with CaLiCaNet.
The parts denoted by the dashed line are only used during network training.

##  III Self-Calibration Problem

###  III-A Problem Formulation

For the camera, we consider the pinhole model following [16], by which lens
distortion is modelled using the Unified Spherical Model (USM). The camera
intrinsic calibration is formulated as recovering intrinsic parameters
$\mathbf{\textit{K}}_{i}$, and distortion coefficient $\xi_{i}$ from $i$ input
images.

Meanwhile, the Lidar extrinsic calibration is described by a rigid 3D
transform consisting of rotation $\mathbf{\textit{R}}_{i}$ and translation
$\mathbf{\textit{t}}_{i}$. For each image $i$ and point cloud $(X,Y,Z)_{i}$, a
point in the 3D world can be projected using matrix $\mathbf{\textit{K}}$ to
camera pixel coordinate $(u,v)$, assuming an undistorted image and no scaling
is needed:

$\left[\begin{array}[]{l}u\\\ v\\\
1\end{array}\right]=\mathbf{\textit{K}}[\mathbf{\textit{R}}\mathbf{\textit{t}}]\left[\begin{array}[]{l}X\\\
Y\\\ Z\\\ 1\end{array}\right]$ (1)

When the intrinsic camera parameters and Lidar-Camera parameters estimation
are treated within a unified optimization problem, we get a better chance of
coming to a globally optimal solution[17]. The advantage is especially
relevant since joint optimization provides the physical constraints recognized
by both Lidar and the camera.

Based on the above formulation, we delineate our goal of end-to-end Lidar-
Camera self-calibration to the acquisition of calibration values
$\theta=\\{{f},\mathbf{\xi},\mathbf{\textit{R}},\mathbf{\textit{t}}\\}$ given
sets of image and 3D point cloud pair from natural scenes. The image sensor is
assumed to be square (focal length $f$ is symmetrical along the $x$\- and
$y$-axis), and the Lidar scan is assumed to be ego-motion-compensated. The
constraint is that the measurement does not need to have an explicit geometric
pattern nor be taken in a specific environment with certain structure features
(e.g., pole or building). Considering the difficulty of modelling such an
environment, we defer to a learning-based approach.

###  III-B Deep Learning by Driving: Generating Training Data

Deep learning approaches for sensor calibration are largely supervised and
thus require labelled ground truth. We refer to the learning-by-driving label
generation strategy previously proposed in [8]. Our training label generation
strategy does not need human annotation nor additional infrastructure not
already existing in a modern car and automotive supply chain. First, we pre-
calibrate a pair of camera and Lidar sensor with a checkerboard-based,
controlled environment setup to get $\theta_{0}$. We consider $\theta_{0}$ as
ground truth calibration values. We then collect the sensor measurements while
the car is driving with the target calibration sensors.

The resulting measurements become a dataset of time-series calibrated frames
(point cloud and image). Analogous to augmentation, realistic (i.e., based on
known calibration deviation over a long-term driving period) $\theta_{label}$
are generated together along with sets of distorted images and miscalibrated
point clouds, as seen in Fig. 1.

##  IV End-to-End Design

###  IV-A Isolated Networks

In this work, we adopt networks with public reference implementation as
benchmarks and inspirations for our end-to-end design. Following (1), the
output of an intrinsic calibration network can be chained directly to an
extrinsic calibration network. In this case, the prediction and training of
each intrinsic and extrinsic network happen in isolation.

For the intrinsic networks, we adopt some part of DSNet[7], a Siamese-network
designed to predict camera intrinsics calibration $f$ and $\xi$ with inputs of
two correlated images from different viewpoints. It was originally designed to
work with panoptic datasets and panning, rotation, and zoom (PTZ) camera, but
we have extended it in [8] to work with vehicle cameras as the target domain.
As for extrinsic calibration, inspired by LCCNet[6], we use a volume-cost-
based network that correlates the volume cost shared by the image and the
projected point cloud depth image. Both isolated networks rely on features
correlation to regress $\theta$.

Figure 2: End-to-End vs Isolated Self-Calibration. Intrinsic: red/green halos
indicate pixel-wise differences. Extrinsic: projected point cloud to an image
representing alignment quality. Red arrows point to misalignment regions of
interest.

###  IV-B CalicaNet

CaLiCaNet (Camera Lidar Calibration Network) aims to enable effective end-to-
end learning by building upon feature extraction networks to associate
features from two correlated sensor measurements and regress them into
calibration parameters.

Recall that the DSNet feature map is extracted based on two-correlated
monocular images to enforce a bi-directionality constraint. While this has
been shown to help the network to focus on relevant features, the features
from monocular cameras are inherently two-dimensional. Making the network
susceptible to the famous ”Wile E. Coyote’s tunnel” (a tunnel imagery being
painted on a wall to induce illusion of depth). We address the problem by
enforcing the constraint from two different sensor modalities: Lidar and
camera. Compared to a single modality approach, environmental attenuations
specific to a camera (e.g., low light or fake depth) are mitigated by a Lidar,
and vice versa (e.g., dark object or sparsity).

On the other hand, LCCNet relies on the assumption that the input image is
perfectly undistorted and that the camera intrinsic matrix $\mathbf{K}$ is
sufficiently accurate. If the lens undistortion is inaccurate, for example,
the Lidar-Camera extrinsic prediction can be overfitted to the centre region
of the image. More importantly, LCCNet requires multi-pass inferences to
achieve sufficient accuracy for autonomous driving. With our CaLiCaNet, we
mitigate this problem by explicitly incorporating intrinsic parameters in the
loss function and implementing the Lidar-Camera constraint by means of weight
sharing. Further details of CaLiCaNet can be found in Fig. 1 as well as in the
following subsections:

####  IV-B1 Pre-processing: ensuring consistent input

We discard featureless images (e.g., frames dominated by sky and rural, empty
road) by computing SIFT descriptors on each frame and passing only frames with
a certain threshold of key-points count (see [18] for details). Additionally,
we also ensure the overlap with the Intersection of Union (IoU)) of $\geq 0.5$
between an RGB image and the corresponding Lidar 2D projection. These measures
are intended so that the network sees more consistent input during training
and inference.

####  IV-B2 Network Architecture

We use ResNet50[19] as a feature extractor and implement the necessary fully
connected layers for generating the correlation between inputs. Since we aim
to use single-pass inference from natural scenes, we choose a deeper network
block. In contrast, DSNet uses Inceptionv3, and LCCNet uses ResNet18. We also
aim to accept an arbitrary resolution on the training images and point cloud,
owing to the diversity of vehicle sensors; therefore, we place an adaptive
average pooling before the fully connected layer. Furthermore, we extend the
volume-cost-based correlation module from [6] to work with a Siamese
structure. Finally, we opt to use Parametrized ReLU (PReLU) for our network
extractor activation function. The ReLU weight, therefore, becomes a trainable
parameter. Our functional network architecture as part of the end-to-end
pipeline is described in Fig. 1.

####  IV-B3 Loss Functions

The network minimizes the L2-smooth loss $L$ between regressed calibration
values of $\theta_{pred}$ and that of label $\theta_{label}$. Except for
$\mathbf{R}$ that is converted to equivalent quaternion representation
$\mathbf{q}$, and therefore quaternion distance is treated as a loss. For
vehicle perception purposes, the misalignment in translation does not affect
vehicle perception as much as rotation (refer to Fig. 2 for an example of
misalignment due to rotation). Therefore, we penalize the network more for
predicting the wrong rotation. The training loss $L_{T}$ thus becomes:

$L_{T}=\lambda_{f}L_{f}+\lambda_{\xi}L_{\xi}+\lambda_{q}L_{q}+\lambda_{t}L_{t}$
(2)

with $\lambda_{q}\geq\lambda_{t}$

##  V Evaluation

We propose two evaluation strategies: the first is evaluating with KITTI
Dataset to check the performance in a realistic situation. This serves as
validation of our approach. The second strategy is an ablation study to
examine closer how an end-to-end architecture enhances the network
performance; this serves as verification of our network design. Such two-step
approach is a necessary first step in developing safety-critical software
running on a vehicle[20].

###  V-A Experimental Setup

####  V-A1 Dataset

To evaluate the calibration accuracy, we use the KITTI Dataset [9], in which
the RGB images captured by the right-side monocular camera are used. Ground-
truth calibration values $\theta_{0}$ from the KITTI dataset are generated as
follows: we use the OpenCV Omnidirectional Calibration library [21] to perform
intrinsic calibration for each daily drive using the corresponding
checkerboard recordings. Following the models stated in Section III-A, we
obtain $\theta_{0}$ and the corresponding baseline reprojection root mean
square (RMS) error $\epsilon_{0}$ [16].

####  V-A2 Label Generation and Training

The first day of the KITTI drive (2011-09-26) is used as training labels,
while the remaining four days are used for evaluation. We generated the labels
by setting the labels’ value to $\theta_{0}$ plus a variable deviation (i.e.,
the miscalibration). The label generation refers to physical changes on the
sensors induced by mechanical and environmental factors during everyday
driving and is set to $\theta_{label}=\\{{f}\pm
100$\mathrm{p}\mathrm{x}$,\mathbf{\xi}\pm 0.48,\mathbf{R}\pm
2.0\degree,\mathbf{t}\pm 0.2$\mathrm{m}$\\}$. The distorted image and
misaligned point cloud pairs are generated according to $\theta_{0}$. In
total, we generated approximately $120\,000\text{\,}\mathrm{l}$abels
($\theta_{label}$, image and point cloud) split into 80:20 ratios for training
and validation datasets. Adam Optimizer is used with an initial learning rate
of $3\text{\times}{10}^{-3}$. The batch size used was 60 with maximum epochs
of 300.

####  V-A3 Evaluation Metrics

For calculating the intrinsic calibration accuracy, we use KITTI’s
checkerboard which contain $i$ corners
$\left\\{p_{1j},p_{2j},\cdots,p_{ij}\right\\}\in\mathbb{R}^{3}$ from $j$-th
checkerboard poses $\mathbf{R}\mathbf{t}$. The purpose is to calculate the
Root Mean Squared Error (RMSE) of the reprojection $\mathbf{\epsilon}$ in
pixels, given as:

$\epsilon=\sqrt{\frac{1}{N}\sum_{i=1}\sum_{j=1}\left\|\mathbf{K}[\mathbf{R}\mathbf{t}]p_{ij}-x_{ij}\right\|_{2}}$
(3)

Note that this reprojection error value obtained based on $\theta_{0}$ is
considered as the baseline reprojection error $\epsilon_{0}$. When our network
infers $\theta_{pred}$, the detected corners and checkerboard poses are reused
to calculate inference reprojection error $\epsilon_{pred}$. We consider
$\epsilon_{pred}$ to be more representative of calibration quality relative to
the comparison of the intrinsic part of $\theta_{0}$ to $\theta_{pred}$. We
refer to [16] for more in-depth reasoning.

On the other hand, for extrinsic calibration, for the sake of comparability,
we opt to follow the convention in prior works [6, 22, 11] by directly
comparing the extrinsic part of $\theta_{0}$ to that of $\theta_{pred}$, that
is the mean error of rotation and translation. In this case, the baseline
extrinsic error is assumed to be zero.

###  V-B KITTI Drives

The evaluation using KITTI drives is intended to show CaLiCaNet accuracy and
robustness when applied to diverse driving scenarios. Referring to Fig. 3, the
mean error below 0.028 $\mathrm{p}\mathrm{x}$ across all four days of driving
shows that our approach is comparable to classical checkerboard-based
intrinsic calibration in terms of reprojection error. The significance of this
low error can be visually inspected in Fig. 2. When the DSNet network is
trained in isolation, we noted an error increase of almost 20% relative to our
proposed end-to-end approach (see drive 2011-09-28 in Fig. 3). Additionally,
the undistortion result in Fig. 2 (second image from the left) corroborates
the increased error. Up to 15 pixels differences can be seen on the edge
region of the image, where the distortion effect is more present.

Figure 3: Intrinsic and extrinsic calibration error. For intrinsic, the baseline is the best achievable RMS reprojection error obtained with the checkerboard-based calibration. For extrinsic, we calculate the mean error of all 3D axes. DSNet[7] and LCCNet[6] were trained isolated and only single pass inference was used. See Fig. 2 for the accompanying visualization and Table I for mean values across all drives. TABLE I: Average Intrinsic and Extrinsic Calibration Errors Calibration Error | Baseline[21] | DSNet [7] | LCCNet[6] | Ours
---|---|---|---|---
Intrinsic [px] | 0.018 | 0.024 | - | 0.021
Extrinsic $\mathbf{t}$ [m] | - | - | 0.082 | 0.060
Extrinsic $\mathbf{R}$ [$\degree$] | - | - | 0.204 | 0.154

A similar effect can be found in the extrinsic calibration: the end-to-end
approach results in better Lidar-Camera alignment. Due to the intended use of
online calibration, we put a constraint that the inference must happen in a
single pass (no iteration). Under this constraint, we achieve a noticeably
better mean rotational accuracy compared to when the LCCNet is trained in
isolation (0.154$\degree$ vs. 0.203$\degree$).

We can infer some robustness by inspecting the statistical fluctuation of
CaLiCaNet prediction in Fig. 3. Across four days of driving with notably
diverse sceneries (city, campus, residential, and road, see[9] for more
details) the maximum intrinsic and extrinsic calibration errors remain below
0.028 $\mathrm{p}\mathrm{x}$ and 0.4$\degree$/0.2$\mathrm{m}$, respectively.
For extrinsic error, our outlier values are decisively lower than that of
LCCNet, which we attribute to the Lidar-Camera measurement constraint realized
by the weight sharing. Notwithstanding, we note the lowered accuracy in drive
2011-10-03 and bigger spread since the drive is dominated by highway sceneries
(i.e., fewer edge and line-like features to extract). With this finding, we
consider it beneficial if the car drives in a feature-rich environment (e.g.,
inner city) when the self-calibration functionality is active.

Overall, we consider the KITTI Drives evaluation result to appropriately
reflects CaLiCaNet fitness for vehicle sensors self-calibration. It fulfils
its purpose of avoiding a miscalibration, as illustrated in Fig. 2; which can
and will compromise the road object localization capability of a road vehicle.

TABLE II: Ablation Study Experiments Exp. | RGB train? | Depth train? | Siamese? |  $f$-loss |  $\xi$-loss |  $\mathbf{R}$-loss |  $\mathbf{t}$-loss |
---|---|---|---|---|---|---|---|---
I | ✗ | ✓ | ✗ | - | - | $6.658\mathrm{E}-4\hskip 8.53581pt$ | 0.0014 |
II | ✓ | ✗ | ✗ | 11.41 | 0.070 | - | - |
II | ✓ | ✓ | ✗ | 11.41 | 0.068 | $6.054\mathrm{E}-4\hskip 8.53581pt$ | 0.0010 |
IV | ✓ | ✓ | ✓ | 11.02 | 0.067 | ${4.758\mathrm{E}-4\hskip 8.53581pt}$ | 0.0010 |

###  V-C Ablation Study

We conducted ablation studies to evaluate the influence of end-to-end training
on Lidar-Camera accuracy. Essentially, this study attempts to pinpoint which
part of the pipeline is responsible for the improved performance. The
evaluation metrics used are the validation terminal loss.

We used the identical setup in Section V-A, but varied our training strategy
and the corresponding network architecture. First, the feature extractors are
no longer sharing weights. In experiment I, we froze the layers of the RGB
extractor before training the network. In experiment II, we froze the layers
of the depth extractor instead. In experiment III, all layers of the feature
extractors are trainable, but the weight sharing is not activated. Finally, in
Experiment IV, the normal end-to-end training with Siamese weight sharing was
performed. The results can be seen in Table II.

It is quite evident that end-to-end training results in lower losses overall.
The extrinsic parameters regressor (especially rotation with a decrease of
21%) benefits the most when trained in both end-to-end and Siamese
configurations (see $\mathbf{R}$-loss and $\mathbf{t}$-loss). However, we have
observed that intrinsic calibration losses ($f$-loss and $\xi$-loss) do not
appear to have a significant effect when trained end-to-end (loss decrease of
less than 1%). However, when the weight sharing is activated, we see a modest
decrease in intrinsic losses (3%). This may suggest that the feature-
extracting networks cannot properly generalize the relevant feature for camera
intrinsic parameters.

The behaviour of the intrinsic calibration network can be potentially
explained by insufficient edge coverage from the KITTI checkerboard recording.
The distortion on the far-edge region of the image is not fully modelled and
may result in suboptimal baseline calibration $\theta_{0}$, and by extension,
$\theta_{label}$. In addition, the USM model used does not consider the
tangential and radial distortion[23]. Based on this study, we learned the
limitation of our label generation strategy, which heavily relies on good
checkerboard recording. For real applications and future works, the training
label should be based on a checkerboard recording of sufficient coverage and
tangential and distortion models should be considered.

##  VI Conclusion

We have proposed a novel end-to-end learning strategy for jointly self-
calibrating camera and Lidar sensor with the accompanying deep neural network
CaLiCaNet. The pipeline proposed a label generation strategy with a normally
operating vehicle and existing calibration infrastructure. It also ensures the
quality of the inputs to the self-calibration pipeline by means of detecting
SIFT key-points and detecting Lidar-Camera measurements overlap. Our main
contribution lies in the design of CaLiCaNet, consisting of two feature
extractor networks (one for camera image and one for Lidar point cloud)
arranged in a Siamese structure. The design is devised to deal with the
limitation of using single sensor modality (i.e., bottlenecked by a single
sensor physical limitation) and mitigating the propagation of error when
estimating extrinsic calibration using unverified intrinsic calibration.

Finally, we performed thorough verification and validation based on an
ablation study and evaluation with realistic driving scenarios. Our approach
is shown to perform well when compared to classical, infrastructure-bounded
checkerboard calibration. Furthermore, compared to other learning-based
approaches, our approach can outperform isolated intrinsic and extrinsic self-
calibration with a single-pass inference, paving its way to adoption in real-
world online applications.

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