11institutetext: Observatorio Astronómico Nacional (IGN), C/Alfonso XII, 3,
E-28014 Madrid, Spain
11email<EMAIL_ADDRESS>22institutetext: Centro de Desarrollos Tecnológicos,
Observatorio de Yebes (IGN), 19141 Yebes, Guadalajara, Spain 33institutetext:
LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne
Universités, 75014 Paris, France 44institutetext: Finnish Centre for Astronomy
with ESO (FINCA), University of Turku, Vesilinnantie 5, 20014 Turku, Finland
55institutetext: Aalto University Metsähovi Radio Observatory, Metsähovintie
114, 02540 Kylmälä, Finland 66institutetext: Argelander-Institut für
Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
77institutetext: Department of Astrophysics, University of Vienna,
Türkenschanzstrasse 17, A-1180 Vienna, Austria 88institutetext: European
Southern Observatory (ESO), Karl-Schwarzschild-Straße 2, 85748 Garching,
Germany 99institutetext: Universität Heidelberg, Zentrum für Astronomie,
Institut für Theoretische Astrophysik, Albert-Ueberle-Strasse 2, 69120
Heidelberg, Germany 1010institutetext: Department of Physics & Astronomy,
University of Wyoming, Laramie, WY USA 1111institutetext: Cardiff Hub for
Astrophysics Research & Technology, School of Physics & Astronomy, Cardiff
University, Queens Buildings, Cardiff, CF24 3AA, UK 1212institutetext:
Haystack Observatory, Massachusetts Institute of Technology, 99 Millstone Rd.,
Westford, MA 01886, USA 1313institutetext: Universität Heidelberg,
Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld
225, 69120 Heidelberg, Germany 1414institutetext: Department of Astronomy, The
Ohio State University, 140 West 18th Ave, Columbus, OH 43210, USA
1515institutetext: Department of Physics, Tamkang University, No.151,
Yingzhuan Road, Tamsui District, New Taipei City 251301, Taiwan
1616institutetext: Institut de Radioastronomie Millimétrique (IRAM), 300 Rue
de la Piscine, F-38406 Saint Martin d’Hères, France 1717institutetext:
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo,
181-8588, Japan 1818institutetext: Max Planck Institute for Astronomy,
Königstuhl 17, D-69117 Heidelberg, Germany 1919institutetext: Sub-department
of Astrophysics, Department of Physics, University of Oxford, Keble Road,
Oxford OX1 3RH, UK

# A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales

M. J. Jiménez-Donaire , A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec
scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant
N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scales<EMAIL_ADDRESS>A. Usero A constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scales I. Bešlić A constant N2H+ (1-0)-to-HCN (1-0)
ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scales M. Tafalla A constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales
A. Chacón-Tanarro A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec
scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales Q. Salomé
A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scales C. Eibensteiner A constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales
A. García-Rodríguez A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec
scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales A. Hacar A
constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scales A. T. Barnes A constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scales F. Bigiel A constant N2H+ (1-0)-to-HCN (1-0)
ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scales M. Chevance A constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales
D. Colombo A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA
constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales D. A. Dale A
constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scales T. A. Davis A constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scales S. C. O. Glover A constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scales J. Kauffmann A constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales
R. S. Klessen A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA
constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scales A. K. Leroy A constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scales L. Neumann A constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales
H. Pan A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant
N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales J. Pety A constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales
M. Querejeta A constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA
constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales T. Saito A
constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scales E. Schinnerer A constant N2H+
(1-0)-to-HCN (1-0) ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN
(1-0) ratio on kiloparsec scales S. Stuber A constant N2H+ (1-0)-to-HCN (1-0)
ratio on kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scales T. G. Williams A constant N2H+ (1-0)-to-HCN (1-0) ratio on
kiloparsec scalesA constant N2H+ (1-0)-to-HCN (1-0) ratio on kiloparsec scales

(Received 30 May 2023; accepted 31 July 2023)

Nitrogen hydrides such as NH3 and N2H+ are widely used by Galactic observers
to trace the cold dense regions of the interstellar medium. In external
galaxies, because of limited sensitivity, HCN has become the most common
tracer of dense gas over large parts of galaxies. We provide the first
systematic measurements of N2H+ (1-0) across different environments of an
external spiral galaxy, NGC 6946. We find a strong correlation
($r>0.98\,,p<0.01$) between the HCN (1-0) and N2H+ (1-0) intensities across
the inner $\sim 8\,\mathrm{kpc}$ of the galaxy, at kiloparsec scales. This
correlation is equally strong between the ratios N2H+ (1-0)/CO (1-0) and HCN
(1-0)/CO (1-0), tracers of dense gas fractions ($f_{\mathrm{dense}}$). We
measure an average intensity ratio of N2H+ (1-0)/HCN (1-0)$\,=0.15\pm 0.02$
over our set of five IRAM-30m pointings. These trends are further supported by
existing measurements for Galactic and extragalactic sources. This narrow
distribution in the average ratio suggests that the observed systematic trends
found in kiloparsec-scale extragalactic studies of $f_{\mathrm{dense}}$ and
the efficiency of dense gas ($\mathrm{SFE}_{\mathrm{dense}}$) would not change
if we employed N2H+ (1-0) as a more direct tracer of dense gas. At kiloparsec
scales our results indicate that the HCN (1-0) emission can be used to predict
the expected N2H+ (1-0) over those regions. Our results suggest that, even if
HCN (1-0) and N2H+ (1-0) trace different density regimes within molecular
clouds, subcloud differences average out at kiloparsec scales, yielding the
two tracers proportional to each other.

###### Key Words.:

galaxies: ISM – ISM: molecules – radio lines: galaxies

##  1 Introduction

Understanding the link between star formation and the available molecular
reservoir is crucial to comprehending how gas density regulates star formation
over cosmic time. While a variety of molecular lines are commonly employed to
infer properties of the sites of current or future star formation, the
rotational transitions of carbon monoxide (CO) have been the preferred tracers
of the bulk molecular ISM in the Milky Way and other galaxies (e.g., Dame et
al., 1987; Bally et al., 1987; Kuno et al., 2007) because of the CO abundance
and low critical density ($\sim 10^{2}\,\mathrm{cm}^{-3}$).

Star formation has been shown to take place in the densest parts of molecular
clouds (e.g., Heiderman et al., 2010; Lada et al., 2010, 2012; Evans et al.,
2014). These dense and compact regions are difficult to resolve in external
galaxies. Lines that are more difficult to excite, such as HCN (1-0), and
their ratio to low-density tracers, such as low-$J$ CO, offer the best way to
routinely probe the physical conditions and distribution of dense gas in
external galaxies (e.g., Leroy et al., 2017). Over the last decade, multiple
extragalactic surveys have mapped the dense gas content at low resolution
(kiloparsec scales; e.g., EMPIRE Bigiel et al., 2016; Jiménez-Donaire et al.,
2019) and high resolution (parsec scales; e.g., Gallagher et al., 2018b;
Querejeta et al., 2019; Bešlić et al., 2021; Sánchez-García et al., 2022;
Eibensteiner et al., 2022). All these studies share a key result: the HCN-to-
IR ratio, an observational proxy for the star formation efficiency of dense
gas (SFEdense), depends on the host galaxy and local environment. While these
observed trends are in contrast to a hypothesized constant SFEdense above some
critical surface density as observed on much smaller scales in the Milky Way
(e.g., Lada et al., 2010, 2012; Evans et al., 2014), this conclusion
critically rests on our ability to estimate dense gas masses.

In addition to dust continuum emission, nitrogen hydrides, such as NH3 and
N2H+, are widely used in Galactic observations to trace cold and dense
regions. N2H+ is a very selective high-density tracer, as it only efficiently
forms once CO is heavily depleted (Bergin & Tafalla, 2007), and therefore
traces cold dense gas. Recent studies in the Galaxy compared the emission of
dense gas tracers including HCN and N2H+, with dust temperatures or visual
extinction (e.g., Kauffmann et al., 2017; Pety et al., 2017; Barnes et al.,
2020; Evans et al., 2020; Patra et al., 2022; Dame & Lada, 2023). The results
in Orion, Perseus, and W49 (Kauffmann et al., 2017; Pety et al., 2017; Tafalla
et al., 2021) show that N2H+ is the only emission line among the dense gas
surveys that remains undetected at low densities ($N_{\mathrm{H_{2}}}<10^{22}$
cm-2, $n_{\mathrm{H_{2}}}$¡$10^{4}$ cm-3), but efficiently emits at higher
densities, becoming the only truly density-selective tracer in their sample.

Because radio observations of NH3 and N2H+ in different extragalactic
environments are very costly and sensitivity limited, previous works have
mainly focused on the brightest targets: active galactic nuclei (AGNs),
starburst galaxies, and ultra-luminous infrared galaxies (ULIRGs) (e.g.,
Mauersberger & Henkel, 1991; Meier & Turner, 2005, 2012; Watanabe et al.,
2014; Aladro et al., 2015; Martín et al., 2021). We present the first
extragalactic observations of N2H+ (1-0) and HCN (1-0) in a wide range of
dynamical conditions and star formation properties sampled across an entire
galaxy disk, NGC 6946. This nearby source ($7.72$ Mpc, Anand et al., 2018) is
a gas-rich, very actively star-forming double-barred spiral galaxy. The five
positions observed cover very diverse environments within a galaxy disk. These
include a central starburst, spiral and interarm regions, as well as the outer
disk. The selected regions span 1.7 dex (a factor of $\sim 50$) in star
formation rate (SFR) surface density. The observations presented here provide
a crucial benchmark to understand how well HCN performs as a tracer of cold
dense gas that favors N2H+ emission, in a representative external galaxy.

Figure 1: NGC 6946 with EMPIRE CO (1-0) contours overlaid on an optical image.
The contours illustrate the IRAM-30m 12CO (1-0) integrated intensities. The
millimeter observations have a resolution of $26$″(${\sim}800\,\rm pc$),
indicated by the orange circle. The contours are drawn at arbitrary intervals
between $5$ and $200$ K km s-1 to highlight the distribution of the molecular
gas disk. The yellow circles indicate the locations of our pointed N2H+ (1-0)
and HCN (1-0) observations, covering a wide range of galactic environmental
conditions: center (0), spiral arms (1 and 2), interarm (4), and outskirts
(3). The size represents the $28\arcsec$ resolution of the IRAM-30m beam at
$\sim 86\,$GHz. Table 1: Main properties of NGC 6946

Parameter | Value | Notes
---|---|---
R.A. (J2000) | $20{:}34{:}52.35$ | (1)
Decl. (J2000) | $+60{:}09{:}14.58$ | (1)
Morphology | SAB(rs)cd | (2)
Nuclear type | star-forming, H II | (3)
Distance |  $7.72$ Mpc | (4)
Inclination | $33\degr$ | (5)
P.A. major axis | $242\degr$ | (6)
VLSR |  $50$ km s-1 | (1)
$\mathrm{SFR}$ |  $6.17$ M⊙ yr-1 | (7)
$\log_{10}(M_{\star})$ | $10.39\,\log_{10}(\mathrm{M}_{\odot})$ | (7)

Notes: (1): Schinnerer et al. (2006); (2) de Vaucouleurs et al. (1991); (3):
Goulding & Alexander (2009); (4): Anand et al. (2018); (5): de Blok et al.
(2008); (6): Crosthwaite (2002); (7): Leroy et al. (2019).

##  2 Observations and data reduction

Table 2: Summary of observations and spectral line parameters for each
observed position.

Position | RA (J2000) | DEC (J2000) | Radius | Line | Int. intensity | FWHM | $V_{lsr}$ | $\langle\rm rms\rangle$ | Obs. time | $\langle T_{\rm sys}\rangle$
---|---|---|---|---|---|---|---|---|---|---
| [hh:mm:ss] | [o:$\arcmin$:$\arcsec$] | [kpc] | ($1-0$) | [K km s-1] | [km s-1] | [km s-1] | [mK] | [hr] | [K]
|  |  |  |  |  |  |  | (1) | (2) | (3)
Pos-0 | 20:34:52.20 | +60:09:14 | 0.0 | HCN | $11.12\pm 0.01$ | $147\pm 15$ | $56\pm 6$ | 0.3 | 44.3 | 92
N2H+ | $1.83\pm 0.01$ | $133\pm 11$ | $55\pm 6$ | 0.2
Pos-1 | 20:35:05.70 | +60:10:52 | 5.1 | HCN | $0.33\pm 0.01$ | $32\pm 2$ | $-32\pm 4$ | 0.5 | 10.0 | 112
N2H+ | $0.040\pm 0.008$ | $21\pm 6$ | $-28\pm 3$ | 0.4
Pos-2 | 20:35:11.88 | +60:08:51 | 5.7 | HCN | $0.39\pm 0.01$ | $31\pm 2$ | $-9\pm 2$ | 0.6 | 8.5 | 116
N2H+ | $0.067\pm 0.011$ | $27\pm 5$ | $-6\pm 2$ | 0.5
Pos-3 | 20:34:22.86 | +60:10:31 | 8.6 | HCN | $0.13\pm 0.01$ | $25\pm 2$ | $114\pm 12$ | 0.5 | 13.9 | 108
N2H+ | $0.026\pm 0.006$ | $30\pm 8$ | $117\pm 15$ | 0.3
Pos-4 | 20:34:47.10 | +60:07:59 | 3.3 | HCN | $0.26\pm 0.01$ | $32\pm 4$ | $126\pm 12$ | 0.4 | 13.2 | 99
N2H+ | $0.031\pm 0.007$ | $20\pm 6$ | $131\pm 15$ | 0.3

Notes: (1) Average rms measured in a 4 km s-1 wide channel. (2) Total on-
source time used to generate the final spectra after reduction. (3) Average
system temperature during observations.

We observed NGC 6946 with the IRAM-30m telescope from 2018 to 2020, using the
3 mm band (E090) of the dual-polarization Eight MIxer Receiver (EMIR, Carter
et al., 2012). The observations presented here correspond to a total $\sim
90\,\mathrm{h}$ of on-source observing time (see Table 2). We achieved an
angular resolution of $\sim 28\arcsec$ at 86 GHz. Additional details on the
observations, as well as N2H+ (1-0) and HCN (1-0) data reduction and analysis
are given in Appendix A.

NGC 6946 belongs to the IRAM-30m survey EMPIRE (Jiménez-Donaire et al., 2019),
and thus a suite of high critical density tracers as well as CO (1-0) and
carbon isotopologs are publicly available. These lines were mapped at $\sim
30\arcsec$ resolution across a total area of $4.5\,\arcmin\times
6.5\,\arcmin$; therefore, a direct comparison to common dense and bulk
molecular gas tracers is possible. For this study we used the CO (1-0) EMPIRE
integrated intensity maps of NGC 6946 (Jiménez-Donaire et al., 2019) to
compare the line ratios derived for each observed position in Fig. 1.

##  3 Results

N2H+ (1-0) and HCN (1-0) were successfully detected at every position with a
peak signal-to-noise ratio higher than or equal to 3. The spectra presented in
Fig. 5 show that the detected line centroid of N2H+ (1-0) agrees with the mean
HCN (1-0) velocity, and the two transition lines show similar line widths
within $\leq 5\,\%$, indicating that the presence of HCN-emitting gas is
always accompanied by gas emitting N2H+ in our $\sim 28\arcsec$ beam ($\sim
1\,$kpc). It is not possible to resolve the hyperfine structure (hfs) of these
lines in the observed positions since the splitting separation ($\sim
12\,\mathrm{km\,s}^{-1}$ for HCN (1-0) and $\sim 15\,\mathrm{km\,s}^{-1}$ for
N2H+ (1-0)) is smaller than the measured line widths. Table 2 reports the
spectral line parameters derived for each emission line, in every observed
position.

Figure 2: Calculated $I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$
ratios (left) from our measurements and $I_{\mathrm{HCN}}/I_{\mathrm{CO}}$
ratios (middle) from EMPIRE survey data (Jiménez-Donaire et al., 2019), as a
function of galactocentric radius in NGC 6946. Each observed position is
indicated. The right panel shows
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ as a function of
$I_{\mathrm{HCN}}/I_{\mathrm{CO}}$, illustrating that
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ remains approximately
constant, while $f_{\mathrm{dense}}$ changes by a factor of $\sim 4$. The
purple region shows the range of previous literature values found for the
central regions of starburst galaxies, AGNs, and ULIRGs (e.g., Mauersberger &
Henkel, 1991; Watanabe et al., 2014; Aladro et al., 2015; Nishimura et al.,
2016b; Eibensteiner et al., 2022, but see also Fig. 4.).

The left panel of Fig. 2 shows our estimated N2H+ (1-0)-to-HCN (1-0) intensity
ratios ($I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$) as a function of
galactocentric radius and observed position. We measure
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ ratios that range between
0.12 and 0.20 in NGC 6946, with a mean value of $0.15\pm 0.02$ and a standard
deviation of 0.03. Our line ratios agree well with the range of observed
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ values in nearby
starbursts, AGNs, and ULIRGs (see also Fig. 4), as reflected by the purple
area in the plot. This compilation is available in Table 3. The middle panel
of Fig. 2 shows the HCN (1-0)-to-CO (1-0) intensity ratios
($I_{\mathrm{HCN}}/I_{\mathrm{CO}}$) from EMPIRE (Jiménez-Donaire et al.,
2019) extracted at the same positions.

While $I_{\mathrm{HCN}}/I_{\mathrm{CO}}$ systematically decreases by a factor
of $\sim 4$ with increasing galactocentric distance,
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ appears to be constant
across the galaxy disk. We compare our observations (yellow stars) to a
compilation of literature measurements in the left panel of Fig. 3, showing
integrated intensities of N2H+ (1-0) as a function of HCN (1-0). This includes
measurements of resolved Galactic clouds (squares, Jones et al., 2012; Pety et
al., 2017; Barnes et al., 2020; Yun et al., 2021) and kiloparsec-scale regions
in nearby galaxies (circles, Mauersberger & Henkel, 1991; Watanabe et al.,
2014; Aladro et al., 2015; Nishimura et al., 2016b; Takano et al., 2019;
Eibensteiner et al., 2022). While these data are taken at different
resolutions (from 0.1 pc to 10 kpc), we note that such measurements are scarce
and constitute our only source of comparison. It is clear from the plot that
the strong correlation between $I_{\mathrm{N}_{2}\mathrm{H}^{+}}$ and
$I_{\mathrm{HCN}}$ is observed across a wide variety of systems. A fit to the
global literature data (including this work) yields

$\mathrm{log}\,I_{\mathrm{N}_{2}\mathrm{H}^{+},\mathrm{global}}=(0.99\pm
0.04)\,\mathrm{log}\,I_{\mathrm{HCN,global}}-(0.87\pm 0.04).$ (1)

We measured the strength and significance of the correlation between the
derived intensities with the Spearman rank correlation coefficient, $r=0.96$,
with a $p-\mathrm{value}<0.001$. We note that this strong correlation between
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}$ and $I_{\mathrm{HCN}}$ holds, while
$I_{\mathrm{HCN}}$ (and $I_{\mathrm{N}_{2}\mathrm{H}^{+}}$) varies across
almost three orders of magnitude.

The right panel of Fig. 3 shows the correlation of
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{CO}}$ as a function of the
commonly used $f_{\mathrm{dense}}$ tracer, $I_{\mathrm{HCN}}/I_{\mathrm{CO}}$,
for the same literature compilation. These plots indicate that the two line
ratios are correlated across several orders of magnitude, suggesting that they
are equally good at tracing variations in $f_{\mathrm{dense}}$. A fit to all
available data yields

$\mathrm{log}\,\frac{I_{\mathrm{N}_{2}\mathrm{H}^{+}}}{I_{\mathrm{CO}}}_{\mathrm{global}}=(1.00\pm
0.05)\,\mathrm{log}\,\frac{I_{\mathrm{HCN}}}{I_{\mathrm{CO}}}_{\mathrm{global}}-(0.84\pm
0.06),$ (2)

with $r=0.99$, and $p-\mathrm{value}<0.001$. Figure 4 shows the observed line
ratio ranges for our literature compilation in detail, targeting nearby
galaxies, starbursts, AGNs and ULIRGs, as well as Milky Way regions. The solid
line in the plot indicates a mean value of $0.15\pm 0.03$ for the global
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ ratio. The dashed lines
represent a $3\sigma$ deviation from this mean. Our average value for NGC 6946
is in good agreement with the literature values, within the uncertainties. In
this broad comparison, the Orion B cloud is the only clear outlier from the
correlation with a $I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ ratio
that is three times smaller than the average value of 0.15 found from the
literature compilation. In contrast, our sampled ratios in NGC 6946 are very
compatible with those found in low-metallicity environments such as the Large
Magellanic Cloud (LMC), where we would expect important differences in the
chemical composition of the clouds due to its deficiency in N-bearing
molecules (Dufour et al., 1982). Lower HCN/HCO+ ratios have been measured in
the low-metallicity galaxies LMC and IC 10 (Nishimura et al., 2016a, b; Patra
et al., 2022), compared to the normal star-forming disks sampled by EMPIRE
(Jiménez-Donaire et al., 2019).

Figure 3: Correlation between N2H+ (1-0), HCN (1-0) and CO (1-0) intensities
and line ratios. Left: Comparison between $I_{\mathrm{N}_{2}\mathrm{H}^{+}}$
and $I_{\mathrm{HCN}}$ in our observed locations (yellow stars), including
available data for nearby galaxies (circles, Aladro et al., 2015; Watanabe et
al., 2014; Mauersberger & Henkel, 1991; Nguyen et al., 1992; Eibensteiner et
al., 2022; Nishimura et al., 2016b; Takano et al., 2019) and Galactic
molecular clouds (squares, Jones et al., 2012; Pety et al., 2017; Barnes et
al., 2020; Yun et al., 2021). The solid line represents a linear fit to the
ensemble data in both panels, showing that the two emission lines are strongly
correlated ($r>0.98$) across almost four orders of magnitude. The dashed lines
indicate a $3\sigma$ dispersion from the mean. Right:
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{CO}}$ as a function of
$I_{\mathrm{HCN}}/I_{\mathrm{CO}}$ for NGC 6946 and the same literature
compilation. The two ratios, both proxies for the dense gas fraction, also
remain well correlated across almost three orders of magnitude.

##  4 Implications for extragalactic dense gas observations

Several studies have questioned whether the low-$J$ transitions of HCN and
HCO+, commonly used high critical density tracers, actually trace cold dense
gas emission (e.g., Pety et al., 2017; Kauffmann et al., 2017; Tafalla et al.,
2021). While these transitions have high critical densities, using this
parameter as an indication of dense gas is an oversimplification. Among other
effects, Shirley (2015) showed that radiative trapping and excitation lower
the effective critical density of these lines by one or two orders of
magnitude (see also Jiménez-Donaire et al., 2017). This causes a large
fraction of the HCN emission to be subthermally excited (Leroy et al., 2017;
Jones et al., 2023; García-Rodríguez et al., 2023). Moreover, electron
excitation can significantly contribute to its emissivity (Goldsmith &
Kauffmann, 2017). In fact, HCN (1-0) has a higher critical density than that
of N2H+ (1-0) ($4.7\times 10^{5}$ vs. $6.1\times 10^{4}\,\mathrm{cm}^{-3}$ at
10 K, respectively), but the effective density is slightly higher for N2H+
(1-0), which is significantly less opaque than HCN (1-0) (Shirley, 2015, see
Table 1), even though both lines show hfs. Some studies focusing on a small
number of local clouds find extended regions with low-density gas
($50-100\,\mathrm{cm}^{-3}$) that contribute significantly to the global
emission seen in HCN and HCO+ (Evans et al., 2020). In particular, Pety et al.
(2017) and Kauffmann et al. (2017) find that the vast majority of the HCN and
HCO+ luminosities come from regions with $A_{V}<8\,\mathrm{mag}$, while most
dense star-forming cores are seen at $A_{V}\geq 8\,\mathrm{mag}$. Both studies
also agree that N2H+ is able to selectively trace gas at much higher densities
in Orion A and Orion B, where at least 50% of its emission comes from regions
with $A_{V}\geq 16\,\mathrm{mag}$.

In addition, N2H+ primarily originates from cold gas ($<20\,\mathrm{K}$). In
contrast, HCN is often found to originate from gas with moderate temperatures
($>35\,\mathrm{K}$) (e.g., Pety et al., 2017; Barnes et al., 2020). One of the
reasons why the N2H+ molecule is a particularly good tracer of cold dense gas
is that it is destroyed by CO via ion-neutral interactions (Meier & Turner,
2005). In addition, N2H+ formation depends on the available amount of
$\mathrm{H}^{+}_{3}$ to react with $\mathrm{N}_{2}$, a process in which
$\mathrm{N}_{2}$ competes with CO. In cold high-density gas, CO freezes out
onto the dust grains, eliminating the main N2H+ destroyer and making available
more $\mathrm{H}^{+}_{3}$ to react with N2 (Caselli & Ceccarelli, 2012). The
combined result is a significant enhancement of the N2H+ abundance in cold and
dense gas. Since the excitation conditions are similar for N2H+ (1-0) and HCN
(1-0), it is the depletion of CO that makes the N2H+ (1-0) emission an ideal
tracer of cold dense gas. Consequently, N2H+ yields a bright emission line in
the densest regions of molecular clouds (Caselli et al., 1999; Bergin &
Tafalla, 2007), while in those same regions carbon-bearing molecules freeze
onto the dust grains (Tafalla et al., 2002; Bergin & Tafalla, 2007). Recent
Galactic work has shown that while HCN (1-0) extends over a significant
fraction of the cloud, it scales linearly with the H2 column density of the
gas (e.g., Tafalla et al., 2021; Dame & Lada, 2023). In NGC 6946, cloud-to-
cloud variations are averaged out at the scales we probe, yielding N2H+
proportional to HCN emission. If indeed N2H+ can selectively pick the densest
and coldest regions (with a typical $A_{V}\geq 16\,\mathrm{mag}$) and HCN
mainly traces gas in regions with $A_{V}\sim 6-8\,\mathrm{mag}$, our
remarkably constant $I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ ratio
across the disk of NGC 6946 suggests that cloud structures are similar and
subcloud differences average out at kiloparsec scales, yielding the two
tracers proportional to each other.

Figure 4: $I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ ratios observed
toward nearby galaxies (Aladro et al., 2015; Watanabe et al., 2014;
Mauersberger & Henkel, 1991; Nguyen et al., 1992; Eibensteiner et al., 2022;
Nishimura et al., 2016b; Takano et al., 2019, and this work) and Galactic
molecular clouds (Jones et al., 2012; Pety et al., 2017; Barnes et al., 2020;
Yun et al., 2021). For galaxies with several measurements, we represent their
mean values with associated uncertainties. The sizes of the circular markers
are given in logarithmic scale. For visual reasons, the markers for the
Galactic regions are scaled with 500 and the markers in the extragalactic
regions with 300. The solid line represents the mean value among all samples,
while the dashed lines indicate a $3\sigma$ dispersion from the mean.

The implications for extragalactic work are large. Over the last two decades,
many studies have extended the strong correlation found by Gao & Solomon
(2004) between HCN luminosity and IR-traced SFRs, from resolved galaxy disks
(e.g., Usero et al., 2015; Gallagher et al., 2018b; Jiménez-Donaire et al.,
2019) to individual cores in the Galaxy (e.g., Wu et al., 2005, 2010; Stephens
et al., 2016; Shimajiri et al., 2017). Using HCN (1-0) to trace dense gas,
Bigiel et al. (2016), Jiménez-Donaire et al. (2019), and Neumann et al.
(2023), among others, find systematic variations in $f_{\mathrm{dense}}$ and
SFEdense as a function of galactic environment. These results, however, are
subject to the interpretation of the HCN emissivity. Our observations show a
clear correlation between $I_{\mathrm{N}_{2}\mathrm{H}^{+}}$ and
$I_{\mathrm{HCN}}$ within a prototypical star-forming galaxy, NGC 6946, over a
wide range of galactic radii, in regions with different morphologies
(circumnuclear regions, spiral arms, and interarm region). These results are
also in agreement with existing measurements for starbursts, ULIRGs, AGNs, and
individual Galactic regions, where we find a constant ratio across more than
three orders of magnitude in integrated intensities. This constant
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{HCN}}$ provides evidence that the
variation of HCN (1-0) emission could be an interesting tool to estimate the
cold dense gas traced by N2H+ (1-0) across approximately kiloparsec-size
regions. Moreover, ratios of other lines with different critical densities
such as $I_{\mathrm{HCN}}/I_{\mathrm{CO}}$ and
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{CO}}$ can interchangeably be used
as good indicators for $f_{\mathrm{dense}}$ in other galaxies, at least at
kiloparsec scales, as previously pointed out by Leroy et al. (2017) and
Gallagher et al. (2018a). Additional kiloparsec-scale observations, however,
are needed to confirm the observed trends in other nearby galaxies, including
galaxy outskirts; high-resolution observations are also needed.

The strong correlation we observe implies that the amount of gas traced by HCN
(1-0) is proportional to the amount of star-forming high-density gas traced by
N2H+ (1-0) at kiloparsec scales. This is expected if the structure of
individual star-forming molecular clouds is similar overall among different
clouds, and is scale-free. The shape of the probability distribution of column
densities ($N$-PDF) in the observed molecular clouds is best described by
power-law functions (Kainulainen et al., 2009; Schneider et al., 2013;
Lombardi et al., 2015; Abreu-Vicente et al., 2015) in their high-density
regimes. Kiloparsec-sized extragalactic observations capture ensembles of
molecular clouds, where we expect the resulting PDF of the ensemble to reflect
this power-law shape if the individual clouds are indeed self-similar (e.g.,
independent of the scale). In that case, the ratio of gas traced by HCN (1-0)
and N2H+ (1-0) (sensitive to moderate to high column density thresholds) is
expected to be relatively constant. N2H+ is selective of cold dense gas, that
is expected to ultimately form stars. It is interesting to find bright N2H+
emission in regions where HCN is also present, because HCN often traces
moderately dense warmer gas (e.g., Barnes et al., 2020). This suggests that
when observing a large region of another galaxy, it is common to find small
highly shielded regions where conditions are conducive for N2H+ to be
abundant, likely similar to the dark clouds found in our Galaxy. Our
observations suggest that these regions have an approximately constant
proportion to the warmer more excited dense gas, perhaps associated with
photodissociation regions (PDRs), that likely dominate HCN emission. This
could make sense given that the PDRs are natural products of star formation,
and the conditions where N2H+ forms may be a relatively universal prerequisite
to form stars. This adds significant weight to the interpretation of
extragalactic HCN measurements in terms of tracing dense star-forming gas and
the associated $f_{\mathrm{dense}}$ and $\mathrm{SFE}_{\mathrm{dense}}$.

##  5 Summary and conclusions

In this work we study the relation between N2H+ (1-0) and HCN (1-0) across the
disk of NGC 6946, based on extensive observations with the IRAM-30m dish at
$\sim 90\,\mathrm{GHz}$. We find that the integrated intensities from the two
emission lines are strongly correlated within the galaxy disk at kiloparsec
scales. Moreover, this correlation is also extended to a wide range of
physical environments including Galactic clouds, nearby starburst galaxies,
AGNs, ULIRGs, and even low-metallicity regimes. Dense gas fractions, as probed
by $I_{\mathrm{HCN}}/I_{\mathrm{CO}}$ and
$I_{\mathrm{N}_{2}\mathrm{H}^{+}}/I_{\mathrm{CO}}$, are also very well
correlated across several orders of magnitude. Our results directly address
recent discussions of varying $f_{\mathrm{dense}}$ and
$\mathrm{SFE}_{\mathrm{dense}}$ within and across galaxies, and indicate that
despite the known caveats for HCN emission, this molecule can be used instead
of canonical Galactic dense gas tracers like N2H+ to estimate
$f_{\mathrm{dense}}$, at least on kiloparsec scales.

###### Acknowledgements.

This work is based on observations carried out with the IRAM-30m telescope.
IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). The
authors would like to E. Pellegrini, D. Harsono and S. Ellison for useful
discussions, and the referee, Neal Evans, for a constructive and helpful
report. AH acknowledges funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement Nos. 851435). RSK and SCOG acknowledge financial support from the
ERC via the ERC Synergy Grant “ECOGAL” (project ID 855130), from the German
Excellence Strategy via the Heidelberg Cluster of Excellence (EXC 2181 -
390900948) “STRUCTURES”, and from the German Ministry for Economic Affairs and
Climate Action in project “MAINN” (funding ID 50OO2206). RSK also thanks for
computing resources provided bwHPC and DFG through grant INST 35/1134-1 FUGG
and for data storage at SDS@hd through grant INST 35/1314-1 FUGG. MJJD and MQ
acknowledge support from the Spanish grant PID2019-106027GA-C44, funded by
MCIN/AEI/10.13039/501100011033. MC gratefully acknowledges funding from the
DFG through an Emmy Noether Research Group (grant number CH2137/1-1). LN
acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - 516405419. AU acknowledges support from the Spanish
grants PGC2018-094671-B-I00, funded by MCIN/AEI/10.13039/501100011033 and by
“ERDF A way of making Europe”, and PID2019-108765GB-I00, funded by
MCIN/AEI/10.13039/501100011033.

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##  Appendix A Observations and data analysis

The data were taken as part of projects 061-18, 055-20, 054-20, and 159-20
(PI: M. J. Jiménez-Donaire). The E090 receiver yielded an instantaneous
bandwidth of 15.6 GHz per polarization, and used the fast Fourier transform
(FFT) spectrometers with 195 kHz spectral resolution (FTS200, $\sim
0.5\,\mathrm{km}\,\mathrm{s}^{-1}$) to record the data. The observations were
performed in five different positions, representative of distinct environments
within the nearby galaxy NGC 6946 (see Fig. 1). These were carried out in
wobbler-switching mode with a total ON-OFF throw of $\pm 110\arcsec$ in
azimuth and a wobbling frequency of 0.5 Hz. Four of these positions were
previously observed by Usero et al. (2015), but those observations were not
sensitive enough to detect N2H+ (1-0). Every position was observed only when
the line emission in the OFF positions along the Azimuth axis could not
produce significant contamination. We did this by checking the integrated
intensity and velocity in the HERACLES CO (2-1) cube of the target galaxy
(Leroy et al. 2009). We tuned EMIR with a local oscillator frequency of $\sim
88.07\,\mathrm{GHz}$, which allowed us to simultaneously observe HCN (1-0) in
the lower-outer subband, and N2H+ (1-0) transition in the lower-inner subband.
Additionally, other fainter transitions, such as dense gas isotopologs (H13CN
(1-0), H13CO+ (1-0), and HN13C (1-0)), are present in the observed band and
will be analyzed in a future paper. The angular resolution of the IRAM-30m at
the frequency of HCN (1-0) is $27.9\arcsec$. This sets our working resolution,
which corresponds to a spatial resolution of $\sim 0.92\,\mathrm{kpc}$ at the
distance of NGC 6946. Because the observations of N2H+ (1-0) and HCN (1-0)
have similar beam sizes, aperture corrections are not applied.

During the course of the observations, the focus of the telescope was checked
on planets (Mars and Saturn) or bright quasars (K3-50A) at the beginning of
each session and then every $\sim 3$ hours. In addition, it was also checked
at sunset and sunrise if needed. We corrected the telescope pointing
approximately every 1–1.5 hours using point-like sources close to NGC 6946,
such as 2037+511, 1226+023, 1928+738, and 1253-055. We obtained a standard
chopper wheel calibration every $\sim 15\,$minutes to place the data on the
antenna temperature scale ($T_{\mathrm{A}}^{*}$). The weather conditions were
good overall, and the mean radiometer opacity at 225 GHz was measured at
around 0.5.

Figure 5: Individual spectra for each observed position as a function of the
LSR velocity. The orange lines show the N2H+ (1-0) spectrum, while the blue
lines show the HCN (1-0) emission divided by a factor of 5. The vertical
dashed lines delimit the windows used to reduce the data and derive the
spectral line parameters. N2H+ (1-0) shows good agreement with the mean HCN
LSR velocity. Table 2 reports the derived spectral line parameters.

The spectra for each observed location were calibrated with
MRTCAL111https://www.iram.fr/IRAMFR/GILDAS/doc/pdf/mrtcal-prog-manual.pdf as
part of the GILDAS222https://www.iram.fr/IRAMFR/GILDAS/ software package. The
data were then converted to main beam temperature ($T_{\mathrm{MB}}$) with the
available beam efficiencies from the IRAM
documentation333https://publicwiki.iram.es/Iram30mEfficiencies
($F_{\mathrm{eff}}=95\%$ and $B_{\mathrm{eff}}=81\%$ at 3 mm) following
equation $T_{\mathrm{MB}}=F_{\mathrm{eff}}/B_{\mathrm{eff}}\times
T_{\mathrm{A}}^{*}$. We extracted the target emission lines using the
Continuum and Line Analysis Single-dish
Software444https://www.iram.fr/IRAMFR/GILDAS/doc/html/class-html/class.html
(CLASS). We subtracted a third-order polynomial baseline in each spectrum, and
avoided fitting the baseline within the velocity range of molecular emission
by using the available CO velocity field maps from the EMPIRE survey. Thus, we
omitted the range of $\pm 150\,\mathrm{km}\,\mathrm{s}^{-1}$ centered around
the galactic mean CO (1-0) velocity. The individual spectra were then re-
gridded to have a $4\,\mathrm{km}\,\mathrm{s}^{-1}$ channel width across the
full bandpass, and averaged to produce a final spectrum per observed location.
For each position, we calculated the integrated intensities by summing over
the velocity range containing bright molecular emission HCN (1-0) (see dashed
limits in Fig. 5). Finally, we measured the root mean square (rms) noise level
of $T_{\mathrm{MB}}$ at line-free channels of the HCN and N2H+ spectra. These
values and detailed information about the observations can be found in Table
2.

Figure 5 presents the resulting individual spectra obtained for the five
different positions shown in Fig. 1, plotted in main-beam brightness
temperatures and smoothed to a common spectral resolution of
$4\,\mathrm{km}\,\mathrm{s}^{-1}$. The average rms noise level achieved by the
observations is 0.46 mK for the HCN (1-0) spectra and 0.34 mK for N2H+ (1-0).

##  Appendix B Literature data

The data presented in Table 3 contains the most up-to-date N2H+ (1-0) and HCN
(1-0) observations in the literature. This compilation includes the data used
to construct Fig. 3 and Fig.4. Upper limits given by Nishimura et al. (2016a)
were not employed in the plots. When this compilation table is used, the
original studies providing the various data sets should be referenced.

Table 3: Galactic and extragalactic N2H+ (1-0) and HCN (1-0) line
measurements.

Source | $I_{\mathrm{N}_{2}\mathrm{H}^{+}}$ | $I_{\mathrm{HCN}}$ | Resol.
---|---|---|---
| [K km s-1] | [K km s-1] |
Orion-A(1) | 0.005 | 0.03 | 0.11 pc
Orion-B(2) | $0.08\pm 0.01$ | $1.5\pm 0.2$ | 0.05 pc
CMZ(3) | $32\pm 2$ | $212\pm 1$ | 2 pc
SgrA(3) | $5.7\pm 0.5$ | $31.4\pm 0.2$ | 2 pc
SgrB2(3) | $9.3\pm 0.2$ | $40.6\pm 0.3$ | 2 pc
SgrC(3) | $2.4\pm 0.2$ | $18.4\pm 0.3$ | 2 pc
G1.3(3) | $17.0\pm 0.2$ | $132.0\pm 0.3$ | 2 pc
W49(4) | $0.17\pm 0.01$ | $1.38\pm 0.06$ | 3 pc
IC 10(5) | $<0.11$ | $0.34\pm 0.01$ | 4.9 pc
LMC-N113(6) | $0.80\pm 0.07$ | $3.15\pm 0.05$ | 9.2 pc
LMC-N159W(6) | $0.32\pm 0.06$ | $1.53\pm 0.04$ | 9.2 pc
IC342(7) | $2.1\pm 0.4$ | $26.4\pm 0.6$ | 340 pc
M82(8) | $1.65\pm 0.04$ | $23.6\pm 0.4$ | 428 pc
NGC 253(8) | $10.6\pm 0.2$ | $68.4\pm 0.7$ | 466 pc
M83(8) | $1.17\pm 0.05$ | $10.2\pm 0.1$ | 554 pc
NGC 6946(9,10) | $0.90\pm 0.02$ | $8.2\pm 0.1$ | 957 pc
NGC 6946(11) | $6.4\pm 0.5$ | $57.4\pm 0.6$ | 100 pc
M51(8) | $0.74\pm 0.04$ | $5.4\pm 0.1$ | 1 kpc
M51-P1(12) | $0.53\pm 0.06$ | $2.7\pm 0.1$ | 1 kpc
M51-P2(12) | $0.24\pm 0.08$ | $1.7\pm 0.1$ | 1 kpc
NGC 1068(8) | $1.99\pm 0.08$ | $23.55\pm 0.28$ | 1.8 kpc
NGC 1266(13) | $0.17\pm 0.03$ | $1.89\pm 0.04$ | 3 kpc
Arp220(8) | $2.36\pm 0.42$ | $11.06\pm 0.24$ | 10.5 kpc

Notes: (1): Yun et al. (2021) do not provide uncertainties; (2) Pety et al.
(2017); (3): Jones et al. (2012); (4): Barnes et al. (2020); (5): Nishimura et
al. (2016a); (6): Nishimura et al. (2016b); (7): Takano et al. (2019); (8):
Aladro et al. (2015); (9): Mauersberger & Henkel (1991); (10): Nguyen et al.
(1992); (11): Eibensteiner et al. (2022); (12): Watanabe et al. (2014); (13):
Davis et al. in prep.


# A Two-Stage Data Association Approach for 3D Multi-Object Tracking

Minh-Quan Dao, Vincent Frémont
Laboratoire des Sciences du Numérique de Nantes (LS2N)
École Centrale de Nantes
Nantes, 44300
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Multi-object tracking (MOT) is an integral part of any autonomous driving
pipelines because it produces trajectories which has been taken by other
moving objects in the scene and helps predict their future motion. Thanks to
the recent advances in 3D object detection enabled by deep learning, track-by-
detection has become the dominant paradigm in 3D MOT. In this paradigm, a MOT
system is essentially made of an object detector and a data association
algorithm which establishes track-to-detection correspondence. While 3D object
detection has been actively researched, association algorithms for 3D MOT seem
to settle at a bipartie matching formulated as a linear assignment problem
(LAP) and solved by the Hungarian algorithm. In this paper, we adapt a two-
stage data association method which was successful in image-based tracking to
the 3D setting, thus providing an alternative for data association for 3D MOT.
Our method outperforms the baseline using one-stage bipartie matching for data
association by achieving 0.587 AMOTA in NuScenes validation set.

##  1 Introduction

Multi-object tracking have been a long standing problem in computer vision and
robotics community since it is a crucial part of autonomous systems. From the
early work of tracking with hand-craft features, the revolution of deep
learning which results in highly accurate object detection models [1, 2, 3]
has shifted the focus of the field to the track-by-detection paradigm [4, 5].
In the framework of this paradigm, tracking algorithms receive a set of object
detection, usually in the form of bounding boxes, at each time step and they
aim to link detection of the same object across time to form trajectories.

While image-based methods of this paradigm have reached a certain maturity, 3D
tracking is still in its early phase where most of the published approaches
are originated from successful 2D exemplars. The most popular attempt at 3D
tracking with established 2D tracking method is [6] which is an extension of
[4] into 3D space. In these works, the tracking algorithm is made of the
Hungarian algorithm [7] and Kalman filter. While the former finds track-to-
measurement correspondences by solving a linear assignment problem, the later
performs prediction and correction of tracks’ state. An improvement of [6] is
proposed by [8] which replaces the 3D IoU [9] with the Mahalanobis distance as
the cost function for the assignment problem. The idea of handling tracking as
a matching problem is also used in the context of end-to-end learning [10, 11,
12]. [10] solves the tracking task in same fashion as [6]; however, this work
trains a sub network for calculating the cost function of the assignment
problem and the correction step is carried out by another sub network instead
of the Kalman filter.[11, 12] train deep models to predict tracks position in
the following frames along with generating detection and the track-to-
detection correspondences are found by greedy matching.

Even though 3D tracking has been progressed rapidly thanks to the availability
of standardized large scale benchmarks such as KITTI [13], NuScenes [14],
Waymo Open Dataset [15], the focus of the field is placed on developing better
object detection models rather than developing better tracking algorithm as
evidenced in the Table.1. There are two trends can be observed in this table.
First, tracking performance experiences significant boost when a better object
detection model is introduced. Second, the method of AB3DMOT [6] is favored by
most recent 3D tracking systems.

Table 1: Summary of tracking methods which details are published in the leader board of NuScenes and Waymo Open Dataset Dataset |  Method Name |  Tracking Method |  AMOTA |  Object Detector |  mAP
---|---|---|---|---|---
NuScenes |  CenterPoint [11] |  Greedy closest-point matching |  0.650 |  CenterPoint |  0.603
PMBM* |  Poisson Multi-Bernoulli Mixture filter [16] |  0.626 |  CenterPoint |  0.603
StanfordIPRL-TRI [8] |  Hungarian algorithm with Mahalanobis distance as cost function and Kalman Filter |  0.550 |  MEGVII [17] |  0.519
AB3DMOT [6] |  Hungarian algorithm with 3D IoU as cost function and Kalman Filter |  0.151 |  MEGVII |  0.519
CenterTrack |  Greedy closest-point mathcing |  0.108 |  CenterNet [18] |  0.388
Waymo |  HorizonMOT [19] |  3-stage data associate, each stage is an assignment problem solved by Hungarian algorithm |  0.6345 |  AFDet [20] |  0.7711
CenterPoint |  Greedy closest-point matching |  0.5867 |  CenterPoint |  0.7193
PV-RCNN-KF |  Hungarian algorithm and Kalman Filter |  0.5553 |  PV-RCNN [21] |  0.7152
PPBA AB3DMOT |  Hungarian algorithm with 3D IoU as cost function and Kalman Filter |  0.2914 |  PointPillars and PPBA[22] |  0.3530

The reason of AB3DMOT’s popularity is that despite its simplicity, it achieves
competitive result in challenging datasets at significantly high frame rate
(more than 200 FPS). However, such simplicity comes at the cost of the MOT
system being vulnerable to false associations due to occlusion or imperfect
detections which is case for objects in a clutter or far away from the ego
vehicle.

Aware of the shortage of a generic 3D tracking algorithm which can better
handle occlusion and imperfect detections so that to limit the false track-to-
detection correspondence, yet remains relatively simple, we adapt the image-
based tracking method proposed by [23] to the 3D setting. Specifically, this
method is a two-stage data association scheme. In this scheme, each tracked
trajectory is called a tracklet and is assigned a confidence score computed
based on how well associated detection matches with tracklet. The first
association stage aims to establish the correspondence between high confident
tracklets and detection. The second stage matches the left over detection with
the low confident tracklets as well as link low confident tracklets to high
confident ones if they meet a certain criterion.

In this paper, we make two contributions

  * •

Our main contribution is the adaptation of an image-based tracking method to
the 3D setting. In details, we exploit a kinematically feasible motion model,
which is unavailable in 2D, to facilitate objects pose prediction. This model
in turn defines the minimal state vector needed to be tracked.

  * •

Extensive experiment carried out in various datasets proves the effectiveness
of our approach. In fact, our better performance, compared to AB3DMOT-style
models, show that adding a certain degree of re-identification can improve the
tracking performance while keeping the added complexity to the minimum.

##  2 Related work

A multi-object tracking system in the track-by-detection paradigm consists of
an object detection model, a data association algorithm and a filtering
method. While the last two components are domain agnostic, object detection
models, especially learning-based methods, are tailored to their operation
domain (e.g images or point clouds). This paper targets autonomous driving
where objects pose are required thus interest in 3D object detection models.
However, developing such a model is not in the scope of this paper, instead we
use the detection result provided by baseline models of benchmarks (e.g.
PointPillars of NuScenes) to focus on the data association algorithm and to
have a fair comparison. Interested readers are referred to [24] for a review
of 3D object detection.

Data association via the Hungarian algorithm was early explored in [25] where
a 2-stage tracking scheme was proposed for offline 2D tracking. Firstly,
detections are linked frame-by-frame to form tracklets. The affinity matrix of
the Hungarian algorithm is established by geometric and appearance cue. While
the geometric cue is the 2D Intersection over Union (IoU), the appearance cue
is the correlation between two bounding boxes. Secondly, tracklets are
associated to each other to compensate trajectory fragments and ID switch due
to occlusion. This association is also carried out the Hungarian algorithm.

Due to its batch-processing nature [25] cannot be applied to online tracking,
[4] overcomes this by eliminating the second stage and let objects which
temporarily left the sensor’s field of view reenter with new IDs. Despite its
simplicity, SORT - the method proposed by [4] achieve competitive result in
MOT15[26] with lightning-fast inference speed (260 Hz). The success of SORT
inspired [6] to adapt it to 3D setting by using 3D IoU as the affinity
function. The performance of SORT in 3D setting is later improved in [8] which
shows the use of Mahalanobis distance is superior to 3D IoU. [27] integrates
the 3D version of SORT into a complete perception pipeline for autonomous
vehicles.

The two-stage association scheme is adapted to online tracking in [23] which
proposes a confidence score to quantify tracklets quality. Based on this
score, tracklets are associated with detections or another tracklets, or
terminated. The appearance model learned by ILDA in [23] is improved by deep
learning in the follow-up work [28]. Recently, this association scheme is
revisited in the context of image-based pedestrian tracking by [29] which
proposed to use the rank of the Hankel matrix as tracklets motion affinity.

Differ from [23] and its related works, this paper applies the two-stage
association scheme to online 3D tracking. In addition, we can provide
competitive result despite relying solely on geometric cue to compute tracklet
affinity thanks to the Constant Turning Rate and Velocity (CTRV) motion model
which can accurately predict objects position in 3D space by exploiting their
kinematic.

##  3 Method

###  3.1 Problem Formulation

Online multi-object tracking (MOT) in the sense of track-by-detection aims to
gradually grow the set of tracklets
$\mathbb{T}_{0:t}=\\{\mathcal{T}^{i}\\}_{i=1}^{|\mathbb{T}_{0:t}|}$ by
establishing correspondences with the set of detections received at every time
step $\mathbb{D}_{t}=\\{d_{t}^{j}\\}_{j=1}^{|\mathbb{D}|}$ and updating
tracklets state accordingly. A tracklet is a collection of state vectors
corresponding to the same object $\mathcal{T}^{i}=\\{\mathrm{x}_{k}^{i}|0\leq
t_{s}^{i}\leq k\leq t_{e}^{i}\leq t\\}$, here $t_{s}^{i},t_{e}^{i}$ are
respectively the starting- and ending-time of the tracklet. A detection
$d_{t}^{j}$ at time step $t$ encapsulates information of a 3D bounding box
including the position of its center in a common reference frame $[x,y,z]$,
heading $\theta$, and size $[w,l,h]$. It is worth to notice that in the
context of autonomous driving, objects are assumed to remain in contact with
the ground; therefore, their detections are up-right bounding boxes which
orientation is described by a single number - the heading angle.

The correspondence between $\mathbb{T}_{0:t}$ and $\mathbb{D}_{t}$ can be
formally defined as finding the set $\mathbb{T}_{0:t}^{*}$ that maximizes its
likelihood given $\mathbb{D}_{t}$.

$\mathbb{T}_{0:t}^{*}=\underset{\mathbb{T}_{0:t}}{\mathsf{argmax}}\text{
}\mathtt{p}\left(\mathbb{T}_{0:t}|\mathbb{D}_{t}\right)$ (1)

Due the exponential growth of possible associations between $\mathbb{T}_{0:t}$
and $\mathbb{D}_{t}$, Equation.(1) is computationally intractable after a few
time steps. In this paper, such a correspondence is approximated by the two-
stage data association proposed by [23] as shown in the following.

###  3.2 Two-stage Data Association

####  3.2.1 Tracklet Confidence Score

The reliability of a tracklet is quantified by a confidence score which is
calculated based on how well associated detections match with its states
across its life span and how long its corresponding object was undetected.

$\mathsf{conf}\left(\mathcal{T}^{i}\right)=\left(\frac{1}{L^{i}}\sum_{k\in[t_{s}^{i},t_{e}^{i}]|v^{i}(k)=1}\Lambda\left(\mathcal{T}^{i},d_{k}^{j}\right)\right)\times\exp\left(-\beta\frac{W}{L_{i}}\right)$
(2)

where $v^{i}(k)$ is a binary indicator which takes 1 if the tracklet has a
detection $d_{k}^{j}$ associated with it at time step $k$, and 0 otherwise.
$L^{i}$ is the number of time step that the traklet gets associated with a
detection. $\Lambda(\cdot)$ is the affinity function which detail will be
presented later. $\beta$ is a tuning parameter which takes high value if the
object detection model is accurate. $W=t-t_{s}^{i}-L^{i}+1$ is the number of
time step that tracklet was undetected (i.e. did not have associated
detection) calculated from its birth to the current time step $t$.

Applying a threshold $\tau^{c}$ this confidence score divides the set
$\mathbb{T}_{0:t}$ into a subset of high confidence tracklets
$\mathbb{T}_{0:t}^{h}=\\{\mathcal{T}^{i,h}|\mathsf{conf}\left(\mathcal{T}^{i}\right)>\tau^{c}\\}$
and a subset of low confidence tracklets
$\mathbb{T}_{0:t}^{l}=\\{\mathcal{T}^{i,l}|\mathsf{conf}\left(\mathcal{T}^{i}\right)\leq\tau^{c}\\}$.
These two subsets are the fundamental elements of the two-stage association
pipeline showed in Figure.1

Figure 1: The pipeline of two-stage data association. The first stage - local
association establish the correspondences between detections at this time step
$\mathbb{D}_{t}$ and high confidence tracklets $\mathbb{T}_{0:t}^{h}$. Then,
global association stage matches each low confidence tracklets
$\mathcal{T}^{i,l}$ with either a high confidence tracklet or a left-over
detection, or terminates it.

####  3.2.2 Local Association

In this association stage, high confident tracklets are extended by their
correspondence in the set of detections $\mathbb{D}_{t}$. This tracklet-to-
detection is found by solving the a linear assignment problem (LAP)
characterized by the cost matrix $\mathbf{L}$ as following

$\mathbf{L}=\begin{bmatrix}l_{i,j}\end{bmatrix}\in\mathbb{R}^{h\times
d},\text{ with }l_{i,j}=-\Lambda\left(\mathcal{T}^{i},d_{t}^{j}\right)$ (3)

where $h,d$ are respectively the number of high confidence tracklets and the
number of detections. The intuition of this association stage is that because
tracklets with high confidence have been tracked accurately for a number of
time steps, the affinity function can identify if a detection is belong to the
same object as the tracklet with high accuracy, thus limiting the possibility
of false correspondences. In addition, low confidence tracklets are usually
resulted from fragment trajectories or noisy detections, excluding them from
this association stage help reduces the ambiguity.

####  3.2.3 Global Association

As shown in Figure.1, the global assocaition stage carries out the following
tasks

  * •

Matching low confidence tracklets with high confidence ones

  * •

Matching low confidence tracklets with detections left over by the local
association stage

  * •

Deciding to terminate low confidence tracklets

These tasks are simultaneously solved as a LAP formulated by the following
cost matrix

$\mathbf{G}_{(l+d^{\prime})\times(h+l)}=\begin{bmatrix}\mathbf{A}_{l\times
h}&B_{l\times l}\\\ \infty_{d^{\prime}\times h}&C_{d^{\prime}\times
l}\end{bmatrix}$ (4)

here, $l,d$ are respectively the number low confidence tracklets and
detections left over by the local association stage. Recall $h$ is the number
of high confidence tracklets. Submatrix $\mathbf{A}$ is the cost matrix of the
event where low confidence tracklets are matched with high confidence ones

$\mathbf{A}=[a_{i,j}]\in\mathbb{R}^{l\times h},\text{ with
}a_{i,j}=-\Lambda\left(T^{i,l},T^{j,h}\right)$ (5)

Submatrix $\mathbf{B}$ represents the event where low confidence tracklets are
terminated.

$\mathbf{B}=[b_{i,j}]\in\mathbb{R}^{l\times l},\text{ with
}b_{i,j}=\begin{cases}-\log\left(1-\mathsf{conf}\left(\mathcal{T}^{i}\right)\right),\text{
if }i=j\\\ \infty,\text{ otherwise}\end{cases}$ (6)

Finally, submatrix $\mathbf{C}$ is the cost of the associating low confidence
tracklets with detections left over by local association stage.

$\mathbf{C}=[c_{i,j}]\in\mathbb{R}^{d^{\prime}\times l},\text{ with
}c_{i,j}=-\Lambda\left(\mathcal{T}^{j},d^{i}_{t}\right)$ (7)

The solution to the LAP in this stage and in the Local Association stage is
the association that minimize the cost and can be either found by the
Hungarian algorithm for the optimal solution or by a greedy algorithm which
interatively pick and remove correspondence pair with the smallest cost until
there is no pair has cost less than a threshold (the detail of this greedy
algorithm can be found in [8]).

Once a detection is assocatied with a tracklet, its position and heading is
used to update the tracklet’s state according to the equation of the Kalman
Filter, while its sizes is averaged with tracklet’s sizes in past few time
steps to result in updated sizes. Detections do not get associated in the
global association stage are used to initialize new tracklets.

####  3.2.4 Affinity Function

Affinity function $\Lambda(\cdot)$ is to compute how similar a detection to a
tracklet or a tracklet to another. As mentioned earlier, due to the lack of
colorful texture in point cloud, the affinity function used in this work is
just comprised of geometric cue. Specifically, it is the sum of position
affinity and size affinity.

$\Lambda(\mathcal{T}^{i},d_{t}^{j})=\Lambda(\mathcal{T}^{i},d_{t}^{j})^{p}+\Lambda(\mathcal{T}^{i},d_{t}^{j})^{s}$
(8)

The scheme for computing position affinity between a tracklet and a detection
or between two tracklets are shown in Figure.2.

Figure 2: The computational scheme of position affinity. The filled triangles
(or rectangles) are subsequent states of a tracklet. The colored arrow
represents the time order: the closer to the tip, the more recent the state.
The triangle (or rectangle) in dash line is the state propagated forward (or
backward) in time. The covariance of these propagated states are denoted by
ellipses with the same color. The two-headed arrows indicate the Mahalanobis
distance. In the subfigure (a), the blue circle denotes a detection.

As shown in Figure.2.a, the position affinity
$\Lambda(\mathcal{T}^{i},d_{t}^{j})^{p}$ between a tracklet $\mathcal{T}^{i}$
and a detection $d_{t}^{j}$ is defined as the Mahalanobis distance between
tracklet’s last state propagated to the current time step $t$ and the
measurement vector $\mathrm{z}_{t}^{j}$ extracted from $d_{t}^{j}$

$\Lambda(\mathcal{T}^{i},d_{t}^{j})^{p}=\left(\mathsf{h}\left(\bar{\mathrm{x}}^{i}_{e}\right)-\mathrm{z}_{t}^{j}\right)^{T}\cdot\mathrm{S}^{-1}\cdot\left(\mathsf{h}\left(\bar{\mathrm{x}}^{i}_{e}\right)-\mathrm{z}_{t}^{j}\right)$
(9)

where $\bar{\mathrm{x}}^{i}_{e}$ is last state of tracklet $\mathcal{T}^{i}$
propagated to the current time step using the motion model which will be
presented below. $\mathsf{h}(\cdot)$ is the measurement model computing the
expected measurement using the inputted state and the measurement vector
$\mathrm{z}_{t}^{j}=[x,y,z,\theta]^{T}$. The matrix $\mathcal{S}$ is the
covariance matrix of the innovation (i.e. the difference between expected
measurement $\mathsf{h}\left(\bar{\mathrm{x}}^{i}_{e}\right)$ and its true
value $\mathrm{z}_{t}^{j}$)

$\mathbf{S}=\mathbf{H}\cdot\mathbf{P}\cdot\mathbf{H}^{T}+\mathbf{R}$ (10)

here, $\mathbf{H}=\delta\mathsf{h}/\delta\mathrm{x}$ is the Jacobian of the
measurement model, $\mathbf{P},\mathbf{R}$ are covariance matrix of
$\bar{\mathrm{x}}^{i}_{e}$ and $\mathrm{z}_{t}^{j}$, respectively.

In the case of two tracklets $\mathcal{T}^{i}$ and $\mathcal{T}^{j}$, assuming
$\mathcal{T}^{j}$ starts after $\mathcal{T}^{i}$ ended, their motion affinity
is, according to Figure.2.b, is the sum of

  * •

Mahalanobis distance between the last state of $\mathcal{T}^{i}$ propagated
forward in time and the first state of $\mathcal{T}^{j}$

  * •

Mahalanobis distance between the first state of $\mathcal{T}^{j}$ propagated
backward in time and the last state of $\mathcal{T}^{i}$

$\Lambda(\mathcal{T}^{i},\mathcal{T}^{j})^{p}=\Lambda(\mathcal{T}^{j},\bar{\mathrm{x}}^{i}_{e})^{p}+\Lambda(\mathcal{T}^{i},\bar{\mathrm{x}}^{j}_{s})^{p}$
(11)

here, $\bar{\mathrm{x}}^{i}_{e}$ is the last state of tracklet
$\mathcal{T}^{i}$ propagated forward in time to the first time step of
tracklet $\mathcal{T}^{j}$, while $\bar{\mathrm{x}}^{j}_{s}$ is the first
state of tracklet $\mathcal{T}^{j}$ propagated backward in time to the last
time step of tracklet $\mathcal{T}^{i}$.

The size affinity $\Lambda(\mathcal{T}^{i},d_{t}^{j})^{s}$ is computed as
following

$\Lambda\left(\mathcal{T}^{i},d_{t}^{j}\right)^{s}=-\frac{|w_{e}^{i}-w_{t}^{j}|}{w_{e}^{i}+w_{t}^{j}}\cdot\frac{|l_{e}^{i}-l_{t}^{j}|}{l_{e}^{i}+l_{t}^{j}}\cdot\frac{|h_{e}^{i}-h_{t}^{j}|}{h_{e}^{i}+h_{t}^{j}}$
(12)

here, $[w_{e}^{i},l_{e}^{i},h_{e}^{i}]$ are the size of the last state of
tracklet $\mathcal{T}^{i}$, while $[w_{t}^{j},l_{t}^{j},h_{t}^{j}]$ are the
size of the detection $d_{t}^{j}$. In the case of two tracklets
$\mathcal{T}^{i}$ and $\mathcal{T}^{j}$, assuming $\mathcal{T}^{j}$ starts
after $\mathcal{T}^{i}$ ended, there size affinity is

$\Lambda\left(\mathcal{T}^{i},\mathcal{T}^{j}\right)^{s}=-\frac{|w_{e}^{i}-w_{s}^{j}|}{w_{e}^{i}+w_{s}^{j}}\cdot\frac{|l_{e}^{i}-l_{s}^{j}|}{l_{e}^{i}+l_{s}^{j}}\cdot\frac{|h_{e}^{i}-h_{s}^{j}|}{h_{e}^{i}+h_{s}^{j}}$
(13)

The subscript $e,s$ in Equation.(13) respectively denote the ending and
starting state of a tracklet.

###  3.3 Motion Model and State Vector

Exploiting the fact that objects are tracked in 3D space of a common static
reference frame which can be referred to as the world frame, motion of objects
can be described by more kinematically accurate models, compared to the
commonly used Constant Velocity (CV) model. In this work, we use the Constant
Turning Rate and Velocity (CTRV) model to predict motion of car-like vehicles
(e.g. cars, buses, trucks), while keep the CV model for pedestrians.

For a car-like vehicles, its state can be described by

$\mathrm{x}=[x,y,z,\theta,\mathtt{v},\dot{\theta},\dot{z}]^{T}$ (14)

here, $[x,y,z]$ is the location in the world frame of the center of the
bounding box represented by the state vector, $\theta$ is the heading angle,
$\mathtt{v}$ is longitudal velocity (i.e. velocity along the heading
direction), $\dot{\theta},\dot{z}$ are respectively velocity of $\theta$ and
$z$.

The motion on x-y plane of car-like vehicles can be predicted using CTRV as
following

$\mathrm{x}_{t+1}=\mathrm{x}_{t}+\begin{bmatrix}\frac{\mathtt{v}}{\dot{\theta}}\left(\sin(\theta+\dot{\theta}\Delta
t)-\sin(\theta)\right)\\\
\frac{\mathtt{v}}{\dot{\theta}}\left(-\cos(\theta+\dot{\theta}\Delta
t)+\cos(\theta)\right)\\\ \dot{z}\Delta t\\\ \dot{\theta}\Delta t\\\ 0\\\ 0\\\
0\\\ \end{bmatrix}$ (15)

where, $\Delta t$ is the sampling time. Note that in Equation.(15), $z$ is
assumed to evolve with constant velocity. In the case of zero turning rate
(i.e. $\dot{\theta}=0$),

$\mathrm{x}_{t+1}=\mathrm{x}_{t}+\begin{bmatrix}\mathtt{v}\cos(\theta)&\mathtt{v}\sin(\theta)&\dot{z}\Delta
t&\dot{\theta}\Delta t&0&0&0&\end{bmatrix}^{T}$ (16)

The state vector of a pedestrian is

$\mathrm{x}=\begin{bmatrix}x&y&z&\theta&\dot{x}&\dot{y}&\dot{z}&\dot{\theta}\end{bmatrix}^{T}$
(17)

The motion of pedestrians is predicted according to CV model

$\mathrm{x}_{t+1}=\mathrm{x}_{t}+\begin{bmatrix}\dot{x}&\dot{y}&\dot{z}&\dot{\theta}&0&0&0&0\end{bmatrix}^{T}\cdot\Delta
t$ (18)

##  4 Experiments

The effectiveness of our method is demonstrated by benchmarking against the
SORT-style baseline model on 3 large scale datasets: KITTI, NuScenes, and
Waymo. In addition, we perform ablation study using NuScenes dataset to better
understand the impact of each component on our system’s general performance.

###  4.1 Tracking Results

Evaluation Metrics: Classically, MOT systems are evaluated by the CLEAR MOT
metrics [30]. As pointed out by [31] and later by [6], there is a linear
relation between MOTA and object detectors’ recall rate, as a result, MOTA
does not provide a well-rounded evaluation performance of trackers. To remedy
this, [6] proposes to average MOTA and MOTP over a range of recall rate,
resulting in two integral metrics AMOTA and AMOTP which become the norm in
recent benchmarks.

Datasets To verify the effectiveness of our method, we benchmark it on 3
popular autonomous driving datasets which offer 3D MOT benchmark: KITTI,
NuScenes, and Waymo. These datasets are collections of driving sequences
collected in various environment using a multi-modal sensor suite including
LiDAR. KITTI tracking benchmark interests in two classes of object which are
cars and pedestrians. Initially, KITTI tracking was designed for MOT in 2D
images and recently [6] adapts it to 3D MOT. NuScenes concerns a larger set of
objects which comprises of cars, bicycles, buses, trucks, pedestrians,
motorcycles, trailers. Waymo shares the same interest as NuScenes but groups
car-like vehicles into meta class.

Public Detection: As can be seen in Table.1, AMOTA highly depends on the
precision of object detectors. Therefore, to have a fair comparison, the
baseline detection results made publicly available by the benchmarks are used
as the input to our tracking system. Specifically, we use MEGVII detection and
PointPillars with PPBA detection for NuScenes and Waymo, respectively.

The performance of our model compared to the SORT-style baseline model in 3
popular benchmarks are shown in Table.2. As can be seen, our model
consistently outperforms the baseline model in term of the primary metric
AMOTA. The main reason of this is the lower ID switches and trajectory
fragments of ours which shows the better ability of establishing track-to-
detection correspondence compared to SORT-style algorithm.

Table 2: Quantitative performance of our model on KITTI, NuScenes, and Waymo validation set. Dataset |  Method |  AMOTA$\uparrow$ |  AMOTP$\downarrow$ |  MT$\uparrow$ |  ML$\downarrow$ |  FP$\downarrow$ |  FN$\downarrow$ |  IDS$\downarrow$ |  FRAG$\downarrow$
---|---|---|---|---|---|---|---|---|---
KITTI |  Ours |  0.415 |  0.691 |  N/A |  N/A |  766 |  3721 |  10 |  259
AB3DMOT[6] |  0.377 |  0.648 |  N/A |  N/A |  696 |  3713 |  1 |  93
NuScenes |  Ours |  0.583 |  0.748 |  3617 |  1885 |  13439 |  28119 |  512 |  511
StanfordIPRL-TRI[8] |  0.561 |  0.800 |  3432 |  1857 |  12140 |  28387 |  679 |  606

###  4.2 Ablation Study

In this ablation study, default method is the method presented in Section.3
which has

  * •

Two stages of data association (local and global). Each stage is formulated as
a LAP and solved by a greedy matching algorithm [8].

  * •

The affinity function the sum of position affinity and size affinity (as in
Equation.(8)).

  * •

The motion model is Constant Turning Rate and Velocity (CTRV) for car-like
objects (cars, buses, trucks, trailers, bicycles) and Constant Veloctiy (CV)
for pedestrians.

To understand the effect of each component on the system’s general
performance, we modify or remove each of them and carry out experiment with
the rest of the system being kept the same as the default method and the same
hyperparameters. The changes and the resulted performance are shown in
Table.3.

Table 3: Ablation study using NuScenes dataset. Method |  AMOTA$\uparrow$ |  AMOTP$\downarrow$ |  MT$\uparrow$ |  ML$\downarrow$ |  FP$\downarrow$ |  FN$\downarrow$ |  IDS$\downarrow$ |  FRAG$\downarrow$
---|---|---|---|---|---|---|---|---
Default |  0.583 |  0.748 |  3617 |  1885 |  13439 |  28119 |  512 |  511
Hungarian for LAP |  0.587 |  0.743 |  3609 |  1880 |  13667 |  28070 |  596 |  573
No ReID |  0.583 |  0.748 |  3616 |  1882 |  13429 |  28100 |  504 |  510
Global assoc only |  0.327 |  0.924 |  2575 |  2244 |  26244 |  38315 |  4215 |  3038
Const Velocity only |  0.567 |  0.781 |  3483 |  1966 |  12649 |  29427 |  718 |  606
No size affinity |  0.581 |  0.748 |  3595 |  1904 |  13423 |  28448 |  512 |  508

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