aainstitutetext: Khon Kaen Particle Physics and Cosmology Theory Group
(KKPaCT),
Department of Physics, Faculty of Science, Khon Kaen University, 123 Mitraphap
Rd.,
Khon Kaen, 40002, Thailandbbinstitutetext: National Astronomical Research
Institute of Thailand, Chiang Mai 50180, Thailand

# Bounds on ALP-Mediated Dark Matter Models from Celestial Objects

Tanech Klangburam a,b Chakrit Pongkitivanichkul<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We have studied the signals from axion-like particles (ALPs) as dark matter
mediators from celestial objects such as neutron stars or brown dwarfs. We
consider the accumulation of dark matter inside the celestial objects using
the multiscatter capturing process. The production of ALP from the dark matter
annihilation can escape the celestial object and decay into gamma-ray and
neutrinos before reaching the Earth. We investigate our model using gamma-ray
observations from Fermi and H.E.S.S and neutrino observations from IceCube and
ANTARES. The effective Lagrangian approach allows us to place constraints on
the ALP-photon and ALP-fermion couplings. In the gamma-ray channel, our
results improve the existing bounds on ALPs by 1-2 orders of magnitude.
Although the constraints from neutrino fluxes rule out a significant portion
of the parameter space, the remaining part of the parameter space is
accessible by future experiments.

##  1 Introduction

After decades of searching for Dark Matter (DM), the models in which DM
interacts directly with the Standard Model (SM) particles are strongly
disfavoured. In particular, the parameter space of the standard WIMP (Weakly
Interacting Massive Particle) models has been severely constrained. This has
slowly led to a great interest in models with DM interactions via mediators
such as Higgs portal model, neutrino portal model, etc Patt:2006fw ; March-
Russell:2008lng ; Djouadi:2012zc ; Falkowski:2009yz ; Escudero:2016tzx ;
Escudero:2016ksa . Among them, one particular model provides a variety of
phenomenological aspects, i.e., a fermion DM with an axion-like particle as
the mediator. Axion-like particles (ALPs) are light pseudoscalar particles
arising from extensions of the standard model. For example, the String/M
theory framework generically provides a plethora of pseudoscalars due to the
compactification process of the extra-dimensions Svrcek:2006yi ;
Arvanitaki:2009fg . Since ALPs mainly interact with gauge bosons and fermions
via dimension-five couplings, the main phenomenological aspects rely on the
ALP-fermion and ALP-photon couplings. One could test the ALP-mediated models
using particle colliders such as the beam dump experiments and rare mesons
decay. Other interesting tests come from astrophysics such as the neutrino
flux coming from supernova SN1987A, effects on stellar evolution of the
horizontal branch of the HR-diagram and the number of relativistic degrees of
freedom around the time of Big Bang Nucleosynthesis (BBN).

Recently, there has been interest in dark matter capturing in celestial
objects such as neutron star (NS) and brown dwarf (BD) Dasgupta:2019juq ;
Ilie:2020vec ; Ilie:2021iyh ; Nguyen:2022zwb . It has been shown that if the
multiscattering effect is taken into account, the celestial objects
accumulating dark matter are appealing targets for observation of DM
annihilation. The DM-overdense objects then could be observed in 2 different
ways depending on the lifetime of the mediator. If the mediator is short-lived
and decays inside the celestial object, an extra energy would change the
energy budget and the DM can be observed through the lifetime and stability of
the celestial objects. On the other hand, if the mediator has a sufficiently
long enough lifetime such that it decays outside the celestial object, the
situation allows us to probe DM via indirect detection.

In this work, we study ALP-mediated DM models using an effective Lagrangian
approach. We then use the DM-nucleon cross-section to calculate the
multiscatter capture rate of the celestial objects in the DM-rich region. We
are interested in the case where ALP produced from annihilation has a
sufficiently long lifetime such that the ALP decays into SM particles. Using
the distribution of celestial objects near the Galactic Center and the DM
generalized Navarro-Frenk-White (NFW) density profile, we are able to produce
gamma-ray fluxes and neutrino fluxes from the DM annihilation in this model.
The fluxes from neutrino produced from DM annihilation inside BDs could be
detected via IceCube experiment and ANTARES neutrino telescope, whereas in the
NS case, the neutrino fluxes are too low for any current observations. The
gamma-ray fluxes from NSs and BDs can be tested against Fermi satellite and
H.E.S.S. data. The constraints from gamma-ray observation using NSs and BDs as
targets are able to rule out a significant portion of parameter space of
coupling between ALP and fermions and the ALP mass. The constraints from
gamma-ray fluxes perform better than the ones from neutrino fluxes due to the
difference in the observed fluxes from the experiments.

##  2 Dark Matter Simplified Model

In this model, the SM is extended by a Dirac fermion, $\chi$, and a
pseudoscalar ALP, $a$. The effective Lagrangian is given by

$\displaystyle\mathcal{L}\supset\frac{1}{2}\partial_{\mu}a\partial^{\mu}a-\frac{1}{2}m^{2}_{a}a^{2}+\sum_{f}\frac{m_{f}}{f_{a}}C_{f}\bar{f}i\gamma_{5}fa+\frac{m_{\chi}}{f_{a}}C_{\chi}\bar{\chi}i\gamma_{5}\chi
a.-\frac{g_{a\gamma\gamma}}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}a,$ (1)

where $f$ is any SM fermion with mass $m_{f}$, the masses of ALP and DM are
$m_{a}$ and $\ m_{\chi}$, respectively.
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the $U(1)_{\rm
EM}$ field strength tensor
($\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$)
and $g_{a\gamma\gamma}$ is an effective ALP-photon coupling. For convenience,
we define the effective ALP couplings by

$\displaystyle g_{aff}=m_{f}C_{f}/f_{a}\quad\text{and}\quad
g_{a\chi\chi}=m_{\chi}C_{\chi}/f_{a}.$ (2)

In this model, we assume that ALP-fermion coupling $C_{f}$ is universal for
any fermion, i.e., $C_{f}$ and $g_{aff}\propto m_{f}$. The Dirac fermion
$\chi$ is assumed to be the DM particle. The ALP, $a$, acts as the mediator
between SM and the hidden sector where $g_{aff}$ acts as the connector
coupling and $g_{a\chi\chi}$ acts as the hidden sector coupling. Figure 1
shows the diagram of the scattering between DM and nucleons which is mediated
by ALP and the diagrams of ALP decay into 2 photons and 2 SM particles
(specifically neutrinos if kinematically allowed), respectively. The different
regimes of interactions were studied for the freeze-in/out scenarios in
Bharucha:2022lty ; Ghosh:2023tyz ; Dror:2023fyd , therefore we will not focus
on the DM production mechanism in this paper.

Figure 1: Diagrams a non-relativistic scattering of dark matter with the
standard model where the ALP propagates an interaction (left), an ALP
effectively decays into gamma-rays (middle) and an ALP effectively decays into
SM particles (right).

As DM condenses around the celestial objects in the DM-rich region, the DM can
fall into the objects by the mechanism called “dark matter multiscatter
capture” Bramante:2017xlb ; Ilie:2020vec . We will discuss this mechanism in
the next section. DM can lose kinetic energy by scattering with the object’s
components and falling into the celestial object. The scattering cross-section
of $\chi f\rightarrow\chi f$ in the low-energy limit is

$\displaystyle\sigma(\chi f\rightarrow\chi
f)=\frac{g_{aff}^{2}g_{a\chi\chi}^{2}}{2\pi{m_{a}^{4}}}\frac{m^{4}_{\chi}+m^{4}_{f}+m^{2}_{\chi}m^{2}_{f}}{(m_{\chi}+m_{f})^{2}}.$
(3)

Each time that DM scatters with the components of the object, DM will lose
kinetic energy and move toward the center of the object. DM will then
accumulate and start to annihilate inside the object producing a long-lived
mediator which is an ALP. The lifetime of ALP decay into gamma-rays is given
by

$\displaystyle\tau^{-1}_{a\rightarrow\gamma\gamma}=\Gamma_{a\rightarrow\gamma\gamma}=\frac{1}{64\pi}g^{2}_{a\gamma\gamma}m_{a}^{3},$
(4)

and the lifetime of ALP decay into SM particles is written as

$\displaystyle\tau^{-1}_{a\rightarrow ff}=\Gamma_{a\rightarrow
ff}=\frac{g_{aff}^{2}n_{f}^{c}m_{a}}{8\pi}\sqrt{1-\frac{4m^{2}_{f}}{m_{a}^{2}}}.$
(5)

The probes on the ALP coupling have been studied in various methods (see
Dolan:2017osp ; Bharucha:2022lty  for a review). For example, the
astrophysical sources such as SN 1987A can provide us the constraints on ALP
coupling that $g_{a\gamma\gamma}<6\times 10^{-9}$ GeV-1 for small ALP masses
Jaeckel:2017tud . The red giant branch in several clusters Straniero:2020iyi
give the upper bounds on the ALP coupled to electrons as $g_{aee}<1.48\times
10^{-13}$, which is also similar to the analysis of the red giant in the
Galactic globular cluster $\omega$ Centauri $g_{aee}<1.3\times 10^{-13}$
Capozzi:2020cbu . The ALP coupled to both SM and DM has been recently studied
in the context of thermal history Bharucha:2022lty ; Balazs:2022tjl ;
Ghosh:2023tyz ; Dror:2023fyd ; Leane:2023woh .

We are interested in the case that ALPs were generated by the DM annihilation
from the celestial objects. We primarily aimed to constrain the ALP-SM
coupling, $g_{aff}$ from the gamma-rays and neutrinos observations. The
celestial objects such as NSs and BDs are good targets since they are densely
distributed around the Galactic Center where the DM density is generically
higher.

##  3 Dark Matter Multiscatter Capture

DM from the Galactic halo can start to fall into the celestial object if the
DM is sped up to the escape velocity of the object at the surface of the
celestial body which is

$\displaystyle v_{\text{esc}}=\sqrt{G_{N}M/R},$ (6)

where $G_{N}$ is the gravitational constant, $M$ and $R$ are the total mass
and the radius of the celestial body, respectively. As DM particles transit
through the celestial body, they can scatter with stellar materials and lose
their kinetic energy. Once the velocity of DM drops below the escape velocity
of the celestial objects, DM is captured. The capturing process can occur via
single or multiple scatters Kouvaris:2010vv ; Bramante:2017xlb ;
Dasgupta:2019juq ; Ilie:2020vec ; Ilie:2021iyh . The probability for a given
DM to undergo $N$ scatter is given by

$\displaystyle p_{N}(\tau)=2\int_{0}^{1}dy\frac{ye^{y\tau}(y\tau)^{N}}{N!},$
(7)

where $\tau=\frac{3}{2}\frac{\sigma_{\chi n}}{\sigma_{\text{sat}}}$ is the
optical dept. The saturate cross-section is defined as
$\sigma_{\text{sat}}\equiv\pi R^{2}/N_{n},$ where $N_{n}$ is the number of
nucleons inside the celestial object. The meaning of saturation cross-section
$\sigma_{\text{sat}}$ is the threshold that guarantees that DM will scatter at
least once. The probability can be approximated in the limit of single scatter
$(\tau\leq 1)$ and multiple scatter $(\tau\gg 1)$ Ilie:2020vec  as

$\displaystyle
p_{N}(\tau)\approx\begin{dcases}\frac{2\tau^{N}}{N!(N+2)}+\mathcal{O}(\tau^{N+1}),&\text{if
}\quad\tau\leq 1\\\ \frac{2}{\tau^{2}}(N+1)\Theta(\tau-N).&\text{if
}\quad\tau\gg 1\end{dcases}$ (8)

The capture rate after DM scattered $N$ times and became trapped in the
celestial object is given by

$\displaystyle C_{N}(\tau)=\pi
R^{2}p_{N}(\tau)\frac{\sqrt{6}n_{\chi}}{3\sqrt{\pi}\bar{v}}\left((2\bar{v}^{2}+3v^{2}_{esc})-(2\bar{v}^{2}+3v^{2}_{N})\text{exp}\left(-\frac{3}{2}\frac{(2v_{N}^{2}+3v^{2}_{esc})}{\bar{v}^{2}}\right)\right),$
(9)

where $n_{\chi}=\rho_{\chi}(r)/m_{\chi}$ is the local number density of DM and
$\bar{v}$ is the DM dispersion velocity. After scattering $N$ times, the
typical velocity of DM which takes into account the energy loss in each
scattering is

$\displaystyle v_{N}=v_{\text{esc}}\left(1-\frac{\beta_{+}}{2}\right)^{-N/2},$
(10)

where $\beta_{+}=4m_{\chi}m_{n}/(m_{\chi}+m_{n})^{2}$. The total capture rate
for a single celestial body is then given by

$\displaystyle C=\sum_{N=1}^{\infty}C_{N}.$ (11)

For sufficiently large $N$, $2\bar{v}^{2}+3v^{2}_{N}$ becomes much larger than
$\bar{v}_{N}^{2}$, therefore, the exponential term in equation 9 can be
neglected and the capture rate approaches $C_{N}\rightarrow p_{N}(\tau)\times
C_{\text{max}}$. $C_{\text{max}}$ is called the maximum capture rate
(geometric capture rate) which is given by

$\displaystyle C_{\text{max}}=\pi
R^{2}n_{\chi}(r)v_{0}\left(1+\frac{3}{2}\frac{v^{2}_{\text{esc}}}{\bar{v}(r)^{2}}\right),$
(12)

where $v_{0}=\sqrt{8/3\pi}\bar{v}$. This guarantees that the particles undergo
a sufficiently large number of $N$ scatters are captured.

Figure 2 shows the mass capture rate for a NS (left) with $M_{\text{NS}}=1.5$
M⊙ and $R_{\text{NS}}=10$ km and a BD (right) with $M_{\text{BD}}=0.0378$ M⊙
and $R_{\text{BD}}=69,911$ km as a function of nucleon scattering cross-
section $\sigma_{\chi n}$. As $\sigma_{\chi n}$ increases, the number of
scatters $N$ also increases, once it goes above the certain number of scatters
that is sufficiently large enough to capture DM, the capture rate in equation
9 (black line) is approximately the maximum capture rate (red dash line)
(equation 12). It can be seen that the BD capture rate is 100 times larger
than the NS capture rate. This is because the effective radius of BD is much
larger than the radius of NS, which causes more DM flux to pass through the
object.

Figure 2: The mass capture rate $m_{\chi}C$ as a function of nucleon
scattering cross-section $\sigma_{\chi n}$ for a NS and a BD. The red dash
line indicates $m_{\chi}C_{\text{max}}$. The DM model is applied with
$m_{\chi}=10^{3}$ GeV, $\rho_{\chi}=0.42$ GeV/cm3 and $\bar{v}=220$ km/s.

In order to calculate the total capture rate for all celestial objects within
a given system (e.g. the Galactic Center or galaxy clusters), we need to take
into account the number density of the objects within radii $r_{1}$ and
$r_{2}$, the total capture rate is

$\displaystyle C_{\text{total}}=4\pi\int_{r_{1}}^{r_{2}}r^{2}n_{\star}(r)C\
dr,$ (13)

where $n_{\star}(r)$ is the number density of the celestial objects and $C$ is
the capture rate of a single object (equation 13). The number density of the
celestial objects will be described in the following section.

##  4 The Galactic Center

In this section, we introduce the specific model of the DM velocity
distribution and the target number density. The DM velocity dispersion and DM
density affect the rate at which DM would fall into the gravitational
potential and eventually fall into the object. At a given radius $r$ from the
Galactic Center, the dispersion velocity is

$\displaystyle\bar{v}=\sqrt{3/2}v_{c}(r),$ (14)

where $v_{c}$ is the circular velocity at radius $r$, which depends on the
total mass of the sphere of radius $r$:

$\displaystyle v_{c}=\sqrt{{G_{N}M(r)}/{r}}.$ (15)

We use the model for the mass distribution following reference Sofue:2013kja .
This model has five components which are the central supermassive black holes
$M_{\text{BH}}=4\times 10^{6}$ M⊙, inner and outer bulges
$(\rho_{\text{inner}}$ and $\rho_{\text{outer}})$, an disk component
$(\rho_{\text{disk}})$ and DM halo $(\rho_{\text{DM}})$. Combining all five
components, the total mass is

$\displaystyle
M(r)=M_{\text{BH}}+4\pi\int_{0}^{r}r^{2}(\rho_{\text{inner}}+\rho_{\text{outer}}+\rho_{\text{disk}}+\rho_{\text{DM}})\
dr.$ (16)

The DM generalized Navarro-Frenk-White (NFW) density profile Navarro:1995iw
is

$\displaystyle\rho_{\text{DM}}=\frac{\rho_{0}}{(r/r_{s})^{\gamma}(1+(r/r_{s})^{1-\gamma})},$
(17)

where $\rho_{0}=0.42$ GeV/cm3 is the local DM density, the scale radius is
$r_{s}=12$ kpc and the slope index is ranging from $\gamma=1-1.5$. The model
of the inner bulge, outer bulge and disk is assumed to be an exponential
sphere model:

$\displaystyle\rho_{i}=\rho_{0,i}e^{-r/a_{i}},$ (18)

where the parameters of each component are shown in table 1.

Mass component | Total mass (M⊙) | Scale radius (kpc) | Center density (M⊙ pc-3)
---|---|---|---
Inner bulge | $5\times 10^{7}$ | 0.0038 | $3.6\times 10^{4}$
Outer bulge | $8.4\times 10^{9}$ | 0.12 | $1.9\times 10^{2}$
Disk | $4.4\times 10^{10}$ | 3.0 | 15
Table 1: Parameters in equation 18 for described the mass distribution of each
component of the Galactic Center, from Sofue:2013kja .

Neutron stars (NS) is the collapsed core of the massive star ($10-25$
$M_{\odot}$), NS is composed almost entirely of degenerate neutrons with a
mass ranging from $1-1.5$ M⊙ and a radius $R_{\text{NS}}\simeq 10$ km. The
number densities of NSs and black holes in regions around the Galactic Center
have been numerically studied with the nuclear star cluster dynamics
Generozov:2018niv . Two types of cluster models were described, ‘Fiducial’ and
‘Fiducial $\times$ 10’, both of which are potentially good candidates for an
NS distribution in the nuclear star clusters. The NS radial number density
distribution from the ‘Fiducial $\times$ 10’ model is given by

$\displaystyle n_{\text{NS}}(r)=\begin{dcases}5.98\times
10^{3}\left(\frac{r}{1\ \text{pc}}\right)^{-1.7}&\text{if }\quad 0.1\
\text{pc}\ <r<2\ \text{pc}\\\ 2.08\times 10^{4}\left(\frac{r}{1\
\text{pc}}\right)^{-3.5}&\text{if }\quad r>2\ \text{pc}\end{dcases}.$ (19)

On the other hand, Brown dwarf (BD) is a failed star that is not massive
enough to process the nuclear fusion of hydrogen into helium in its core. A
huge amount of BDs are expected to be presented in our galaxy. The Milky Way
may contain $25-100$ billion BDs 10.1093/mnras/stx1906 . To obtain the radial
distribution function of BDs, we follow the works in references Kroupa:2011aa
; Amaro-Seoane:2019umn . They extended the Kroupa Initial Mass Function (IMF)
into the BD IMF which is described by the broken power law function,
$\frac{dN_{\text{BD}}}{dm}\propto m^{-\alpha}$, where $\alpha=0.3$,
$N_{\text{BD}}$ is the number of BD and $m$ is the mass of BD. The number
density of the BDs with the mass range from $0.01-0.07$ M⊙ is given by

$\displaystyle n_{\text{BD}}(r)=7.5\times 10^{4}\left(\frac{r}{1\
\text{pc}}\right)^{-1.5}.$ (20)

In our calculation, we will use the average mass of BDs as
$M_{\text{BD}}=0.0378$ M⊙ and their radius is assumed to be equal to the
Jupiter radius $R_{\text{BD}}=R_{\text{J}}=69,911$ km.

##  5 Detection

Once DM becomes trapped in the celestial bodies (NS or BD), it has two
possible fates. First, if DM does not self-annihilate, its density will rise
near the core of the celestial object and it can lead to black hole formation
and collapse. The second case is that the DM is annihilated with each other
inside the celestial body. The evolution of the number of DM particles
Kouvaris:2010vv  inside an object $N(t)$ is governed by an interplay between
DM capture rate and DM annihilation rate, i.e.,

$\displaystyle\frac{dN(t)}{dt}=C_{\text{total}}-C_{A}N(t)^{2},$ (21)

where $C_{\text{total}}$ is the total capture rate (equation 13) and
$C_{A}=\langle\sigma_{A}v\rangle/V_{\text{eff}}$ is the average thermal
annihilation cross-section over the effective volume of the celestial object
body $V_{\text{eff}}=4\pi R_{\star}^{3}/3$. The solution of equation 21 reads

$\displaystyle
N(t)=\sqrt{\frac{C_{\text{total}}}{C_{A}}}\tanh{\frac{t}{t_{\text{eq}}}},$
(22)

where $t_{\text{eq}}=1/\sqrt{C_{A}C_{\text{total}}}$ is the time scale
required for the celestial object body to reach the DM equilibrium between
capture and annihilation. Once it reaches the equilibrium, $t>t_{\text{eq}}$,
the number of DM becomes time-independent, and the annihilation rate
$\Gamma_{\text{ann}}$ is simply

$\displaystyle\Gamma_{\text{ann}}=\frac{C_{\text{total}}}{2},$ (23)

where the factor $1/2$ comes from the fact that the DM annihilation process
involves 2 DM particles. Since it has been shown that within the lifetime of
the universe, the equilibrium between DM capturing rate and DM annihilation
rate has already been reached for most of the parameter space Nguyen:2022zwb ,
we will assume the above condition for our study. From equation 9, we know
that the annihilation rate is proportional to the local DM density,
$\Gamma_{\text{ann}}\propto n_{\chi}$, and the total capture rate (equation
13) is proportional to the number density of the celestial body,
$\Gamma_{\text{ann}}\propto n_{\star}$, thus, the total annihilation rate is
$\Gamma_{\text{ann}}\propto n_{\chi}n_{\star}$.

We consider the case in which DM annihilates into a long-lived ALP which
weakly interacts with the matter such that it can escape from the celestial
body and subsequently decay into a pair of SM particles. DM captured in
celestial objects is expected to be annihilated with a small relative
velocity, the mediator may be produced with the Lorentz boost factor
$\eta=m_{\chi}/m_{a}$, where $m_{a}$ is the mass of ALP. In order to calculate
the sensitivities for the possible signals, we assume that the lifetime of the
mediator, $\tau$, is sufficiently long such that the decay length exceeds the
radius of the object $R$:

$\displaystyle L=\eta c\tau>R.$ (24)

The differential energy flux of gamma-ray arriving at Earth is given by
Leane:2017vag

$\displaystyle E^{2}\frac{d\Phi}{dE}=\frac{\Gamma_{\text{ann}}}{4\pi
D^{2}}\times E^{2}\frac{dN}{dE}\times\text{BR}(a\rightarrow\text{SM})\times
P_{\text{surv}},$ (25)

where $D$ is the average distance from the target to Earth, in this case, the
target is located around the Galactic Center, so $D\approx 8$ kpc.
BR$(a\rightarrow\text{SM})$ is the branching ratio for the mediator into SM
particles. For simplicity, we are assuming the unity of the branching ratio
and the analysis is then separated into 2 cases, i.e., the spectrum of DM
annihilates into a mediator and then decays into a pair of either 100% gamma-
rays or 100% neutrinos. Their number distributions can be described by a box-
shaped spectrum Ibarra:2012dw  which is given by

$\displaystyle\frac{dN}{dE}=\frac{4}{\Delta E}\Theta(E-E_{-})\Theta(E_{+}-E),$
(26)

where $\Theta$ is Heaviside step function, $\Delta E=E_{+}-E_{-}$ is the width
of the spectrum and
$E_{\pm}=(m_{\chi}/2)\left(1\pm\sqrt{1-m_{a}^{2}/m_{\chi}^{2}}\right)$. The
probability of the mediator, $a$, decaying outside the celestial object is

$\displaystyle P_{\text{surv}}=e^{-R/\eta c\tau}-e^{-D/\eta c\tau}.$ (27)

Figure 3: The plot shows the probability of the SM particle surviving from the
NS to the detector at the Earth.

An example of the survival probability as a function of the decay length $\eta
c\tau$ is shown in figure 3. The plot shows the survival probability of the SM
particle produced by the mediator at the distance $\eta c\tau$ from a NS. The
probability is taken with the assumption that the SM particle does not reach
the Earth if the mediator decays inside the star (red dash line), i.e., for
$\eta c\tau<R_{\star}$, the probability is assumed to be zero. In this work,
we will consider the maximally optimistic case where we take
$P_{\text{surv}}=1$ for both gamma-rays and neutrinos cases. Furthermore, we
also calculate the bounds of an effective ALP-photon coupling and an effective
ALP-SM coupling for ALPs that decay in the distance between the object’s
surface and the Earth. The bound on $g_{a\gamma\gamma}$ are calculated from
the lifetime in equation 4 and the result is shown in the left panel of figure
4. We found that for the lower mass of ALP, the $g_{a\gamma\gamma}$ has
already been excluded by the constraint from the helioscope CAST (CERN Axion
Solar Telescope) CAST:2008ixs ; CAST:2011rjr ; CAST:2013bqn ; CAST:2017uph .
Therefore for the ALP decaying into 100% photons case, we will aim to study
for ALP mass range higher than $10^{-9}$ GeV. In the 100% decaying into
neutrinos case, the bound on $g_{aff}$ is calculated with the lifetime from
equation 5. The result is shown in the right panel of figure 4 where
$m_{f}=m_{\nu}=0.1$ eV is assumed. We only show the bound from BD because it
is the only target that can generate sufficient fluxes of neutrinos that are
detectable by the experiments, which we will discuss shortly.

Figure 4: The left panel shows the bounds on an effective ALP-photon coupling
for an ALP decay between the object’s surface and the Earth. We also display
the bounds from CAST helioscopes in the grey region. The right panel shows the
bounds on an effective ALP-SM coupling for an ALP decay between the object’s
surface and the Earth, where in this case ALP decays into neutrinos.

In order to set the limits on the ALP coupling, We use the Galactic Center
measured fluxes from various observations. The gamma-ray fluxes are taken from
Fermi and H.E.S.S. and the details of the data analysis are given in
Malyshev:2015hqa . Fermi data is used for the model fluxes with energy less
than 100 GeV Fermi-LAT:2015bhf , and we use H.E.S.S. HESS:2016pst  to set
limits around TeV scale. The high energy neutrino fluxes were taken from
IceCube IceCube:2017trr  and ANTARES ANTARES:2016mwq  where they have the
upper limits on the muon neutrinos flux from the direction of the Galactic
Center ($-40^{\circ}<l<40^{\circ}$ and $-3^{\circ}<b<3^{\circ}$ in Galactic
Coordinate) in the energy range between 1 TeV and 1 PeV. We set the limits by
requiring that the maximum value of the DM fluxes must not exceed the
measurable fluxes from the observation. This can be done by determining the
ALP coupling $g_{aff}$ for each energy bin $(E\approx m_{\chi})$ which gives
the highest possible scattering cross-section (equation 3) providing the
energy flux (equation 25) below the measured flux.

To generate the gamma-ray energy spectra, we use Asteria Leane:2023woh  to
calculate the total DM capture rate. We use the number density of NSs from
equation 19 and the number density of BDs from equation 20. We integrate both
densities over the spherical shell between radius $r_{1}=0.1$ pc and
$r_{2}=100$ pc. Choosing $r_{1}=0.1$ pc to avoid poor modeling of the DM cusp-
like profile at the Galactic Center. The outer radius $r_{2}=100$ pc is chosen
due to the fact that NSs/BDs number density falls rapidly outside this region.
We assume that the ALP decays into 100% photons and $P_{\text{surv}}=1$ for
simplicity. We show the largest possible signal for different choices of
density slope index $\gamma=1,1.5$ in the NFW DM profile in figure 5 along
with the measured gamma-ray flux from Fermi (black) and H.E.S.S. (blue). For
NSs, one could expect that only the gamma-rays from the NFW DM profile with
$\gamma=1.5$ are detectable by H.E.S.S. data. For BDs, both gamma-rays from
slope index $\gamma=1,1.5$ are higher than the measured fluxes from Fermi and
H.E.S.S., which would give us strong constraints to the ALP coupling.

Figure 5: Maximum differential fluxes of gamma-ray of the NSs or BDs from the
Galactic Center. The gamma-ray fluxes from the Galactic Center observed by
Fermi and H.E.S.S. are shown in black and blue, respectively.

To generate the neutrino spectra, we follow the same procedure as the gamma-
rays spectra. The largest possible fluxes for $\gamma=1,1.5$ are shown in
figure 6 along with the measured neutrino fluxes from IceCube (blue) and
ANTARES (black). We find that the maximum flux from the NS is lower than the
observation fluxes by approximately 2 orders of magnitude. As a result, we
will study only BDs as the potential sources of neutrino flux.

Figure 6: Maximum differential fluxes of neutrino from the NSs in the Galactic
Center. The neutrino fluxes from the Galactic Center observed by IceCube and
ANTARES are shown in blue and black, respectively.

##  6 Results

Figure 7 shows our constraints on an effective ALP coupling for DM-SM
scattering mediated by ALP. The gamma-rays were produced by the production of
DMs that annihilated inside the object, $\chi\chi\rightarrow a$,
$a\rightarrow\gamma\gamma$. The limits were placed on $E^{2}d\Phi/dE$ with the
gamma-ray observatories (Fermi and H.E.S.S.). Our calculation on gamma-ray
flux has been described in the previous section and the other parameters in
the plot are $g_{a\chi\chi}=1\times 10^{-3}$ and $m_{a}=m_{\chi}/10^{5}$. The
bounds from our works are displayed along with the other constraints. The
Fermi upper limit is shown in black for BD targets and the H.E.S.S. upper
limit using NSs and BDs as targets are shown in blue and red, respectively. We
show the results of BDs for $\gamma=1$ in the dash line and $\gamma=1.5$ in
the solid line. Since we concentrate on the simplest case where $C_{f}$ is
universal, therefore the result we present is in the form of
$g_{aff}/m_{f}=C_{f}/f_{a}$, where $m_{f}$ is the mass of the fermion. In the
gamma-rays final state, we use proton and neutron mass for calculating the
scattering cross section, therefore $m_{f}=m_{n}\approx m_{p}$. For NSs, we
show only $\gamma=1.5$ since the gamma-ray flux is detectable only at H.E.S.S.
Our results show that the limits in this work are significantly stronger than
previously existing constraints. For higher masses, NSs provide stronger
bounds than BDs since NSs require a lower cross-section to capture the DM
(figure 2).

Figure 7: The constraints plot on the ALP-fermion coupling, $g_{aff}$ as a
function of $m_{a}$. The results of our work are shown in black, red and blue
for BDs focused on Fermi, BDs focused on H.E.S.S. and NSs focused on H.E.S.S.,
respectively. Other constraints were briefly discussed in the text (see
Bharucha:2022lty  for a review).

The other constraints in the figure 7 are the following. The constraints of
the study of the supernova SN 1987A can provide strong constraints
Jaeckel:2017tud  and recently updated in reference Hoof:2022xbe . Considering
the cooling of the horizontal branch (HB) stars provides the parameters space
of ALPs Cadamuro:2011fd . The exclusion bounds for the decay of
$K^{+}\rightarrow\pi^{+}+$ inv. from NA62 NA62:2021zjw  are given in the form
of branching ratio. The strong constraints come from the electron beam dump
experiment SLAC E137 Bjorken:1988as , ALPs can be produced by the
bremsstrahlung, positrons annihilation, or Primakoff effect.
$K^{+}\rightarrow\pi^{+}a(\rightarrow e^{+}e^{-})$ from NA 48/2 NA482:2009pfe
provide C.L. 90% upper limits. $B^{+}\rightarrow
K^{+}a(\rightarrow\mu^{+}\mu^{-})$ from LHCb LHCb:2016awg  provide C.L. 95%
upper limits on the branching ratio as a function of the lifetime of $a$.
$B\rightarrow K^{*}a(\rightarrow\mu^{+}\mu^{-})$ from LHCb LHCb:2015nkv  also
provide the 95% C.L. upper limit on the branching ratio as a function of the
mass and lifetime of $a$. The limits from $B\rightarrow
K^{*}a(\rightarrow\mu^{+}\mu^{-})$ at fixed target come from CHARM
Dobrich:2018jyi . $K^{+}\rightarrow\pi^{+}a(\rightarrow\mu^{+}\mu^{-})$ from
NA 48/2 NA482:2016sfh  provide C.L. 90% upper limits on the branching ratio as
a function of lifetime of $a$. From the work of Bharucha:2022lty  where they
have studied the model of the production of ALP-mediated DM in the thermal
scenarios, our work (small $g_{a\chi\chi}$ and $g_{aff}$) corresponds to the
case that dark matter is produced by freeze-in and the ALP may or may not be
in equilibrium with the SM. Therefore, viable parameter space for the freeze-
in scenario is not ruled out by our study.

We also show results when we vary the ratio of the mass of DM and ALP in
figure 8. The constraints on $g_{aff}$ NS as the targets are calculated with
$g_{a\chi\chi}=1\times 10^{-3}$ and the mass ratios $m_{a}=m_{\chi}/10^{4}$,
$m_{a}=m_{\chi}/10^{5}$ and $m_{a}=m_{\chi}/10^{6}$ are shown in blue, red and
black, respectively. The results show that the decreasing of the mass ratio
gives stronger bounds in the lower masses region of ALP.

Figure 8: The constraints plot on the ALP-fermion coupling where we fixed the
$g_{a\chi\chi}=1\times 10^{-3}$. We only focused on the NSs as the only
sources. We show the results for the mass $m_{a}=m_{\chi}/10^{4}$,
$m_{a}=m_{\chi}/10^{5}$ and $m_{a}=m_{\chi}/10^{6}$ in blue, red and black,
respectively.

Figure 9 shows the upper limits on the ALP-fermion coupling from the neutrino
observations (IceCube and ANTARES). The neutrino fluxes were generated by only
focusing on BD as the targets as discussed in the earlier section. The bounds
are weaker than the gamma-ray case but still stronger than existing
constraints, especially in the higher ALP mass region. Under the universal
ALP-fermion coupling assumption, the region where ALP decays en route (figure
4) can be constrained to be $m_{a}\gtrsim 0.6$ GeV.

Figure 9: The constraints plot from the neutrino observations on the ALP-
fermion coupling, $g_{aff}$ as a function of $m_{a}$ where we fixed the ALP-DM
coupling as $g_{a\chi\chi}=1\times 10^{-3}$ and $m_{a}=m_{\chi}/10^{5}$. The
NFW DM profiles with $\gamma=1,1.5$ are represented in dash and solid lines,
respectively.

##  7 Conclusion

In this work, we have studied the constraints of the ALP-mediated DM models
from celestial objects such as neutron stars (NS) and brown dwarfs (BD). The
DM can be captured in celestial objects where the DM accumulation process is
controlled by the dark matter multiscatter capture mechanism. The annihilation
of DM inside the objects produces the long-lived ALPs which will decay into
the SM particles outside the objects. We study the case that ALP decays into
gamma-rays and neutrinos separately and compare the fluxes with data from
observation. For gamma-ray fluxes, we present our results on $g_{aff}/m_{f}$
in figure 7. The results show the strong bounds that can rule out a
significant portion of the parameter space. We point out that by using NSs as
targets the model provides a stronger bound than BD ones due to the lower
fluxes required to match with H.E.S.S. data. For Fermi data with the lower DM
mass range, only BDs are detectable via the gamma-ray channel. We also present
the upper bounds for different mass ratios in figure 8 and show that our
limits are generally stronger than existing constraints across the mass range
of ALP. The results for neutrino fluxes are present in figure 9. We found that
the only possible targets that can produce sufficient neutrino flux are BDs.
With data from IceCube and ANTARES, we can probe our model with the higher
mass range. Similar to the gamma-ray channel, the bounds from neutrino fluxes
are also generally stronger than the previous constraints on ALP. Provided
that ALP decays en route towards the Earth, the allowed parameter space for
this model is strictly constrained with $m_{a}\gtrsim 0.6$ GeV. This
emphasizes the importance of the next-generation probes of the neutrino fluxes
at higher energies such as ARIANNA, ARIA, IceCube-Gen2 IceCube-Gen2:2020qha ;
Ishihara:2019aao , KM3Net KM3Net:2016zxf , ANITA-IV ANITA:2019wyx , PUEO
PUEO:2020bnn , RNO-G RNO-G:2020rmc  and Auger PierreAuger:2019ens .

For future works, we are interested in exploring the phenomenological
consequences of the possibility that ALP decays inside celestial objects
providing additional energy to their internal processes. We also would like to
investigate the effects of ALP-mediated interactions on the nuclear equation
of state (EoS) and neutron star stability.

## Acknowledgement

TK and CP have been supported by the National Astronomical Research Institute
of Thailand (NARIT). CP is supported by Fundamental Fund 2566 of Khon Kaen
University and Research Grant for New Scholar, Office of the Permanent
Secretary, Ministry of Higher Education, Science, Research and Innovation
under contract no. RGNS64-043. We thank Areef Waeming and Daris Samart for
their helpful comments and discussions.

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# Which to Match? Selecting Consistent GT-Proposal Assignment for Pedestrian
Detection

Yan Luo1, Chongyang Zhang$~{}^{1,2,}$111*, Muming Zhao1, Hao Zhou1, Jun Sun1
1Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

Accurate pedestrian classification and localization have received considerable
attention due to its wide applications such as security monitoring, autonomous
driving, etc. Although pedestrian detectors have made great progress in recent
years, the fixed Intersection over Union (IoU) based assignment-regression
manner still limits their performance. Two main factors are responsible for
this: 1) the IoU threshold faces a dilemma that a lower one will result in
more false positives, while a higher one will filter out the matched
positives; 2) the IoU-based GT-Proposal assignment suffers from the
inconsistent supervision problem that spatially adjacent proposals with
similar features are assigned to different ground-truth boxes, which means
some very similar proposals may be forced to regress towards different
targets, and thus confuses the bounding-box regression when predicting the
location results. In this paper, we first put forward the question that
Regression Direction would affect the performance for pedestrian detection.
Consequently, we address the weakness of IoU by introducing one geometric
sensitive search algorithm as a new assignment and regression metric.
Different from the previous IoU-based one-to-one assignment manner of one
proposal to one ground-truth box, the proposed method attempts to seek a
reasonable matching between the sets of proposals and ground-truth boxes.
Specifically, we boost the MR-FPPI under R75 by 8.8% on Citypersons dataset.
Furthermore, by incorporating this method as a metric into the state-of-the-
art pedestrian detectors, we show a consistent improvement.

##  1 Introduction

Figure 1: IoU-based GT-Proposal assignment may result in inconsistent
regression supervision: the two proposals $\mathbf{P}_{1}$ and
$\mathbf{P}_{2}$ (in the left picture), which have close location, area, and
similar features, are assigned to two much different ground-truth boxes:
$\mathbf{G}_{1}$ and $\mathbf{G}_{2}$, respectively. This inconsistent
supervision may confuse the regression training, and thus one low-quality
regression box, $\mathbf{R}_{1}$ in the right picture, will be predicted. With
the proposed DRNet, $\mathbf{P}_{1}$ will be assigned to $\mathbf{G}_{2}$ due
to less depth cost, and then one correct predicted box, $\mathbf{R}_{2}$ in
the right, will be got.

Pedestrian detection is a key problem in a number of real-world applications
including auto-driving systems and surveillance systems, and is required to
have both high classification and localization accuracy. Driven by the success
of general object detection, many of the proposed pedestrian detectors [75,
39, 43, 22] still follow the basic practice such as Faster R-CNN [46] and SSD
[38]. Therefore, Intersection over Union (IoU), which is required to assign
proposals, has been one indispensable metric in the recent frameworks.

However, the commonly used IoU metric has two main drawbacks. (1) On one hand,
it is difficult to set a proper threshold for IoU during training. A lower IoU
threshold (e.g. 0.5) could keep adequate number of positive samples but will
result in many ”close but not correct” false positives during inference [39].
A relative higher threshold (e.g. 0.7) can reject low-quality proposals but
will reduce a large number of matched positives. Although the Cascade R-CNN
[5] and ALFNet [39] have provide solutions that gradually refine the proposals
by several stages with a set of IoU thresholds, the hand-crafted IoU
threshold, whether single or multiple, is still not the best choice. (2) On
the other hand, the IoU-based GT-Proposal assignment suffers from the
inconsistent supervision problem that spatially adjacent proposals with
similar features are assigned to different ground-truth boxes, which means
some very similar proposals may be forced to regress towards different
targets, and thus confuses the bounding-box regression when predicting the
location results. This disadvantage is more prominent in the scene with large
scale variation and occlusion. As illustrated in Figure 1, overlapped
proposals with almost the same features are assigned to different ground-truth
boxes, which results in the confusion during training and thus brings about
low-quality regression results during inference (Figure 1 red solid box).

The above analyses motivate us to address the weakness of IoU-based GT-
Proposal matching mechanism and propose one new assignment and regression
metric for pedestrian detection. Different from the previous one-to-one
assignment manner of one proposal to one ground-truth box, we conduct the
assignment in a set-to-set process of finding a reasonable matching between
the two sets (proposals and ground-truth boxes). This searching-based method
does not depend on a fixed hyper-parameter (e.g. IoU threshold), instead it
constantly searches for the matched pairs along the dynamic training
procedure. Furthermore, the search algorithm consists of one cost function to
answer the question of ”which to match” for each proposal. Motivated by the
discovery that the distribution of pedestrian scales is highly connected with
the depth-variant direction, the introduced cost function starts with the
depth estimation, and gradually assigns all proposals step by step, pushing
more proposals closer to ground-truth boxes with smaller depth (scale)
variance. By this way, the regressor will be optimized towards consistent
location direction: regress to the direction with smaller depth variance. On
top of this, we are the first to point out that ”Regression needs Direction”
to the best of our knowledge, and a novel pedestrian detection architecture is
thus constructed, denoted as Directional Regression Network (DRNet).

To demonstrate the generality of the proposed depth-guided search algorithm,
we evaluate various CNN-based pedestrian detectors on both Citypersons [74]
and Caltech [13] dataset including OR-CNN [77], PBM [22] and CSP[61].To sum
up, the main contributions of this work lie in:

(1) We first attempt to put forward the inconsistent supervision problem of
IoU-based assignment mechanism and propose one search algorithm as a new
assignment metric, which is both dynamic and direction-sensitive during
training procedure.

(2) We first attempt to point out that ”Regression needs Direction” and
propose one directional regression network, named DRNet, to tackle with the
challenging problem of large scale variation for accurate pedestrian
localization.

(3) We achieve state-of-the-art accuracy on widely-used datasets including
Caltech and Citypersons, especially under the challenging setting of R75 by
8.8% on Citypersons dataset. We incorporate the proposed method into the most
popular pedestrian detection algorithms such as CSP and PBM [22], and show
further performance improvement.

Figure 2: An example of the matching process of the proposed method. The
second row shows matching cost of different proposal to GTs. In each step, the
ground-truth box is assigned with several proposals with minimum matching
cost. Besides, if exists any conflict, which means the proposal has already
assigned to one ground-truth box, the matching process would be rerun.
Detailed procedure is shown in Algorithm 1.

##  2 Related Work

###  2.1 Pedestrian Detection

Driven by the success of general object detection, many pedestrian detectors
follow the anchor-based/anchor-free paradigm. Specifically, the anchor-based
detectors are proposed in the two-stage/one-stage framework. RPN+BF [71]
discusses the issue that improves performance of Faster R-CNN for pedestrian
detection with a RPN followed by shared boosted forests. MS-CNN [3] attempts
to introduce various receptive fields that match different object scales to
tackle with multi-scale problem in pedestrian detection. Recently, methods
such as PBM [22] use both part and full-body features to more accurately
classify and localize pedestrians especially in crowded scenes. However,
aforementioned detectors focus more on feature representation, and less
attention is paid to the inherent drawbacks of IoU-based assignment method
during the training procedure.

###  2.2 Increment-based Label Assignment

Most recently, similar to our intuition, some researchers attempt to revisit
the role of IoU for object detection. First, methods such as Cascade R-CNN [5]
and ALFNet [39] try to replace the single threshold with an incremental IoU
setting. Although these detectors have provided solutions that gradually
refine the proposals, they are still not free from the hand-crafted setting
and perform somewhat inflexible during training. Most recently, a progressive
network [63] is proposed with three-phase progression to gradually refine
anchors following human annotation process. However, the problem is also
obvious. On the one hand, although the gradual strategy could bring
performance improvements, it also brings double the computational cost. On the
other hand, some incremental settings are not flexible. For example, Cascade
R-CNN and ALFNet both use manually defined IoU (e.g, 0.5, 0.6 and 0.7) to
measure the quality.

###  2.3 Statistics-based Label Assignment

Different from previous methods that choose a fixed number of best scoring
anchors, statistics-based label assignment detectors model the anchor
selection procedure as a likelihood estimation for a probability distribution.
ATSS [76] proposes an adaptive training sample selection method that uses the
sum of mean and standard deviation as the IoU Threshold. PAA [26] proposes a
probabilistic anchor assignment strategy that adaptively separates a set of
anchors into positive and negative samples for a GT box according to the
learning status of the model associated with it. These methods have taken a
step towards more accurate assignment strategies. Nevertheless, to the best of
our knowledge, these still get caught into the IoU-based assignment, which
means they still not escape from the inconsistent problem our work raised.

###  2.4 Prior-based Label Assignment

To relief the extra computation cost, some works introduce human prior to
improve label assignment. FreeAnchor [64] proposes one detector that
guarantees at least one anchor’s prediction close to the ground-truth for each
object. GuideAnchoring [57] and MetaAnchor [67] try to dynamically change the
shape of anchors to fit various distributions of objects. The aforementioned
methods, whether prior-related (e.g., GuideAnchoring) or instance-related
(e.g., FreeAnchor), ignore the inconsistent supervision problem existing
during the assignment process. Therefore, they are still not free from the
confusion caused by ”similar proposals, different regression targets”, and
thus bring about false regression results during inference. Compared to their,
ours is not only dynamic and flexible, but also ensures the consistency of
matching, so it better answers the question of ”which to match” and thus shows
further improvement for accurate pedestrian localization. On top of above,
combined with depth estimation (z axis), our work is also the first attempt
that tries to solve the assignment problem from a 3D perspective compared with
other 2D assignment (x-y axis), e.g. ATSS [76] or PAA [26].

##  3 Proposed Method

Generally speaking, the proposed method consists of one search algorithm and
one depth-guided function to solve the main task: ”which to match” for each
proposal. The overall process is shown in Figure 2.

###  3.1 How to Design Depth-guided Assignment?

”Which to Match” is an important question especially for pedestrian detection.
It is essentially a matching problem, that is how to match reasonable ground-
truth boxes for each proposal so as to push the network training under a
better direction. From the above analysis, the IoU metric performs as one-to-
one assignment manner of one proposal to one ground-truth, but brings about
inconsistent supervision problem during training. Different from IoU, we
regard the assignment as the process of finding a reasonable matching between
two sets (proposals and ground-truth boxes). We formulate this process in the
following. The ground-truth boxes and proposals are denoted as
$\mathbf{G}=\left\\{g_{i}\right\\}_{i=1}^{M}$ and
$\mathbf{B}=\left\\{b_{j}\right\\}_{j=1}^{N}$, in which $M$ and $N$ represent
the number of ground-truth boxes and proposals, respectively.

Figure 3: The example of the same shortest path length but with several
possible paths. The red, yellow, green and blue paths all have the same
shortest Manhatten length of 14. The two dark red points represent the start
and end, respectively.

To compute the relevance between one proposal and one ground-truth box, the
proposed method defines one search function, which can be divided into two
parts. The first one is Manhattan distance, denoted as
$\mathbf{D}(g_{i},b_{j})$, which measures the distance between the paired
proposal and ground-truth box. It is worth noting that in the view of paths,
there exist several shortest paths with the smallest Manhattan distance, as is
shown in Figure 3. The red, green, yellow and blue paths all have the same
shortest path length, and this also brings some uncertainty to our second part
of depth estimation, denoted as $\mathbf{Z}(g_{i},b_{j})$. From the start of
one proposal to the end of one ground-truth box, Manhattan distance proposes
several possible paths, denoted as
$\mathbf{P}=\left\\{p_{k}\right\\}_{k=1}^{T}$, in which $p_{k}$ represents the
k-th path and T is the total number of paths. Various paths in $\mathbf{P}$
shows various depth variance, denoted as
$\mathbf{V}=\left\\{v_{k}\right\\}_{k=1}^{T}$, in which $v_{k}$ is the sum of
the depth changes along the k-th path, formulated as:

$v_{k}=\mathbf{A}(p_{k})=\sum_{q=1}^{s}d_{q}$ (1)

where $\mathbf{A}(\cdot)$ represents the aggregation function to calculate the
depth variance, $s$ represents the total number of points included in the path
$p_{k}$, and $d_{q}$ represents the q-th depth variance between two adjacent
points. We select the path with the minimal depth variance among different
paths to calculate the depth variance cost $\mathbf{Z}(g_{j},b_{i})$ along the
matching process. It canbe defined as the following:

$\mathbf{Z}(b_{i},g_{j})=\mathop{\min}\ (\mathbf{V})$ (2)

Consider the example in Figure 2, which schematically shows a detector with
five generated proposals and three ground-truth boxes, the overall matching
cost $\tilde{l}_{mc}(g_{i},b_{j})$ can be formulated as:

$\tilde{l}_{mc}(g_{i},b_{j})=\mathbf{D}(g_{i},b_{j})+\mathbf{Z}(g_{i},b_{j})$
(3)

Algorithm 1 Search Algorithm

Input:
$\mathbf{B}$ is a set of ranked proposals generated by RPN
$\mathbf{G}$ is a set of ground-truth boxes on the image
$M$ and $N$ are the number of GTs and proposals
$N_{p}$ and $N_{e}$ are the number of positives and negatives
$\tilde{l}_{mc}(g_{i},b_{j})$ is the matching cost with $g_{i}$ and $b_{j}$.
Output:
$\mathcal{P}$ and $\mathcal{N}$ are sets of positives and negatives

1: build an empty set for positive samples:
$\mathcal{P}\leftarrow\varnothing$;

2: build an empty set for negative samples:
$\mathcal{N}\leftarrow\varnothing$;

3: build an empty set for pending proposals:
$\mathcal{T}\leftarrow\varnothing$;

4: for each box $g_{i}\in\mathbf{G}$ do

5: $\left\langle g_{i},b_{j}\right\rangle\leftarrow$select the proposal
$b_{j}\in\mathbf{B}$ with the minimal cost $\tilde{l}_{mc}(g_{i},b_{j})$ as
the assigned pair;

6: if the number of assigned pairs of $g_{i}$ $\leqslant[\frac{N_{p}}{M}]$
then

7: $\mathcal{P}=\mathcal{P}\cup\left\\{b_{j}\right\\}$;

8: else

9: re-compare the cost $\tilde{l}_{mc}$ of all proposals that have assigned to
$g_{i}$;

10: if the proposal with the largest cost is $b_{j}$ then

11: $\mathcal{T}=\mathcal{T}\cup\left\\{b_{j}\right\\}$;

12: else

13: $b_{l}\leftarrow$ proposal with the largest cost;

14:
$\mathcal{P}=\mathcal{P}\cup\left\\{b_{j}\right\\}$;$\mathcal{P}=\mathcal{P}-\left\\{b_{l}\right\\}\leftarrow$
abandon $b_{l}$;

15: $\mathcal{T}=\mathcal{T}\cup\left\\{b_{l}\right\\}\leftarrow$ put it into
the pending set;

16: end if

17: end if

18: $\mathbf{B}=\mathbf{B}-\left\\{b_{j}\right\\}$;

19: end for

20: if $(|\mathcal{N}|+|\mathcal{T}|)<N_{e}$ then

21: compute IoU between $\mathbf{G}$ and $\mathbf{B}$;

22: randomly assign negative samples to fill $\mathcal{N}$;

23: end if

24: return $\mathcal{P}$, $\mathcal{N}$, $\mathcal{T}$;

where $\mathbf{D}$ and $\mathbf{Z}$ represent the value of Manhatten distance
and depth variance of all proposals. In the following, we consider a simple
assignment strategy based on the estimated matching cost
$l_{mc}(g_{i},b_{j})$. We sort the proposals with the corresponding confidence
estimated by Region Proposal Network (RPN). The proposed matching strategy
sequentially assigns proposals to the ground-truth boxes. In details, we
define the maximum number of assignment of each ground-truth box as $N_{p}/M$,
in which $N_{p}$ represents the number of positive samples and $M$ represents
the number of ground-truth boxes of the input image. For each proposal in
$\mathbf{B}=\left\\{b_{j}\right\\}_{j=1}^{N}$, we sequentially assign the
ground-truth box with the smallest $l_{mc}$ during its matching. Detailed
matching process is shown in Algorithm 1.

It is worth noting that (1) each ground-truth boxes can assign up to
$[N_{p}/M]$ proposals; (2) the proposal that was previously assigned to
ground-truth box but was removed latter is sampled as hard negative samples;
(3) if the negative samples are still not full at the end, IoU is also
introduced to fill the negative set. Besides, in order to enrich the diversity
of negative samples, half of negative proposals are randomly sampled from the
negative set $\mathcal{N}$, and the remaining half are from pending set
$\mathcal{T}$. The matching algorithm assigns a unique proposal to each
ground-truth box. Given this algorithm, we define a loss function on pairs of
sets $\mathbf{G}$ and $\mathbf{B}$ as:

$\displaystyle L(\mathbf{G},\mathbf{B})$
$\displaystyle=\alpha\sum_{j=1}^{N_{p}}l_{reg}(b_{j},\tilde{g})+\beta\sum_{j=1}^{N_{p}+N_{e}}l_{cls}(b_{j},\tilde{g})$
(4)

where $l_{reg}$ is a regression loss, $l_{mc}$ is the matching cost and
$l_{cls}$ is a cross-entropy loss on a proposal’s confidence that it would be
matched to a ground-truth box. The label for this cross-entropy loss is
provided by $\tilde{g}$. $\alpha$, $\beta$ and $\eta$ are the parameters
trading off between the three parts. Note that for the proposed matching, we
can update the network by back-propagating the gradient of this loss function.

###  3.2 Implement Details

The above algorithm attempts to answer the question ”which to match” for each
proposal. In this section, we hope to further improve the proposed algorithm.
We focus on the most important two aspects: (1) how to assign proposals with
various scales to different ground-truth boxes? (2) how to relief
inconsistency between depth-guided assignment and IoU-based performance
evaluation?

To answer the first question, we follow the practice in FCOS [53], which
introduces multi-level prediction with FPN [32]. With the use of multi-level
feature maps defined as $\left\\{C_{3},C_{4},C_{5},C_{6},C_{7}\right\\}$,
proposals and ground-truth boxes with different scale would be allocated to
the specific feature level. The specific feature level is only used to assign
proposals with the specific scale. During experiments, the size range is (0,
64] for $C_{3}$, (64, 128] for $C_{4}$, (128, 256] for $C_{5}$, (256, 512] for
$C_{6}$ and (512, $+\infty$] for $C_{7}$. Since proposals with different sizes
are allocated to different feature levels, the proposals and ground-truth
boxes on the same level would be closer with larger overlap, which further
enhances our algorithm.

In order to ensure our consistency between training and evaluation, we attempt
to incorporate assignment into the confidence score of each predicted bounding
box. We add one parallel branch to the detection head of classification and
regression to predict the matching cost ${l}_{mc}$ of each proposal. The
aforementioned calculated matching cost $\tilde{l}_{mc}$ servers as the
estimation target to supervise the whole process. Therefore, the total loss in
Equation 4, redefined as:

$\tilde{L}(\mathbf{G},\mathbf{B})=L(\mathbf{G},\mathbf{B})+\gamma\sum_{j=1}^{N_{p}}\mathbf{Logistic}(l_{j,mc},\tilde{l}_{j,mc})$
(5)

where Logistic($\cdot$) means the loss function using in standard Logistic
Regression. In this way, we try to reduce the confidence of some detected
boxes, which regress different from our matching procedure. In details, taking
the proposal $b_{t}$ as an example, the regressed box of $b_{t}$ is denoted as
$\tilde{b}_{t}$. The actual matching cost between $b_{t}$ and $\tilde{b}_{t}$
can be calculated following Equation 4, denoted as
$l^{*}_{mc}(b_{t},\tilde{b}_{t})$. The confidence score $s_{t}$ of $b_{t}$ is
re-measured, formulated as:

$\tilde{s}_{t}=s_{t}\cdot[1.5-\mathbf{Sigmoid}(|l^{*}_{mc}-l_{mc}|)]$ (6)

Table 1: Comparisons with the state-of-the-art methods on Citypersons validation dataset. Results are the MR-2 evaluation metric of the corresponding methods, in which lower is better. Boldface indicates the best performance. ex. means whether the method introduces extra annotations. For example, PBM and PRNet use extra part annotation, and TLL+MRF considers the sequence input. Method | Year | ex. | Backbone | Scale | R50 | R75 | Heavy | Inference Time
---|---|---|---|---|---|---|---|---
Faster R-CNN+ATT [75] | CVPR2018 | $\checkmark$ | VGG-16 | $\times\ 1.0$ | 16.0 | 48.1 | 56.7 | -
TLL [49] | ECCV2018 | $\times$ | ResNet-50 | $\times\ 1.0$ | 15.5 | 44.2 | 53.6 | -
TLL+MRF [49] | ECCV2018 | $\checkmark$ | ResNet-50 | $\times\ 1.0$ | 14.4 | - | 52.0 | -
ALFNet [39] | ECCV2018 | $\times$ | ResNet-50 | $\times\ 1.0$ | 12.0 | 36.5 | 51.9 | 0.27s / img
RepLoss [60] | CVPR2018 | $\times$ | ResNet-50 | $\times\ 1.3$ | 11.6 | - | 55.3 | -
OR-CNN [77] | ECCV2018 | $\times$ | VGG-16 | $\times\ 1.0$ | 12.8 | - | 55.7 | -
MGAN [43] | ICCV2019 | $\times$ | VGG-16 | $\times\ 1.0$ | 11.5 | - | 51.7 | -
PBM [22] | CVPR2020 | $\checkmark$ | VGG-16 | $\times\ 1.0$ | 11.0 | - | 53.3 | -
CSP [61] | CVPR2019 | $\times$ | ResNet-50 | $\times\ 1.0$ | 11.0 | 34.7 | 49.3 | 0.33s / img
PRNet [63] | ECCV2020 | $\checkmark$ | ResNet-50 | $\times\ 1.0$ | 10.8 | - | 53.3 | 0.22s / img
DRNet (ours) | - | $\times$ | VGG-16 | $\times\ 1.0$ | 10.2 | 29.7 | 46.9 | 0.12s / img
|  |  |  | $\times\ 1.3$ | 9.5 | 27.0 | 43.6 | 0.14s / img
DRNet (ours) | - | $\times$ | ResNet-50 | $\times\ 1.0$ | 10.1 | 25.9 | 46.2 | 0.15s / img
|  |  |  | $\times\ 1.3$ | 9.1 | 25.4 | 42.1 | 0.18s / img

##  4 Experiments

We assess the effectiveness of our proposed method for pedestrian detection on
widely used datasets Caltech [14] [13] and Citypersons [74].

###  4.1 Experimental Setup

The proposed method is based on the Faster R-CNN baseline [46], pre-trained on
the ImageNet. We optimize the network using the Stochastic Gradient Descent
(SGD) algorithm with 0.9 momentum and 0.0005 weight decay, which is trained on
1 1080Ti GPU with the mini-batch involving 1 image per GPU. For Caltech
dataset, we train the network for $40k$ iterations with the initial learning
rate of $10^{-3}$ and decay it to $10^{-4}$ for another $20k$ iterations. For
Citypersons dataset, we train the network for $20k$ iterations with the
initial learning rate of $10^{-3}$ and decay it to $10^{-4}$ for another $10k$
iterations. All images are in the original $1\mathrm{x}$. scale during
training and testing. Other parameters $\alpha$, $\beta$ and $\gamma$ are set
to 1, 1 and 0.01, respectively. $N_{p}$ and $N_{e}$ are both equal to 256 as
usual. Specifically, our depth estimation model is VNL [68]. Better depth
estimation model may have better performance, but it is not in the main scope
of ours method.

###  4.2 Evaluation Metrics

In experiments, we used the standard average-log miss rate (MR) on False
Positive Per Image (FPPI) in [$10^{-2}$, $10^{0}$]. This kind of metric is a
bit similar to Average Precision (MAP) and refers more to the object not
detected.

On Caltech and Citypersons, we report results across different occlusion
degrees: Reasonable, Heavy and Partial. The visibility ratio is (0.65, 1], (0,
0.65) and (0.65, 1), respectively. In all subsets, the height of pedestrian
over 50 pixels is taken for evaluation. It is worth noting that Heavy is
designed to evaluate performance in case of severe occlusions. To further
demonstrate our performance, we have designed two special settings. The first
one is to validate the localization accuracy. We not only test on Reasonable
set under IoU$=0.5$ ($\mathbf{R}_{50}$), but also on Reasonable set under
IoU$=0.75$ ($\mathbf{R}_{75}$). The second one is that we split the
Citypersons into a new subset called Large Height Variation (LHV), in which
each image contains height variation of pedestrian instances larger than 50
pixels with some occlusion, in which the visibility ratio is [0.2, 0.9].

Table 2: Comparisons with different modules on Cityperons validation dataset. Results are the MR-2 evaluation metric of the corresponding methods, in which lower is better. Boldface indicates the best performance. Directional Assignment is described in Sec 3.1 and Refinement is described in Sec 3.2 Backbone | Scale | IoU Assignment | Direction Assignment | Refinement | R50 | Heavy | Partial | Bare
---|---|---|---|---|---|---|---|---
VGG-16 | $\times\ 1.0$ | $\checkmark$ |  |  | 15.8 | 53.2 | 16.9 | 9.7
VGG-16 | $\times\ 1.0$ |  | $\checkmark$ |  | 11.8 | 49.6 | 15.5 | 9.3
VGG-16 | $\times\ 1.0$ |  | $\checkmark$ | $\checkmark$ | 10.2 | 46.9 | 14.5 | 6.6
Figure 4: The visualization results of our DRNet and the state-of-the-art
methods, e.g., CSP [61] and OR-CNN [77]. The box with the check mark or cross
represents the correct and false results, respectively. The red and yellow
cross show false positives of OR-CNN and CSP, while ours performs better.

###  4.3 Main Results

We compare DRNet with corresponding methods on Citypersons and Caltech dataset
in Table 1 and Table 4. For fair comparisons, we report results in terms of
backbone, scale, inference time and all challenging subsets.

(1) Citypersons Dataset. Table 1 reports the results compared to state-of-the-
arts on Citypersons. First of all, our algorithms has significant performance
improvement under various fairness settings. For example, when leveraging
ResNet-50, ours achieves the highest accuracy with an improvement of 0.9% MR-2
from the closet competitor CSP [61] on R50 and 8.8% MR-2 on R75. Furthermore,
it is worth noting that from Table 1, ours has also demonstrated the self-
contained ability to handle occlusion issues in crowded scenes. Especially on
Heavy occlusion subset, ours reports the brand-new state-of-the-arts of 42.1%
MR-2. This is probably because harder samples are mined with the introduced
directional assignment metric, and thus training a more discriminant
predictor. And we mainly improve the training matching process, there is no
additional modification to the network structure. Therefore, there is no extra
computation consumption during testing, and our inference time is comparable
to others.

(2) Caltech Dataset. We also test our method on Caltech and the comparison
with state-of-the-arts on this benchmark is shown in Table 4. Our method
achieves MR-2 of 3.08% under the IoU threshold of 0.5, which is comparable to
the best competitor (3.80% of CSP [61]). Besides, in the case of a stricter
occlusion level of Heavy subset, our method achieves 30.45% MR-2,
outperforming all previous state-of-the-arts with an improvement of 6.05% MR-2
over CSP [61]. It indicates that our method has a substantially better
localization accuracy.

###  4.4 Ablation

To show the effectiveness of each proposed component, we report the overall
ablation studies in Table 2.

(1) Search and Assignment. As analyzed above, it can be seen that the IoU
metric are suboptimal primarily because it is difficult to answer ”which to
match” question for each proposal. The performance is summarized in Table 2.
When evaluated respectively, the search and assignment strategy shows the
improvement of 4.0% MR-2 compared with the original IoU assignment.
Furthermore, by incorporating with the refinement module, we show a consistent
improvement of 1.6% MR-2. In addition, we can also see that from the
visualization results in Figure 4, our algorithm effectively reduces false
detection results when there are large height variance of pedestrians in each
image compared with other algorithms, e.g., CSP [61] or OR-CNN [77]. Taking
the first image in Figure 4 as an example, the red dotted box failed to
regress toward the green box (boy with white T-shirt), and thus brings about
”close but not correct” false positives. Moreover, our algorithm is
particularly effective in two challenging scenarios. One is the scene with
large height variance, such as the example in Figure 4 (first row, third
column). The confusion in IoU makes it harder to train a high quality
regressor, and thus brings about ”close but not correct” false positives
during testing, while ours can effectively eliminate this kind of low-quality
regression. The second is the occlusion scene. In this case, our algorithm has
more consistent matching targets and therefore has better performance (Figure
4, second row, second column).

Table 3: Comparisons with the state-of-the-art methods on Citypersons validation dataset. Boldface indicates the best performance. $\dagger$ means our re-implementation of referred method, and $\dagger\dagger$ means the corresponding detector combined with our assignment manner. Method | Backbone | R50 | LHV
---|---|---|---
BiBox [83] | VGG-16 | 11.2 | -
BiBox† | VGG-16 | 11.2 | 30.6
BiBox†† | VGG-16 | 10.5 | 24.7
PBM [22] | VGG-16 | 11.1 | -
PBM† | VGG-16 | 11.0 | 30.3
PBM†† | VGG-16 | 10.2 | 23.9
CSP [61] | ResNet-50 | 11.0 | -
CSP† | ResNet-50 | 10.9 | 29.0
CSP†† | ResNet-50 | 10.2 | 23.7
Table 4: Comparisons with the state-of-the-art methods on Caltech testset. Boldface indicates the best performance. Method | Backbone | R50 | Heavy
---|---|---|---
DeepParts [40] | - | 12.90 | 60.42
MS-CNN [3] | - | 8.08 | 59.94
RPN+BF [71] | - | 7.28 | 74.36
SDS-RCNN [1] | ResNet-50 | 6.44 | 58.55
ALFNet [39] | ResNet-50 | 4.50 | 43.40
RepLoss | ResNet-50 | 4.00 | 63.36
CSP [61] | ResNet-50 | 3.80 | 36.50
DRNet (ours) | ResNet-50 | 3.08 | 30.45
Figure 5: The performance of MR-2 along training process under different
epoch. LHV means the maximum pedestrian height variance per image.

(2) Large Height Variance. In order to better verify our algorithm, we
specially set a Large Height Variance subset (LHV). The separation of this
subset is based on the maximum variance in pedestrian height per image. On one
hand, in Figure 5, we show that the change of miss rate during training
process under different settings. We found that our method can accurately
detect under large height variance (LHV). On the other hand, we also integrate
the algorithm with other algorithms and verify its performance on the LHV
subset (Table 3). In order to make a fair comparison, we reproduced the
relevant algorithm and report a comparison between our reproduced performance
and the performance reported in corresponding paper. When integrated with
other algorithms, the performance has also been greatly improved, especially
on the LHV subset. However, we have also noticed that although our algorithm
has improved different detectors (e.g., CSP or PBM), results report in Table 3
just show competitive with the bare search algorithm in Table 1. We think this
is because there exist some incompatibility between the previous algorithms
and ours method. For example, PBM [22] and BiBox [83] both introduce part
annotations, which leads to some confusion in our assignment procedure.

Table 5: Comparisons with the state-of-the-art methods on COCO minival set. $\dagger$ means the corresponding detector combined with our assignment manner. Boldface indicates the best performance. Method | Backbone | AP | AP75
---|---|---|---
FPN [32] | ResNet50 | 33.9 | -
RetinaNet [33] | ResNet50 | 36.3 | 38.8
FCOS [53] | ResNet50 | 36.6 | 38.9
FreeAnchor [64] | ResNet50 | 38.7 | 41.6
ATSS [76] | ResNet50 | 39.3 | 42.8
PAA [26] | ResNet50 | 41.1 | 44.3
FPN† | ResNet50 | 37.5 | 40.9
FCOS† | ResNet50 | 41.4 | 44.7
RetinaNet† | ResNet50 | 41.5 | 45.0

(2) Extended Experiments. In order to further validate our algorithm, we also
report results on COCO minival dataset compared with other detectors. We use a
COCO training setting which is the same as [26] in the batch size, frozen
Batch Normalization, learning rate, etc. For ablation studies, we use
ResNet-50 backbone and run 135 iterations of training. We notice that the
performance of the proposed label assignment is improved by adding it to other
widely-used frameworks, such as 3.6% on FPN [32], 5.2% on RetinaNet [33] and
4.8% on FCOS [53]. Furthermore, compared with other label assignment
strategies, ours also show competitive performance in Table 5. Although our
algorithm is designed specially for pedestrian detection, it still has some
performance improvement for more general detection dataset such as COCO. We
believe that the performance will be further improved if there is a more
refined design for the general purpose.

##  5 Conclusion

In this paper, we present a simple yet effective pedestrian detector with a
novel assignment strategy, achieving competitive accuracy while performing
competitive inference time with the state-of-the-art methods. On top of a
backbone, the proposed method can serve as a metric incorporated into the
other pedestrian detectors, and experimental results show a consistent
improvement on some popular pedestrian detection benchmarks, e.g. Citypersons
and Caltech. This novel design is flexible and independent of any backbone
network, without being limited by the two-stage detection framework.
Therefore, it is also interesting to incorporate the proposed assignment
strategy with other detectors like FCOS [53] and YOLO [45], which will be
studied in future. Furthermore, we also consider extending this method, not
only for pedestrian detection, but also study the proposed method on more
general object detection tasks.

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