# Enhanced signature of vacuum birefringence in a plasma wakefield

Feng Wan  Ministry of Education Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Ting Sun  Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China  Bai-Fei Shen  Department of Physics, Shanghai Normal
University, Shanghai 200234, China  Chong Lv  Department of Nuclear Physics,
China Institute of Atomic Energy, P. O. Box 275(7), Beijing 102413, China
Qian Zhao  Ministry of Education Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Mamutjan Ababekri  Ministry of
Education Key Laboratory for Nonequilibrium Synthesis and Modulation of
Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and
Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University,
Xi’an 710049, China  Yong-Tao Zhao  Ministry of Education Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China  Karen Z.
Hatsagortsyan<EMAIL_ADDRESS>Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Christoph H. Keitel
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany  Jian-Xing Li<EMAIL_ADDRESS>Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,
Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China

###### Abstract

Vacuum birefringence (VB) is a basic phenomenon predicted in quantum
electrodynamics (QED). However, due to the smallness of the signal,
conventional magnet-based and extremely intense laser-driven detection methods
are still very challenging. This is because in the first case the interaction
length is large but the field is limited, and vice versa in the second case.
We put forward a method to generate and detect VB in a plasma bubble
wakefield, which combines both advantages, providing large fields along large
interaction lengths. A polarized $\gamma$-photon beam is considered to probe
the wakefield along a propagation distance of millimeters to centimeters in
the plasma bubble. We find via plasma particle-in-cell simulations that the VB
signal in terms of Stokes parameters can reach about $10^{-5}$
($10^{-3}$-$10^{-2}$) for tens of MeV (GeV) probe photons with moderately
intense lasers ($10^{20}$-$10^{21}~{}\mathrm{W/cm^{2}}$). The main source of
noise from plasma electrons is mitigated, in particular, by a choice of
$\gamma$-photon polarization and by proper modulation of the plasma density.
The proposed method represents an attractive alternative for the experimental
observation of VB via laser-plasma interaction.

In quantum electrodynamics (QED) the vacuum is fluctuating with the virtual
creation and annihilation of electron-positron pairs due to the Heisenberg
uncertainty principle which, in particular, leads to the vacuum polarization
[1, 2, 3]. The latter in external electromagnetic fields transforms the vacuum
into a refractive medium [4, 5, 6, 7], and the anisotropy induced by the
strong field renders the vacuum birefringent [8]. As one of the basic
predictions of nonlinear QED, the vacuum birefringence (VB) effect is a covet
for experimental observation, and in recent years it has been extensively
studied in various strong-field configurations; see e.g. [9, 10, 11, 12].

The experimental attempts, though yet not successful, to measure the VB effect
have been mostly connected with the effect in a static magnetic field of a
stationary magnet in the laboratory (PVLAS project [13], and its
modifications: BFRT [14], and BMV [15]), which provides a field strength of
the order of Tesla within a rather large scale (about centimeters to meters).
The VB signal of an optical probe wave as a rotation angle of the wave
polarization amounts to $\delta\phi\sim 10^{-11}$-$10^{-10}$, which is still
beyond the current accuracy of measurements (about $10^{-9}$ for optical
photons) [13]. Here, $\delta\phi\propto B_{\perp}^{2}L/\lambda_{\gamma}$,
where $L$ is the total distance of the probe photon traversing the transverse
magnetic field $B_{\perp}$, and $\lambda_{\gamma}$ the wavelength of the probe
photon. As the VB signal is proportional to the energy of the probe photons,
there are proposals to use high-energy $\gamma$-photons [16], e.g., tens of
GeV $\gamma$-photons traversing through a several-meter-long large-scale
superconducting magnet, where $\delta\phi$ may reach $10^{-4}$ [17]. However,
this would demand a $\gamma$-photon beam with a challenging high collimation.
While the VB effect is magnified via multiple reflections in a magnetized
cavity, the latter introduces considerable systematic errors.

With advancement of strong laser technique in recent years [18, 19, 20, 21],
there are justified hopes to detect the VB effect with laser fields, being the
largest fields in a laboratory. In fact, the lasers provide $\sim 10^{5}$
times larger fields than via laboratory static magnets. The main disadvantage
of the setup with ultrastrong laser fields is that the high field interaction
region is limited to the focal size of the laser beam of several micrometers.
VB schemes with the use of lasers have been widely investigated theoretically,
with probe photon energies spanning from eV (optical photons) [22, 23, 24,
25], tens of keV (X-rays) [26, 27, 28, 29, 30] to GeV ($\gamma$-rays) [31, 32,
33]. Since the pulse duration of multi-PW lasers is limited to tens of
femtoseconds, for keV X-ray photons, the signal may reach $\delta\phi\simeq
10^{-5}$. Owing to the low flip rate of polarization and instability of PW
laser facilities ($\sim 20\%$ [21]), the signal measurements usually require
$10^{2}$-$10^{3}$ shots to accumulate sufficient statistics [26]. For GeV
probe $\gamma$-photons the polarization flip rate is much higher
($10^{-3}$-$10^{-1}$), but the final signal is accompanied by electron-
positron pair production (vacuum dichroism), and thousands of shots are
required to reach a high confidence level [31, 32]. Thus, the VB effect is
still not verified directly in experiments and, an efficient, stable, and
compact detection method is still in great demand.

Figure 1: Interaction scenario. Upper column: the formation process of the VB
signals in a plasma bubble. Lower column: the schematic evolution of the
polarization vector of a probe photon in the bubble wakefield.

In this Letter, we put forward such an efficient and compact scheme for a VB
measurement employing currently available experimental techniques. In this
scheme, a laser beam of moderate-intensity
($10^{20}$-$10^{21}~{}\mathrm{W/cm^{2}}$) generates a bubble wakefield in
plasma. The VB induced by the ultrastrong field of the plasma bubble is probed
by $\gamma$-photons which propagate in plasma synchronized with the bubble
over a long distance; see the interaction scenario in Fig. 1. The whole
interaction process can be divided into three stages. In stage I, a driving
beam excites a wakefield in plasma in the bubble regime. In stage II, linearly
polarized (LP) $\gamma$-photons are injected into the plasma bubble
experiencing the stable and ultrastrong transverse electromagnetic field of
the bubble. Due to VB in the ultrastrong field region, a small circular
polarization (CP) of $\gamma$-photons arises determined by the phase
retardation $\delta\phi$. The characteristic signature of VB may form an
angular pattern of a four- (eight-)leaf clover (see Figs. 1 and 2). In stage
III, the plasma bubble begins to extinguish, and the probe photons are
extracted and detected. We have carried out three-dimensional (3D) QED
particle-in-cell (PIC) simulations of the plasma and tracked $\delta\phi$ for
$\gamma$-photons, taking into account the explicit plasma field along the
$\gamma$-ray. The effective transverse field in the bubble is shown to reach
the order of $10^{3}$-$10^{4}$ T, within the interaction region of $1$ cm,
allowing for $\delta\phi$ up to about $10^{-5}$ ($10^{-3}$-$10^{-2}$), with
tens of MeV (GeV) probe photons and moderately intense laser fields. The main
limitation of the scheme is the noise because of plasma electrons. The Compton
scattering (CS) of probe photons and the plasma emission in a strong field are
included in our simulations and taken into account in the evaluation of the VB
signal. The first effect appears to be not significant because of the
judiciously chosen linear polarization for the probe, when the small CP of the
$\gamma$-photons during interaction can arise only due to VB but not due to CS
[34]. The effect of the plasma radiation, which is limited in spectral range,
is also shown not to distort significantly the VB signal.

Let us introduce our simulation method. Relativistic units with $c=\hbar=1$
are used throughout. The VB effect of probe photons in a bubble wakefield is
described by the polarization flip of the Stokes parameters
$\bigg{(}\begin{array}[]{c}S_{1}^{f}\\\ S_{2}^{f}\\\
\end{array}\bigg{)}=\bigg{(}\begin{array}[]{cc}\cos\delta\phi&-\sin\delta\phi\\\
\sin\delta\phi&\cos\delta\phi\end{array}\bigg{)}\bigg{(}\begin{array}[]{c}S_{1}^{i}\\\
S_{2}^{i}\end{array}\bigg{)},$ where
$S^{i}=\left(S_{0}^{i},S_{1}^{i},S_{2}^{i},S_{3}^{i}\right)$ and
$S^{f}=\left(S_{0}^{f},S_{1}^{f},S_{2}^{f},S_{3}^{f}\right)$ are the initial
and final Stokes parameters of the probe photons, respectively, $S_{1}$ and
$S_{3}$ indicate the linear polarization, $S_{2}$ the CP, and $S_{0}$ the
intensity [35]. The VB effect is determined by the phase retardation between
two eigenstates of polarization $\delta\phi=\int_{0}^{L}{\rm
d}l\frac{2\pi}{\lambda_{\gamma}}\Delta n(\omega_{\gamma})$, with $\Delta
n=n_{\parallel}-n_{\perp}$, with $\parallel,\perp$ denote the polarization
modes parallel to the two polarization eigenstats $\hat{e}_{1}$ and
$\hat{e}_{2}$, respectively, $\hat{e}_{1}\parallel{\bf E}_{\rm red\perp}$,
$\hat{e}_{2}=\hat{k}\times\hat{e}_{1}$, $\mathbf{E}_{\rm
red\perp}\equiv(\mathbf{E}+\hat{k}\times\mathbf{B})_{\perp}$ is the transverse
reduced field, and $\hat{k}$ the unit vector along the propagation direction
of the probe photon. The VB refractive index $n(\omega_{\gamma})$ in an as
here sufficiently slowly varying bubble wakefield can be treated as in a
constant field. To suppress pair production and the consequent vacuum
dichroism, we limit the quantum nonlinearity parameter to
$\chi_{\gamma}=|e|\sqrt{-(F_{\mu\nu}k_{\gamma}^{\nu})^{2}}/m_{e}^{3}\ll 1$
($S_{0}$), with the electron charge $-e$, mass $m_{e}$, frequency
$\omega_{\gamma}$, four wave vector $k_{\gamma}^{\mu}$ of the probe photon,
field tensor $F_{\mu\nu}$, and use a simple expression for
$n(\omega_{\gamma})$ (see Sec. I of [36] and Refs. [37, 38, 39]) which is
identical to that of the low-frequency limit [40, 41]:

$\displaystyle{\rm
Re}[n(\omega_{\gamma})]=1+\frac{\alpha}{90\pi}\overline{\chi}^{2}_{\gamma}\left\\{\begin{array}[]{c}4_{\parallel}\\\
7_{\perp}\\\ \end{array}\right\\},$ (3)

where $\overline{\chi}_{\gamma}\equiv\chi_{\gamma}/\omega_{\gamma}$. The VB
effect via the phase delay $\delta\phi$ has been implemented into the 3D PIC
code EPOCH [42, 43]; see details in Sec. III of [36].

The signatures of the VB effect in the wakefield and its development during
the interaction are illustrated in Fig. 2. Here we use as the driving beam a
10-cycle LP laser pulse with the invariant field parameter
$a_{0}=|e|E_{0}/m_{e}\omega_{0}=40$, the wavelength $\lambda_{0}=0.8~{}\mu$m
(with the corresponding laser peak intensity $I_{0}\simeq 1.37\times
10^{18}a_{0}^{2}\left(\frac{1~{}\mu{\rm m}}{\lambda_{0}}\right)^{2}~{}{\rm
W/cm^{2}}\approx 3.44\times 10^{21}~{}{\rm W/cm^{2}}$), frequency
$\omega_{0}=2\pi/\lambda_{0}$, and the focal radius $w_{0}=12~{}\mu\mathrm{m}$
(the cases of $a_{0}=5,10$, and 20 are discussed in Sec. V of [36]). The
moderate laser intensity is beneficial for noise suppression. The background
plasma is pure hydrogen gas plasma, which can be generated by the field
ionization using the driving laser beam [44], or by capillary discharge [45].
The plasma density $n_{e}$ linearly increases from zero (at $z=0$) to
$0.002n_{c}$ (at $z=100~{}\mu\mathrm{m}$) and then to $0.007n_{c}$ (at
$z=1.0~{}$mm), where $n_{c}=m_{e}\omega_{0}^{2}/4\pi e^{2}$ is the critical
plasma density (other cases with different plasma densities are discussed in
Sec. V of [36]). The probe beam is composed of $10^{7}~{}\gamma$-photons with
an average energy of $\varepsilon_{\gamma}$ = 20 MeV, energy spread
$\Delta\varepsilon_{\gamma}$ = 1 MeV, and angular spread $\Delta\theta=10$
mrad [such $\gamma$-beam can be delivered by ELI-NP within several years [46],
single-shot all-optical or beam-foil nonlinear Compton scattering (NCS) [47,
43, 48]]. The spatial distribution of $\gamma$-photons is a Gaussian with a
longitudinal radius $r_{l}=7~{}\mu$m, and transverse radius $r_{t}=15~{}\mu$m.
The probe photon beam propagates along the $\hat{z}$ direction and is LP with
the Stokes parameters $S^{i}=(1,1,0,0)$ (the case of circularly polarized
probe is discussed in Sec. IV of [36]). The simulation box of the moving
window is set as $\Delta x\times\Delta y\times\Delta z$ = $90~{}\mu{\rm
m}\times 90~{}\mu{\rm m}\times 100~{}\mu{\rm m}$, with the spatial grid size
$180\times 180\times 1600$. The macro-particles per cell are 5 and 1 for
electron and proton, respectively, and the total number of macro-particles for
probe $\gamma$-photons is set as $1.5\times 10^{6}$. To ensure that probe
photons can experience strong fields and stay in the bubble as long as
possible, the probe beam is synchronized with the driving beam with a relative
delay of $d_{\gamma}=27~{}\mu$m (cf. with the radius of the bubble
$\rho_{b}\simeq(2/k_{p})\sqrt{a_{0}}\simeq 20~{}\mu{\mathrm{m}}$, where
$k_{p}=\sqrt{4\pi n_{e}e^{2}/m_{e}}$ and $n_{e}\simeq 0.01n_{c}$ [36, 49]).

Figure 2: Signatures of the VB effect with a LP $\gamma$-photon probe: (a) 3D
distribution of $\overline{S_{2}^{f}}$ of injected probe $\gamma$ photons; (b)
evolution of the asymmetry parameters $\mathcal{A}_{i}$, where $i=2,3$ refer
to $S_{2,3}^{f}$, respectively; black and red lines are calculated with
photons from cones centered at $p_{1}$ and $p_{2}$ with both radii of
$r=1~{}\mathrm{\mu m}$. Light-blue solid and blue dash-dotted lines are
calculated by including and excluding the NCS photons (in “A”-“D”) with the
transverse position $r\geq 10~{}\mathrm{\mu m}$, respectively; (c), (d), and
(e): the Stokes parameters $S_{1}^{f}$, $S_{2}^{f}$ and $S_{3}^{f}$ of
$\gamma$-photons with $\omega_{\gamma}\in 20\pm 5$ MeV and azimuthal angle
$\Delta\theta\leq 10$ mrad [including radiated photons (within the black dash-
dotted circles) and injected probe photons] in the transverse $x$-$y$ plane
(averaged along $z$ direction), respectively. Green arrows in (c) indicate the
orientation of the transverse reduced field $\mathbf{E}_{\mathrm{red\perp}}$,
while $p_{1}$ at ($-19.5~{}{\rm\mu m},0$) and $p_{2}$ at ($0,19.5~{}{\rm\mu
m}$) denote two cylinders along $\hat{z}$ with both radii of $r$. Other
parameters are given in the text.

For the initially LP probe, a final CP emerges due to VB, with
$S_{2}^{f}\simeq 7\times 10^{-6}$, at the interaction length $1$ mm. The
spatial distribution of $S_{2}^{f}$ forms a unique structure of a four-leaf
clover (eight leaves for $S_{3}^{f}$) [Figs. 2(a), (d), (e)], while the change
in $S_{1}$ is negligible [Fig. 2(c)]. For detection convenience, here, we
define an asymmetry parameter
$\mathcal{A}\equiv\overline{S^{f}_{2}}(A)+\overline{S^{f}_{2}}(C)-\overline{S^{f}_{2}}(B)-\overline{S^{f}_{2}}(D)=|\overline{S^{f}_{2}}(A)|+|\overline{S^{f}_{2}}(B)|+|\overline{S^{f}_{2}}(C)|+|\overline{S^{f}_{2}}(D)|$,
where $\overline{S^{f}_{2}}$ is the averaged Stokes parameter in regions
“A”-“D” with each one containing a leaf of the clover in Fig. 2(d). Compared
with PVLAS, the plasma wakefield can maintain a stronger field $B\gtrsim
10^{3}$ T, and the corresponding VB signal $B^{2}L\simeq 10^{3}$-$10^{4}$
${\rm T^{2}m}$, will be one to two orders larger than in the planned PVLAS-FE
[13].

Since the driving laser is LP, the low-energy X-ray from betatron radiation
and $\gamma$-photons from NCS are LP, i.e., they can affect $S_{1}^{f}$ and
$S_{3}^{f}$, but not CP; see signals within the black dash-dotted circles of
Figs. 2(c) and (e). Theoretically, $S_{2}$, i.e., the CP will not be affected,
because CS of LP photons cannot produce circularly polarized photons [34] as
is confirmed in Fig. 2(d) (within the black dash-dotted circle) and more
details in Sec. IV of [36]. As the VB signal is proportional to the photon
energy $|S_{2,\mathrm{LP}}^{f}|~{}\sim\delta\phi\propto\omega_{\gamma}$,
higher-energy probe photons can produce stronger signals, for instance, at
$\omega_{\gamma}\sim 1$ GeV, $\delta\phi\sim 10^{-3}$, with noise still
insignificant [Figs. 3 (e), (f)]. Thus, the CP signal of VB in the plasma
wakefield setup is of the order of $|S_{2}^{f}|\simeq 3.5\times 10^{-6}$ and
can be increased using a longer interaction length in the plasma. For the
detection of such a weak CP signal of a $\gamma$-photon beam we could follow
the principles of sensitive circular polarimetry methods in [50, 51, 52, 53];
and for the detection of the linear polarization, the polarization-dependent
Bethe-Heitler (BH) pair production method can be employed [31]. The number of
detected $\gamma$-photons ($N_{\gamma}$) should be large enough to suppress
the statistical error $1/\sqrt{N_{\gamma}}$ below the required accuracy
$|S_{2}^{f}|\sim 3\times 10^{-6}$, i.e. $N_{\gamma}\sim 10^{11}$. With
$10^{7}$ as number of $\gamma$-photons in a beam, one will need $10^{3}\sim
10^{4}$ shots for statistical accuracy. These requirements can be fulfilled by
all-optical NCS [47, 43] or beam-foil radiation [48]. We underline that the
considered setup is realized with tabletop 100s TW or PW laser systems, which
provide much more stable fields in the plasma wakefield, compared with the
fields of 10-100 PW laser systems.

The development of the VB signal is elucidated in Fig 3(a). In the plasma
wakefield, the magnitude of the transverse field $\mathbf{E}_{{\rm red}\perp}$
is proportional to $k_{p}\propto\sqrt{n_{\rm e0}}$ [54, 55]. In stage I, the
excited wakefield in the low-density plasma region is rather weak, and the VB
effect is negligible. In stage II, due the applied density gradient,
$|\mathbf{E}_{\rm red\perp}|$ linearly rises up to $|\mathbf{E}_{\rm
red\perp}|\simeq 6000$ T (at $z\simeq 1~{}{\rm mm}$), and the phase
retardation of probe photons increases as $\delta\phi\propto|\mathbf{E_{\rm
red\perp}}|^{2}L\propto L^{3}$, yielding the VB signal
$S^{f}_{2}\simeq\frac{\alpha}{45}\frac{|\mathbf{E}_{\rm
red\perp}(1~{}\mathrm{mm})|^{2}}{E_{cr}^{2}}\frac{L}{\lambda_{\gamma}}$
amounting to $5\times 10^{-6}$ at $L=1$ mm [cf. Fig. 3(a)], where
$E_{cr}=m^{2}/e$ is the QED critical field. In stage II, $\mathbf{E}_{\rm
red\perp}$ is radially aligned [see Fig. 2(c)], i.e., $\mathbf{E}_{\rm
red\perp}\parallel(\cos\theta\hat{x}+\sin\theta\hat{y})$, with the azimuthal
angle $\theta=\arctan(y/x)$. Defining eigenstates
$\hat{e}_{1}=(\cos\theta\hat{x}+\sin\theta\hat{y})$ and
$\hat{e}_{2}=\hat{k}\times\hat{e}_{1}=(-\sin\theta\hat{x}+\cos\theta\hat{y})$,
the initial Stokes parameters in this frame are $S^{{\rm LP},i}=(1,\cos
2\theta,0,\sin 2\theta)$ and $S^{{\rm CP},i}=(1,0,1,0)$; see in Sec. II of
[36]. After traversing the polarized vacuum, the final Stokes parameters are
given by $S^{{\rm LP},f}=\bigg{[}1,\frac{\cos
4\theta+1}{2}(\cos\delta\phi-1)+1,\cos 2\theta\sin\delta\phi,\frac{\sin
4\theta}{2}(\cos\delta\phi-1)\bigg{]}$ (see Sec. II in [36]). Therefore,
$S^{{\rm LP},f}_{2}=\cos 2\theta\sin\delta\phi\propto\cos
2\theta\cdot\rho^{2}\exp(-2\rho^{2}/\rho_{b}^{2})$, where we use
$\delta\phi\propto|\mathbf{E}_{\perp}(\mathbf{B}_{\perp})|^{2}\propto\exp(-2\rho^{2}/\rho_{b}^{2})\rho^{2}$,
phenomenologically describing the local transverse bubble field as
$\propto\exp(-\rho^{2}/\rho_{b}^{2})\bm{\rho}$, with
$\bm{\rho}=x\hat{x}+y\hat{y}$, and $\rho=|\bm{\rho}|$. Thus, the VB signature
forms a unique structure with four isolated poles due to the term $\cos
2\theta$, with a maximum at $\rho=\rho_{b}$, which is consistent with Figs.
2(a) and (d). The interaction length with the bubble structure is limited by
the dissipation of the laser energy,
$L_{dp}\lesssim\left(\frac{n_{e}}{n_{c}}\right)^{-3/2}\lambda_{0}\frac{\sqrt{2}}{\pi}a_{0}\simeq
1.8$ cm for $n_{e}=0.01n_{c}$ [49] ($S_{2}^{f}\sim 6\times 10^{-5}$, and only
25 shots are required to suppress the statistical error), but the wakefield
scales as ($|\mathbf{E}|\propto\sqrt{n_{e}}$), such that for the VB signal
$\delta\phi\propto|\mathbf{E}|^{2}L_{pd}\propto\frac{a_{0}}{\sqrt{n_{e}}}$, an
intense driver with low-density plasma is beneficial.

Figure 3: (a) Stokes parameter $S_{2}$ (black) and the corresponding
experienced reduced field $\mathbf{E}_{\rm red\perp}$ (blue) vs the
interaction path $L$ (mm) of two sampled particles with solid and dash-dotted
lines denoting particles from $p_{1}$ and $p_{2}$ in Fig. 2(d), respectively.
(b) Energy spectra of the plasma emission (NCS, blue-dotted) and injected
probe photons (black-solid). (c) Number density of electrons of the bubble,
where the gray dash-dotted circle indicates the probe beam position. (d) The
experienced plasma densities of sampled photons from “A′” and “B′” in (c),
with solid and dash-dotted lines denoting electrons ($n_{e}$) and ions
($n_{H}$), respectively. (e) Scattering probability log${}_{10}(P_{\rm
scatt.})$ vs photon energy log${}_{10}(\varepsilon_{\gamma})$ (MeV) of
$\gamma$-photons traversing a hydrogen plasma of 2 mm, where blue and red
lines are taken from the center ($n_{e}=6\times 10^{-6}n_{c}$) and the sheath
($n_{e}=0.012n_{c}$) of the bubble, respectively. Dot, dash-dotted and thick
solid lines indicate CS, total pair production (BH and trident pair
productions) and the total scattering (CS + pair production), respectively.
(f) Total scattering probabilities log${}_{10}(P_{\mathrm{scatt.}})$ vs time
$t$ for probe photons (20 MeV) located at “A′” and “B′”, respectively, where
$P_{\mathrm{scatt.}}(t)\equiv
1-\prod_{i=1}^{n}\exp\left\\{-\left[n_{e}(t_{i})(\sigma_{\mathrm{CS}}+\sigma_{\pm,\mathrm{tri.}})+n_{H}(t_{i})\sigma_{\pm,\mathrm{BH}}\right]c\Delta
t\right\\}$, $t_{i}=i\Delta t$, $t=n\Delta t$ and $\Delta t$ is the time step
size. Figure 4: Impact of the probe $\gamma$-photon beam parameters on the VB
signal via the parameter $\mathcal{A}$ vs: (a) spatial delay $d_{\gamma}$, (b)
angular spread $\Delta\theta$, (c) energy spread $\sigma$, and (d) temperature
$T$ of the probe beam. In (a) and (b), the beam energy is assumed to follow a
Gaussian distribution with the energy spread of 1 MeV. In (a), (c) and (d),
the angular spread is 10 mrad. The probe beam energy in (c) is a Gaussian
distribution
$f(\omega;\omega_{0},\sigma)\propto\frac{1}{2\pi\sigma}\exp\left(-\frac{(\omega-\omega_{0})^{2}}{2\sigma^{2}}\right)$
with the central energy $\omega_{0}$ and RMS $\sigma$, respectively, and in
(d) a Maxwellian distribution $f(\omega)\propto\exp\left(-\omega/T\right)$
with temperature $T$. Black (Blue) lines indicate the signal collected from
photons with transverse position $r\geq 10~{}(15.7)~{}\mathrm{\mu m}$. All
other parameters are the same as those in Fig. 2.

In the considered setup noise from plasma electrons may affect the VB signal.
Such noise stems from: 1) self-generated photons of the plasma electrons, and
2) CS of the probe photons off the plasma electrons. Both the self-generated
photons and the scattered ones mix with the probe and thus affect the final VB
signal. During propagation of the wakefield, electrons are self-injected into
the wakefield, accelerated to high energies, and can emit (low-energy X-rays
via betatron radiaiton [56] do not affect the VB signal) high-energy photons
via NCS in the strong external field. These newborn photons via NCS are LP
[57, 58, 47] and they do not change the number of circularly polarized photons
generated via VB, however, they would reduce the average CP of the photon beam
impinging on the detector. Nevertheless, the radiation spectrum of the NCS
peaks around $\omega_{\rm NCS}\sim 0.44\gamma_{e}m_{e}\chi_{e}\ll\omega_{\rm
probe}$ as $|\mathbf{E}_{\rm red\perp}|\simeq 6000$ T, $\gamma_{e}\lesssim
2000$ and $\chi_{e}\lesssim 0.001$ [59, 60, 61, 62]; see the energy spectra of
radiated photons in Fig. 3(b), and they are naturally separated from the probe
beam in angular distribution [Figs. 2(c)-(e)]. The applied setup with a
linearly increasing smooth gradient also suppress the self-injection of
electrons and subsequent NCS.

As for the second problem, the high-energy probe photons ($\omega_{\gamma}\gg
2m_{e}$) propagating in the hydrogen plasma wakefield, can scatter off
background electrons (CS) and produce pairs in the electron (trident process)
and proton field (BH process [63, 36, 64, 65, 66, 67]), and in the strong
wakefield via the nonlinear Breit-Wheeler process. The pair production process
could deplete the probe beam, however, the probability at the given conditions
with $\chi_{\gamma}\sim 0.001$ is $P_{\pm}\lesssim 10^{-8}$, and the probe
depletion due to pair production is negligible; see Sec. IV of [36].

The CS of LP probe photons cannot create CP for the scattered photon [34],
however, it could also deplete the probe beam. The probability of CS for a
probe photon is maximal when it traverses the bubble sheath [where the
electron number density reaches the maximum $n_{e}^{(max)}\simeq
10^{-2}n_{c}$] and can be estimated as $P_{\rm CS}\simeq n_{e}\sigma_{\rm
cs}L\simeq 3\times 10^{-8}$, for $L=2$ mm; see Figs. 3(d) and (e). This
estimation is consistent with the real-time calculated scattering
probabilities of the two sampled photons at “A′” and “B′”; see Fig. 3(f).

For the experimental feasibility, we also study the impact of the spatial
delay $d_{\gamma}$, angular divergence $\Delta\theta$, energy spread, and
transverse radius of the probe beam on the VB signatures. Since the maximum
distance a $\gamma$-photon can propagate in the wakefield is given by
$L_{\mathrm{max}}\simeq\frac{c}{c-v_{\mathrm{w}}}d_{\gamma}$, where
$v_{\mathrm{w}}$ is the velocity of the wakefield ($v_{\rm w}<c$), the total
phase retardation $\delta\phi$ is thus limited by $d_{\gamma}$. With current
laser and plasma parameters, one obtains $d_{\gamma,\rm optimal}\simeq
27~{}\mu{\rm m}$. When $d_{\gamma}<d_{\gamma,\rm optimal}$, $L<L_{\rm max}$,
and $\delta\phi$ and $\mathcal{A}$ will be smaller. When
$d_{\gamma}>d_{\gamma,\rm optimal}$, the probe $\gamma$-photons are located
near the tail of the bubble. Thus the average wakefield experienced by them
will be weaker, and the VB signal smaller [Fig. 4(a)]. The quality of the
probe beam may also influence the VB signal. The angular spread $\Delta\theta$
will reduce the VB signal [Fig. 4(b)], however, even at $\Delta\theta\simeq
20$ mrad, one still obtains $\mathcal{A}\simeq 10^{-5}$. In the case of the
beam-like energy distribution around the central energy, e.g. [68, 47], the
energy spreading does not disturb much the VB signal [Fig. 4(c)], while in the
case of a Maxwellian distribution, $\mathcal{A}$ is proportional to the
“temperature” of the probe beam [Fig. 4(d)]. The impact of beam radii $r_{t}$
and $r_{l}$ on the signatures of VB are discussed in Sec. V of [36].

We note finally that the bubble wakefield can be created also by
ultrarelativistic electron or proton beams [69, 70]. However, in the first
case NCS is significant and should be spatially separated from the probe
photons. In the second case, the large wakefield requires larger proton
densities presently not available with conventional schemes [71, 72].

In conclusion, we put forward a competitive VB detection method based on a
plasma wakefield and using moderate laser intensities, which can provide a VB
signal $\sim 10^{-5}$ for MeV-level probe photons and $\sim 10^{-3}$-$10^{-2}$
for GeV level within several millimeters of the interaction length. The noise
because of plasma electrons is shown to be reducible the signal level. The
method is robust with respect to laser and plasma parameters and can be
realized with currently feasible laser facilities. Furthermore, the method has
the potential to apply to the search for axion-like particles, as the VB could
be mediated via weakly interacting slim particles (WISPs) rather than
electron-positrons.

Acknowledgment: The work is supported by the National Natural Science
Foundation of China (Grants Nos. 12022506, 11874295, 11875219, 11905169,
12005305), the National Key R&D Program of China No. 2019YFA0404900, Open
Foundation of Key Laboratory of High Power Laser and Physics, Chinese Academy
of Sciences (SGKF202101), the China Postdoctoral Science Foundation (Grant No.
2020M683447), the Open Fund of the State Key Laboratory of High Field Laser
Physics (Shanghai Institute of Optics and Fine Mechanics) and the Fundamental
Research Funds for the Central Universities.

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# A Novel Wide-Area Multiobject Detection System with High-Probability Region
Searching

Xianlei Long1, Hui Zhao2,1, Chao Chen1, Fuqiang Gu1∗, and Qingyi Gu3 This work
is supported by China Postdoctoral Science Foundation (No. 2023M740402), the
National Natural Science Foundation of China (No. 42174050, 62322601), Venture
& Innovation Support Program for Chongqing Overseas Returnees (No. cx2021047),
Chong Startup Project for Doctorate Scholars (No. CSTB2022BSXM-JSX005), the
Excellent Youth Foundation of Chongqing (No. CSTB2023NSCQJQX0025), and
Fundamental Research Funds for the Central Universities (No. 2023CDJXY-038,
2023CDJKYJH034). Corresponding author: Fuqiang Gu. (e-mail:
gufq@cqu.edu.cn)1College of Computer Science, Chongqing University, Chongqing
400044, China. 2College of Computer Science, China University of Geoscience,
Wuhan 430000, China. 3Institute of Automation, Chinese Academy of Sciences,
Beijing 100190, China.

###### Abstract

In recent years, wide-area visual surveillance systems have been widely
applied in various industrial and transportation scenarios. These systems,
however, face significant challenges when implementing multi-object detection
due to conflicts arising from the need for high-resolution imaging, efficient
object searching, and accurate localization. To address these challenges, this
paper presents a hybrid system that incorporates a wide-angle camera, a high-
speed search camera, and a galvano-mirror. In this system, the wide-angle
camera offers panoramic images as prior information, which helps the search
camera capture detailed images of the targeted objects. This integrated
approach enhances the overall efficiency and effectiveness of wide-area visual
detection systems. Specifically, in this study, we introduce a wide-angle
camera-based method to generate a panoramic probability map (PPM) for
estimating high-probability regions of target object presence. Then, we
propose a probability searching module that uses the PPM-generated prior
information to dynamically adjust the sampling range and refine target
coordinates based on uncertainty variance computed by the object detector.
Finally, the integration of PPM and the probability searching module yields an
efficient hybrid vision system capable of achieving 120 fps multi-object
search and detection. Extensive experiments are conducted to verify the
system’s effectiveness and robustness.

##  I INTRODUCTION

Wide-area visual detection systems, leveraging computer vision technology,
have found extensive application across diverse domains, encompassing densely
populated public areas [1], airport security surveillance [2], sports event
tracking [3], and intelligent transportation systems [4]. In these varied
contexts, the role of multiobject detection emerges as pivotal, enabling the
recognition and interpretation of complex actions and behaviors. This
capability, in turn, underpins numerous critical tasks, including surveillance
monitoring [5, 6], transportation detection [7], et al.

However, the adoption of wide-area visual detection system faces three
challenges: the pursuit of high-definition imaging for individual targets
within a wide-area field of view (FOV) presents a formidable hurdle. The
utilization of a wide-angle camera to cover large areas inevitably results in
reduced resolution for individual objects, potentially causing the system to
lose small targets. Consequently, this limitation decreases the overall
accuracy of detection outcomes. (2) Existing object searching methods are not
optimal when dealing with large scenes. Most of them require global random
sampling of the image during the initial detection stage, which leads to low
search efficiency as most sampling areas contain irrelevant backgrounds [8].
This limitation hampers real-time object detection and imposes constraints on
the system’s overall performance. (3) Target localization is often inaccurate
in wide-area FOV due to the small size of object in an image, making accurate
localization difficult. Existing image capture methods, which often use wide-
angle [9] or fish-eye lenses [10], exacerbate the difficulty of target
localization by introducing image distortion and quality degradation [11]. To
address the intricate challenges in exists wide-area visual systems, numerous
methods have been developed. Some researchers have sought to attain high-
definition imaging within a wide FOV by employing camera arrays comprising
multiple cameras [12] or by implementing switching mirrors to simulate the
functionality of multiple cameras [13, 14, 15, 16]. Nevertheless, the former
approach entails intricate camera configurations, substantial costs, and
manufacturing complexities. The latter approach necessitates either random
scanning of the entire area or relies on wide-angle cameras for target
localization, thereby introducing difficulties in achieving precise target
localization.

There are many semantic probability map-based object detection paradigms [17,
18], such as salient object detection (SOD) and attention mechanism-based
object detection. SOD aims to detect the most visually conspicuous object in
an image [19]. However, SOD heavily relies on high-level semantic extraction;
when objects are relatively smaller in a panoramic image, the SOD often hard
to extract salient features about interest objects [20], besides, the massive
irrelevant background regions would hinder the high-probability information
aggregation, resulting in a detection performance degradation in wide areas.
Similarly, the attention mechanism has a prominent ability to extract the most
relevant region through weighted combinations, which can integrate layer-wise
attention, channel-wise attention, and spatial attention to form semantic
feature maps [21, 22]. Nevertheless, the probability feature map generated by
the attention method can easily lose discriminating information related to the
objects that are captured in a panoramic image [19]. Despite many advanced
prior probability-guided methods having been proposed, the probability maps
are often coarsely computed by a CNN through the original image, which ignores
the relationship between foreground objects and background regions and misses
the corresponding probability of desired objects. Different from the existing
probability map generation methods, which just parse the high-probability
regions. In this research, we introduce a hybrid multi-camera detection
framework to generate a coarse-to-fine semantic probability map and
cooperatively detect desired objects.

In recent years, uncertainty estimation has been widely explored in object
searching. However, Nguyen et al. [23] pointed out that uncertainty measures
may not be a reliable stand-in for a reasonable uncertainty estimate. Miller
et al. [24] proposed an uncertainty estimation and bounding box merging
strategy to refine the object coordinate. However, they ignore the
relationship of objects to the environment. It’s better to combine the
uncertainty and the object’s state to jointly optimize the search process.

Consequently, to solve the above-mentioned problems, we propose a panoramic
probability map (PPM) guided detection system based on two cameras, where the
proposed system can achieve high-speed object detection in a wide surveillance
area. Specifically, a wide-angle camera is used to provide prior information
about the working scene based on the segmentation results of the captured
panoramic image. Next, we use the prior information to guide the search camera
in capturing detailed images of the targets. To achieve efficient target
detection and accurate localization, we introduce two components: PPM and the
probability searching module. The PPM estimates high-probability regions where
targets are likely to appear and provides approximate location estimation for
some targets. Subsequently, the probability searching module utilizes the
uncertainty variance obtained from the object detector to dynamically adjust
the sampling range and refine the coordinates of the detected targets. This
approach not only enhances the search efficiency but also improves the
accuracy of target localization. Finally, by combining the PPM and probability
searching modules, we develop a wide-area object detection system.

The main contributions of this research are as follows:

  * •

We propose a novel PPM prior information generation method based on
segmentation results from a wide-angle camera. The scanning process can focus
on the generated high-probability regions to reduce cost.

  * •

We introduce a probability searching module based on the PPM to improve
detection efficiency. The searching module can adjust the searching range by
estimating the uncertainty of related objects, which shows adaptive searching
ability in wide-area multiobject searching.

  * •

We implemented a hybrid wide-area multiobject detection system and conducted
extensive indoor/outdoor experiments to verify the performance of the system.

##  II Hybrid Wide-Area Multiobject Detection System with Probability
Searching

Figure 1: Illustration of the proposed hybrid wide-area multiobject detection
system. (a) shows the generation process of PPM, which includes three steps:
panoramic image $I_{w}$ capture, semantic segmentation, and mask probability
mapping. (b) is the proposed probability searching method, it includes
galvano-mirror searching and scanning.

In Fig.1-(a), drawing insights from prior research[25, 26], we utilize the
segmentation outcomes to categorize each semantic region into distinct
probability regions based on its label and spatial location, subsequently
assigning varying probability values accordingly. Following this, the particle
sampling process is initiated, generating a set of particles primarily within
high-probability regions conducive to target detection. To facilitate this, a
galvano-mirror scanning module functionaly switch the search camera to capture
the environmental information within the particle FOV. However, it is
noteworthy that the initial object localization may exhibit deviations from
the actual object position by several pixels. In response, a probability
searching module assumes a crucial role in refining the object boundaries to
rectify such errors. As illustrated in Fig. 1-(b), we introduce a probability
searching module in our proposed system. This module can dynamically adjust
sampling range and refine the particle coordinates. These adjustments are made
based on the uncertainty variance computed by the object detector, enhancing
the precision and accuracy of object localization within our wide-area visual
detection system.

###  II-A PPM Generation for Region Sampling

####  II-A1 Semantic Segmentation-based PPM Generation

As shown in Fig. 1-(a), PPM generation first performs semantic segmentation on
the images captured by the wide-angle camera to obtain a segmentation map of
the entire area. This serves as the foundation for assigning different
sampling probabilities to different regions in the subsequent steps. This
approach is based on the prior that the probability of targets appearing in
different regions is different.

We capture a panoramic image $I_{w}$ with the wide-angle camera. Then we fed
this image into a semantic segmentation model (SemSegModel) [27] to obtain a
segmentation map of the entire area:

$\begin{gathered}S_{1}\cup S_{2}\cup\cdots\cup
S_{n}=\operatorname{SemSegModel}\left(I_{w}\right)\\\ \qquad S_{r}\cap
S_{j}=\phi,\forall r\neq j\end{gathered}$ (1)

where $S_{r},r=1,2,...,n$ represents the different regions after segmentation.
Then, we assign a sampling probability $F(S_{r},\mathcal{T})$ to each
segmented region $S_{r}$.

$\begin{gathered}{F}\left(S_{r},\mathcal{T}\right)=\frac{P\left(S_{r}\right)\times
P\left(\mathcal{T}\mid S_{r}\right)}{\sum_{r=1}^{n}P\left(S_{r}\right)\times
P\left(\mathcal{T}\mid S_{r}\right)}\\\ P\left(S_{r}\right)=\frac{\text{ Area
}_{S_{r}}}{\text{ Area }_{I_{w}}},r=1,2,\ldots,n\end{gathered}$ (2)

where $P(S_{r})$ represents the relative area of the current region $S_{r}$ to
the entire image $I_{w}$. $P(\mathcal{T}|S_{r})$ is the semantic probability
of target-region, which is obtained by statistical analysis of object-region
relationships in the Visual Genome dataset [26]. Through the steps above, the
semantic segmentation model divides the search scene into several regions and
determines the sampling probability $F(S_{r},\mathcal{T})$ for each region
$S_{r}$.

Building upon panoramic area segmentation, we introduce panoramic object
detection to generate the PPM. The PPM refines the sampling probabilities
within each region, aiming to minimize searching in ineffective areas and
enhance the accuracy of target localization. To achieve this, we use the PPM
segmentation results to select ‘obejcts’ as the initial results, to get
$B_{\boldsymbol{O}}$, where $o_{i}$ is the initial detected object,
$O=(o_{1},o_{2},...,o_{n})$ is the collection of detected objects during
semantic segmentation in Eq. 1, and $n$ is the number of segmentation objects.
To simplify the estimation of uncertainty probability, we use the output of
the segmentation results’ confidence as the uncertainty
$\sigma_{O}=B_{\boldsymbol{O}}$ [28].

If a target $o$ is detected in region $S_{r}$, a new sampling region $S_{ro}$
is defined around the target $o$, and the corresponding sampling probability
is denoted as $F(S_{ro},\mathcal{T})$. Assume the number of particles
allocated to region $S_{r}$ is determined as
$x_{r}=\mathcal{F}\left(S_{r},\mathcal{T}\right)\times N$. The number of
particles assigned to the region $S_{ro}$ is denoted as $x_{ro}$, while the
remaining region $S_{rm}$ within $S_{r}$ receives $x_{rm}=x_{r}-x_{ro}$
uniformly distributed particles. The calculation for $x_{ro}$ is as follows:

$\begin{gathered}x_{ro}=\frac{F\left(S_{ro},\mathcal{T}\right)\times
S_{ro}}{F\left(S_{ro},\mathcal{T}\right)\times
S_{ro}+F\left(S_{r},\mathcal{T}\right)\times S_{rm}}\cdot x_{r}\\\
S_{ro}=\pi\left(r\sigma_{o}\right)^{2}\\\ S_{rm}=S_{r}-S_{ro}\\\
\end{gathered}$ (3)

where $F(S_{ro},\mathcal{T})$ represents the sampling probability of the
detected target $o$, which is set to 100% in this paper. The hyperparameter
$r$ is set to 50, and $\sigma_{o}$ denotes the uncertainty variance of object
$o$ in Eq. 1.

Therefore, by dynamically adjusting the search area, the refined PPM optimizes
the search process. It focuses on refining the localization of targets with
small uncertainty while expanding the search for targets with high
uncertainty.

####  II-A2 Particle Sampling in High-Probability Regions

The particle sampling module searches for the target of interest by generating
a set of particles $\boldsymbol{x}_{k}$. The entire particle sampling process
is based on a pre-selected importance sampling proposal distribution function
$q(\boldsymbol{x}_{k}^{i}|\boldsymbol{x}_{k-1}^{i},z)$:

$\boldsymbol{x}_{k}^{i}\sim
q(\boldsymbol{x}_{k}^{i}|\boldsymbol{x}_{k-1}^{i},z),i=1,2,...,N_{k},$ (4)

where $N_{k}$ is the number of particles at stage $k$, $z$ is the whole wide-
area FOV, and $\boldsymbol{x}_{k}^{i}$ is the $i^{th}$ particle at stage $k$.
The visualization of particle sampling based on the PPM high-probability
regions is shown in Fig. 2-(a), where several particles (indicated by green
circles) are sampled in segmentation regions.

During the normal iteration process, the proposal distribution function
$q(\boldsymbol{x}_{k}^{i}|\boldsymbol{x}_{k-1}^{i},z)$ is updated based on the
dynamically updated particles and their weights:

$q(\boldsymbol{x}_{k}^{i}|\boldsymbol{x}_{k-1}^{i},z)=\sum_{i=1}^{\hat{N}_{k-1}}\tilde{w}_{k-1}^{i}*\mathcal{N}(\tilde{w}_{k-1}^{i},\sigma_{k-1}^{2}I),$
(5)

These parameters are obtained from the weight update module at time $k-1$.
where $\tilde{w}_{k-1}^{i}$ is the normalized weight of particle $i$ (Eq. 12),
$\hat{N}_{k-1}$ is the number of particles retained after removing redundant
particles. $\sigma_{k-1}$ represents the uncertainty variance of object
detection (Eq. 6), which determines the sampling range for each particle.

In the initial detection stage, we use the PPM to allocate sampling
probabilities. As shown in the particle sampling part in Fig. 2-(a), we first
assign sampling probabilities
$q(\boldsymbol{x}_{0}^{i})=\mathcal{F}\left(S_{r},\mathcal{T}\right)$ to each
region based on the semantic map. Subsequently, we perform fine-grained
allocation of sampling probabilities within each region, based on the detected
targets and their corresponding uncertainties.

Figure 2: Visulization of high-probability region sampling and galvano-mirror
scanning using particle filter.

####  II-A3 High-probability Region Scanning Based on PPM

By moving the galvano-mirror, it determines whether the target is within the
FOV of the particle and updates the particle’s weight accordingly. If particle
$\boldsymbol{x}_{k}^{i}$ detects the object of interest, the probability of
the target existing in the current visual image is calculated:

$p\left(z\mid
x_{k}^{i}\right),\sigma_{k}=\text{Detector}\left(\operatorname{Img}\left(\boldsymbol{x}_{k}^{i}\right)\right)$
(6)

$\sigma_{k}$ represents the uncertainty variance generated by the object
detector, where hard samples such as small or occluded objects often exhibit
large uncertainty. This process is shown in Fig. 2-(b), the system controls
the galvano-mirror to scan the particles that lie in different high-
probability regions.

The weight of each particle is updated based on the measurement model of the
likelihood function, which is derived from the probability of detecting the
object. The specific formula used for this calculation is:

$w_{k}^{i}=w_{k-1}^{i}p(z|\boldsymbol{x}_{k}^{i}),i=1,2,...,N_{k}.$ (7)

where the weight of each particle $w_{k}^{i}$, is determined by the previous
weight and the likelihood probability $p(z|\boldsymbol{x}_{k}^{i})$. The
likelihood probability is calculated by the object detection model (as shown
in Eq. 6).

###  II-B Probability Searching for Object’s Coordinate Refinement

As shown in Fig. 3, we propose to use probability searching to adjust the
coordinates of the retained objects for more accurate positioning and enable
efficient searching.

####  II-B1 Coordinate Refinement

The image coordinate system of particles is transformed into the galvano-
mirror scanning coordinate system using the following equations:

$\left\\{\begin{array}[]{l}\hat{g}_{\theta_{h}}=\theta_{h}+\alpha\cdot\left(t_{x}-W/2\right)\\\
\hat{g}_{\theta_{v}}=\theta_{v}+\alpha\cdot\left(t_{y}-H/2\right)\end{array}\right.$
(8)

where $(\theta_{h},\theta_{v})$ is the coordinates of the particle $x_{k}^{i}$
in the galvano-mirror scanning system, $(t_{x},t_{y})$ is the coordinates of
the target $\boldsymbol{b}$ in the image coordinate system,
$(\hat{g}_{\theta_{h}},\hat{g}_{\theta_{v}})$ represent the refined
coordinates of the searched object in the galvano-mirror system, $\alpha$ is a
transform coefficient, which is 0.002 in this paper, and $(W/2,H/2)$ is the
image center.

Then, we perform NMS on the refined detection objects. If there are multiple
nearby objects, only the one with the highest confidence score is retained;
otherwise, both objects are kept. After performing NMS on all refinement
searched objects $\hat{\boldsymbol{b}}$, the remaining search windows
$\widehat{SW}_{k}$ are:

$\widehat{SW}_{k}\cup\text{NMS}(\hat{\boldsymbol{b}}^{i},\hat{\boldsymbol{b}}^{j}),\forall
i,j\in\left[1,2,\ldots,N_{k}\right]$ (9)

Figure 3: Coordinate refinement $m$ by the probability searching. (a) The red
rectangle on the left is the object’s boundary after adjustment. (b) The red
circle on the right is the sampling range during probability searching.

####  II-B2 Probability Searching

After adjusting the coordinates of the retained object, the search particle
center is more accurate. Initially, for each retained object
$\hat{\boldsymbol{b}}^{i}$, we compute the coordinates and uncertainty
variances $p_{i}$ of each object $\hat{\boldsymbol{b}}^{i}$ that overlap with
it.

$p_{i}=e^{-\left(1-\mathrm{IoU}\left(\hat{\boldsymbol{b}}^{i},\hat{\boldsymbol{b}}^{m}\right)\right)^{2}/\sigma_{t}},\quad\forall
i,\operatorname{IoU}\left(\hat{\boldsymbol{b}}^{i},\hat{\boldsymbol{b}}^{m}\right)>0$
(10)

We then improve the coordinates of object $m$ through a voting process during
NMS as follows:

$\hat{b}_{\theta_{h}}^{m^{\prime}}=\frac{\sum_{i}p_{i}\hat{b}_{\theta_{h}}^{i}/\sigma_{\theta_{h},i}^{2}}{\sum_{i}p_{i}/\sigma_{\theta_{h},i}^{2}},\quad\hat{b}_{\theta_{v}}^{m^{\prime}}=\frac{\sum_{i}p_{i}\hat{b}_{\theta_{v}}^{i}/\sigma_{\theta_{v},i}^{2}}{\sum_{i}p_{i}/\sigma_{\theta_{v},i}^{2}},\\\
$ (11)

where
$\hat{\boldsymbol{b}}^{m^{\prime}}=\left(\hat{b}_{\theta_{h}}^{m^{\prime}},\hat{b}_{\theta_{v}}^{m^{\prime}}\right)$
is the refined coordinates of object $m$,
$\operatorname{IoU}\left(\hat{\boldsymbol{b}}^{i},\hat{\boldsymbol{b}}^{m}\right)>0$
represents all objects that overlap with object $m$,
$(\sigma_{\theta_{h}},\sigma_{\theta_{v}})$ is the uncertainty variance in the
$\theta_{h}$ and $\theta_{v}$ dimensions of object $i$, and $\sigma_{t}$ is a
hyper-parameter set to 0.025 in this study. As illustrated in Fi. 3, left red
rectangle is the refined object boundary,
$(\sigma_{\theta_{h},m},\sigma_{\theta_{v},m})$ is the radius of the adjusted
sampling range. As shown in Fig. 3-(b), the following search procedure only
needs to be processed in the optimized range. During the voting process, two
types of adjacent objects receive lower weights: (1) objects with high
uncertainty variance, and (2) objects with small IoU with the selected object.

After removing potential duplicate sampled particles, we update the weight of
each particle. All particles in $\widehat{SW}_{k}$ are weight-normalized as
follows:

$\widehat{w}_{k}^{i}=\frac{w_{k}^{\prime
i}}{\sum_{i=1}^{\widehat{N}_{k}}w_{k}^{\prime
i}},i=1,2,\ldots,\widehat{N}_{k}$ (12)

Based on the normalized weights, particles enter a new round of particle
sampling to obtain a new iterative estimate of the target state (as shown in
Eq. 5).

##  III Experiments

Finally, the detection system is built up by combining the proposed PPM method
and probability searching module. We conducted several experiments to verify
its performance.

###  III-A Experimental Configuration

Figure 4: Construction of the system, which includes a wide-angle camera, a
high-speed search camera, and a pan-tilt galvano-mirror.

During our experiments, we employ a hybrid detection system by integrating a
wide-angle camera (A5201CU150, Dahua, 1920$\times$1200 @ 150 fps) and a search
camera (A7040CU000, Dahua, 640$\times$480 @ 437 fps). The focal length of
wide-angle camera and search camera is 6 mm and 50 mm, respectively. To
facilitate high-speed scanning, we utilize a dual-axis galvano-mirror
(TSH8310A, Sunny Technology) to switch the FOV in a wide area. An overview of
this system is presented in Fig. 4. We control the whole system in a hardware-
software integration manner by using a PC equipped with an Intel Core i7-8700K
CPU @ 3.70 GHz, 16-GB DDR-4 memory, 256 GB of RAM, and a single Nvidia RTX
2080 Ti GPU. Cameras and the galvano-mirror are controlled by an FPGA control
board. The image size of panoramic is 1440$\times$1200, while the search
camera is 264$\times$224\. The galvano-mirror’s scanning FOV ranges from
$-20^{\circ}$ to $20^{\circ}$, and its step response time is 0.25 ms.

###  III-B Object Detection Analysis of the Proposed System

Figure 5: Indoor object detection setup for performance verification.

To conduct a rigorous performance evaluation and ensure a fair comparison with
the MPF, we conduct real-time object detection experiments in indoor
environments. The experimental setup, as illustrated in Fig. 5, involves
several toy cars within a 4-meter radius of the detection system. The captured
image size is 224$\times$224 pixels. In the captured images, the three types
of toy cars are varying pixel dimensions: 94$\times$29, 101$\times$43, and
144$\times$43, respectively. These toy cars exhibit dynamic movements,
including outlier toy cars, to comprehensively verify the system’s detection
performance.

We conduct outdoor experiments to show the PPM-based wide-area detection
system’s performance. As shown in Fig. 6, the detected objects lie in the
center of the search camera, which indicates the probability searching method
has object refinement ability and can search fast-moving objects clearly and
efficiently. The detection results of the MPF and our proposed algorithm for
hard cases are shown in Fig. 7. The hard cases in the experiment include
large-sized vehicles that are difficult to capture in a single FOV, cars that
are occluded, and cars that are outliers and have a side-facing pose. The
detection results captured by the wide-angle camera enable our method to
locate large-sized vehicles accurately. Moreover, due to the improved search
efficiency brought by the PPM and the probability searching module, our method
can locate small targets with occlusion and pose changes better.

Figure 6: The multiobject detection results in a transportation monitoring
scene.

###  III-C Recall Comparison of Different Searching Methods

To validate the efficiency of our algorithm, we present our sampling process
and compare it with MPF [14] and RPM-guided PF [25] algorithms. Compared with
these methods, our advantages mainly lie in the PPM and the probability
searching modules. To verify their effectiveness, we deploy the PPM and the
probability searching module to obtain the final result (Ours). As shown in
Fig. 7, by adding the probability searching module, our method can locate
targets faster than MPF and RPM-guided PF, achieving a recall of 0.60 with 300
sampled particles and a recall of 0.67 with 400 sampled particles. The PPM can
locate partial targets based on a wide-angle camera and obtain a region
probability distribution map before sampling. Thus, before the searching
process, our method has an initial detection result that is superior to other
region sampling-based methods, e.g., MPF and RPM. As shown in Fig. 7, the PPM
curve reaches a recall of 0.33 at a particle number of 0, and by detecting
difficult targets, its recall reaches 1.00 at a particle number of 800. By
combining the PPM and probability searching, our method always reaches the
highest recall rate at different particle settings. Fig. 7 indicates our
approach outperforms all the comparison methods throughout the entire sampling
process and can detect all objects in the scene when the number of particles
is 800. In contrast, MPF needs to perform random scanning of the entire scene,
resulting in low scanning efficiency and unstable results. For example, when
the number of sampled particles increases from 300 to 400, the recall value of
MPF decreases from 0.47 to 0.40.

Figure 7: Recall comparison of different methods. We conduct separate evaluations of the MPF [14], RPM-guided PF [25], and the final performance achieved by combining the PPM and probability searching modules (Ours). TABLE I: Recall comparison of different methods under different proportions of high-probability region settings Method Proportion | 27% | 35% | 41% | 49% | 63%
---|---|---|---|---|---
Omni-Pano [15] | 0.51 | 0.62 | 0.66 | 0.72 | 0.73
MPF [14] | 0.63 | 0.72 | 0.76 | 0.82 | 0.84
RPM-Guided [25] | 0.89 | 0.87 | 0.84 | 0.83 | 0.85
PPM (Ours) | 0.93 | 0.90 | 0.87 | 0.86 | 0.85

To further demonstrate the superiority of the proposed PPM prior information
with probability searching, we compare the recall rate of different
multiobject searching methods under different proportions of high-probability
regions. The results are listed in Table I. In this experiment, we compare the
PPM method against three region probability-based algorithms, which include
the Omni-Pano [15], MPF searching [14], and RPM-guided detection method [25].
During the experiment, we changed the proportion of high-probability regions
from 27% to 63%. As can be seen from Table I, owing to the PPM prior
information generation module, our method gets the best recall rates at
different proportions. When the high-probability region is about 27%, the
searching particles are sampled in a compact region, resulting in dense
searching, and the recall rate is the highest, reaching 0.93. When the high-
probability region enlarges, the search performance is always better than
other methods.

###  III-D Ablation Study of PPM and Probability Searching

Since PPM gives the search process prior information, we conduct an ablation
study to analyze the role of the PPM module. We record the results when
several object detection models are deployed in the system in Table II.
Specifically, we separately employ YOLOv3 [29], YOLOv8-M [30] and YOLO-NAS-M
[31] as the CNN detector in Eq. 6. Then, the ablation study is classified into
a searching framework with and without PPM guidance. We statistic the recall
rate, the AP value, and the processing speed, respectively. These three
detection models suffer severe performance degradation when there is no PPM
module (w/o). E.g., the proposed object searching method with YOLOv3 which
contains PPM (w/) gets a 0.88 recall rate and 0.92 AP. However, when PPM is
missed, the recall rate and AP decrease to 0.53 and 0.64, respectively. The
best performance is the searching framework with YOLO-NAS-M, the recall rate
reaches 0.94, and the AP is 0.96. It is noted that because of the panoramic
segmentation process in PPM generation, there is a slight decrease in speed
when the system employs the PPM method; e.g., YOLOv8-M decreases from 122.46
fps to 100.16 fps, and YOLO-NAS-M decreases from 133.27 fps to 120.98 fps.

Through the above ablation study, we can conclude that the PPM prior
information generation is critical in the wide-area multiobject search
framework, which generates high probability guidance for the following
searching module.

TABLE II: Ablation study of the multiobject search framework without (w/0) and with (w/) PPM prior generation Object Detector | Recall | AP | speed (fps)
---|---|---|---
YOLOv3 [29] (w/o) | 0.53 | 0.64 | 88.18
YOLOv3 [29] (w/) | 0.88 | 0.92 | 79.37
YOLOv8-M [30] (w/o) | 0.60 | 0.75 | 122.46
YOLOv8-M [30] (w/) | 0.93 | 0.95 | 100.16
YOLO-NAS-M [31] (w/o) | 0.64 | 0.79 | 133.27
YOLO-NAS-M [31] (w) | 0.94 | 0.96 | 120.98
Figure 8: Ablation study of the probability searching module. The $x$ and $y$
are pixel deviations of each target from the image center after being
calibrated by probability searching (represented as w/ ps-x, w/ ps-y) and
without probability searching (represented as w/o ps-x, w/o ps-y). The dashed
line is the mean of each axis. The right part shows the uncertainty before and
after uncertainty searching, which gives less uncertainty after searching.

Then, we investigate the coordinate refinement performance of the probability
searching module. In Fig. 8, the pixel deviations of each target from the
image center are depicted to compare the sustained gaze effect on the target.
We compare the performance of our method using probability searching (w/ ps-x,
w/ ps-y) to refine the target boundary (Eq. 11) with that of not using the
uncertainty module (w/o ps-x, w/o ps-y). As depicted in Fig. 8, our method
achieves smaller positioning deviations by utilizing probability searching. In
particular, for moving targets (target 7, target 8, and target 9), probability
searching can significantly reduce the positioning deviation. For target 7,
without using the probability searching module, the positioning deviation in
the x-axis and y-axis are 95 pixels and 69 pixels, respectively. By adding
probability searching, the positioning deviation in the x-axis decreases to 43
pixels and the positioning deviation in the y-axis decreases to 23 pixels.
Besides, the uncertainties are reduced after estimation and iterative
searching, as shown in the right part of Fig. 8. This experiment demonstrates
the effectiveness of the proposed probability searching method.

##  IV CONCLUSIONS

This paper introduces a novel wide-area hybrid vision system designed to
address the challenges of multiobject detection. The system integrates a wide-
angle camera and a search camera, thereby facilitating the capture of high-
resolution images of objects and achieving precise object detection. To
overcome the limitations of prior methods, we propose two pivotal components:
the PPM prior information generation and the probability searching module. The
former is dedicated to estimating regions of high probability for target
appearance, significantly enhancing search efficiency. The latter dynamically
adjusts the sampling range and refines target coordinates through uncertainty
variances. The successful integration of these two modules culminates in the
realization of a wide-area multiobject detection system. Comprehensive
experiments demonstrate the system’s exceptional capability to detect
multiobject in large scenarios.

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