# Emergent Ashkin-Teller criticality in a constrained boson model

Anirudha Menon1, Anwesha Chattopadhyay2, K. Sengupta1, and Arnab Sen1 1 School
of Physical Sciences, Indian Association for the Cultivation of Science,
Kolkata 700032, India
2 Department of Physics, School of Mathematical Sciences, Ramakrishna Mission
Vivekananda Educational and Research Institute, Belur, Howrah 711202, India.

###### Abstract

We show, via explicit computation on a constrained bosonic model, that the
presence of subsystem symmetries can lead to a quantum phase transition where
the critical point exhibits an emergent enhanced symmetry. Such a transition
separates a unique gapped ground state from a gapless one; the latter phase
exhibits a broken $Z_{2}$ symmetry which we tie to the presence of the
subsystem symmetries in the model. The intermediate critical point separating
these phases exhibits an additional emergent $Z_{2}$ symmetry which we
identify; this emergence leads to a critical theory in the Ashkin-Teller,
instead of the expected Ising, universality class . We verify this scenario
via explicit exact-diagonalization computations, provide an effective Landau-
Ginzburg theory for such a transition, and discuss the connection of our model
to the PXP model describing Rydberg atom arrays.

Symmetries play a pivotal role in shaping the properties of phases and phase
transitions in generic many-body systems [1]. These symmetries can be global
or local; the former can be spontaneously broken leading to typical symmetry-
broken phases of matter such as magnets or superconductors. The local or gauge
symmetries, in contrast, are always preserved [2]; however they can lead to
local constraints shaping properties of low-energy phases. A well-known
example of such a phenomenon involves spin-ice systems [3].

Figure 1: Schematic representation of a few classical Fock states of the
bosons model. The longitudinal ($d_{\ell}$) and the transverse ($d_{t}$)
diagonals are shown by dashed red and black lines respectively. A typical set
of horizontal and vertical connected links are shown by the green and orange
ellipses. (a) Ground state at $\lambda/w\to-\infty$ for $N_{p}=2n=6$
plaquettes and total boson number $N=N_{p}+1$. The state has exactly one boson
in any of the connected horizontal and vertical links. (b) One of the
classical states forming the ground state manifold for $\lambda/w\to\infty$.
(c) A state with two bosons on a vertical and a horizontal link; such states
are dynamically disconnected from those shown in (a) and (b). (d) Schematic
phase diagram of the model as a function of $\lambda/w$ showing gapped and
gapless $Z_{2}$ symmetry broken phases separated by a quantum critical point
(yellow circle) with enhanced $Z_{2}\times Z_{2}$ symmetry at
$\lambda=\lambda_{c}$. See text for details.

More recently, a class of symmetries which are intermediate between these two
types have been identified. They manifest on ($d>0$)-dimensional subsystems of
a ($D>d$)-dimensional system and are, therefore, called subsystem symmetries.
In this method of classification, the gauge (global) symmetries corresponds to
$d=0(D)$. Such subsystem symmetries play a key role in field theories which
feature dimensional reduction and in the physics of fractons [4, 5, 6]. It has
been shown that such symmetries can be spontaneously broken [7].

The above-mentioned features elicit a natural question regarding role of
subsystem symmetries in shaping properties of quantum phases and transitions
between them. This question has been partially addressed in Ref. [8], where a
quantum critical point with fractal symmetry was identified in the Newman-
Moore model [9]. In addition, disordered Ising system with subsystem
symmetries are shown to have first order transitions whose properties are
shaped by the presence of such symmetries [10, 11].

In this work, we demonstrate, using a constrained boson model, that the
presence of subsystem symmetries may change the universality class of a
quantum phase transition in an unexpected manner. The model hosts a gapped and
a gapless phases separated by a quantum critical point as schematically shown
in Fig. 1. The gapless phase has a $Z_{2}$ symmetry-broken ground state which
we show to be a consequence of the subsystem symmetry of the model.
Remarkably, the intermediate critical point has an enhanced $Z_{2}\times
Z_{2}$ symmetry and belongs to the Ashkin-Teller, rather than the expected
Ising, universality class [12, 13, 14, 15, 16]. The additional $Z_{2}$
symmetry at the critical point is emergent; we identify the root of this
symmetry using an effective Landau-Ginzburg theory and discuss its relation
with the subsystem symmetries of the model. We explicitly compute, using
exact-diagonalization (ED) on finite quasi one-dimensional arrays, the
dynamical critical exponent $z$ and the correlation length exponent $\nu$ of
the transition. In addition, we also compute entanglement entropy $S_{E}$ and
determine the central charge $c$ of the conformal field theory governing the
critical point. Our results indicate $z=1$, $\nu\simeq 1.56$ and $c=1$ showing
that the critical theory belong to Ashkin-Teller universality class [17, 15].
We also discuss the relation of the boson model with the well-known PXP model
describing experimentally realizable Rydberg atom arrays [18, 19]. Our
analysis reveals that the model allows a realization of Askhin-Teller
criticality in a one-dimensional model with Hilbert space dimension (HSD)
$\zeta^{L}$ for $L$ sites, where $\zeta=(1+\sqrt{5})/2$ is the golden ratio.
This is much smaller than HSD for other realizations of the Ashkin-Teller
model [12, 13, 14, 15]. We therefore expect it to provide significant
numerical advantage in simulating physics of Ashkin-Teller criticality.

The model: We consider a model of hardcore bosons on an array of square
plaquettes, whose number is denoted by $N_{p}$, as shown in Fig. 1. In
addition, the bosons have a constraint; any neighboring sites $j$ and $j+n$
can not be simultaneously occupied. The longitudinal (transverse) diagonals of
these plaquettes are designated by $d_{l}$ ($d_{t}$). We define two-boson
states which contain bosons on these diagonals as
$|\ell_{j}\rangle=b_{j}^{\dagger}b_{j+d_{l}}^{\dagger}|0_{j};0_{j+d_{l}}\rangle$
and
$|t_{j}\rangle=b_{j}^{\dagger}b_{j+d_{t}}^{\dagger}|0_{j};0_{j+d_{t}}\rangle$.
Here $b_{j}^{\dagger}$ denotes the boson annihilation on site $j$ of the
plaquettes, $|..0_{j}..\rangle$ indicates a Fock state having zero bosons on
site $j$, and we define the boson number operator to be
$\hat{n}_{j}=b_{j}^{\dagger}b_{j}$. In terms of $\hat{n}_{j}$, the constraint
condition can be written as $\hat{n}_{j}\hat{n}_{j+n}=0$ for all $j$. The
constrained boson Hamiltonian can then be written as $H=H_{0}+H_{1}$ where

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle-w\sum_{j}(|t_{j}\rangle\langle\ell_{j}|+{\rm h.c.})$
$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\lambda\sum_{j}(|\ell_{j}\rangle\langle\ell_{j}|-|t_{j}\rangle\langle
t_{j}|),$ (1)

where $w>0$ and $\lambda$ can have either sign. We note that $H_{0}$ denotes
the familiar ring-exchange term for the bosons; it has been extensively
studied and is known to show strong Hilbert space fragmentation [20]. $H_{1}$,
in contrast, is unconventional in the sense that it has a different sign for
bosons on transverse and longitudinal diagonals. In this work, we seek the
phase diagram of this model as a function of $\lambda/w$ having a fixed even
number of plaquettes $N_{p}$; the boson number $N=\sum_{j}n_{j}$ is also fixed
to $=N_{p}+1$ for an array with open boundary condition. In what follows, we
shall set $w=1$ and all energy scales will be measured in units of $w$.

Symmetries and Phases: It is straightforward to check that $[H,N]=0$ leading
to preservation of the total boson number in the model. Eigenstates of $H$ can
be labelled by momentum $k$ along the longitudinal direction for periodic
boundary conditions. For open chains, they can be labelled by $r=\pm 1$ where
$r=+1$ (-1) denote odd/even eigenstates $|n_{\pm}\rangle$ satisfying
$R|n_{\pm}\rangle=\pm 1|n_{\pm}\rangle$. Here $R$ represents reflections about
an axis along the transverse direction dividing the system in two equal
halves. Importantly, $R$ is the only global symmetry that can be spontaneously
broken for open chains; thus, it can lead to a $Z_{2}$ broken phase. Moreover,
$n_{0}=\sum_{j\in q}n_{j}$, where $q$ denotes any vertical or horizontal link
with three sites as shown in Fig. 1(a)(except the links involving the end
sites of an open chain that contain only two sites), remains invariant for all
links. This constitutes subsystem symmetries of the model. In what follows we
shall work in the sector $n_{0}=1$ for all such links.

For $\lambda<0$ and $|\lambda|\gg w$, the ground state of $H$ constitutes
maximum number of bosons arranged on the longitudinal diagonals as
schematically shown in Fig. 1(a). The first excited state can be created by
action of $w$ on one of the end sites; this has an energy cost $\Delta
E_{1}=3|\lambda|$ and leads to two end states as shown in Fig. 2(a). In
contrast, action of $w$ on any of the bulk sites lead to higher energy cost
$\Delta E_{2}=4|\lambda|$. Thus we have a gapped ground state. The end states
are protected from the rest of the bulk states by a finite energy gap and are
therefore stable against small perturbations. The entanglement entropy of the
ground state, $S_{E}$, plotted as a function of the subsystem size $l_{p}$ in
Fig. 2(b), shows expected area-law behavior; $S_{E}$ increases with decreasing
$|\lambda|/w$.

Figure 2: (a) Plot of eigenenergies of $H$ for $\lambda/w=-10$ showing the end
states and the first excited state manifold. (b) Entanglement entropy $S_{E}$
as a function of the subsystem size $l_{p}$ showing area-law behavior for
$\lambda/w=-10$ (red dashed line) and $0.7$ (orange dots). (c) Same as in (a)
but for $\lambda/w=10$ showing dispersing band of eigenstates; the ground
state is two-fold degenerate and gapless in thermodynamic limit. (d) Plot of
$S_{l_{p}}$ as a function of $l_{p}$ for $\lambda/w=10$ showing logarithmic
behavior. For all plots $N-1=N_{p}=28$. See text for details.

In contrast, for $\lambda\gg w$, the classical ground state manifold is
$N_{p}/2$ fold degenerate and comprises of Fock states with maximal bosons
across the transverse diagonals in a plaquette (Fig. 1(b)). The conservation
of $n_{0}$ ensures that two such plaquettes can not be neighbors; such states
belong to a sector with $n_{0}=2$ on at least one vertical and horizontal
links (Fig. 1(c)) and are dynamically disconnected from the $n_{0}=1$ sector.
These states can be characterized by the position of the single bosons
$|j_{0}\rangle$ (Fig. 2(b)). We note that $j_{0}$ is either always even or
always odd and can take $N_{p}/2$ values; this leads to a $N_{p}/2$-fold
degenerate classical ground state manifold.

This classical degeneracy is lifted by quantum fluctuations induced by
$H_{0}$. The effect of $H_{0}$ can be understood within perturbation theory
which is detailed in [21]. This lead to an effective Hamiltonian

$\displaystyle H_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{-w^{2}}{3\lambda}\sum_{j_{0}}(|j_{0}\rangle\langle
j_{0}+2|+{\rm h.c.})+{\rm O}(w^{4}/\lambda^{3})$ (2)

Note that the leading term in $H_{\rm eff}$ connects classical states where
the position of the single boson is two lattice sites apart. To analyze this
further, we note that chains with periodic boundary conditions and $N=N_{p}-1$
shows identical behavior to open chains with $N=N_{p}+1$ [21]. For the former,
the energy dispersion corresponding to $H_{\rm eff}$ is [21]

$\displaystyle E(k)$ $\displaystyle=$
$\displaystyle-\frac{2w^{2}}{3\lambda}\cos 2k+{\rm O}(w^{4}/\lambda^{3}),$ (3)

where $k$ is the momentum along $\ell_{d}$ and we have set the length of the
diagonals of the plaquette to unity. We note that $E(k)=E(\pi-k)$ leading to a
two-fold degeneracy. This degeneracy is lifted only by terms of ${\rm
O}(w^{N_{p}}/\lambda^{N_{p}-1})$ as discussed in [21]; it is protected by
subsystem symmetries. It stems from the fact that in the $n_{0}=1$ sector, the
dynamically connected states always have $j_{0}$ either even or odd. For open
boundary condition, the presence of the subsystem symmetry ensures a two-fold
reflection symmetry $R$ in the ground state manifold which protects this
degeneracy [21] with small energy splittings between $R=\pm 1$ states for
finite $N_{p}$ (see Fig. 2(c)) that decrease with $N_{p}$. Thus overall we
find a $Z_{2}$ symmetry-broken gapless ground state for $\lambda\gg w$. The
gapless nature of the ground state manifests itself in the logarithmic
dependence of $S_{E}$ on $\ell_{p}$ as shown in Fig. 2(d) for $\lambda/w=10$.

Figure 3: (a) Plot of $\Delta N_{p}$ as a function of $\lambda$; the curves
cross at $\lambda=\lambda_{c}\simeq 0.7$ indicating $z=1$. (b) The scaling
collapse obtained from a plot of $\Delta N_{p}$ as a function of
$N^{-1/\nu}(\lambda-\lambda_{c})$ leading to optimal collapse for $\nu\simeq
1.55495\pm 0.00155$ and $\lambda_{c}\simeq 0.7$. (c) Plot of
$\lambda_{c}(N_{p})$ obtained from crossing of $\Delta N_{p}$ for systems
sizes $N_{p}$ and $2N_{p}$ as a function of $1/N_{p}$ yielding
$\lambda_{c}/w=0.704\pm 0.000721$ in the thermodynamic limit. (d) Plot of
$S_{E}$ as a function of $l_{p}$ showing logarithmic dependence of $S_{E}$ on
$l_{p}$; the inset show a plot of $S_{E}$ as a function of
$\ln[(N_{p}/\pi)\sin(\pi l_{p}/N_{p})]$ leading to $c=1\pm 0.04971$. See text
for details.

Quantum critical point: The $Z_{2}$ symmetry-broken ground state at
$\lambda\gg w$ ensures an intermediate phase transition which separates it
from the ground state at $\lambda<0$. To investigate the nature of this
transition, we perform ED on the boson chain with open boundary condition for
different $N_{p}$ and carry out standard finite-size scaling analysis. The
results obtained from such a study is shown in Fig. 3. It is well-known that
the smallest energy gap $\Delta\equiv\Delta_{10}=(E_{1}-E_{0})$ (where $E_{n}$
denotes the energy of $n^{\rm th}$ eigenstate), near a critical point, for a
system with finite $N_{p}$, satisfies
$\Delta=N_{p}^{-z}f(N_{p}^{1/\nu}(\lambda-\lambda_{c}))$, where $f$ denotes a
scaling function and $\lambda_{c}$ is the critical value of $\lambda$. Our
results, shown in Figs. 3(a), and (b) indicate an excellent match with the
above form of $\Delta$ for $z=1$ and $\nu=1.55495\pm 0.00155$.

The position of the critical point is found to be around $\lambda_{c}\simeq
0.7$. To determine this better, we consider plot the crossing points of the
curves $\Delta N_{p}$ for $N_{p}=N_{0}$ and $2N_{0}$ for several $N_{p}$. Such
crossings yield $\lambda_{c}(N_{p})$ which is then plotted as a function of
$1/N_{p}$ as shown in Fig. 3(c). A standard $1/N_{p}$ extrapolation of
$\lambda_{c}(N_{p})$ shown in Fig. 3(c) yields $\lambda_{c}=0.704\pm 0.00072$
in the thermodynamic limit.

The value of $\nu$ demonstrates that the transition does not belong to the
expected Ising universality class; in contrast, it seems to indicate a
transition with Ashkin-Teller universality. To further verify this conjecture,
we plot $S_{E}$ as a function of $l_{p}$ at the critical point as shown in
Fig. 3(d). It is well-known that $S_{E}=(c/6)\ln[(N_{p}/\pi)\sin(\pi
l_{p}/N_{p})]$ [17]; this indicates a logarithmic dependence of $S_{E}$ as a
function of $l_{p}$ which is consistent with behavior of $S_{E}$ in Fig. 3(d);
moreover, it allows one to extract the central charge $c$ of the conformal
field theory governing the critical point. Our results, shown in the inset of
Fig. 3(d), indicates $c=1\pm 0.04971$ (and not close to $c=1/2$ expected for
Ising criticality) indicating that the critical point is governed by a
conformal field theory with $c=1$ consistent with Ashkin-Teller universality
[16].

Figure 4: Plot of (a) $\Delta_{20}N_{p}$ (a) and (b) $\Delta_{30}N_{p}$ as a
function of $\lambda$ showing crossings at $\lambda_{c}/w=0.70925\pm 0.00636$
and $0.74544\pm 0.0273375$ respectively. (c) and (d) In contrast, Ising model
shows such crossing only for $\Delta_{10}N_{p}$ as a function of $J/B$ but not
for $\Delta_{20}N_{p}$. See text for details.

To further distinguish this transition from its Ising counterpart we define
the gap $\Delta_{n0}=E_{n}-E_{0}$ and plot $\Delta_{20}N_{P}$ and
$\Delta_{30}N_{p}$ as a function of $\lambda$ in Figs. 4(a) and (b). These
show crossing similar to $\Delta$ shown in Fig. 3(a) indicating the presence
of three gapless modes in the thermodynamic limit. This indicates the presence
of four low-energy states in the critical manifold consistent with
$Z_{2}\times Z_{2}$ symmetry. Figs. 4 (c) and (d) show analogous results for
Ising criticality starting from the well-known Ising chain with $H_{\rm
Ising}=-J\sum_{\langle
ij\rangle}\sigma_{i}^{z}\sigma_{j}^{z}-B\sum_{j}\sigma_{j}^{x}$. We find that
for the Ising model $\Delta_{20}$ does not show any crossing as a function of
$J/B$ while $\Delta_{10}$ does at $J/B=1$, consistent with Ising criticality
with $Z_{2}$ symmetry. This analysis clearly distinguishes the present
critical point from its Ising counterpart.

The origin of the additional emergent $Z_{2}$ symmetry at criticality can be
understood from the Landau-Ginzburg theory. We note that the ground state at
$\lambda>\lambda_{c}$ is two-fold degenerate. Thus the critical theory of the
transition, which occurs when a state from the excited state manifold crosses
the ground state, involves low energy bosonic fields near the crossing points.
The degeneracy ensures that if such a crossing occurs, in a periodic chain, at
a momentum $k_{0}$ in the first Brillouin zone ($-\pi\leq k_{0}\leq\pi$),
there must be an analogous crossing at $\pi-k_{0}$. The critical theory is
therefore governed by a field [22, 23]

$\displaystyle\Phi(x,t)$ $\displaystyle=$ $\displaystyle
e^{ik_{0}x}\varphi_{1}(x,t)+e^{i(\pi-k_{0})x}\varphi_{2}(x,t)$ (4)

where $\varphi_{1}$ and $\varphi_{2}$ are the low-energy fluctuations around
$k_{0}$ and $\pi-k_{0}$ respectively. The invariance of the $\Phi(x,t)$ under
$k_{0}\to\pi-k_{0}$ demands that $\varphi_{1(2)}\to\varphi_{2(1)}$ under such
a transformation. The fields $\varphi_{1(2)}$ controls the low-energy
properties of the critical theory. An effective Landau-Ginzburg theory for the
transition which respects the above-mentioned symmetry therefore necessarily
needs two low-energy fields and can be written as

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1,2}(|\partial_{\mu}\varphi_{\alpha}|^{2}+r|\varphi_{\alpha})|^{2})+c_{1}(|\varphi_{1}|^{4}+|\varphi_{2}|^{4})$
(5)
$\displaystyle+c_{2}|\varphi_{1}|^{2}|\phi_{2}|^{2}+c_{3}(\varphi_{1}^{\ast}\varphi_{2}+h.c.)^{2},$

where $c_{1}>0$ and we have suppressed the space time labels of
$\varphi_{1,2}$ for clarity. At the critical point, $r=0$ and the critical
theory involves condensation of both fields if either $c_{2}+2c_{3}<0$ and
$c_{3}<0$ or if $c_{2}<0$. In either case, ${\mathcal{L}}$ is two-fold
degenerate since it remains invariant under $\theta_{0}\to\pi-\theta_{0}$,
where $\theta_{0}$ is the relative phase between $\varphi_{1}$ and
$\varphi_{2}$. Its minima corresponds to $\theta_{0}=0,\,{\rm\pi}$ for
$c_{3}<0$ and $\theta_{0}=\pi/2,\,3\pi/2$ if $c_{3}>0$. Thus the presence of
this term leads to an additional emergent $Z_{2}$ symmetry. We note that our
analysis indicates that the microscopic $Z_{2}$ symmetry of the gapless phase
at $\lambda>0$, which derives its root from the subsystem symmetry of the
model, necessitates a ${\mathcal{L}}$, with two low-energy fields; it is
therefore a necessary (but not sufficient) requirement for the additional
emergent $Z_{2}$ symmetry involving their relative phase at the critical
point.

Relation to the PXP model: Interestingly, the bosonic Hamiltonian $H$ can be
mapped into a variant of the well-known spin model [24, 25] used to understand
the physics of Rydberg atoms [18]. This can be understood as follows. Consider
a plaquette with two bosons occupying a longitudinal diagonal; we map such a
state to a down spin located at the centre of the plaquette:
$|\ell_{j}\rangle\equiv|\downarrow_{x_{j}}\rangle$ where $x_{j}$ denotes the
coordinate of the center of the plaquette. Similarly we map
$|t_{j}\rangle\equiv|\uparrow_{x_{j}}\rangle$ and define Pauli matrices
$\tau_{x_{j}}^{z}=|t_{j}\rangle\langle
t_{j}|-|\ell_{j}\rangle\langle\ell_{j}|$ and
$\tau_{x_{j}}^{x}=(|t_{j}\rangle\langle\ell_{j}|+{\rm h.c.})$ which acts on
these spin states. The presence of subsystem symmetry then ensures that two
up-spins can not occupy centers of neighboring plaquettes; this is analogous
to constraint on spins in a Rydberg system and can be implemented by a local
projection operator $P_{x_{j}}=(1-\tau_{x_{j}}^{z})/2$. In language of spins,
$H$ can be mapped exactly to $H_{s}$, disregarding an irrelevant constant
term, given by [21]

$\displaystyle
H_{s}=\sum_{j}[-w\tilde{\tau}^{x}_{x_{j}}+\frac{\lambda}{2}(\tau^{z}_{x_{j}}-P_{x_{j}-1}P_{x_{j}}P_{x_{j}+1})].$
(6)

where $\tilde{\tau}_{x_{j}}=P_{x_{j}-1}\tau_{x_{j}}P_{x_{j}+1}$. The first two
terms of $H_{s}$ mimics the PXP Hamiltonian in an magnetic field which
provides an effective descriptions for Rydberg atoms and is known to have an
Ising critical point at $\lambda_{c}/w\simeq 1.3$ [25, 23]. The third term
provide an additional three-site interaction. The addition of such a multi-
spin term therefore provides a way to realize Ashkin-Teller criticality in
these experimentally realizable Rydberg systems [18]; engineering such terms
in Rydberg arrays requires further work which we leave for future studies.
Moreover, since the Hilbert space dimensions of the PXP model is known to
scale as $\zeta^{L}$, the present model allows one to realize the physics of
Ashkin-Teller criticality with a much smaller HSD; this feature leads to
obvious advantage for numerical simulation of Ashkin-Teller physics in finite-
sized systems [26, 27].

In conclusion, we have demonstrated the emergence of Ashkin-Teller criticality
in a constrained bosonic model with subsystem symmetry. We point out the
crucial role of subsystem symmetries in determining the universality class of
the critical theory. Our analysis also shows that the model is closely related
to the well-known PXP model relevant for Rydberg atom arrays.

Acknowledgements: KS thanks DST, India for support through SERB project
JCB/2021/000030. AM thanks DST for support through SERB NPDF grant
PDF/2022/000124.

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# Effective Temperature and Einstein Relation for Particles in Mesoscale
Turbulence

Sanjay CP and Ashwin Joy<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Madras, Chennai - 600036

###### Abstract

From the smallest scales of quantum systems to the largest scales of
intergalactic medium, turbulence is ubiquitous in nature. Often dubbed as the
last unsolved problem of classical physics, it remains a time tested paradigm
of dynamics far from equilibrium. The phenomenon even transcends to self-
propelled fluids such as dense bacterial suspensions that can display
turbulence at mesoscale even though the constituent particles move at Reynolds
number below unity. It is intensely debated whether such fluids possess an
effective temperature and obey fluctuation-dissipation relations (FDR) as they
are generally marred by a lack of detailed balance. In this letter, we answer
this question and report an exact expression of the effective temperature for
a distribution of interacting particles that are advected by a mesoscale
turbulent flow. This effective temperature is linear in particle diffusivity
with the slope defining the particle mobility that is higher when the
background fluid exhibits global polar ordering, and lower when the fluid is
in isotropic equilibrium. We believe our work is a direct verification of the
Einstein relation —the simplest FDR, for interacting particles immersed in a
mesoscale turbulence.

Fluid turbulence has been fascinating to us since time immemorial. From the
deluge drawings of Leonardo da Vinci [1] dating almost five centuries ago to
the relatively modern works on complex fluids [2], plasmas [3] and atmospheric
systems [4], fluid turbulence is ubiquitous in nature. Despite many decades of
exhaustive research, this chaotic state of fluid motion continues to remain as
the oldest unsolved problem in classical physics. Put simply, there is
currently no rigorous theory that starts from the Navier-Stokes equation and
leads to the experimentally observed laws on energy dissipation and velocity
structure functions in turbulent flows [5, 6]. Nevertheless, turbulence is
considered as a classical paradigm of nonlinear dynamics far from
thermodynamic equilibrium [7, 8, 9]. An exciting area of mesoscale turbulence
has recently emerged where the fluid displays spatio-temporal chaos even
though its constituent units are self-propelled and move at Reynolds number
below unity. Examples include dense bacterial suspensions [10, 11],
mircotubule networks [12], artificial swimmers [13] and active liquid crystals
[14], to mention a few. It is a matter of intense debate whether these self-
propelled fluids should satisfy fluctuation-dissipation relations (FDR) as the
constituent particle dynamics generally lack detailed balance [15, 16]. A
related and important question is whether one can describe an effective
temperature of particles dispersed in such fluids [17, 18]. In this report, we
address these fundamental issues by predicting an exact expression of the
effective temperature for a distribution of interacting particles that are
advected by an active turbulent flow in the background. This effective
temperature is linear in particle diffusivity with the slope characterizing
the particle mobility, that is higher when the background fluid exhibits
global polar ordering, and lower when the fluid is in isotropic equilibrium.
Our work is therefore a direct verification of the celebrated Einstein
relation - the simplest FDR, for interacting particles advected in mesoscale
turbulence. The results reported here are valid across four decades of
variation in the damping coefficient, thereby putting a large number of active
systems within the ambit of our work —from dense suspensions of microswimmers
that are traditionally overdamped [19, 20, 21] to the lesser understood
underdamped suspensions where inertial effects are significant, notable
examples being ciliates, planktons, copepods and other macroswimmers moving in
a background media of low viscosity [22]. In what follows, we provide the
details of the phenomenological model and methods used in our work.

Model/Methods: To mimic a generic active suspension, we use a minimal
continuum model of an incompressible active fluid in two dimensions [23, 24],
whose streamfunction $\Psi(x,y)$ evolves according to the following equation

$\displaystyle\frac{\partial}{\partial
t}(\nabla^{2}\Psi)+\lambda_{0}\frac{\partial(\Psi,\nabla^{2}\Psi)}{\partial(x,y)}=-\Gamma_{0}\nabla^{4}\Psi-\Gamma_{2}\nabla^{6}\Psi-\mu\nabla^{2}\Psi$
(1)

here the non-dimensional parameter $\lambda_{0}$ decides the type of active
unit, meaning they are either pusher or a puller, corresponding respectively
to the case $\lambda_{0}>0$ or $\lambda_{0}<0$. The model was used recently to
investigate pattern formation [25] and transport coefficients in dense active
liquids [26]. We set the value of $\lambda_{0}=3.5$ throughout our work
implying that we have a pusher type of active units. We keep $\Gamma_{0,2}>0$
to enable energy injection into the active liquid via fluid instabilities. The
scalar field $\mu=\alpha+\beta|\bm{u}|^{2}$ depends on the local velocity
$\bm{u}=\bm{\nabla}\times\Psi\hat{z}$, and was first introduced by Toner and
Tu to model the “flocking” behavior in self-propelled rod-like objects [28,
29]. The parameter $\alpha$, henceforth referred to as the Ekman friction,
acts at intermediate scales and can either lead to a damping of energy when
$\alpha>0$ or an injection of energy when $\alpha<0$. Former leads the fluid
to an isotropic equilibrium and the latter yields a globally ordered polar
state with mean velocity $\sqrt{|\alpha|/\beta}$. To non-dimensionalize eq.
(1), we normalize all distances to
$\sigma_{0}=5\pi\sqrt{2\Gamma_{2}/\Gamma_{0}}$ and all times to
$t_{0}=5\pi\sqrt{2}\Gamma_{2}/\Gamma_{0}^{2}$. In terms of these reduced
units, we fix the values of the model parameters as
$\Gamma_{0}=(5\pi\sqrt{2})^{-1},\Gamma_{2}=(5\pi\sqrt{2})^{-3}$ and
$\beta=0.5$, in order to remain consistent with earlier works [30, 26].
Equation (1) is then numerically solved using a pseudo-spectral approach over
a square grid of $512^{2}$ points in a doubly periodic box of size $2\pi$
[31]. A time step of $\Delta t=2\times 10^{-4}$ is employed for time marching
$\Psi$ with the Crank-Nicholson scheme that is sufficient to maintain
numerical stability in the entire range of parameters explored here.

Into the active turbulent flow, we throw $N$ particles that follow the
dynamics:

$\displaystyle\frac{\text{d}\bm{r}_{i}}{\text{d}t}$ $\displaystyle=$
$\displaystyle\bm{v}_{i}$ $\displaystyle\frac{\text{d}\bm{v}_{i}}{\text{d}t}$
$\displaystyle=$
$\displaystyle-\gamma(\bm{v}_{i}-\bm{u}(\bm{r}_{i}))-\bm{\nabla}_{i}\Phi$ (2)

where $\gamma$ is the damping coefficient and $(\bm{r}_{i},\bm{v}_{i})$
correspond respectively to position and velocity of an $i^{\text{th}}$
particle. We realize $\bm{u}(\bm{r}_{i})$ as the turbulent flow field
projected at the location of this particle, and $-\bm{\nabla}_{i}\Phi$ as the
mass normalized force acting on it. The limit $\gamma\longrightarrow\infty$
naturally corresponds to an overdamped dynamics of passive tracers advected by
a background flow. In this limit, the tracers can exhibit an intervening
anamalous diffusive regime as the Ekman friction is pushed to negatively
larger values, say $\alpha<-6$ [27]. The limit $\gamma\longrightarrow 0$ on
the other hand, corresponds to the Newtonian dynamics of particles without any
background fluid. At intermediate values of $\gamma$, both inertial and
damping effects coexist thereby rendering generality to the particle dynamics.
We set the interaction between particles as the repulsive Weeks-Chandler-
Andersen potential energy $\Phi=\sum_{j<k}\phi(|\bm{r}_{j}-\bm{r}_{k}|)$,
where the potential

$\phi(r)=\begin{cases}4\epsilon\bigg{[}\bigg{(}\dfrac{\sigma_{0}}{r}\bigg{)}^{12}-\bigg{(}\dfrac{\sigma_{0}}{r}\bigg{)}^{6}\bigg{]}+\epsilon,&r<2^{1/6}\sigma_{0}\\\
0,&r>2^{1/6}\sigma_{0}\end{cases}$ (3)

All measurements on the particles as well as the turbulent fluid are carried
out only after the latter attains a steady state. To improve our statistics,
we average these measurements over 100 independently prepared realizations of
the initial state. Below we show that the statistics of the Eulerian velocity
$\bm{u}$ is indeed profitable to the prediction of an effective temperature of
our particles.

Statistics of $\bm{u}$: We start with a measurement of the Eulerian flow field
projected at the particle positions, i.e $\bm{u}(\bm{r}_{i})$. In Fig. (1), we
show the distribution of a component of this velocity in the steady state and
realize that it is well approximated by a Gaussian. In the inset, we show its
temporal auto-correlation that is clearly an exponential with a relaxation
time $\tau$. Combining these two facts, we can claim that the fluid velocity
projected at particle locations is a Gaussian colored noise of zero mean and
variance given as -

$\langle
u_{i\chi}(t)u_{j\psi}(t^{\prime})\rangle=(1/2)\delta_{ij}\delta_{\chi\psi}u^{2}_{\text{rms}}\text{exp}(-|t-t^{\prime}|/\tau)$
(4)

where $u_{\text{rms}}$ is a velocity scale that sets the kinetic energy of the
background fluid in the steady state. We use the Latin symbols $i,j$ to denote
particle labels and the Greek symbols $\chi,\psi$ to denote spatial indices.
In the light of all these arguments, we realize that the stochastic dynamics
in Eq. (2) is essentially non-Markovian and its corresponding steady state
Fokker-Planck equation must be carefully solved (see next) to derive an exact
expression of the stationary probability distribution. Factorization of this
distribution into position and momentum parts will yield an exact expression
of the effective temperature of our particles. This is done next.

Figure 1: Distribution of Eulerian velocities projected on the particles fits well to a Gaussian. Data shown here corresponds to $\alpha=3$; results are similar for other $\alpha$. Inset: Temporal auto-correlation of $\bm{u}$ decays exponentially with a relaxation time $\tau$. We can thus visualize $\bm{u}$ as a Gaussian colored noise - a headway that will be used to formulate an effective temperature for our particles. |
---|---
|
Figure 2: Evidence of linear relationship between effective temperature
$T_{\text{eff}}$ and particle diffusivity $D$ is presented here at various
value of damping coefficient $\gamma$. The numerical value of $T_{\text{eff}}$
is extracted by fitting $\bm{u}$ to a Gaussian and the theoretical value is
predicted by Eq. (9). Dashed line is a linear fit to our numerical data and
has a slope that indicates the inverse of particle mobility $\mu$. This is a
direct verification of the Einstein’s relation, the simplest FDR. Particle
diffusivity $D=\lim_{t\to\infty}\langle\Delta r^{2}\rangle/(4t)$ is varied by
varying the fluid friction $\alpha$. For each panel that corresponds to a
specific $\gamma$, the mobility of particles is higher when the background
fluid is in a globally ordered polar state ($\alpha<0$), and lower when it is
in isotropic equilibrium ($\alpha>0$).

Theoretical Prediction of $T_{\text{eff}}$: To systematically derive an
expression for the effective temperature $T_{\text{eff}}$ in terms of the
fluid properties, we will now set up a Fokker-Planck equation for the
governing dynamics of the interacting particles and solve it to get a
stationary velocity distribution. As stated earlier, the dynamics shown in Eq.
(2) is essentially non-Markovian due to the Gaussian colored noise $\bm{u}$.
An exact solution of the corresponding Fokker-Planck equation, with the
exception of harmonic interactions, is intractable in the general case and
approximations are always required [32]. To circumvent this difficulty, we
reduce the original non-Markovian dynamics in Eq. (2) to an equivalent
Markovian dynamics by an appropriate enlargement of the phase space [33, 34].
In the component form, this simply reads

$\displaystyle\frac{\text{d}r_{i\chi}}{\text{d}t}$ $\displaystyle=$
$\displaystyle v_{i\chi}$ $\displaystyle\frac{\text{d}v_{i\chi}}{\text{d}t}$
$\displaystyle=$
$\displaystyle-\gamma(v_{i\chi}-u_{i\chi})-\nabla_{i\chi}\Phi$
$\displaystyle\frac{\text{d}u_{i\chi}}{\text{d}t}$ $\displaystyle=$
$\displaystyle-\frac{1}{\tau}u_{i\chi}+\sqrt{\frac{u^{2}_{\text{rms}}}{\tau}}\xi_{i\chi}$
(5)

where $\bm{\xi}_{i}$ is a Gaussian white noise with zero mean and correlation
$\langle\xi_{i\chi}(t)\xi_{j\psi}(t^{\prime})\rangle=\delta_{ij}\delta_{\chi\psi}\delta(t-t^{\prime})$.
The reader should note that the dynamics in Eq. (5) is consistent with the
expected correlation properties of $\bm{u}$ that we already know from Eq. (4).
The Fokker-Planck equation for the probability distribution
$\mathcal{P}(r_{\chi},v_{\chi},t)$ corresponding to the dynamics in (5) can be
written using a decoupling approximation as [35]:

$\frac{\partial}{\partial
t}\mathcal{P}(r_{\chi},v_{\chi},t)=\mathcal{L}\;\mathcal{P}(r_{\chi},v_{\chi},t)$
(6)

where the operator $\mathcal{L}$ is defined as

$\mathcal{L}=\bigg{[}-v_{\chi}\frac{\partial}{\partial
r_{\chi}}+\frac{\partial}{\partial v_{\chi}}\bigg{(}\gamma
v_{\chi}+\frac{\partial\Phi}{\partial
r_{\chi}}\bigg{)}+\dfrac{\gamma^{2}u^{2}_{\text{rms}}\tau}{2\bigg{(}1+\gamma\tau+\tau^{2}\bigg{\langle}\dfrac{\partial^{2}\Phi}{\partial
r_{\chi}^{2}}\bigg{\rangle}\bigg{)}}\dfrac{\partial^{2}}{\partial
v^{2}_{\chi}}+\dfrac{\gamma^{2}u^{2}_{\text{rms}}\tau^{2}}{2\bigg{(}1+\gamma\tau+\tau^{2}\bigg{\langle}\dfrac{\partial^{2}\Phi}{\partial
r_{\chi}^{2}}\bigg{\rangle}\bigg{)}}\dfrac{\partial^{2}}{\partial
x_{\chi}\partial v_{\chi}}\bigg{]},$ (7)

The steady state solution of the above Fokker-Planck equation readily
factorizes into position and velocity parts as:

$\mathcal{P}_{\text{s}}(r_{\chi},v_{\chi})=\frac{1}{\mathcal{Z}}\exp\bigg{(}-\frac{v^{2}_{\chi}}{2\sigma^{2}_{\chi}}\bigg{)}\exp\bigg{[}-\frac{\Phi}{\sigma^{2}_{\chi}(1+\gamma\tau)}\bigg{]}$
(8)

with the velocity variance realized as

$\sigma^{2}_{\chi}=\frac{u^{2}_{\text{rms}}\gamma\tau}{2\bigg{(}1+\gamma\tau+\tau^{2}\bigg{\langle}\dfrac{\partial^{2}\Phi}{\partial
x^{2}_{\chi}}\bigg{\rangle}\bigg{)}}\equiv T_{\text{eff}}$ (9)

the effective temperature of our inertial particles, with $\chi$ being $x$ or
$y$. We take a moment of pause to appreciate the beauty of this exact
expression. There are no fitting parameters here and we are able to completely
predict the effective temperature of particles once the background fluid
properties ($u_{\text{rms}},\tau,\gamma$) and particle interactions ($\Phi$)
are specified. The predicted $T_{\text{eff}}$ is in excellent agreement with
the numerically estimated value that is obtained by fitting the velocity
distribution to a Gaussian. This is shown in Fig. 2 where we plot
$T_{\text{eff}}$ vs. $D$, the particle diffusivity. For each $\gamma$, we
calculate the diffusivity as $D=\lim_{t\to\infty}\langle\Delta
r^{2}\rangle/(4t)$ and its variation $D_{\text{min}}\rightarrow
D_{\text{max}}$ comes from varying the fluid friction $\alpha$ in the range
$4.0\rightarrow-4.0$. It is evident from each panel of this figure that
$T_{\text{eff}}$ is linear in $D$ with a slope yielding the inverse of
particle mobility $\mu$. This is a direct verification of the Einstein’s
relation, the simplest formulation of FDR. We also observe that the particle
mobility is higher when the background fluid is in a globally ordered polar
state, i.e $\alpha<0$, and lower when the fluid is in isotropic equilibrium,
or $\alpha>0$. Our results are valid across four decades of variation in
damping coefficient effectively covering the entire range
$1<\gamma\tau<10^{4}$, where the lower and upper limit corresponding
respectively to the inertial and overdamped regime. This puts a large number
of active systems within the purview of our work—from overdamped suspensions
of microswimmers [36, 37] to the underdamped suspensions of macroswimmers such
as ciliates, planktons, copepods moving in low viscosity media [22].

Summary: We have demonstrated that a distribution of interacting particles
immersed in an active turbulent fluid can be described by an effective
temperature. To achieve this, we solved the steady state Fokker-Planck
equation corresponding to the stochastic governing equations of the particles.
The result is an exact relation between the effective temperature, properties
of the background flow and particle interactions. Finally we show this
effective temperature is linear in particle diffusivity with the slope
characterizing particle mobility. This is a direct verification of the
Einstein’s relation - the simplest FDR, for particles advected by mesoscale
turbulence. The mobility is observed to be higher when the background fluid is
in a globally ordered polar phase, and lower when the fluid is in isotropic
equilibrium. The results reported here apply to active liquids prepared under
a broad spectrum of particle damping, and are therefore general in nature.

###### Acknowledgements.

The authors thank V Balakrishnan, Sumesh Thampi and Abhijit Sen for
discussions and comments on the manuscript. All simulations were done on the
HPC-Physics cluster of our group. Support from the SERB core grant, DST and
New Faculty Seed Grant (NFSG), IIT Madras, is gratefully acknowledged.

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# A Lightweight and Accurate Recognition Framework for Signs of X-ray Weld
Images

Moyun Liu11footnotemark: 1<EMAIL_ADDRESS>Jingming Xie22footnotemark: 2
<EMAIL_ADDRESS>Jing Hao33footnotemark: 3<EMAIL_ADDRESS>Yang
Zhang44footnotemark: 4<EMAIL_ADDRESS>Xuzhan Chen55footnotemark: 5
<EMAIL_ADDRESS>Youping Chen66footnotemark: 6<EMAIL_ADDRESS>School
of Mechanical Science and Engineering, Huazhong University of Science and
Technology, Wuhan, 430074, China  School of Mechanical Engineering, Hubei
University of Technology, Wuhan, 430068, China

###### Abstract

X-ray images are commonly used to ensure the security of devices in quality
inspection industry. The recognition of signs printed on X-ray weld images
plays an essential role in digital traceability system of manufacturing
industry. However, the scales of objects vary different greatly in weld
images, and it hinders us to achieve satisfactory recognition. In this paper,
we propose a signs recognition framework based on convolutional neural
networks (CNNs) for weld images. The proposed framework firstly contains a
shallow classification network for correcting the pose of images. Moreover, we
present a novel spatial and channel enhancement (SCE) module to address the
above scale problem. This module can integrate multi-scale features and
adaptively assign weights for each feature source. Based on SCE module, a
narrow network is designed for final weld information recognition. To enhance
the practicability of our framework, we carefully design the architecture of
framework with a few parameters and computations. Experimental results show
that our framework achieves 99.7% accuracy with 1.1 giga floating-point of
operations (GFLOPs) on classification stage, and 90.0 mean average precision
(mAP) with 176.1 frames per second (FPS) on recognition stage.

###### keywords:

Convolutional neural network , signs recognition , X-ray weld images,
lightweight.

††journal: Computers in Industry

##  1 Introduction

Manufacturing industry relies on X-ray images to monitor weld quality in daily
production, because they have ability to reflect the internal condition of
artifacts (Malarvel et al., 2017). Some signs printed on X-ray images include
cross mark and weld information such as the date of the photograph, the serial
number of the artifact and the mark of image quality indicator (IQI). Cross
mark is used to show the pose of images, and weld information needs to be
stored into digital system for tracing images. Therefore, an automatic signs
recognition framework is vital for an advanced digital X-ray weld image
system. These signs are produced by some moveable types whose material is
plumbum, and they are selectively placed on the top of weld metal manually.
Finally, these signs would be projected into image through X-ray detection
apparatus.

Table 1: List of abbreviations Abbreviation | Full name
---|---
BDB | basicblock-downsample-basicblock
BN | batch normalization
CBR | convolutional-batch normalization-rectified linear unit
CNNs | convolutional neural networks
Conv | convolution
FN | false negative
FP | false positive
FPS | frames per second
GFLOPs | giga floating-point of operations
GRNet | group convolution-based resnet
GYNet | greater-YOLO network
IQI | image quality indicator
Madds | multiply-add operations
mAP | mean average precision
Params | parameters
ReLU | rectified linear unit
SCE | spatial and channel enhancement
SGD | stochastic gradient descent
TP | true positive

To better save and observe, original weld photographs are scanned to digital
images as shown in Fig. 1. There is a cross mark on each image, and it is
usually unordered. Only the mark showing right&up means the correct direction,
and it is necessary to redirect the image based on the classification result
of cross mark. Weld information printed on image is required to be further
recognized after completing the forementioned classification task. The
categories of these information are mainly numbers, letters and some marks. We
regard the information recognition as object detection task, which has been
studied for many years in deep learning field.

In recent years, deep learning approaches are facing vigorous development,
among which convolutional neural networks (CNNs)-based methods have made
excellent achievements in various image tasks. To exploit the potential of
CNNs, researchers have proposed many superior network structures for image
classification (Simonyan and Zisserman, 2014; He et al., 2016) and object
detection (Ren et al., 2015; Redmon et al., 2016). However, to our knowledge,
these methods are more used for defects on weld images (Yaping and Weixin,
2019; Duan et al., 2019; Dong et al., 2021), but not for signs recognition.
Besides signs, the foreground contents also include weld region and noises,
and they have a variety of scales. Hence, achieving recognition on X-ray weld
images is a challenging task because the context of image is complex. In
general, multi-scale features fusion (Bell et al., 2016; Lin et al., 2017a) is
introduced to address the scale diversity problem. This strategy allows the
extracted feature maps to obtain features at different scales simultaneously,
and it is also helpful to predict objects of different sizes (Redmon and
Farhadi, 2018). However, since size distribution of weld information is
consistent and single, the features at one scale are often more important than
at other scales, so it is crucial to assign weights for different feature
sources at weld information recognition task. There are many existing methods
designed for enhancing the multi-scale representation ability of network (He
et al., 2015; Chen et al., 2017). However they usually simply add or
concatenate feature maps from different scale sources, without ranking their
importance and assigning them with different weights.

Figure 1: Digitization of X-ray weld image. (a) Movable types are placed on
the top of weld metal, and original X-ray weld photograph is produced by X-ray
detection apparatus. Finally, the photograph is transformed to digital image
by scanner. (b) Weld information mainly contains the date of the photograph,
the serial number and IQI. (c) Cross mark can reflect the pose of weld image.

In this paper, we propose a signs recognition framework for X-ray images to
accomplish the above tasks. Our framework is compact and high-performant,
consisting of two CNNs, i.e., Group convolution-based ResNet (GRNet) for cross
mark classification and Greater-YOLO network (GYNet) for weld information
recognition. Based on the residual block of ResNet, we design a shallow
backbone for GRNet, and group convolution (Howard et al., 2017) is introduced
to reduce the parameters and computations. Inspired by the efficient structure
of Tiny-YOLO v3 (Redmon and Farhadi, 2018), we propose a more narrow GYNet
based on a novel spatial and channel enhancement (SCE) module. SCE module
firstly integrates features from multiple scales, and then adaptively weights
them according to their contributions. To validate the effectiveness of our
framework, we conduct extensive experiments on our datasets. Experimental
results show that our framework achieves high performance with fast speed and
a few parameters, compared with the state-of-the-art methods.

In summary, this work makes the following contributions.

  * 1.

We design a compressed and accurate framework to fulfill the signs recognition
of weld images with fast speed and high performance.

  * 2.

A elaborate backbone for GRNet is proposed, and it is designed with a few
layers based on group convolution.

  * 3.

We propose a narrow and light GYNet, in which a novel SCE module is introduced
to complement feature information at different scales and weight them
adaptively.

  * 4.

The experimental results show that our methods achieve fast speed and accurate
prediction compared with state-of-the-art models.

The rest of this paper is organized as follows. Section 2 introduces some
related works about CNNs of classification, detection and multi-scale features
fusion methods. Section 3 presents our framework in detail. All experimental
results are shown and discussed in Section 4. Finally, we conclude this paper
in Section 5. A list of abbreviations is listed in Table 1.

##  2 Related works

To the best of our knowledge, there is no research on signs recognition of
weld images in the past. We regard it as the task of image classification and
detection, which has been studied for many years, and many excellent works
have been proposed.

###  2.1 Image Classification and Object Detection

The use of convolutional neural network for image classification can be traced
back to the 1990s, when the proposed LeNet-5 (LeCun et al., 1998) laid the
foundation of CNNs. AlexNet (Krizhevsky et al., 2012) won the first prize
ImageNet competition in 2012, and triggered the research of CNNs. VGG
(Simonyan and Zisserman, 2014) reduces the size of the filter to $3\times 3$
and deepens the network depth, greatly improving the classification accuracy
on the ImageNet dataset. ResNet (He et al., 2016) increases the potential of
CNNs by introducing residual connection, which solves the problem of gradient
disappearance in the training process and makes it possible to design deeper
networks.

As for object detection, CNNs can be divided into one-stage and two-stage
detector. The biggest difference between them is that the latter generates
regional proposal, while the former does not. The classical two-stage
detectors, such as Faster R-CNN (Ren et al., 2015), Cascade R-CNN (Cai and
Vasconcelos, 2018) and Libra R-CNN (Pang et al., 2019) firstly generate a set
of region proposals, and they will be classified and regressed bounding box at
the end. Two-stage methods usually can achieve more accurate prediction
results, but they also need more computation resources, and are not
satisfactory on detection speed. One-stage models are more applicable when
tasks have requirements on inference speed. YOLO (Redmon et al., 2016; Redmon
and Farhadi, 2017, 2018), SSD (Liu et al., 2016), RetinaNet (Lin et al.,
2017b) are typical one-stage CNNs. Although they have higher efficiency, the
lack of region proposal step makes them not accurate enough compared with two-
stage network in most cases.

In addition, many lightweight classification and detection networks are
designed to enhance the practicability of CNNs. A novel Fire Module was
proposed in SqueezeNet (Iandola et al., 2016). This module reduces parameters
by using $1\times 1$ convolution to replace $3\times 3$ convolution. The
MobileNet (Sandler et al., 2018) series networks proposed depthwise separable
convolution that can reduce the model complexity. ShuffleNet (Zhang et al.,
2018; Ma et al., 2018) changes the channel order of feature maps, which
enables cross-group information flow. A cheap operation is introduced in
GhostNet (Han et al., 2020), which has fewer parameters while obtaining the
same number of feature maps.

###  2.2 Multi-scale Features Fusion

Feature fusion is a very effective strategy to achieve feature complementarity
among different layers of CNNs. The original fusion way is simply adding or
concatenating multi-scale features (Bell et al., 2016), which achieves
improvement on performance to some extend. To obtain better approaches, more
fusion strategies are exploited. SSD (Liu et al., 2016) and MS-CNN (Cai et
al., 2016) directly combine prediction results of the feature hierarchy.
Feature Pyramid Networks (Lin et al., 2017a) constructs a top-down structure
to fuse feature layers of different scales, and produces enhanced feature maps
for final classification and localization prediction. Recently, some
researches have found that multi-scale fusion based on different receptive
fields can greatly improve the performance of CNNs. For example, SPP (He et
al., 2015) generates multiple groups of feature maps through pooling
operations with different kernel size. Similarly, ASPP (Chen et al., 2017)
achieves above goal by using atrous convolutions with different dilation
rates. In spite of success, the current fusion methods do not consider which
scale is more important for the final prediction. The essence of these
strategies is to treat all scales equally.

An incredible recognition framework requires an outstanding baseline, and
advanced feature fusion method. To achieve this goal, we design our
classification network based on residual structure (He et al., 2016) and the
convolution method used in MobileNet (Sandler et al., 2018). Inspired by the
fast speed of one-stage model, we propose our recognition network based on
Tiny-YOLO v3 (Redmon and Farhadi, 2018). Moreover, a new feature map fusion
method named SCE is proposed, and it is used to improve the multi-scale
representation ability of recognition network.

##  3 Method

The proposed framework consists of two CNNs, i.e., GRNet for cross mark
classification and GYNet for weld information recognition. The architecture of
our framework is represented in Fig. 2. GRNet is a lightweight yet effective
classifier with only 14 convolution (Conv) layers. GYNet is a compressed but
high-performing network designed by a few number of channels on high-level
layers. In this section, we will explain the detailed structures of GRNet and
GYNet.

Figure 2: The architecture of weld information recognition framework. Our
framework contains two networks, GRNet for cross mark classification and GYNet
for weld information recognition. The image is first input into GRNet, and the
category of the cross mark is obtained through 14 layers of convolution. Based
on this classification result, the image is adjusted to correct pose. And
then, image is extracted feature by lightweight GYNet which owns two
prediction heads, to output the information content printed on X-ray weld
image.

###  3.1 GRNet for Cross Mark Classification

The final purpose of our framework is to recognize the information of X-ray
weld images. However, the pose of these digital images is random and casual in
actual production. Thus we need to classify the direction mark, i,e, the cross
mark at first, and then adjust image to correct pose. A compact and efficient
classification network can redirect faster, improving the overall efficiency
of recognition framework. The obvious insight is to build the classifier with
a few layers and lightweight convolution way. ResNet (He et al., 2016) is a
successful series of classification CNN, but there is still room for further
optimization in terms of the number of layers and internal modules. To achieve
this goal, we propose a novel GRNet.

The backbone of our GRNet has 9 modules with only 14 Conv layers, and its
architecture is shown in Table 2. The input images with $224\times 224\times
3$ resolution are fed into a Conv-BN-ReLU (CBR) module. CBR contains Conv
layer with the kernel of $3\times 3$ followed by Batch Normalization (BN)
(Ioffe and Szegedy, 2015) and Rectified Linear Unit (ReLU) (Nair and Hinton,
2010). And then GRNet employs a BasicBlock which is a residual module
including a CBR module and $3\times 3$ Conv followed by BN layer. The input
and output feature maps of BasicBlock will be fused by element-wise adding.
BasicBlock-Downsample-BasicBlock (BDB) is used to downsample the feature maps,
and it has two branches. One of them is a BasicBlock whose stride of the Conv
layer in CBR module is 2. Another branch employs a MaxPool layer whose size is
$2\times 2$ and stride is 2, followed by a $3\times 3$ Conv with stride of 1
and BN layer. Finally, the obtained feature maps by two branches are added
element-wise as well.

Table 2: Detailed architecture of GRNet Layer name | Input size | Output size
---|---|---
CBR | $224\times 224\times 3$ | $224\times 224\times 64$
BasicBlock | $224\times 224\times 64$ | $224\times 224\times 64$
BasicBlock | $224\times 224\times 64$ | $224\times 224\times 64$
BDB | $224\times 224\times 64$ | $112\times 112\times 256$
BasicBlock | $112\times 112\times 256$ | $112\times 112\times 256$
BDB | $112\times 112\times 256$ | $56\times 56\times 512$
BasicBlock | $56\times 56\times 512$ | $56\times 56\times 512$
AvgPool | $56\times 56\times 512$ | $1\times 1\times 512$
Fully Connected Layer | $1\times 1\times 512$ | $1\times 1\times 4$
SoftMax

To further cut down the parameters and model size of GRNet, we use group
convolution (Howard et al., 2017) to replace all $3\times 3$ normal
convolution. In general, we define a $D\times D\times C_{1}\times C_{2}$
convolution filter, where $D$ is the spatial dimension of the kernel assumed
to be square. $C_{1}$ is the number of input channels, and $C_{2}$ is the
number of output channels. The normal convolution can be defined as follows:

$O_{m,n,c_{2}}=\sum_{i,j,c_{1}}D_{i,j,c_{1},c_{2}}\cdot
I_{m+i-1,n+j-1,c_{1}},$ (1)

where $I$ and $O$ are input and output feature maps, respectively.

Group convolution splits the filters and feature maps into the same number of
group in the channel direction. When the number of group is $g$, it is defined
as:

$\hat{O}_{m,n,c_{2}/g}=\sum_{i,j}\hat{D}_{i,j,g}\cdot
I_{m+i-1,n+j-1,c_{1}/g}\;,$ (2)

where $\hat{D}$ is the group convolution kernel of size $D\times D\times
C_{1}/G\times C_{2}/G$, and $\hat{O}$ is the output feature map.

In this paper, we define the $g$ as the greatest common divisor of input and
output channel numbers. If the dimensions of input and output are the same,
the number of parameters in normal convolution is g times that of group
convolution (Howard et al., 2017).

###  3.2 GYNet for Weld Information Recognition

There are many foreground contents in weld image, including weld information,
weld region and noises. The context of image is complicated because the scales
of these contents vary greatly. To achieve accurate weld information
recognition, a novel SCE module is proposed to enhance the contextual
representation ability of extracted feature maps. In addition, we propose a
narrow and efficient recognition network GYNet based on SCE module.

SCE Module. To better utilize different scale features, we propose a novel SCE
module and its detailed structure is shown in Fig. 3. SCE is composed of
spatial integration block (Bochkovskiy et al., 2020) and channel weighted
block (Hu et al., 2018), and the feature maps are processed by these two
blocks successively. Spatial integration block uses multiple pooling
operations with different kernel sizes to obtain a spatial pooling pyramid,
and it owns local and global features from different spatial scales. These
features from diverse scales will be fused by concatenation at last, and the
whole process can be formulated as follow:

$\mathcal{O}=\mathcal{C}\left(\left.M_{i}(\mathcal{I})\right|_{i=1,5,9,13}\right),$
(3)

where $\mathcal{O}$ and $\mathcal{I}$ represent output and input feature map
of spatial pooling pyramid, respectively. $\mathcal{C}(\cdot)$ is
concatenation operation, and $M_{i}(\cdot)$ is maxpool with kernel size
$i\times i$ ($i=1$ representing the identity branch). Spatial integration
block fuses different receptive field information, which makes the obtained
feature map capable of capturing diverse spatial features.

Figure 3: The internal structure of proposed SCE module. SCE is composed of
spatial integration block and channel weighted block, and image is processed
by them successively. The first block produces a spatial pooling pyramid and
concatenates feature map among pyramid. The second block can obtain the
importance of each channel and update feature maps based on weights.

Channel weighted block is a type of attention mechanism method. It can learn
the relationship between different channels and obtain the weight of each
channel. Firstly, it uses global average pooling to generate channel
descriptor $\mathcal{Z}\in\mathbb{R}^{1\times 1\times C}$ across spatial
dimensions. $\mathcal{O}\in\mathbb{R}^{H\times W\times C}$ is input data, and
the $c$-th element of $\mathcal{Z}$ is obtained as follow:

$\mathcal{Z}_{c}=\frac{1}{H\times
W}\sum_{i=1}^{H}\sum_{j=1}^{W}\mathcal{O}_{c}(i,j),$ (4)

where $\mathcal{O}_{c}$ is the $c$-th feature map of $\mathcal{O}$. Then, the
channel descriptor $\mathcal{Z}$ is excited to redefine the importance of each
channel. Specifically, we employ a linear layer followed by ReLU layer and a
linear layer followed by sigmoid layer, and the process can be described as:

$\mathbf{S}=\sigma\left(\mathbf{W}_{2}\delta\left(\mathbf{W}_{1}\mathcal{Z}\right)\right),$
(5)

where $\delta$ refers to the ReLU function, and $\sigma$ is Sigmoid function,
$\mathbf{W}_{1}\in\mathbb{R}^{\frac{C}{r}\times C}$ and
$\mathbf{W}_{2}\in\mathbb{R}^{C\times\frac{C}{r}}$. $r$ is a hyper-parameter
that controls the model complexity, and it is set as 4 in this paper.

The final output of channel weighted block is calculated as:

$\mathbf{X}=\mathfrak{\mathbf{F}}\left(\mathcal{O},\mathbf{S}\right)=\mathcal{O}\cdot\mathbf{S},$
(6)

where $\mathbf{X}\in\mathbb{R}^{H\times W\times C}$ and $\mathbf{F(\cdot)}$
refers to channel-wise multiplication.

Channel weighted block is an adaptive adjuster whose function is to learn the
importance of each channel information, and further can show which scale
feature is more significant. Although multi-scale information is the basis of
effective feature map, different scales make different contributions to the
results. Especially when the sizes of recognized objects are similar, there is
only one scale that is essential for final prediction theoretically. Compared
with other foreground contents, the scale distribution of weld information is
relatively consistent. Hence, channel weighted block is designed to weight
different scale adaptively during network learning, and more significant
channel, in other words, more meaningful scale feature would be assigned more
weight.

Overall, the proposed SCE module improves the contextual representation
ability of feature maps through integrating more information sources, and
further weight them adaptively based on their importance. The effect of SCE
will be discussed detailedly in Section 4.2.

Architecture. Inspired by Tiny-YOLO v3, we design a recognition network for
weld information, and its detailed architecture is given in Table 3. GYNet has
the same numbers of Conv and MaxPool layer compared with Tiny-YOLO v3. More
narrow model can decrease the parameters and FLOPs more directly. To obtain a
smaller width backbone, we strictly limit the number of channels in each
layer. Almost all layers are below 512 channels, and this design strategy
makes network bring few burden on computation device. We embed SCE module at
the tail of backbone to ensure it process more meaningful information, and
make the enhanced features closer to the output layer for more accurate
recognition results.

Table 3: Detailed architecture of GYNet Layer name | Type | Filters | Size | Stride | Output size
---|---|---|---|---|---
S1 | Conv | 16 | 3 | 1 | $416\times 416\times 16$
S2 | MaxPool | – | 2 | 2 | $208\times 208\times 16$
S3 | Conv | 32 | 3 | 1 | $208\times 208\times 32$
S4 | MaxPool | – | 2 | 2 | $104\times 104\times 32$
S5 | Conv | 64 | 3 | 1 | $104\times 104\times 64$
S6 | MaxPool | – | 2 | 2 | $52\times 52\times 64$
S7 | Conv | 64 | 3 | 1 | $52\times 52\times 64$
S8 | MaxPool | – | 2 | 2 | $26\times 26\times 64$
S9 | Conv | 128 | 3 | 1 | $26\times 26\times 128$
S10 | MaxPool | – | 2 | 2 | $13\times 13\times 128$
S11 | Conv | 256 | 3 | 1 | $13\times 13\times 256$
S12 | MaxPool | – | 2 | 1 | $13\times 13\times 256$
S13 | Conv | 512 | 3 | 1 | $13\times 13\times 512$
S14 | SCE | – | – | – | $13\times 13\times 2048$
S15 | Conv | 128 | 1 | 1 | $13\times 13\times 128$
S16 | Conv | 256 | 1 | 1 | $13\times 13\times 256$
S17 | Conv | 135 | 1 | 1 | $13\times 13\times 135$
S18 | Conv | 128 | 1 | 1 | $13\times 13\times 128$
S19 | UpSample | – | – | – | $26\times 26\times 128$
S20 | Concat | – | – | – | $26\times 26\times 256$
S21 | Conv | 256 | 3 | 1 | $13\times 13\times 256$
S22 | Conv | 135 | 3 | 1 | $13\times 13\times 135$

##  4 Experiments

To show the superiority of our framework for X-ray weld image signs
recognition, experiment results and analysis are represented in this section.
Firstly, the experimental setup including datasets, the implementation details
and the evaluation metrics, is introduced in Section 4.1. Then, we validate
the effectiveness of SCE module. Specifically, ablation studies are designed
to show its necessity, and we visualize the weight values to prove
aforementioned weight assignment mechanism. Finally, aiming at classification
subtask and recognition subtask, we compare our proposed methods with the
state-of-the-art models.

###  4.1 Experimental Setup

Datasets. We have obtained 897 digital X-ray weld images from the actual
production workshops of special equipment companies. All images have been
annotated carefully by professionals. We build two datasets for
training/testing classification and recognition.

For the classification subtask, to make the cross mark more eye-catching in
the image, we divide the image into $416\times 416$ pixels multiple sub-images
as input. We use the flip and minor operation to augment our dataset for
obtaining a more robust network. At the end, we have 3588 images in cross mark
classification dataset, and it is randomly divided into training set,
validation set and testing set according to the ratio of 8:1:1. Cross mark
classification dataset has four classes to represent the direction of images.

For the recognition subtask, we resize the whole image to $416\times 416$
pixels for adapting to the normal input size of YOLO. The number of weld
information classes is 40, which is relatively large, and the condition of
whole image is very complex. So, we use more complicated augmentation methods
by combining changing brightness and contrast, using Gaussian blur and
rotating way. Each original image is augmented two or three times randomly,
and finally obtain 3550 images, which are randomly divided into 3145 images in
training set and 355 images in test set.

Implementation Details. We conduct our all experiments on a i7-8700K CPU and a
single NVIDIA GeForce GTX1070Ti GPU. All models are based on deep learning
framework PyTorch. In cross sign classification experiments, we choose
stochastic gradient descent optimizer with 0.9 momentum parameter and 0.0005
weight decay. The initial learning rate and total epochs are set as 0.1 and
80, respectively. The step policy is used to divide initial learning rate in
10 by each 50 epochs. Label smoothing strategy is introduced to optimize the
classification training process. In information recognition experiments, the
series of YOLO networks are trained by SGD optimizer with 0.9 momentum
parameter and 0.0005 weight decay as well. The initial learning rate and total
epochs are set as 0.001 and 300, respectively. Learning scheduler utilizes
LambdaLR strategy. The state-of-the-art methods are trained and tested based
on MMdetection (Chen et al., 2019) which is an open source 2D detection
toolbox, and all related hyper-parameters are adjusted to the optimal.

Evaluation Metrics. We adopt mean average precision (mAP), Recall, floating-
point of operations (FLOPs), parameters (Params) and frame per second (FPS) as
evaluation metrics to evaluate the proposed network comprehensively. mAP and
Recall are used to show the detection performance, while the rest of metrics
are used to represent the computation complexity and speed property. Relevant
metrics are defined as:

$Precision=\frac{TP}{TP+FP},$ (7) $Recall=\frac{TP}{TP+FN},$ (8)

and the involved concepts TP, FP, FN (Padilla et al., 2020) are explained as
follow.

  * 1.

True positive (TP): The number of objects that are detected correctly.

  * 2.

False positive (FP): The number of incorrect detection including nonexistent
and misplaced predictions.

  * 3.

False negative (FN): The number of objects that are undetected successfully.

mAP is a metric used to evaluate comprehensive ability over all classes. It is
simply the average AP over all classes, and AP can be computed as the area of
Precision $\times$ Recall curve with axles.

Moreover, to compare the computation complexity of different networks, time
complexity, i.e., FLOPs and space complexity, i.e., Params are chosen to show
the difference between different methods. In addition, we use FPS to show the
speed during inference stage, and the results of FPS are the average of 350
testing images in this paper.

###  4.2 Effectiveness of the SCE module

Ablation Studies. To explore the importance of each module in our GYNet, we
design a series of ablation studies, and the obtained results are shown in
Table 4. Based on similar backbone, all combinations are much the same in
terms of recognition speed and computations. Although the introduction of both
two blocks brings a slight increase in the number of parameters and model
size, the recognition ability of our method has a great improvement, attaining
90.0 mAP and 88.8$\%$ recall. However, the result improvement is extremely
limited when either of them is used alone. We attribute the superior
performance of our SCE module to its ability of weighting the information
after feature fusion.

Table 4: The comparison results from different combinations of spatial integration block (SIB), channel weighted block (CWB). SIB | CWB | mAP | Recall($\%$) | FPS | FLOPs(G) | Params(M) | Size(MB)
---|---|---|---|---|---|---|---
– | – | 86.4 | 87.4 | 195.0 | 2.8 | 2.6 | 10.7
$\surd$ | – | 87.2 | 87.5 | 165.8 | 2.8 | 2.8 | 11.5
– | $\surd$ | 87.0 | 87.5 | 191.5 | 2.8 | 2.8 | 11.2
$\surd$ | $\surd$ | 90.0 | 88.8 | 176.1 | 2.8 | 4.9 | 19.9
Figure 4: The scale distribution of weld information.

Validation of Weight Assignment Mechanism. To intuitively observe the scale
distribution of weld information, we normalize the width and height of weld
information relative to weld image size, and its scale distribution is shown
in Fig. 4. The scale of weld information is consistent, while widths and
heights are almost all less than 0.25, and it means that weld information
belongs to a relatively small scale space. To validate the reasonability of
weight assignment mechanism, we visualize the weights produced by channel
weighted block as shown in Fig. 5. We divide channels into four parts,
corresponding to four features scales of the spatial pooling pyramid. The blue
dots indicate the weights assigned to each channel by channel weighted block.
Red dots indicate the average weight of each channel interval, and color
deepness reflects the size of the average value. It can be observed that SCE
module assigns almost average weight of 0.4 to the channels of first three
intervals, and about average weight of 0.7 to the last interval which comes
from the identity branch. Maxpool operation with large kernel size would
weaken local feature, and it is not favorable for the recognition of small
scale object. Hence, SCE module assigns more weights to identity branch, while
treating other scale sources as less important contributions. This results
show that the proposed SCE module can adaptively weight each feature source
based on multi-scale feature fusion.

Figure 5: The weight values produced by channel weighted block of SCE module.
The blue dots represent the weight values of each channel, and the red dots
show the average weight values of each channel interval.

###  4.3 Comparisons with State-of-the-art Models

The Classification of Cross Mark. In order to validate the performance of our
GRNet in cross mark classification dataset, we introduce many advanced
classification networks, including classical models like ResNet-18 (He et al.,
2016) and ResNet-34 (He et al., 2016), and lightweight classification networks
such as ShuffleNetV1 (Zhang et al., 2018), ShuffleNetV2 (Ma et al., 2018),
MobileNetV2 (Sandler et al., 2018), SqueezeNet (Iandola et al., 2016),
GhostNet (Han et al., 2020). The loss curves of above networks during training
process are presented in Fig. 6, and we can observe that the performance of
all models tends to be stable after 50 epochs. It is worth noting that the
loss values of our GRNet have less fluctuation at the beginning, and come down
to very low and similar loss values eventually compared with other methods.

Figure 6: The training loss curve comparison of different classification methods. Table 5: The classification results in comparison with state-of-the-art methods. Models | Accuracy($\%$) | FLOPs(G) | Madds(G) | Params(M) | Size (MB)
---|---|---|---|---|---
ShuffleNetV1 (Zhang et al., 2018) | 99.7 | 2.2 | 4.3 | 0.9 | 3.9
ShuffleNetV2 (Ma et al., 2018) | 99.4 | 2.2 | 4.4 | 1.3 | 5.2
MobileNetV2 (Sandler et al., 2018) | 99.7 | 2.4 | 4.8 | 2.3 | 9.2
SqueezeNet (Iandola et al., 2016) | 99.7 | 2.7 | 5.2 | 0.7 | 3.0
GhostNet (Han et al., 2020) | 96.1 | 0.2 | 0.3 | 3.9 | 15.8
ResNet-34 (He et al., 2016) | 100.0 | 56.9 | 113.8 | 21.3 | 85.2
ResNet-18 (He et al., 2016) | 99.7 | 27.3 | 54.5 | 11.2 | 44.7
GRNet(ours) | 99.7 | 1.1 | 2.1 | 0.2 | 0.8

From Table 5, it can demonstrate that our GRNet outperforms all other methods,
while it achieves 99.7$\%$ accuracy with the least 0.2M Params and 0.8 MB
model size. Although the accuracy of ResNet-34 is 100%, its parameters are 112
times that of our method, and its FLOPs and Madds are the most, 52 and 53
times that of our method, respectively. ShuffleNetV1, MobileNetV2, SqueezeNet
and ResNet-18 also obtain the same accuracy compared with our method, but they
are not lightweight enough. Specifically, ResNet-18 has $25\times$ FLOPs and
$59\times$ Params than our GRNet. In comparison with the other three
lightweight models, our GRNet also achieves greater lightweight performance,
while it only has about 50$\%$ FLOPs, and 8$\%$$\sim$26$\%$ Params. Therefore,
we can conclude that our method can classify cross mark accurately with lower
computational complexity and fewer resources. GhostNet has lowest FLOPs and
multiply-add operations (Madds) because of its particular design of reusing
feature maps. But its accuracy is only 96.1$\%$ and the number of parameters
is $20.6\times$ than our method.

Table 6: The recognition results in comparison with state-of-the-art methods. $\dagger$ indicates that network combines with SCE module. Methods | mAP | FPS | FLOPs(G) | Madds(G) | Params(M) | Size(MB)
---|---|---|---|---|---|---
Tiny-YOLO v3 (Redmon and Farhadi, 2018) | 87.3 | 143.3 | 5.5 | 11.0 | 8.8 | 35.1
YOLO v3 (Redmon and Farhadi, 2018) | 90.4 | 25.6 | 65.8 | 131.6 | 61.7 | 247.3
RetinaNet (Lin et al., 2017b) | 87.3 | 20.1 | 37.5 | 75.0 | 36.9 | 148.8
Faster R-CNN (Ren et al., 2017) | 90.2 | 18.1 | 46.7 | 93.4 | 41.3 | 166.5
Cascade R-CNN (Cai and Vasconcelos, 2018) | 90.1 | 12.2 | 74.4 | 148.8 | 69.1 | 277.4
Libra R-CNN (Pang et al., 2019) | 90.0 | 17.2 | 46.9 | 93.8 | 41.6 | 167.5
Dynamic R-CNN (Zhang et al., 2020) | 89.5 | 18.2 | 46.9 | 93.8 | 41.5 | 166.5
Tiny-YOLO v3$\dagger$ | 87.9 | 114.5 | 5.8 | 11.6 | 17.9 | 71.8
YOLO v3$\dagger$ | 91.0 | 25.0 | 66.4 | 132.8 | 71.7 | 287.2
GYNet(ours) | 90.0 | 176.1 | 2.8 | 5.6 | 4.9 | 19.9

The Recognition of Weld Information. Based on the MMdetection, we compare our
method with many state-of-the-art models using ResNet-50 as backbone, such as
RetinaNet (Lin et al., 2017b), Faster R-CNN (Ren et al., 2017), Cascade R-CNN
(Cai and Vasconcelos, 2018), Libra R-CNN (Pang et al., 2019), Dynamic R-CNN
(Zhang et al., 2020), and all related parameters have been set to make all
models perform best. The comparison results are shown in Table 6. The Params
of our GYNet is only 4.9M, which is 55.7$\%$ of Tiny-YOLO v3 and 7.9$\%$ of
YOLO v3. Such small number of parameters makes the model size only 19.9 MB,
15.2 MB smaller than the famous compressed network Tiny-YOLO v3. In addition,
GYNet has the fastest speed with 176.1 FPS, which is $1.2\times$ faster than
Tiny-YOLO v3 and $6.9\times$ faster than YOLO v3. Under such deep lightweight
optimization, our method still achieves 90.0 mAP, and is much higher than its
baseline Tiny-YOLO v3 and famous one-stage network RetinaNet, while only 0.4
points lower than YOLO v3. Classical two-stage CNN models Faster R-CNN,
Cascade R-CNN, and Libra R-CNN have similar performance compared with our
GYNet, but their recognition speeds are all below 30 FPS, far from meeting the
actual requirements. Furthermore, their Params and model size are overweight
for normal hardware. We combine SCE module with YOLO v3 and Tiny-YOLO v3, and
their performance has been improved, which validates the effectivenss of SCE
module as well. The visualization recognition results of GYNet are shown in
Fig. 7, and we can observe that our GYNet is capable of dealing with weld
images with various types and lighting conditions.

Figure 7: Visualization results of weld information recognition produced by
GYNet. GYnet can accurately recognize weld information for different types of
X-ray images, even when the brightness of image is extremely low.

##  5 Conclusion

In this paper, we propose a high-performing lightweight framework for signs
recognition of weld images, in which GRNet and GYNet are connected to complete
whole task. Aiming at classification of cross mark, GRNet is presented with a
well-designed backbone to compress model. For weld information, a new
architecture with a novel SCE module is designed for GYNet, in which the SCE
module integrates multi-scale features, and assigns adaptively weights to
different scale sources. Experiments show that our signs recognition framework
obtains high prediction accuracy with tiny parameters and computations.
Specifically, GRNet achieves 99.7% accuracy with only 0.8 MB model size and
1.1 GFLOPs, which is 1.8% and 4% of ResNet-18, respectively. GYNet achieves
90.0 mAP on recognition dataset, 2.7 points higher than Tiny-YOLO v3, and its
FPS is 176.1, $1.2\times/6.9\times$ faster than Tiny-YOLO v3/ YOLO v3. In the
future, we will focus on the further optimization of algorithm, and the
application on embedded platforms (Raspberry Pi and Jetson Nano) to reduce
hardware costs.

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