# Density dependent gauge field inducing emergent SSH physics, solitons and
condensates in a discrete nonlinear Schrödinger equation

W. N. Faugno1, Mario Salerno2, and Tomoki Ozawa1 1Advanced Institute for
Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan
2Dipartimento di Fisica “E.R. Caianiello,” CNISM and Istituto Nazionale di
Fisica Nucleare—Gruppo Collegato di Salerno, Universita dí Salerno, Via
Giovanni Paolo II, 84084 Fisciano (SA), Italy

###### Abstract

We investigate a discrete non-linear Schrödinger equation with dynamical,
density-difference-dependent, gauge fields. We find a ground-state transition
from a plane wave condensate to a localized soliton state as the gauge
coupling is varied. Interestingly we find a regime in which the condensate and
soliton are both stable. We identify an emergent chiral symmetry, which leads
to the existence of a symmetry protected zero energy edge mode. The emergent
chiral symmetry relates low and high energy solitons. These states indicate
that the interaction acts both repulsively and attractively.

Introduction - The non-linear Schrödinger equation (NLSE) is an essential
model for a variety of physical systems [1, 2]. In ultracold atomic physics,
the NLSE describes Bose-Einstein condensates (BECs) in the dilute limit where
only the leading order interaction is relevant [3, 4, 5, 6]. In optical
systems, the NLSE describes light propagating in nonlinear media, originally
employed to describe self-focusing beams of light [7, 8, 9]. Since these
initial investigations a wide range of studies into the NLSE and its
extensions have been developed, identifying a variety of non-linear phenomena
[10, 11, 12, 13, 14, 15, 16]. Additionally, extensions have been made to
simulate gauge theories [17, 18, 19, 20, 21]. Much work has gone into
understanding the stability of NLSE solutions. This is in part because the
onset of instability can often be associated with a phase transition to a
stable non-dispersive bound state, termed a soliton [22, 23, 24, 25, 26, 27].

Solitons and other localized states have been predicted and verified in the
NLSE and its extensions [28, 29, 30, 31]. These bound states result from the
competition between the non-linearity and the kinetic dispersion. In the case
of the discrete NLSEs [32, 33], the existence of bound states termed discrete
breathers has been shown to be generic [34]. Such self-stabilizing solutions
are widely studied for both their mathematical properties as well as their
potential applications in developing communications technology. Recently,
matter waves breathers have been experimentally observed by quenching the
strength of the attractive interactions in a quasi one-dimensional Bose-
Einstein condensate [35]. Engineering and manipulation of soliton modes has
been achieved through various methods, including time-dependent modulation and
spatial modulation [36, 37, 38, 39, 40].

An interesting avenue explored in recent years has been to consider NLSE which
contains a dynamical, density-dependent, gauge field. Here, in the context of
discrete (lattice) models, the gauge field is a nontrivial phase factor one
has upon hopping between adjacent sites, and density-dependent gauge field
means that the phase factor depends on the density of sites involved in the
hopping [41, 42, 43, 44, 45, 46]. It has been shown that density-dependent
hopping induced by sinusoidal temporal modulations of the nonlinearity in the
discrete NLSE leads to the formation of compactons, i.e., nonlinear bound
states that, unlike breathers, do not have exponential decay tails [31].
Recently, two of us proposed a novel type of density-dependent gauge field, in
the context of interacting non-Hermitian quantum physics, where the hopping
depends on the difference of the density of sites involved in hopping [47];
such a gauge field has a unique feature that the sign of the interaction is
not determined a priori but dynamically depends on the density gradient of the
wavefunction profile. In this paper, we find that the NLSE derived from such a
density-difference-dependent gauge field results in a variety of unique
phenomena, including the coexistence of plane-wave condensate state and
soliton state, the coexistence of attractive and repulsive solitons, and the
emergent SSH-like physics with a symmetry-protected zero-energy mode. Our
paper opens an avenue to explore nonlinear physics of this exotic form of
gauge field.

The Discrete Density Difference Dependent Non-Linear Schrodinger Equation
(D4NLSE) - We consider a one dimensional discrete NLSE derived from the
following classical Hamiltonian:

$H=\sum_{j}\Psi^{*}_{j+1}\bigg{[}-J+\gamma(n_{j+1}-n_{j})\bigg{]}\Psi_{j}+c.c.,$
(1)

where $\Psi_{j}$ is the complex-valued amplitude of the wavefunction at site
$j$, $n_{j}=\Psi_{j}^{*}\Psi_{j}$ is the corresponding density, $J$ is the
density-independent part of hopping between adjacent sites, $\gamma$ is the
coupling constant of the density-difference-dependent hopping which can
generally be a complex number. We assume that the wavefunction is normalized
as $\sum_{j}|\Psi_{j}|^{2}=1$. A crucial aspect of this Hamiltonian is that
the sign of the interaction, either repulsive or attractive, is not determined
a priori, but rather determined dynamically through the density profile of the
wavefuntion.

The corresponding NLSE can be obtained by computing the Poisson bracket
$i\frac{d\Psi_{j}}{dt}=\left\\{\Psi_{j},H\right\\}=\frac{\delta
H}{\delta\Psi_{j}^{*}}$:

$\displaystyle i\frac{d\Psi_{j}}{dt}$
$\displaystyle=\left\\{-J+\gamma(2n_{j}-n_{j-1})\right)\\}\Psi_{j-1}-\gamma\Psi_{j}^{2}\Psi_{j+1}^{*}$
$\displaystyle+\left\\{-J-\gamma^{*}(2n_{j}-n_{j+1})\right)\\}\Psi_{j+1}+\gamma^{*}\Psi_{j}^{2}\Psi_{j-1}^{*},$
(2)

which is the one-dimensional Discrete Density Difference Dependent (D4) Non-
linear Schrödinger equation we are going to explore in this paper.

We note that the Hamiltonian in Eq. 1 can be also thought of as quantum
Hamiltonian by identifying $\Psi^{*}_{j}$ and $\Psi_{j}$ as creation and
annihilation operators of a particle at site $j$. Then the Poisson bracket
mentioned above should be replaced by a commutator, and D4 NLSE describes a
situation where the mean-field theory is applicable and the Gross-Pitaevskii
treatment of the condensate wavefunction is justified.

Ground state phase diagram:— We explore the stationary state of the D4 NLSE
evolving in time as $\Psi_{j}(t)=e^{-i\mu t}\Psi_{j}(0)$, where $\mu$ is the
chemical potential. Of particular interest is the ground state, which is the
state with the lowest energy. We can numerically explore the ground state by
the method of imaginary-time propagation, which is to start from a random
initial state and simulate the time evolution of the D4 NLSE with an imaginary
time, that is, to consider the equation obtained after setting $t=-i\tau$ and
simulate the evolution with respect to $\tau$. After evolving long enough time
in $\tau$, one converges to the ground state (if the ground state is unique)
[48, 49]. Assuming periodic boundary condition and applying imaginary time
evolution method to various values of $\gamma$, we see that the ground state
is either a plane-wave condensate state or a sharply localized soliton-like
state, which we explain now.

We first note that, for states with a uniform density, $n_{j}=n_{j+1}$,
$\gamma$-dependence in the Hamiltonian vanishes. The plane-wave state
$e^{ikx}$ with $k=2\pi(\mathrm{integer})/L$, where $L$ is the number of
lattice sites, is thus a valid stationary state with chemical potential
$\mu=-2J\cos(k)$. Among the plane wave solutions, the ground state is the
$k=0$ state, and the corresponding chemical potential as well as the energy is
$\mu=E=-2J$.

Now the question to ask is if one can obtain a state with lower energy by
allowing the density to vary. We indeed find that the system can host
localized stationary states which we call solitons. Essential features of the
soliton can be captured through the following ansatz where the soliton is
spread over only three sites around a site $n$:

$\displaystyle\Psi_{n\pm 1}$
$\displaystyle=e^{i\phi_{\pm}}\sqrt{(1-\alpha)/2},$ $\displaystyle\Psi_{n}$
$\displaystyle=\sqrt{\alpha},$ (3)

where $\phi_{-},\phi_{+},\alpha$ are variational parameters which we take as
real numbers. We expect that the interaction energy dominates in such soliton
states. The transition between the $k=0$ plane wave and the localized ansatz
can be solved analytically for purely real $\gamma$ and is predicted to occur
at $|\gamma|=2.53J$. For complex $\gamma$, the critical $|\gamma|$ weakly
depends on its phase, decreasing slightly as the phase goes towards purely
imaginary $\gamma$ where the transition occurs around $|\gamma|\approx 2.5J$.
We find from imaginary time evolution that the transition occurs at
$|\gamma|\approx 2.04J$ for fully real $\gamma$ and $|\gamma|\approx 1.9J$ for
fully imaginary $\gamma$; these values are smaller than the prediction from
the three-site ansatz we obtained above because the soliton state around the
transition point can be more spread and/or asymmetric. We numerically confirm
that, typically, more than 95% of the weight of the soliton lies on three
sites, and the three-site ansatz becomes more and more accurate as $|\gamma|$
becomes larger.

Figure 1: Left Panel. Energy (green solid curve) and the chemical potential
(blue dotted curve) of the localized soliton state, in units of $J$, obtained
from the imaginary time propagation method, as a function of the gauge
coupling which we assume to be purely imaginary, $\gamma=i\gamma_{I}$. The red
dashed lines correspond to the chemical potential of the localized states
obtained when setting $J=0$. The dashed horizontal black lines correspond to
the lowest and highest plane wave condensate states at $\mu=E=\pm 2J$. The
inset highlights the transition point and soliton disappearance. Right Panels.
Ground state and highest energy state wavefunction for $J=1$, $\gamma=-4.0i$
and length $L=29$. The blue, green and red curves are the wave function
amplitude, real part and imaginary part, respectively.

We provide a more detailed analysis for the case of purely imaginary
$\gamma=i\gamma_{I}$. The results are qualitatively the same for any fixed
phase of $\gamma$ with slight quantitative differences as mentioned above. In
Fig. 1, we plot the energy and the chemical potentials for the lowest and
highest energy localized soliton states as a function of $|\gamma_{I}|$
obtained from imaginary time evolution. These results were found in excellent
agreement also with the ones obtained by means of self-consistent numerical
diagonalizations of the nonlinear eigenvalue problem associated with Eq. 2
[30]. The symmetry around $E=0$ is a consequence of an emergent chiral
symmetry which we discuss later. At $|\gamma_{I}|>1.9J$, the solitons are the
lowest energy states. In the region $1.67J<|\gamma_{I}|<1.9J$, the soliton
solution still exists as a stationary state but its energy is higher than the
energy of the plane wave condensate state. As $|\gamma_{I}|$ approaches
$\approx 1.67J$ from above, the chemical potential of the soliton changes
rapidly, and at $|\gamma_{I}|<1.67J$, the soliton state no longer exists as a
stationary state but rather merges with the extended modes. The energy and the
chemical potential of the soliton state just above $|\gamma_{I}|\approx 1.67J$
are $E\approx-1.9J$ and $\mu\approx-2.3J$. Note that we have studied this
model in the continuum and found that in solving for a localized state, the
energy is unbounded from below, suggesting that soliton formation is a genuine
lattice effect (see supplemental material).

Stability of the condensate:— We have just seen that the ground state changes
from plane-wave condensate state to localized soliton state as one increases
the coupling $|\gamma|$. The mechanism behind the formation of soliton ground
state is qualitatively different from more conventional soliton formation in,
for example, NLSE with a cubic nonlinearity, where the development of
modulational instability of plane-wave condensate state leads to the formation
of solitons [50]. In contrast, in the D4 NLSE, there can be regions in the
parameter space where the zero momentum plane wave condensate and the soliton
both exist as stable stationary states. Depending on the value of $\gamma$,
the condensate can become unstable, as we show soon, but the instability of
condensate and the formation of soliton appear to be not directly related. We
now explore this linear stability of the condensate phase and present a
complete phase diagram, Fig 2, including the stability analysis. (In Appendix
A, we also present the continuum model and demonstrate that its plane wave
solutions have the same stability condition.)

Figure 2: Phase Diagram of the D3NLSE as a function of the complex gauge field
coupling $\gamma$. The complex gauge coupling is defined as
$\gamma_{R}+i\gamma_{I}$, $J$ is the single particle hopping parameter, and
$L$ is the length of the chain. Note this phase diagram only considers the
linear stability. Numerically we have observed that the instability can occur
for completely real $\gamma$, likely due to higher order instabilities.

To investigate the linear stability of a condensate upon small perturbation,
we introduce the Fourier transformed Hamiltonian. Defining the wavefunction in
momentum space with momentum $k$ by $\tilde{\Psi}_{k}$ as

$\Psi_{j}=\frac{1}{\sqrt{L}}\sum_{k}\tilde{\Psi}_{k}e^{-ijk}.$ (4)

the Hamiltonian is

$\displaystyle
H=\sum_{k}-2J\cos{k}\tilde{\Psi}^{*}_{k}\tilde{\Psi}_{k}+\frac{1}{L}\sum_{k,k^{\prime},q}V(q,k)\tilde{\Psi}^{*}_{k}\tilde{\Psi}^{*}_{k^{\prime}}\tilde{\Psi}_{k+k^{\prime}-q}\tilde{\Psi}_{q}$
$\displaystyle
V(q,k)=2i\gamma_{R}(\sin{k}-\sin{q})+2i\gamma_{I}(\cos{q}-\cos{k}).$ (5)

where we have introduced the notation $\gamma=\gamma_{R}+i\gamma_{I}$. The
time evolution equation in momentum space is then

$\displaystyle i\dot{\tilde{\Psi}}_{k}=-2J(\cos{k}-1)\tilde{\Psi}_{k}$ (6)
$\displaystyle+\sum_{q,k^{\prime}}V(q,k)\tilde{\Psi}^{*}_{k^{\prime}}\tilde{\Psi}_{k+k^{\prime}-q}\tilde{\Psi}_{q}+\sum_{q,k^{\prime}}V(q,k^{\prime})\tilde{\Psi}^{*}_{k^{\prime}}\tilde{\Psi}_{k+k^{\prime}-q}\tilde{\Psi}_{q}$

We assume that the condensate takes place at zero momentum, and add small
perturbation at momentum $p$ to investigate its stability; we set
$\tilde{\Psi}_{0}\approx 1$, $\tilde{\Psi}_{\pm p}=\delta\tilde{\Psi}_{\pm p}$
which should be kept up to linear order, and $\tilde{\Psi}_{k}=0$ otherwise.
Inserting them into Eq 6 and ignoring terms of higher order in
$\delta\tilde{\Psi}_{\pm p}$, we obtain the equation for the time derivative
of $\delta\tilde{\Psi}_{p}$ as

$i\frac{d\delta\tilde{\Psi}_{p}}{dt}=-2J(\cos{p}-1)\delta\tilde{\Psi}_{p}+\frac{4i\gamma_{I}}{L}(\cos{p}-1)\delta\tilde{\Psi}^{*}_{-p},$
(7)

and a similar equation for the time evolution of $\delta\tilde{\Psi}_{-p}$.
This set of equations can be diagonalized, in a manner analogous to the
Bogoliubov transformation, to obtain the dispersion relation, which is

$\epsilon(p)=2J\sqrt{\left(1-\frac{4\gamma_{I}^{2}}{J^{2}L^{2}}\right)}(1-\cos{p}).$
(8)

This dispersion relation contains unusual features. First, at small momentum
$p$, the dispersion is quadratic rather than more standard linear excitation
spectrum in NLSE. The effect of the interaction $\gamma$ is to modify the
overall multiplicative factor, which is analogous to modifying the effective
mass of a particle. The second point to notice is that only the imaginary part
of $\gamma$ enters the dispersion relation. The real part does not affect the
excitation properties of the zero momentum condensate phase. The third point
is that the dispersion becomes imaginary, implying the appearance of
modulational instability, above a threshold value of $\gamma_{I}>JL/2$;
notably the transition depends on the length of the system $L$. The threshold
to the instability of this condensate is drawn in the phase diagram Fig 2. We
point out that the stability condition is solely determined by $\gamma_{I}$
whereas the transition from condensate ground state to soliton ground state
depends on $|\gamma|$, implying that these two transitions occur from
different mechanisms. (The stability condition for a condensate at arbitrary
momentum $k$ is presented in Appendix B derived from Bogoliubov analysis of a
bosonic system.)

Chiral Symmetry and Zero Mode:— In Fig. 1, we see that the figure is symmetric
around $E=0$. This symmetry is a consequence of the chiral symmetry present in
the system. By examining the equation of motion, we see that for a stationary
state $\\{\Psi_{j}\\}$, we obtain a state with the opposite chemical potential
by multiplying the state by $e^{i\pi j}$, i.e. a factor of $-1$ to every other
site. In Fig. 1, the wavefunctions of the ground state and the highest energy
state are plotted for $\gamma=-4i$ to demonstrate that they are precisely
related by the transformation $e^{i\pi j}$. Note that the existence of these
two bound states is a signature of the attractive and repulsive nature of our
interaction. Typically the appearance of low energy bound states is a
signature of an attractive interaction [51, 25] whereas the existence of high
energy bound states is a signature of a repulsive interaction [52].

The presence of such a chiral symmetry suggests the existence of a symmetry
protected zero mode analogous to the Su-Schrieffer-Heeger (SSH) model. The SSH
model is the one-dimensional tight-binding model with alternating weak and
strong hopping strengths, and its momentum-space topology is nontrivial when
the hopping from the edge starts with a weak hopping. The topologically
nontrivial SSH model hosts a zero energy edge state protected by the chiral
symmetry whose wavefunction takes nonzero values in every other site. Our D4
NLSE does not have an a priori alternating hopping, however, since the density
dependence affects the hopping strength, one can expect an emergent SSH-like
physics induced dynamically by an appropriate density distribution. To explore
such emergent SSH-like physics, we look for a zero energy edge mode. Such a
zero energy edge mode must have nonzero wavefunction in every other site. We
consider an open boundary condition with $j\geq 1$, and set the wave function
on sites with even indices to zero $\Psi_{2n}=0$. The D4 NLSE taking zero
chemical potential leads to the following condition to be satisfied by the
wavefunctions in odd sites:

$\displaystyle\left(J+\gamma|\Psi_{2n-1}|^{2}\right)\Psi_{2n-1}=\left(-J+\gamma^{*}|\Psi_{2n+1}|^{2}\right)\Psi_{2n+1}.$
(9)

One can iteratively solve this equation for decreasing solution starting from
some value of $\Psi_{1}$ for $\gamma_{R}<0$ or $\Psi_{L}$ for $\gamma_{R}>0$.
When the total number of sites is odd, the iteratively-constructed solution is
the exact zero energy solution of the D4 NLSE. When the total number of sites
is even, this construction fails as the final even site with only one adjacent
odd site cannot properly satisfy the above equation. We note that there is a
zero-energy mode completely localized at one site on the edge, i.e $n=1$, when
$\gamma/J=-1$, which we call a singleton state. (This state exists regardless
of the number of sites). As $\gamma$ moves from $\gamma/J=-1$, the state
becomes more and more delocalized as shown in Fig 3, which plots the ratio of
the $|\Psi_{3}|^{2}/|\Psi_{1}|^{2}$ in the complex $\gamma$ plane as a measure
of localization. The predominant trends show that as $|\gamma|$ moves from 1
and as $\gamma_{I}$ grows the state spreads further. In fact for a purely
imaginary $\gamma$, the state is evenly spread on odd numbered sites.

Figure 3: Localization ratio $|\Psi_{3}|^{2}/|\Psi_{1}|^{2}$ in the complex
$\gamma$ plane. The coloring corresponds to this ratio. Note that ratio
vanishes at $\gamma=-1$ where the singleton state stabilizes.

Floquet Realization:— Finally, we present one method using Floquet engineering
to realize the D4 NLSE, which can be implemented in systems such as Bose-
Einstein condensates of ultracold quantum gases and waveguide arrays. We
derive an effective Hamiltonian of a three step Floquet protocol as derived in
the formalism presented in Ref. [53]. We consider the time dependent quantum
Hamiltonian

$\displaystyle H(t)$
$\displaystyle=-J\sum_{j}\hat{\Psi}^{\dagger}_{j+1}\hat{\Psi}_{j}+\hat{\Psi}^{\dagger}_{j}\hat{\Psi}_{j+1}+V(t)$
(10) $\displaystyle V(t)$ $\displaystyle=\begin{cases}V_{1},&0\leq
t<\frac{T_{\omega}}{3}\\\ V_{2},&\frac{T_{\omega}}{3}\leq
t<\frac{2T_{\omega}}{3}\\\ -V_{1}-V_{2},&\frac{2T_{\omega}}{3}\leq
t<T_{\omega}\end{cases}$ (11)

which corresponds to a free boson with pulsed modulation of the hopping
parameter
$V_{1}=\sum_{j}J_{1}\hat{\Psi}^{\dagger}_{j+1}\hat{\Psi}_{j}+\mathrm{H.c.}$
and a pulsed onsite interaction
$V_{2}=U\sum_{j}\hat{\Psi}^{\dagger}_{j}\hat{\Psi}^{\dagger}_{j}\hat{\Psi}_{j}\hat{\Psi}_{j}$
in three steps of equal duration with total period $T_{\omega}=2\pi/\omega$.
In the limit of large $\omega$ (i.e. $U/\omega,J_{1}/\omega\ll 1$), the
effective Hamiltonian to first order in $1/\omega$ is given as

$\begin{split}H_{\mathrm{eff}}=\sum_{j}\hat{\Psi}^{\dagger}_{j+1}\left[-J+\frac{2i\pi
UJ_{1}}{9\omega}(\hat{n}_{j}-\hat{n}_{j+1})\right]\hat{\Psi}_{j}\\\
+\hat{\Psi}^{\dagger}_{j}\bigg{[}-J-\frac{2i\pi
UJ_{1}^{*}}{9\omega}(\hat{n}_{j}-\hat{n}_{j+1})\bigg{]}\hat{\Psi}_{j+1}\end{split}$
(12)

which maps directly onto Eq. 1 with $\gamma=2i\pi UJ_{1}/(27\omega)$ and
identifying the quantum operators by classical variables assuming mean-field
theory. We control the phase of the complex coupling $\gamma$ by changing the
phase of the modulated hopping $J_{1}$. Complex hopping of $J_{1}$ can be
engineered through modulation such as in Struck et al. [54], which can be
combined with our proposed Floquet protocol, or through Raman-assisted
tunneling [55]. Similar Floquet interactions have been previously studied in
various ultracold and optical systems [56, 41, 43, 44].

Conclusion - In this article we have presented the D4NLSE and its phase
diagram. We have derived the conditions for plane wave stability as well as a
phase transition from the extended $k=0$ plane wave ground state to a
localized ground state. Contrary to conventional NLSE, this model admits
coexistence of a stable $k=0$ plane wave and a soliton ground state. We have
further demonstrated that the interaction of the D4 NLSE acts both either
attractively or repulsively depending on the state in question. Although the
D4 NLSE itself is fully translationally invariant, the density distribution
results in emergent SSH-like physics, and as a consequence, zero energy edge-
localized mode exists. Finally we derived a possible method of Floquet
engineering to realize the D4 NLSE.

Many of the characteristic features of D4 NLSE we found in this paper stem
from the fact that the sign of the interaction is not determined a priori, but
rather dynamically determined by the density distribution of the nonlinear
wave. The emergent chiral symmetry analogous to the one discussed in this
paper can be a versatile mechanism to obtain interaction-induced topology.

The dynamical dependence of the sign of the interaction represents a new
emerging paradigm in the field of nonlinear physics opening up unexpected and
unexplored scenarios at both the classical and quantum levels. In this paper
we have focused on properties of the classical ground state of the mean-field
problem under this exotic form of dynamical gauge field. Ground-state phase
diagram of the corresponding quantum many-body problems as well as its non-
Hermitian generalizations are left for future studies.

###### Acknowledgements.

Acknowledgments.— This work is supported by JSPS KAKENHI Grant No. JP20H01845,
Grant No. JP21H01007, and JST CREST Grant No.JPMJCR19T1.

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##  Appendix A Continuum Model

We now derive the continuum limit of our lattice model and study its
properties. Here we explicitly introduce the lattice spacing $\alpha$, and
consider the limit $\alpha\to 0$ to find the coutinuous model. The Fourier
space Hamiltonian of our model, with $\alpha$ explicitly written, is

$H=\sum_{k}-2J\cos{(k\alpha)}\tilde{\Psi}_{k}\tilde{\Psi}_{k}+\frac{1}{L}\sum_{k,k^{\prime},q}V(q\alpha,k\alpha)\tilde{\Psi}^{*}_{k}\tilde{\Psi}^{*}_{k^{\prime}}\tilde{\Psi}_{k+k^{\prime}-q}\tilde{\Psi}_{q}-\mu\sum_{k}\tilde{\Psi}^{*}_{k}\tilde{\Psi}_{k},$
(13)

where

$\displaystyle
V(q\alpha,k\alpha)=\gamma(e^{-iq\alpha}-e^{-ik\alpha})-\gamma^{\prime}(e^{iq\alpha}-e^{ik\alpha}).$
(14)

In the continuum (long-wavelength) limit $k\alpha,q\alpha\rightarrow 0$, the
momentum dependent coefficients can be approximated through the Taylor
expansion as

$\begin{split}-2J\cos{k\alpha}&\approx-2J+J(k\alpha)^{2},\\\
V(q\alpha,k\alpha)&\approx
2i\gamma_{R}\alpha(k-q)+i\gamma_{I}\alpha^{2}(k^{2}-q^{2}).\end{split}$ (15)

To obtain the continuum limit, we use these expansions in Eq. 13 and perform
the inverse Fourier transform

$\tilde{\Psi}_{k}=\frac{1}{\sqrt{L}}\sum_{n}\Psi_{n}e^{in\alpha k}.$ (16)

We define the continuum field operators at position $x=n\alpha$ by

$\displaystyle\phi(x)=\Psi_{n}/\sqrt{\alpha},$ (17)

with the identification $\int dx=\alpha\sum_{n}$. The continuum limit of the
kinetic and chemical potential parts of the Hamiltonian are then obtained as:

$\displaystyle\sum_{k}\left(-2J-\mu+J(k\alpha)^{2}\right)\tilde{\Psi}^{*}_{k}\tilde{\Psi}_{k}=(-2J-\mu)\sum_{n}\Psi^{*}_{n}\Psi_{n}+\frac{1}{L}\sum_{k}\sum_{n,m}J(k\alpha)^{2}\Psi^{*}_{n}\Psi_{m}e^{ik\alpha(m-n)}$
$\displaystyle=(-2J-\mu)\int dx\phi^{*}(x)\phi(x)+\frac{\alpha}{L}\sum_{k}\int
dxdx^{\prime}Jk^{2}\phi^{*}(x)\phi(x^{\prime})e^{ik(x^{\prime}-x)}$
$\displaystyle=(-2J-\mu)\int dx\phi^{*}(x)\phi(x)+\frac{\alpha}{L}\sum_{k}\int
dxdx^{\prime}J\phi^{*}(x)\phi(x^{\prime})\frac{\partial^{2}}{\partial
x\partial x^{\prime}}e^{ik(x^{\prime}-x)}$ $\displaystyle=(-2J-\mu)\int
dx\phi^{*}(x)\phi(x)+\frac{\alpha}{L}\sum_{k}\int
dxdx^{\prime}J\left(\partial_{x}\phi^{*}(x)\right)\left(\partial_{x^{\prime}}\phi(x^{\prime})\right)e^{ik(x^{\prime}-x)}$
$\displaystyle=(-2J-\mu)\int dx\phi^{*}(x)\phi(x)+J\alpha^{2}\int
dxdx^{\prime}\left(\partial_{x}\phi^{*}(x)\right)\left(\partial_{x^{\prime}}\phi(x^{\prime})\right)\delta(x-x^{\prime})$
$\displaystyle=(-2J-\mu)\int dx\phi(x)^{*}\phi(x)+J\alpha^{2}\int
dx\left(\partial_{x}\phi^{*}(x)\right)\left(\partial_{x}\phi(x)\right).$ (18)

Upon derivation, we have used integration by parts and ignored the boundary
term (which is justified because we use the periodic boundary condition).
Similarly, we can derive the continuum limit of the interaction part of the
Hamiltonian as:

$\frac{1}{L}\sum_{k,k^{\prime},q}\bigg{[}2i\gamma_{R}\alpha(k-q)+i\gamma_{I}\alpha^{2}(k^{2}-q^{2})\bigg{]}\tilde{\Psi}^{*}_{k}\tilde{\Psi}^{*}_{k^{\prime}}\tilde{\Psi}_{k+k^{\prime}-q}\tilde{\Psi}_{q}.$
(19)

After some algebra, one can find that the term proportional to $\gamma_{R}$ is
a total derivative and thus drops off. The continuum limit of the term
proportional to $\gamma_{I}$ is

$\displaystyle-i\gamma_{I}\alpha^{3}\int
dx\left[\phi^{*}(x)^{2}\left(\partial_{x}\phi(x)\right)^{2}-\left(\partial_{x}\phi^{*}(x)\right)^{2}\phi(x)^{2}\right].$
(20)

Combining with the kinetic term, the total continuum Hamiltonian of the model
is

$\displaystyle H=\int
dx\left[(-2J-\mu)\phi(x)^{*}\phi(x)+J\alpha^{2}\left(\partial_{x}\phi^{*}(x)\right)\left(\partial_{x}\phi(x)\right)-i\gamma_{I}\alpha^{3}\left\\{\phi^{*}(x)^{2}\left(\partial_{x}\phi(x)\right)^{2}-\left(\partial_{x}\phi^{*}(x)\right)^{2}\phi(x)^{2}\right\\}\right].$
(21)

We can rewrite it in the following form to make it explicitly look as the
density-gradient-dependent gauge field:

$\displaystyle H=\int dx$
$\displaystyle\left[(-2J-\mu)\phi(x)^{*}\phi(x)+J\alpha^{2}\left\\{\left(i\partial_{x}+\frac{\gamma_{I}\alpha}{J}\partial_{x}|\phi(x)|^{2}\right)\phi^{*}(x)\right\\}\left\\{\left(-i\partial_{x}+\frac{\gamma_{I}\alpha}{J}\partial_{x}|\phi(x)|^{2}\right)\phi(x)\right\\}+\mathcal{O}(\gamma_{I}^{2})\right],$
(22)

where the density gradient term
$\frac{\gamma_{I}\alpha}{J}\partial_{x}|\phi(x)|^{2}$ couples to the momentum
$-i\partial_{x}$ analogous to the minimum coupling of the gauge field.

Let us now consider linearization around the zero momentum plane wave
condensate state. Since the chemical potential of the plane wave condensate
state is $\mu=-2J$, we use this value for the chemical potential, thereby
dropping the first term in the Hamiltonian Eq. (21) as: Firstly, we derive the
equation of motions for the conjugate momenta $\pi$ and $\pi^{*}$ from the
Hamilton equation $\dot{\pi}=\partial H/\partial(\phi)-\partial_{x}[\partial
H/\partial(\partial\phi)]$ as:

$\begin{split}\dot{\pi}=-J\alpha^{2}\partial_{x}^{2}\phi^{*}+2i\gamma_{I}\alpha^{3}\bigg{[}\phi(\partial_{x}\phi^{*})^{2}+2\phi^{*}\partial_{x}\phi^{*}\partial_{x}\phi+{\phi^{*}}^{2}\partial_{x}^{2}\phi\bigg{]},\\\
\dot{\pi}^{*}=-J\alpha^{2}\partial_{x}^{2}\phi-2i\gamma_{I}\alpha^{3}\bigg{[}\phi^{*}(\partial_{x}\phi)^{2}+2\phi\partial_{x}\phi^{*}\partial_{x}\phi+\phi^{2}\partial_{x}^{2}\phi^{*}\bigg{]}.\end{split}$
(23)

Then we assume that the field takes the form of
$\phi(x)=\frac{1}{\sqrt{L\alpha}}+\delta e^{ipx}$, which is a small plane-wave
perturbation with momentum $p$ on top of a zero momentum plane wave state
$1/\sqrt{L\alpha}$. Keeping up to the linear order in $\delta$, we obtain

$\begin{pmatrix}\dot{\pi}\\\
\dot{\pi}^{*}\end{pmatrix}=\begin{pmatrix}J\alpha^{2}p^{2}&-2i\gamma_{I}\alpha^{2}p^{2}/L\\\
2i\gamma_{I}\alpha^{2}p^{2}/L&J\alpha^{2}p^{2}\end{pmatrix}\begin{pmatrix}\delta^{*}\\\
\delta\end{pmatrix}.$ (24)

The dispersion relation of the elementary excitation of the system is obtained
by symplectic transformation of this equation, which is to diagonalize the
two-by-two matrix in the above equation after multiplying $\sigma_{z}$ from
the left. The dispersion relation thus found is

$\pm(\alpha p)^{2}\sqrt{J^{2}-\frac{4\gamma_{I}^{2}}{L^{2}}},$ (25)

which is exactly the same as the Bogoliubov excitation obtained in the main
text in the long-wavelength limit. (Note that $p$ in the main text corresponds
to $\alpha p$ here, writing out explicitly the lattice spacing $\alpha$.)

Finally, we show that the energy of this continuum model is not bounded from
below as may be expected for a BEC with an attractive interaction. This can be
seen assuming the following Gaussian ansatz wavefunction with variational
parameters $a$ and $b$

$\phi(x)=\bigg{(}\frac{2a}{\pi}\bigg{)}^{1/4}e^{-(a+ib)x^{2}}$ (26)

Inserting this ansatz into the continuum Hamiltonian $H_{\mathrm{cont}}$, we
find the energy of this state to be

$E(a,b)=J\alpha^{2}\frac{a^{2}+b^{2}}{a}+2\gamma_{I}\alpha^{3}\sqrt{\frac{a}{\pi}}b$
(27)

In minimizing the energy, we obtain the following relation between our two
parameters from setting the derivative with respect to $b$ to 0

$b=-\frac{\alpha\gamma_{I}}{J\pi^{1/2}}a^{3/2}.$ (28)

Plugging this back into Eq. 27, we find that the energy can be written as a
function of $a$ as

$E=J\alpha^{2}a-\frac{\alpha^{4}\gamma_{I}^{2}}{J\pi}a^{2}$ (29)

from which one can immediately see that the energy goes to $-\infty$ as
$a\rightarrow\infty$, namely by making the Gaussian narrower an narrower, and
thus the energy of the continuum model is not bounded from below. This result
implies that in the continuous case there is no analog of the soliton ground
state of the lattice model. The localized soliton ground state is, therefore,
a consequence of having a discrete lattice.

##  Appendix B Condensate Stability for Arbitrary $k$

In this appendix we calculate the linear stability condition about an
arbitrary momentum $k$ using the quantum Bogoliubov method. Starting from the
quantum analogue of Eq. 5, we perform the Bogoliubov analysis about an
arbitrary momentum $k$ to determine how the stability depends upon which
momentum condensation occurs. We denote the quantum annihilation and creation
operators corresponding to the classical operators $\tilde{\Psi}_{p}$ and
$\tilde{\Psi}_{p}^{*}$ by $\tilde{a}_{p}$ and $\tilde{a}_{p}^{\dagger}$,
respectively. The Hamiltonian is then

$\begin{split}H=\sum_{p}-2J(\cos{p}-\cos{k})\tilde{a}^{\dagger}_{p}\tilde{a}_{p}+\frac{1}{L}\sum_{p,p^{\prime},q}V(q,p)\tilde{a}^{\dagger}_{p}\tilde{a}^{\dagger}_{p^{\prime}}\tilde{a}_{p+p^{\prime}-q}\tilde{a}_{q}\end{split},$
(30)

where we have set the chemical potential to $\mu=-2J\cos{k}$ as is required
for condensation at this momentum. For a condensate, we assume that
$\tilde{a}^{\dagger}_{k}=\tilde{a}_{k}=\sqrt{N_{k}}\approx\sqrt{N}$ where
$N_{k}$ is the number of particles of the $k$ momentum state and $N$ is the
total number of particles. We linearize the quartic term by keeping only terms
proportional to $N_{k}^{2}$ or $N_{k}$, i.e. interactions where at least two
particles are in the momentum $k$ state. The quartic term thus contributes

$\displaystyle\frac{N_{k}}{L}$
$\displaystyle\sum_{p}V(k,p+k)\tilde{a}^{\dagger}_{k+p}\tilde{a}^{\dagger}_{k-p}+V(p+k,k)\tilde{a}_{k-q}\tilde{a}_{k+q}$
$\displaystyle=$
$\displaystyle\frac{2iN_{k}}{L}\sum_{p}\bigg{[}\gamma_{R}\bigg{(}\sin(p+k)-\sin(k)\bigg{)}+\gamma_{I}\bigg{(}\cos(k)-\cos(k+p)\bigg{)}\bigg{]}\tilde{a}^{\dagger}_{k+p}\tilde{a}^{\dagger}_{k-p}$
$\displaystyle-\bigg{[}\gamma_{R}\bigg{(}\sin(p+k)-\sin(k)\bigg{)}+\gamma_{I}\bigg{(}\cos(k)-\cos(k+p)\bigg{)}\bigg{]})\tilde{a}_{k-p}\tilde{a}_{k+p}$
$\displaystyle=$ $\displaystyle
4in_{k}\bigg{(}\gamma_{R}\sin(k)-\gamma_{I}\cos(k)\bigg{)}\sum_{p>0}\bigg{[}(\cos(p)-1)\bigg{]}\tilde{a}^{\dagger}_{k+p}\tilde{a}^{\dagger}_{k-p}-\bigg{[}(\cos(p)-1)\bigg{]}\tilde{a}_{k+p}\tilde{a}_{k-p}$
$\displaystyle=$ $\displaystyle
4in_{k}\gamma_{k}\sum_{p>0}\bigg{[}(\cos(p)-1)\bigg{]}\tilde{a}^{\dagger}_{k+p}\tilde{a}^{\dagger}_{k-p}-\bigg{[}(\cos(p)-1)\bigg{]}\tilde{a}_{k+p}\tilde{a}_{k-p},$
(31)

where we have used $V(p,k)=-V(k,p)$ to cancel the diagonal term and defined
$\gamma_{k}=\gamma_{R}\sin{k}-\gamma_{I}\cos{k}$. The resulting Bogoliubov
spectrum is

$\epsilon(p)^{2}=4J^{2}(\cos(p)-\cos(k))^{2}-16n_{k}^{2}\gamma_{k}^{2}(\cos(p)-1)^{2},$
(32)

which results in the instability condition

$\frac{(\cos(p)-\cos(k))^{2}}{(\cos(p)-1)^{2}}<\frac{4n_{k}^{2}\gamma_{k}^{2}}{J^{2}}.$
(33)

The right hand side of this inequality will be zero when the condensation
momentum $k$ satisfies the relation $\tan{k}=\gamma_{I}/\gamma_{R}$, and so
condensation at this momentum will be stable while all other condensates will
be unstable. Observe that the condition agrees with the calculation presented
in the text for the $k=0$ condensate.


# Separate Scene Text Detector for Unseen Scripts is Not All You Need

Prateek Keserwani1, Taveena Lotey1, Rohit Keshari2, and Partha Pratim Roy1
1 Indian Institute of Technology Roorkee, India
2 Indraprastha Institute of Technology Delhi, India
{pkeserwani<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Text detection in the wild is a well-known problem that becomes more
challenging while handling multiple scripts. In the last decade, some scripts
have gained the attention of the research community and achieved good
detection performance. However, many scripts are low-resourced for training
deep learning-based scene text detectors. It raises a critical question: _Is
there a need for separate training for new scripts?_ It is an unexplored query
in the field of scene text detection. This paper acknowledges this problem and
proposes a solution to detect scripts not present during training. In this
work, the analysis has been performed to understand cross-script text
detection, i.e., trained on one and tested on another. We found that the
identical nature of text annotation (word-level/line-level) is crucial for
better cross-script text detection. The different nature of text annotation
between scripts degrades cross-script text detection performance.
Additionally, for unseen script detection, the proposed solution utilizes
vector embedding to map the stroke information of text corresponding to the
script category. The proposed method is validated with a well-known multi-
lingual scene text dataset under a zero-shot setting. The results show the
potential of the proposed method for unseen script detection in natural
images.

###### Index Terms:

Cross-script Detection, Scene text detection, Unseen script detection

##  I Introduction

The writing system is one of the elemental inventions of human ancestors,
which is a necessary component of the current civilized world. The writing
system evolved in different human civilizations and arose as multiple scripts.
With trade and migration of humans from one culture to another, the scripts
got influenced by each other. It led to the environment of multiple scripts in
the same geographical location. A country such as India has eleven official
scripts [1]. These days, the multi-script text is not confined to the written
content of literature and science. It is also present on the signboards,
posters, t-shirts, shop front, packed products, etc. These written content is
known as scene text [2, 3, 4]. These scene texts may contain text instances of
multiple scripts. Localizing [5, 6] and recognizing multiple scripts [7, 8, 9,
10, 11] in natural images is a substantial problem.

In the last decade, many state-of-the-art methods have been proposed for
multi-script scene text detection [12, 13]. All the existing methods have
assumed that the targeted script has an abundance of well-labeled datasets for
the training purpose of deep learning algorithms. Many countries have several
scripts, and among them, some are low-resourced. Therefore, adding new scripts
to the existing scene text detector needs additional training with new
scripts. Also, for the case of low-resourced scripts, the annotated training
data is an additional challenge. Hence, in place of separate training for each
script, it is beneficial to devise an approach that detects the text for
unseen scripts with the help of a one-time trained model. The proposed work
considers the point mentioned above as a motivation. This approach helps to
eliminate the unnecessary burden of separate training for unseen scripts
(i.e., zero-shot setting).

In object detection, the above-mentioned scenario, i.e., detection of the
unseen object category while testing, has been analyzed, and various works are
available in the literature [14, 15, 16]. This setting of testing is known as
zero-shot setting. However, text detection under the zero-shot setting still
needs to be explored. All these existing object detection work has considered
some object categories as unseen during the training phase and while testing
predicts the bounding box for these objects. Like zero-shot object detection,
unseen script detection in natural images also needs two essential elements.
First is the bounding box regression, and second is identifying the script for
the bounding box. At first glance, unseen script detection looks similar to
zero-shot object detection. However, from the perspective of bounding box
regression, scene text has additional challenges. In scene text, a single
character is a valid instance for the bounding box, and a set of characters is
also a valid instance for the bounding box. Hence, the primary entity is the
character, but the detection requires word-level bounding boxes. It makes
bounding box regression for scene text different from the bounding box
regression for objects. In some scripts, such as English, the space is
considered a hint for word separation. Whereas some other scripts, such as
Chinese, is a non-alphabetic language that does not use spaces between words
[17]. Hence, different scripts have different assumptions for bounding box
annotation. Additionally, the similarity of multiple characters with the
background and the absence of such scripts during the training phase makes
this problem very challenging.

From a script identification perspective, this problem looks similar to an
object classification problem. Still, in many aspects, script identification
is different from object classification. First, text classification is
challenging due to the low-resolution text instances, complex background, and
uneven illumination. Second, many scripts share a common set of characters,
and only due to the difference of some characters, one script differs from
another, such as Greek and English. In contrast, some scripts are entirely
different from others based on a holistic view. For example, Chinese and
Bangla scripts are perceptually different. To encounter such variability, a
model that utilizes the global features and the local stroke information to
identify the script is needed. The sequential model has shown their potential
for script classification [18]. In our proposed model, sequential features are
used to obtain the sequential dependency of strokes for script identification.
At the same time, the global features are used in the proposed system for
script identification from a holistic view.

Among different approaches of zero-shot learning, semantic embedding has been
widely adopted [19]. We assume all classes (seen and unseen) must share a
common embedding space. The semantic embedding of classes is used as guided
information for classification at test time. For semantic embedding, most
approaches adopted the word2vec model [20]. The same approach has been
followed in our work, and word2vec embedding is utilized. The reason to adopt
this idea is that in the literature on the writing system, if two language
writing styles match, we notice that script names are present together in the
text. From this analysis, we have used the word2vec model trained on the
writing system corpus as a semantic embedding for our work.

A preliminary version of this work was earlier presented in [21]. This work
has added a significant number of findings and contributions. First, zero-shot
script identification is extended to text detection under the zero-shot
setting, and a method is presented that shows no need to re-train the model
for new scripts. Second, the in-depth analysis of the cross-script text
detection and understanding of the feasibility of text detection for unseen
scripts. The third is the experimental validation of unseen script detection
on a public dataset.

Figure 1: Script removal from the multi-script images using [22] for
generation of the mono-lingual text images. The upper row shows the images
with multi-lingual text, whereas the second row shows images with mono-lingual
text using the text removal technique. The text region, which is required to
be concealed for creating a single script dataset, is shown in yellow
rectangles.

##  II Related Work

Scene text detection is a well-known computer vision problem and has achieved
state-of-the-art performance for most public datasets. Still, for a new
script, there is always a need to retrain the model. For retraining,
annotation of training data for the new script, is required. In the case of
low-resource scripts, data annotation for training is an additional issue.
Hence, zero-shot scene text detection is a crucial problem that is still
unexplored. However, a few works are present in the literature for zero-shot
object detection [14].

In contrast to traditional object detection, where only the seen object
category is considered a target to be detected, the zero-shot approach detects
the unseen object category during testing. In [14], the authors used visual-
semantic embedding for zero-shot detection. They proposed two approaches based
on the bounding box generated by edge-boxes [23]. The first approach was to
consider a fixed background class, and the second was the background-aware
approach. In [15], authors proposed to jointly learn the recognition and
localization of the object by extending the Faster-RCNN [24] in a zero-shot
object detection setting. They introduced a loss function that combines max-
margin learning and semantic space clustering loss. In [25], authors proposed
a hybrid method by extending YOLOv2 [26], using a convex combination of
embedding and detection. In [27], along with joint learning using RetinaNet
[28], the authors proposed a polarity loss to align the visual and semantic
information through max-margin constraint and refined noisy semantic
representation. In [16], the authors used the textual description to obtain
the semantic embeddings of seen and unseen classes for zero-shot object
detection. The authors proposed joint learning of visual-unit and word-level
attention. In [29], authors used unlabelled unseen class data during training
and introduced pseudo-labeling for unlabeled samples. In [30], authors
synthesized the visual features based on conditional variational auto-encoder
to incorporate unseen object detection. In [31], authors proposed a generative
transfer network consisting of an object detector and knowledge transfer
module. The knowledge transfer module synthesized the features to generate
unseen object features. The IoU-Aware Generative Adversarial Network was used
as a feature synthesizer.

---
Arabic

---
Bangla

---
Chinese

---
Hindi

---
Japanese

---
Korean

---
Latin

Figure 2: Proximity graph for the closeness of one script with other scripts.
The circles with red and green colors represent the closeness of scripts
respective to a single script at the f-measure threshold of $0.6$ and $0.3$,
respectively. The scripts are aligned with six axes (shown by cyan lines). The
shorter lines represent closeness to the respective script. [Annotations: Ar:
Arabic, La: Latin, Hi: Hindi, Ba: Bangla, Ko: Korean, Ja: Japanese, Ch:
Chinese]

##  III Cross-Script Text Detection

Text detection methods trained for a single script or multi-scripts exist in
the literature. However, a few fundamental questions require consideration:
_Does a text detector achieve enough performance for the cross-scripts? Is it
mandatory to train a separate text detector for cross-scripts?_ This section
tries to find out the answers to the above-raised questions.

(a) One Train - Test Rest

(b) One Test - Train Rest

Figure 3: Box plot to compile the performance of cross-script text detection.
The red line marker shows the mean value. (a) Trained on one script (shown on
the vertical axis) and tested on the rest of the scripts, (b) Tested on one
script (shown on the vertical axis) and trained on the rest of the scripts.

In text detection, two scripts are considered similar in two ways. Firstly,
the visual similarity, in which the character set of one script looks similar
to another script. For example, Hindi and Bangla scripts are visually similar.
The second criterion for similarity is the nature of bounding boxes for text
annotation. Some scripts need word-level bounding boxes, whereas some need
line-level bounding boxes. For example, English, Hindi, and Korean scripts
need word-level bounding boxes for text annotation, whereas Chinese script
needs line-level bounding boxes. These similarities motivate us to investigate
the performance of cross-script text detection.

A text detector [6] is chosen for this analysis and is trained on one script
and tested on other scripts. A well-known multi-script text detection dataset
known as MLT2019 [32] is used to conduct experiments. Seven scripts of the
MLT2019 dataset, namely, Arabic, Bangla, Chinese, Hindi, Japanese, Korean, and
Latin, are considered. Since English is a script used internationally, it is
present with other scripts in many images. Hence, we have used the text
removal technique [22] to convert the multi-script text images to single-
script. Some sample images before the text removal and after text removal for
the cross-script analysis are shown in Fig. 1. For the analysis purpose, we
considered the f-measure [2] metric. The closeness of scripts is shown with
the proximity graphs (Fig. 2). The proximity graph presents a visual way to
show the closeness of a script with other scripts. The cross-scripts are
represented by axes (cyan-colored lines). The closest cross-script has the
smallest line.

Two f-measure thresholds are utilized for this analysis. The first threshold
is set to $0.6$, which shows that the scripts are similar, and cross-script
text detection can be recommended for such a set of scripts. The analysis
shows that the model trained on Latin script can be used directly for the
Bangla and Hindi scripts during inference. Likewise, the model for Hindi
script can be used for evaluating Bangla script and vice-versa. The second
f-measure threshold is set to $0.3$. It helps to find scripts that are similar
but loosely coupled. Chinese and Japanese scripts fall into this category. The
behavior of Korean script demonstrates its closeness with Hindi, Bangla, and
Latin scripts more than visually similar Japanese and Chinese scripts. The
reason is a difference in the nature of the bounding box annotation of Korean
script from Chinese and Japanese scripts. The Chinese and Japanese scripts use
line-level bounding box annotation, and the Korean script uses word-level
bounding box annotation in the ground truth. Hence, the model trained on
Korean script favors the word-level bounding boxes while evaluating Hindi,
Bangla, and Latin scripts. Therefore, despite the visual similarity of Korean
with Japanese and Chinese scripts, the model trained in Korean script shows
better text detection for Bangla, Hindi, and Latin scripts. The proximity
graph also reveals that the Arabic script is close to Hindi and Latin scripts.

To further analyze the distribution of cross-script text detection
performance, the box plots of f-measures of the text detectors are shown in
Fig. 3. In Fig. 3 (a), the box plot shows the distribution of the performance
of a model trained on a single script and tested on the rest of the scripts.
From Fig. 3(a), it is observed that the model trained in Korean script has the
best mean f-measure among others. In contrast, the Chinese script has the
worst mean f-measure. The Hindi script has Bangla as an outlier and vice-
versa, as shown by the dotted points. In the case of Bangla script, other than
Hindi, the performance of the rest of the scripts is far inferior. The same
trend is observed in Hindi. Chinese and Japanese scripts have shown similar
behavior to Hindi and Bangla scripts. In Fig 3 (b), the box plot shows the
distribution of performance on a single script by models trained on the rest
of the scripts. Among them, the mean f-measure of Hindi is the best, and
Arabic is the worst.

This cross-script text detection analysis helps to lay the foundation for
unseen script detection. The cross-script text detection demonstrates the
capability of a text detector without getting trained on or fine-tuned to a
target script. From this analysis, it has been concluded that cross-script
text detection is feasible for similar scripts. Based on the above analysis,
the following section proposes a baseline method for unseen script detection
in scene images.

##  IV Unseen Script Detection

Suppose the text detection for multi-scripts is composed of $n$ scripts and
$n=n_{0}+n_{1}$, where $n_{0}$ is the number of classes for seen scripts
during training, and $n_{1}$ is the number of the classes for the unseen
scripts during training. Under the hypothesis of the paper, only the bounding
box of the seen scripts is provided during the training phase, whereas testing
is performed on the unseen scripts. For an image $I\in\mathbb{R}^{P\times
Q\times 3}$, there are $b$ number of bounding boxes
$B=\\{b_{i}|i\in\\{1,...|b|\\}\\}$ with scripts
$C=\\{y_{i}|i\in\\{1,...|b|\\}\\}$.

The fundamental approach adopted for unseen script detection in natural images
consists of two parts, namely, (a) cross-script bounding box prediction and
(b) unseen script identification. The block diagram for the proposed baseline
approach for unseen script detection is shown in Fig. 4. The proposed approach
has three basic building blocks. These are:

  1. 1.

_Bounding Box Prediction :_ Based on the cross-script analysis, the prediction
of the bounding box for unseen categories can be made using an on-the-shelf
text detector. In this work, Quadbox text detector [6] has been used. The
Quadbox text detector has been trained on the seen category of scripts. The
Quadbox returns the set of bounding boxes
$\hat{B}=\\{b_{i}|i\in(1,\hat{n})\\}$. The text detector generates the
bounding box irrespective of the script categories.

Figure 4: Fundamental pipeline for the unseen script detection.

  2. 2.

_Text Region Extraction :_ The bounding boxes generated from the bounding box
prediction phase are cropped out from the image in the text region extraction
phase. The bounding box predicted by the bounding box prediction phase
consists of four coordinates $b_{t}=\\{(x_{i},y_{i})|i\in(1,4)\\}$. A minimum
area rectangle has been computed for the bounding box represented by the four
coordinates. Then the image is rotated with the angle of the minimum area
rectangle. The rotated image rectangle is cropped for the text region
extraction.

  3. 3.

_Unseen Script Recognition :_ The extracted text regions $R$ are considered as
the candidate regions for script recognition. The unseen script identification
uses the model trained on seen script categories and identifies the category
of text region $r_{t}$. The unseen script identification part is taken from
[21].

##  V Experiment

###  V-A Dataset Description

In scene text detection, a well-known multi-script dataset known as the
MLT2019 dataset [32] consists of seven scripts, namely Chinese, Japanese,
Korean, Latin, Arabic, Bangla, and Hindi. In addition, this dataset has some
’symbols’ and ’mixed’ categories. Seven scripts mentioned above are considered
for this work. For unseen script detection, a subset of the seen and unseen
script categories from the zero-shot script identification has been taken as
the seen and unseen script categories from the MLT2019 dataset. Hence, from
the MLT2019 dataset, the seen and unseen categories for unseen script
detection are {Arabic, Latin, Bangla, Japanese} and {Chinese, Korean, Hindi},
respectively. However, some images have scripts from both seen and unseen
categories. Hence, we consider only those images that contain scripts from
seen or unseen categories. The statistics of the dataset used for this work
are given in Table I.

TABLE I: Statistics of the used dataset for the unseen script detection.  Category | Scripts | #Images
---|---|---
Seen | Latin | 3996
Bangla | 1000
Arabic | 1000
Japanese | 940
Unseen | Chinese | 454
Korean | 241
Hindi | 816

###  V-B Implementation Details

The proposed model for unseen script detection requires two separately trained
models. It requires one model for script identification and one for cross-
script text bounding box regression. Both models are implemented in
Pytorch111https://pytorch.org/ and trained on GTX 1070 Ti GPU. For the unseen
script detection in natural images, the text detector [6] is trained on the
unseen category for $400$ epochs with a batch size of $4$. The model was
trained with an initial learning rate of $10^{-4}$, and the learning rate
decays by $0.1$ after $200$ epochs. For training, the data augmentations used
are image rotation, image scaling, and random cropping. During inference,
vanilla non-maximal suppression post-processing has been employed to filter
out the bounding boxes. For unseen script identification, we have taken the
trained model of the paper [21] on which this paper is extended.

###  V-C Evaluation Protocol

Figure 5: Quantitative performance of the unseen script detection. Figure 6:
Performance of bounding box regression by unseen script detection. The model
performance has been computed over a combined image set of Chinese, Korean,
and Hindi scripts (Combined) and individual scripts (Chinese, Korean, and
Hindi).

The same protocol as in zero-shot object detection [33] has been followed for
the unseen script detection to evaluate natural images. The recall is defined
as the ratio of all positive examples rated above a given threshold. Precision
is the ratio of all examples above that rank from the positive class. The
precision/recall curve is computed from a method’s ranked output for a given
task and class. The average precision (AP) summarises the shape of the
precision/recall curve and is defined as the mean precision at eleven equally
spaced recall levels [0, 0.1,…, 1]. The average precision (AP) has been
computed for each unseen category, and subsequently, the overall performance
has been computed using the mean average precision (mAP) metric.

Figure 7: Unseen Chinese script detection in natural images. The yellow color
bounding box shows the Chinese script. The smaller text regions are cropped
out and shown at the corner of each image at a higher resolution for better
visualization.  Figure 8: Unseen Korean script detection in natural images.
The cyan color bounding box shows the Korean script. The yellow color bounding
box shows the Chinese script, which is miss-classified.  Figure 9: Unseen
Hindi script detection in natural images. The pink color bounding box shows
the Hindi script.

###  V-D Quantitative Results

Two factors decide the performance of unseen script detection: (a) the
classification performance to identify the cross-scripts, (b) predicting
bounding box which possesses sufficient overlap with the ground truth (in this
experiment, the intersection over the union of $0.5$ has been taken into
consideration). The quantitative performance of the unseen script detection in
natural images is shown in Fig. 5. These results show the combined impact of
the classification and box regression. The achieved performance for the
proposed pipeline is $22.97\%$ mAP. Hindi script is the best-performing among
Chinese, Korean, and Hindi scripts. From the perspective of unseen script
(zero-shot) recognition, the visual similarity among the scripts plays an
important role. Bangla and Hindi scripts have a close visual appearance.
Bangla script has been used in the training phase, and it helps to achieve
good performance for Hindi in the testing phase. Similarly, the visual
similarity of Japanese with Chinese and Korean scripts should have helped
their recognition. However, the visual closeness of Chinese and Korean scripts
hinders the classification performance.

Another perspective is the proper bounding box regression. Bounding box
regression performance has been computed using well-known precision, recall,
and f-measure text detection metrics to understand this perspective. This
bounding box performance evaluation has been conducted in two different ways.
Firstly, the bounding box performance evaluation has been done for combined
scripts (i.e., Chinese, Hindi, and Korean). Secondly, it has been done for
individual scripts. The precision, recall, and f-measure for both ways are
summarized in Fig. 6. From the Fig. 6, it has been observed that combined
scripts testing has achieved the precision, recall, and f-measure of $0.7701$,
$0.7709$, and $0.7705$, respectively. These results show that the cross-script
bounding box prediction is plausible. Bounding box performance evaluation for
individual scripts shows that the model has performed best for Hindi. The
performance of Hindi script is better due to the high visual similarity
between Hindi and Bangla scripts. Since Bangla is used in training, it helps
the model predict good bounding boxes for Hindi during testing. However, for
the Chinese, the model performance is comparatively inferior. The need for
line-level bounding box prediction performance is the reason behind the low
performance of the model trained on the Chinese script. Since most training
scripts have word-level bounding box ground truth, the model is more inclined
toward predicting word-level bounding box regression.

###  V-E Qualitative Results

The bounding box predicted along with the script categories are visually shown
in Fig. 7, Fig. 8, and Fig. 9. The Chinese, Korean, and Hindi bounding boxes
are shown in qualitative visualization with yellow, cyan, and pink colors.
From Fig. 9 and Fig. 8, it has been observed that the bounding box appears
much tighter and more appropriate. It is due to the word-level bounding box
requirement of Korean and Hindi. In Fig. 7, it has been observed that the
predicted bounding box appears shorter in length due to the requirement of the
line-level bounding box for the Chinese script. It results in a lower value of
the f-measure (shown in Fig. 6) than the scripts that need word-level bounding
boxes, such as Hindi and Korean. This observation also correlates with the
f-measure values for Korean and Hindi (shown in Fig. 6).

###  V-F Comparative Analysis

A comparision has been performed by combining the bounding box prediction
model with various zero-shot script identification models. The proposed
pipeline has been computed by combining the quadbox (QB) with four variants of
zero-shot script identification models (mentioned in [21]). These zero-shot
script identification models are (a) VGG-16 + Global Average Pooling (GAP)
[BL1], (b) VGG-16 + LSTM [BL2], and (c) VGG-16 + SEB + GAP [V]. Among them,
the proposed pipeline with SEB and GAP is the best performing. The ablation is
summarized in Table II.

TABLE II: Comparative analysis of the proposed pipeline with various zero-shot script identification methods. Methods | Average Precision | mAP
---|---|---
Chinese | Hindi | Korean
QB + BL1 | 19.59 | 28.43 | 6.03 | 18.02
QB + BL2 | 23.85 | 28.93 | 4.95 | 19.24
QB + V | 23.03 | 27.01 | 4.56 | 18.20
QB + SEB + GAP | 28.53 | 35.16 | 5.21 | 22.97

##  VI Conclusion

In this work, the analysis to understand the impact of text detectors on
cross-script is performed. The analysis concludes that an on-the-shelf text
detector can be used for unseen scripts that are either visually similar or
possess similar annotation level requirements (i.e., word-level or line-
level). This analysis helps to develop the fundamental pipeline for the zero-
shot learning approach for unseen script detection. It questioned the need for
separate training for unseen scripts. With the combination of the cross-script
text bounding box prediction and unseen (zero-shot) script identification,
unseen script detection can be accomplished. The proposed method is highly
dependent on the effectiveness of the word2vec model, which can be improved in
the future by using a better and more extensive corpus.

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