# Route to chaos in two-dimensional discrete parametric maps with bistable
potentials

Alain M. Dikandé  Laboratory of Research on Advanced Materials and Nonlinear
Science (LaRAMaNS), Department of Physics, Faculty of Science, University of
Buea P.O. Box 63 Buea, Cameroon<EMAIL_ADDRESS>
###### Abstract

The texture of phase space and bifurcation diagrams of two-dimensional
discrete maps describing a lattice of interacting oscillators, confined in on-
site potentials with deformable double-well shapes, are examined. The two
double-well potentials considered belong to a family proposed by Dikandé and
Kofané (A. M. Dikandé and T. C. Kofané, Solid State Commun. vol. 89, p. 559,
1994), whose shapes can be tuned distinctively: one has a variable barrier
height and the other has variable minima positions. However the two
parametrized double-well potentials reduce to the $\phi^{4}$ substrate,
familiar in the studies of structural phase transitions in centro-symmetric
crystals or bistable processes in biophysics. It is shown that although the
parametric maps are area preserving their routes to chaos display different
characteristic features: the first map exhibits a cascade of period-doubling
bifurcations with respect to the potential amplitude, but period-halving
bifurcations with respect to the shape deformability parameter. On the other
hand the first bifurcation of the second map always coincides with the first
pitchfork bifurcation of the $\phi^{4}$ map. However, an increase of the
deformability parameter shrinks the region between successive period-doubling
bifurcations. The two opposite bifurcation cascades characterizing the first
map, and the shrinkage of regions between successive bifurcation cascades
which is characteristic of the second map, suggest a non-universal character
of the Feigenbaum-number sequences associate with the two discrete parametric
double-well maps.

  * March 2021

##  1 Introduction

Real physical systems exhibit extremely complex structures, and consequently a
broad range of unexpected behaviors due to the collective motion of their
constituants mediated by competing effects of many-body and one-body
interactions. Fig. 1 represents a discrete chain of identical atoms of mass
$m$, interacting via springs of a spring constant $K$.

Figure 1: (Color online). Sketch of a $1D$ chain of oscillators interacting
via identical springs. $X_{n}$ denotes the displacement of the $n^{th}$
oscillator with respect to the equilibrium.

The figure is representative of a large variety of physical systems observed
in nature, ranging from solid-state physics to chemistry, biophysics and
chemistry [1, 2, 3, 4]. In these physical systems fig. 1 provides simple
representation of the lattice structures of atomic or molecular crystals, the
structures of organized cellular populations in tissues and embryonic cells,
the organization of glycolitic oscillations in suspensions of yeast cells in
unison, the structure of an assembly of pacemaker cells in the sino-atrial
node, to name these few physical contexts [3]. Although they share in common
the mutual interactions of their elementary constituants (e.g. atoms,
molecules, cells in tissues, chemical species, etc.), every physical system
possesses a unique feature that stems from feedback functions characterizing
the system [3]. These feedback functions are governed by one-body forces which
are typically nonlinear, in the specific context of Hamiltonian systems they
are conservative forces that derive from one-body potentials.

Several potentials are found in the literature, each of them resulting from
efforts to provide a realistic description of a specific physical process. But
the two most common types of potential used in nonlinear Hamiltonian dynamics
are the periodic potentials with infinitely degenerate extrema, and piecewise
potentials with finite numbers of extrema. Of the first type the sine-Gordon
potential is best known and probably the most studied [5, 6], it was proposed
to model systems in which ordered states are governed by periodic forces. This
is for instance the case of the emergence of incommensurate orders due to
dislocations in Frenkel-Kontorova crystals [7, 8], commensurate and
incommensurate orders in chiral liquid crystals [9], the incommensurate
charge-density-wave order mediated by phasons in low-dimensional conductors
[10], the formation and propagation of fluxons in Josephson-Junction
superconductors [11], the equilibrium configuration of finite masses suspended
on periodically aligned coupled pendula, etc. As for potentials with finite
extrema, polynomial-type potentials are most popular among which the canonical
$\phi^{4}$ potential [12, 13, 14]. It was introduced initially in the study of
ferroelectric transitions in one-dimensional ($1D$) centro-symmetric crystals
such as perovskites, and later on generalized to second-order phase
transitions [12, 15, 16]. In this last context the $\phi^{4}$ potential stands
for the Ginzburg-Landau energy functional, whose two degenerate minima
separated by a potential barrier is assumed to mimic orderings of
atoms/molecules in crystals with an inversion (or point) symmetry [15, 16].
But the applicability of the $\phi^{4}$ potential goes beyond second-order
phase transitions in $1D$ crystals, in fact the $\phi^{4}$ model is a paradigm
for bistable systems which can be found in biophysics [17, 18], chemistry
[18], economy [18, 19], population dynamics [18] and so on.

Despite their popularity, the ability of the sine-Gordon and $\phi^{4}$
potentials to provide a fair description of physical processes for which they
were proposed has been questioned in a great number of physical contexts. In
crystal lattices for instance, structural transitions occur in an environment
where characteristic features of the lattices are subjected to frequent
changes. These changes are caused by constraints that are manifest in
variations of the equilibrium configurations of atoms/molecules along the
crystal lattice, or of their characteristic energies at equilibrium (i.e. the
energy barrier) and so on. Some concrete physical situations are the
variations in hydrostatic pressures (due for instance to isotopic
substitutions or chemical reactions), change in temperature of the system,
etc. that induce changes in bond lengths (bond stretching or compressions), in
the activation energy in barrier-crossing processes (lowering or increase of
the potential barrier), and so on [20, 21, 22, 23, 24, 25]. To account for the
later processes, improved variants of the sine-Gordon and $\phi^{4}$
potentials with tunable shape profiles have been proposed. Thus, Remoissenet
and Peyrard introduced a familty of periodic potentials (the well-known
Remoissenet-Peyrard potentials) [2], whose characteristic features such as the
period, the potential amplitude and the steepness of the potential walls, can
be controlled by a deformability parameter. Concerning the $\phi^{4}$
potential, parametrized double-well (DW) potentials have been proposed
including polynomial potentials [26, 27] and hyperbolic DW potentials [28, 29,
30]. While these parametrized DW potentials have a deformable shape profiles,
their link with the standard $\phi^{4}$ model is not always easy to establish.
To effectively deal with a parametrized DW potential that can be reduced
smoothly to the $\phi^{4}$ model, in some previous works we proposed a family
of hyperbolic potentials whose DW shapes could be tuned distinctively,
admitting the canonical $\phi^{4}$ potential as a specific limit [31, 32, 33,
34]. Peculiar features of some of these parametrized DW potentials (hereafter
referred to as Dikandé-Kofané double-well (DKDW) potentials) have been pointed
out in recent studies [35, 36, 37, 38].

In ref. [39], Jensen et al. established that although the $\phi^{4}$ model on
a lattice is not integrable, the associate continuum periodic-kink solution
could be generic of a chaos-free $2D$ discrete map with the Schmidt substrate
potential [27]. Later on this result was extended [40, 41] to two members of
the family of DKDW potentials, leading to two chaos-free maps [40, 41] that
turned out to be two parametrized forms of the Schmidt map [27] and
remarkably, belong to the well-known family of Quispel-Roberts-Thompson maps
[42] characterized by discreteness, low dimensionality and complete
integrability [43]. Yet, even if members of the family of DKDW models on a
lattice are non intgrable in the sense of Quispel–Roberts–Thompson
integrability criteria [42, 43, 44], the fact that the models are parametrized
introduces the perspective of a possible control of the occurance of unstable
and chaotic orbits in the associate phase space. In particular the
deformability parameter could be chosen in such a way that the map is always
chaos free, similar with the recent study [38] in which we established that
exact eigenstates (eigenfunctions and corresponding energy eigenvalues) of the
transfer operator associated with the statistical mechanics of one of the DKDW
models, could be found for specific values of the deformability parameter. In
the present context, the possibility to control the occurance of unstable
trajectories and chaotic orbits would imply a non-universal character of route
to chaos in the discrete DKDW maps, non universal as opposed to the universal
picture related to the discrete $\phi^{4}$ map [13]. The non-universality of
the route to chaos will mean a Feigenbaum sequence [14] and the structure of
the bifurcation diagram dependent on the deformability parameter.

Our objective in the present work is to explore the phase-space texture and
the structure of bifurcation diagrams for $2D$ discrete bistable maps,
considering two different parametrized DKDW potentials. One has fixed minima
but a variable barrier height, and the other has fixed barrier height but
variable minima. We examine analytically the first bifurcations around the
trivial fixed point, and establish conditions (where applicable) for a
controllable route to chaos. While the first bifurcation point is continuously
shifted to larger values of the ratio of the $\phi^{4}$-potential amplitude to
the spring constant $K$ in the case of the discrete map with variable barrier
and fixed minima, we find that for the second map the first pitchfork
instability occurs at the same value of the potential amplitude obtained [14]
in the $\phi^{4}$ context. However, the area separating the first pitchfork
bifurcation and the cascade of bifurcations marking the transition from
period-two to period-four orbits, is gradually shrinked with increasing values
of the deformability parameter. A global analysis of the two maps in phase
space and with the help of bifurcation diagrams, will be carried out
numerically for selected values of the deformability parameter.

##  2 Discrete Hamiltonian and the family of $2D$ parametric maps

Consider a $1D$ periodic lattice of $N$ identical oscillators (which can be
atoms, molecules, macromolecules, cells, etc.) of mass $m$, interacting them
and others via identical linear springs of a spring constant $K$. The
equilibrium configuration of the system under the linear two-body
interactions, sketched in fig. 1, is one in which oscillators are equally
separated (hereafter the lattice spacing will be set to unity). In addition to
the two-body interactions each oscillator is assumed to experience a one-body
force, generated by an on-site potential created by the specific background
substrate the physical system. The substrate potential will tend to keep
individual oscillators at their equilibrium positions on the lattice sites
$n$, thus opposing to their displacements due to two-body forces (springs here
can be interatomic or intermolecular bonds, etc.).

Let $u_{n}$ be the displacement of the $n^{th}$ oscillator relative to its
equilibrium position at the bottom of one of the two energetically degenerate
potential wells. For such discrete systems, the total Hamiltonian can be
expressed:

$H=\sum_{n=1}^{N}{\left[\frac{m}{2}\,\left(\frac{du_{n}}{dt}\right)^{2}+\frac{K}{2}\left(u_{n+1}-u_{n}\right)^{2}+V_{\mu}(u_{n})\right]},$
(1)

where $t$ is time variable and $V_{\mu}(u_{n})$ is the substrate potential.
Here we are interested in a substrate potential with a double-well shape,
corresponding to physical systems with bistable energy landscapes. The most
familiar of such potential energy is the $\phi^{4}$ potential, its canonical
form is (see e.g. [2, 5, 12]):

$V(u)=\frac{a}{4}\left(u^{2}-1\right)^{2},$ (2)

where the potential amplitude $a$ is real and constant. The $\phi^{4}$ field
in eq. (2) possesses two degenerate minima $u_{1,2}=\pm 1$, and a maximum
$u_{0}=0$ where the potential $V(0)=a/4$. As its characteristic parameters
(i.e. its extrema and its maximum value) indicate, the $\phi^{4}$ field has
two fixed minima and an extremum for which the value of the potential depends
only on the real parameter $a$. In the context of second-order phase
transition $a$ depends on temperature and is expected to vanish at the
critical point. However, in real physical contexts the parameter $a$ will not
always depend only on temperature. Indeed in most real contexts of atomic or
molecular crystals, the equilibrium positions of atoms or molecules will
constantly change as a result of variations of bond lengths and strengths, and
hence of characteristic bond energies. Such changes are induced by a variation
of hydrostatic pressures in the crystals, or interactomic/intermolecular
pressures caused by isotopic substitutions, the presence of chemical
catalyzers or chemical reactions occuring concomitantly with the structural
transition [45]. Ideal systems in which such processes are expected include
hydrogen-bonded atomic and molecular crystals prone to ferroelectric and
antiferroelectric phase transitions [46, 47, 48, 49]. To take into
consideration the possible variations of the potential amplitude and positions
of the two potential minima, a family of parametrized double-well potentials
was proposed in refs. [31, 32, 33, 34]. In one version the two degenerate
minima can be tuned by varying a deformability parameter without changing the
height of the potential barrier [31], in the other version [32] the barrier
height can be changed continuously without changing positions of the two
degenerate minima, and in the third version both the dgenerate minima and the
barrier height can change simultaneously [34]. This family of parametrized
double-well potential will be referred to as Dikandé-Kofané potentials. The
DKDW potential with a variable barrier height but fixed minima, was considered
recently by Bazeia et al [36] in their study of oscillons induced by kink-
antikink scatterings in double-well system with complex double-well shape
profiles. In ref. [35], Zhou et al. established that quantum Hamiltonians with
the DKDW potential with fixed minima and variable potential barrier, provide
an ideal model for bistable systems that can display simultaneously first-
order and second-order transitions in quantum tunnellings.

In this work we shall focus on two particular members of the family of DKDW
potentials, i.e. the one with variable barrier height but fixed minima and the
one with variable minima but fixed barrier height. Starting, let us write down
the general expression of the family of DKDW potentials [34]:

$V_{\mu}(u)=\frac{a_{\mu}}{4}\left[\frac{1}{\mu^{2}}\sinh^{2}\left(\alpha_{\mu}u\right)-1\right]^{2},$
(3)

where $a_{\mu}$ and $\alpha_{\mu}$ are two real functions. For the two members
we have chosen to study, these functions are defined as:

  * •

DKDW potential with variable potential barrier but fixed positions of the two
degenerate minima [31]:

$a_{\mu}=a\,\frac{\mu^{2}}{(1+\mu^{2})\,arsinh^{2}(\mu)},\hskip
8.5359pt\alpha_{\mu}=arsinh(\mu).$ (4)

  * •

DKDW potential with fixed barrier but variable positions of the two degenerate
minima [32]:

$a_{\mu}=a,\hskip 8.5359pt\alpha_{\mu}=\mu.$ (5)

In these functions the quantity $\mu$ is the deformability paremeter, it is
real and defined such that eq. (3) reduces to the $\phi^{4}$ potential (2)
when $\mu\rightarrow 0$. The two different DKDW potentials $V_{\mu}(u)$ given
in eqs. (4) and (5), are plotted in fig. 2, for some arbitrary values of the
deformability parameter $\mu$ listed in the caption. The left graph shows the
DKDW potential with variable barrier height but fixed positions of the
potential minima i.e. $u_{1,2}=\pm 1$, while the right graph represents the
DKDW potential with fixed barrier height but variable positions of the two
degenerate minima given by:

$u_{1,2}=\pm\frac{arsinh(\mu)}{\mu}.$ (6)

Figure 2: (Color online) Parametrized double-well potential $V_{\mu}(u)$
versus $u$, for different values of $\mu$. Left panel: DKDW potential with
variable barrier and fixed minima (from highest to lowest potential barriers,
$\mu=0$, 0.5, 2, 4). Right panel: DKDW potential with variable minima and
fixed barrier height (from the minima nearest to $u=0$ to the farthest,
$\mu=4$, 2, 0.5, 0).

As our main interest is the spatial configurations of the system of N
oscillators favored by the competition between the two-body and one-body
forces, we shall ignore the kinetic energy term in total Hamiltonian. With
this consideration the equilibrium configurations of the system will be
determined by minimizing the discrete Hamiltonian eq. (1) with respect to
$u_{n}$, which yields:

$u_{n+1}-2u_{n}+u_{n-1}=\frac{1}{K}\frac{\partial{V_{\mu}(u_{n})}}{\partial{u_{n}}},$
(7)

which is a nonlinear difference equation. Instructively, this nonlinear
difference equation stands in principle as a generalization of the $\phi^{4}$
model and hence is not expected to be integrable, even in the continuum limit.
Note that in this last limit the nonlinear difference equation (7) possesses
[35, 41] a periodic-kink solitary-wave solution. However, to efficiently
address the issue of integrability of the current problem in the presence of
deformability, it is more enriching to resort to an analysis of the texture of
trajectories in phase space as well as of the structure of bifurcation
diagrams characterizing routes to chaos in the system. To this end we
transform the nonlinear difference equation (7) into a $2D$ discrete map of
the form:

$\displaystyle{\it\mathcal{T}_{\mu}}:\hskip 5.69046ptu_{n+1}$ $\displaystyle=$
$\displaystyle
2u_{n}-v_{n}+\frac{1}{K}\frac{\partial{V_{\mu}(u_{n})}}{\partial{u_{n}}},$
$\displaystyle v_{n+1}$ $\displaystyle=$ $\displaystyle u_{n}.$ (8)

In the language of dynamical system theory, the last system represents a $2D$
discrete map with the DKDW potential. Since the underlying nonlinear function
is parametrized, we shall refer to the $2D$ system as a parametric map. Note
that when $\mu\rightarrow 0$, the discrete parametric map reduces to the map
studied by Bak and Jensen in ref. [14], and by Bak and Prokovsky in ref. [13]
in the context of metal-insulator transition in half-filled Peierls systems.
In the next section we examine fixed points of the map including their
stability, and the influence of shape deformability of the potentials on the
two first bifurcations for the two maps. Although the parametrization
complexifies the problem and hence restricts the ability for analytical
formulation of aspects such as the generation of Feigenbaum number sequences
associated with successive period-doubling bifurcations [14], we will be
satisfied with the analytical formulation of the parametric dependence of
characteristic values of the ratio $a/K$ forming the Feigenbaum sequences.
This parametric dependence clearly indicates a non-universal character for
these sequences in the particular case of the two parametric maps studied in
this work.

##  3 Stability of fixed points and the two first sequences of bifurcations

The transformation $\mathcal{T}_{\mu}$ maps a point in the phase space
$(u,v)$, onto another point in the same phase space. The map
$\mathcal{T}_{\mu}$ possesses a fixed point at $(u,v)=(0,0)$ where the
substrate potential $V_{\mu}(u_{n})$ is maximum whatever the value of the
deformability parameter $\mu$, we call it a trivial fixed point. In addition
there are two fixed points at $(u,v)=\pm(1,1)$, corresponding to minima of the
parametrized substrate potential. In the previous study dealing with the
$\phi^{4}$ map [14], it was pointed out that the most interesting and richest
dynamical processses associated with bistable systems were those associated
with the fixed point $(0,0)$. In this previous study it was shown that the
$\phi^{4}$ map was area preserving, and that for $a=4K$ the fixed point
$(0,0)$ undergoes a transition from an elliptical fixed point to an hyperbolic
fixed point, via a pitchfork bifurcation. At this pitchfork instability,
simple-periodic trajectories in phase space become unstable given birth to
period-two orbits involving two new fixed points. These new fixed points are
determined by two transformations generating the following equation:

$\displaystyle(u,v)$ $\displaystyle=$
$\displaystyle\mathcal{T}_{0}\left(\mathcal{M}_{0}(u,v)\right)$ (9)
$\displaystyle=$ $\displaystyle\mathcal{T}^{2}_{0}(u,v),$

which for the $\phi^{4}$ map admits two pairs of solutions namely [14]:

$(u,v)=\pm\left(\sqrt{1-\frac{4}{\tilde{a}}},-\sqrt{1-\frac{4}{\tilde{a}}}\right),\hskip
7.11317pt\tilde{a}=a/K.$ (10)

Consider the same problem for the $2D$ parametric map given by eq. (8),
assuming the fixed point $(0,0)$. In this goal we start by observing that the
first bifurcation associated with this fixed point is best formulated by
linearizing eq. (8) around $(0,0)$, and analyzing the set of tangent space
orbits [50] $(\delta u,\delta v)$ around the fixed point generated by the
$2\times 2$ linear matrix equation:

$\left(\delta u_{n+1},\delta v_{n+1}\right)=\mathcal{M}_{\mu}\left(\delta
u_{n},\delta v_{n}\right),$ (11)

where:

$\mathcal{M}_{\mu}=\left(\begin{array}[]{cc}2+\frac{1}{K}\frac{\partial^{2}{V_{\mu}(u_{n})}}{\partial{u^{2}_{n}}}&-1\\\
1&0\par\end{array}\right)$ (12)

is the $2\times 2$ transformation matrix. This matrix possesses two
eigenvalues which can be expressed most generally, in terms its trace
$\mathrm{Tr}\left[\mathcal{M}_{\mu}\right]$ and its Jacobian
$\det\left[\mathcal{M}_{\mu}\right]$ as follows:

$\lambda_{1,2}=\frac{\mathrm{Tr}\left[\mathcal{M}_{\mu}\right]\pm\sqrt{\left(\mathrm{Tr}\left[\mathcal{M}_{\mu}\right]\right)^{2}-4\det\left[\mathcal{M}_{\mu}\right]}}{2},$
(13)

According to Greene [50] any $2D$ discrete mapping for which
$\det(\mathcal{M}_{\mu})\leq 1$, is area preserving. This is the case for the
$2\times 2$ matrix eq. (12), for which $\det(\mathcal{M}_{\mu})=1$. It follows
that eigenvalues of this matrix will depend only on its trace, i.e. the pair
of eigenvalues given by eq. (13) simplifies to:

$\lambda_{1,2}=\frac{\mathrm{Tr}\left[\mathcal{M}_{\mu}\right]\pm\sqrt{\left(\mathrm{Tr}\left[\mathcal{M}_{\mu}\right]\right)^{2}-4}}{2},$
(14)

where the trace:

$\mathrm{Tr}\left[\mathcal{M}_{\mu}\right]=2+\frac{1}{K}\frac{\partial^{2}{V_{\mu}(u)}}{\partial{u^{2}}}|_{u=0}.$
(15)

Learning from formula (14), it follows that period-one orbits around the fixed
point $(0,0)$ will be stable provided the condition:

$\left(\mathrm{Tr}\left[\mathcal{M}_{\mu}\right]\right)^{2}-4\leq 0.$ (16)

For the discrete DKDW map with variable barrier height and fixed minima, the
stability condition (16) implies that the first bifurcation will occur for
values of $\tilde{a}$ determined by the relation:

$\mu^{2}=\frac{\tilde{a}}{4}-1,$ (17)

with $a$ and $\mu$ positives. For the DKDW map with fixed barrier height and
variable minima, the stability condition (16) suggests that the first
bifucation will occur at $\tilde{a}=4$ as with the $\phi^{4}$ map [14].

When starting this section we recalled that for the $\phi^{4}$ map, the two
fixed points of period-two orbits generated from instability of period-one
orbits around the fixed point $(0,0)$, were obtained from two iterations (i.e.
eq. (9)) and were given by the two pairs of points obtained analytically in
eq. (10). Finding similar analytical expressions for the two parametric maps
under study is not an easy task, nevertheless we can exploit symmetry
properties of the map, which reveal that these two new fixed points, that we
denote $(u_{0},v_{0})$, obey $(u,v)=\pm(y,-y)$ where $y$ is a real and
positive function of $\tilde{a}$ and $\mu$. With this assumption, the period-
two fixed-point equation (9) leads to the following transcendental equation;

$2y+\frac{1}{K}\frac{\partial{V_{\mu}(y)}}{\partial{y}}=0,$ (18)

the roots of which determine $y$. To extract these roots we make use of the
Brent algorithm [51], a robust root-finding rule that combines the bracketing,
bisection and inverse quadratic interpolation methods [51]. Brent’s scheme
turns out to be very efficient when seeking for roots of non-polynomial
functions, which may display multiple dsingularities. Fig. 3 show the
numerical solution to eq. (18), for the parametric map with the DKDW
potentials (3)-(4) (left graph) and (3)-(5) (right graph).

Figure 3: (Color online) Variation of $y$ extracted numerically from eq. (18),
as a function of the ratio $a/K$ for different values of $\mu$. Left panel
corresponds to the DKDW potential with variable barrier height and fixed
minima, and the left panel corresponds to the DKDW potential with variable
minima but fixed barrier height. Left graph, from left to right curves:
$\mu=0$, 0.2, 0.5, 0.9, 1.2. Right graph, from top to bottom curves: $\mu=0$,
0.2, 0.5, 0.9, 1.2.

More explicitly the two graphs of fig. 3 represent $y$ as a function of
$\tilde{a}$, for $\mu=0$, 0.5, 1, 1.5. The left graph shows that when $\mu=0$,
the variation of $y$ with the ratio $\tilde{a}$ reproduces the analytical
expression $y=\sqrt{1-\frac{4}{\tilde{a}}}$ present in eq. (10). As the
deformability parameter $\mu$ is increased, fixed points of period-two orbits
are gradually shifted forward. This is also the case for values of $\tilde{a}$
for which the first bifurcation is expected, indeed eq. (17) clearly indicates
that values of $\tilde{a}$ at the first bifurcation will be increased as we
increase $\mu$. On the other hand, the right graph of fig. 3 suggests that
fixed points of period-two orbits for this map will always vanish at
$\tilde{a}=4$ whatever the value of $\mu$. However when $\tilde{a}>4$,
increasing $\mu$ contracts the region between the instability points of
period-one orbits (i.e. $\tilde{a}=4$ for any $\mu$) and period-two orbits.

##  4  phase-space texture and structures of bifurcation diagrams

In the previous section we discussed characteristic features of the fixed
point $(0,0)$ for the two $2D$ discrete parametric maps, as well its stability
and the influence of shape deformability of the double-well potentials on the
two first bifurcations of the corresponding parametric maps. However, a global
evolution of the two maps is required to gain a full undertanding of the
overall texture of the phase space, and the structure of bifurcation diagrams
of these maps taking into account their distinct parametrizations. We first
consider the phase-space dynamics of the two maps, starting from $\mu=0$ and
increasing the ratio $\tilde{a}$ around its critical value $\tilde{a}_{c}=4$
corresponding to the first period-doubling bifurcation point of the $\phi^{4}$
map.

Fig. 4 represents trajectories around the fixed point $(0,0)$ for the DKDW map
with variable barrier height but fixed mimina, when $\mu=0$ corresponding to
the $\phi^{4}$ map.

(a)

(b)

(c)

(d)

Figure 4: (Color online) Trajectories in phase space for the $2D$ discrete
parametric map with DKDW having a variable barrier height but fixed potential
minima. Graph (a): Elliptic orbits ($\tilde{a}=3.5$), graph (b): unstable
elliptic orbits ($\tilde{a}=3.9$), graph (c): nascent hyperbolic orbit
($\tilde{a}=4.1$), graph (d): hyperbolic orbits around two well separated
fixed points ($\tilde{a}=4.35$). Here $\mu=0$ coinciding with the $\phi^{4}$
map.

Graph (a) features a stable elliptic fixed point ($\tilde{a}=3.5$), in graph
(b) elliptical trajectories are nearly decaying ($\tilde{a}=3.9$), graph (c)
shows a nascent hyperbolic orbit around two distinct fixed points
($\tilde{a}=4.1$), in graph (d) the fixed point $(0,0)$ has now given birth to
two new elliptical fixed points ($\tilde{a}=4.35$). In the graph these two new
fixed points are approximately at $\pm(0.2,-0.2)$, in agreement with the
symmetry considerations that led to the transcendental equation in $y$ given
(18), the solution of which was obtained numerically using Brent’s algorithm.
It is remarkable that in graph (d) of fig. 4, the separatrix consists of dense
orbits intersected by forbidden islands which enclose sparse, isolated orbits
trapped inside the islands. Many other interesting motifs can be generated in
the texture of phase space, as obtained in ref. [14] for higher values of
$\tilde{a}$. However this is not our objective here, let us rather see how the
texture in fig. 4 is affected by the deformability of the double-well
substrate.

To point out the effect of the deformability parameter on the phase space
texture for this first map, in fig. 5 we reploted the graph 4(d) for $\mu=0.2$
(left graph) and $\mu=0.4$ (right graph). One clearly sees that when
$\mu=0.2$, the texture of phase space is reminiscent of the nascent hyperbolic
orbit observed in graph fig. 4(c) for $\mu=0$ and $\tilde{a}=4.1$. As the
deformability parameter is increased, namely when $\mu=4$, the system is taken
back to the phase-space texture dominated by elliptic orbits around the fixed
point $(0,0)$.

Figure 5: (Color online) Replot of graph (d) in fig. 4, now for $\mu=0.2$
(left graph) and $\mu=0.4$ (right graph).

Turning to the second map (i.e. the DKDW map with variable mimima but fixed
barrier height), the associated phase space texture will be similar to the one
of the first map when $\mu=0$ (a value at which the two maps reduce together
to the $\phi^{4}$ map). Therefore we shall examine only the influence of
nonzero values of $\mu$ on the texture of phase space, focusing on the point
$\tilde{a}=4$ and $\mu=0$ which is actually the graph in fig. 4(d). Thus, in
fig. 6 we represent the phase space of the DKDW map with variable minima but
fixed barrier height for $\tilde{a}=4$, with $\mu=0.2$ (left graph) and
$\mu=0.4$ (right graph). We see that a variation of $\mu$ has relatively less
effect on the phae space texture compared, at least with what we observed for
the DKDW map with variable barrier height but fixed minima.

Figure 6: (Color online) Texture of phase space for the DKDW map with variable
minima and fixed barrier, for $\tilde{a}=1$ and $\mu=0.2$ (left graph),
$\mu=0.4$ (right graph).

The last aspect we shall examine is the structure of bifurcation diagrams of
the two maps, with emphasis on their distinct parametrizations. Fig. 7
displays the bifurcation diagram with respect to $\tilde{a}$, of the DKDW map
with variable barrier height for four different values of $\mu$ i.e. $\mu=0$
(a), $\mu=0.2$ (b), $\mu=0.5$ (c) and $\mu=1$ (d). The main insight from this
series of graphs is the observed forward shift of values of $\tilde{a}$ at the
first period-doubling bifurcation, as $\mu$ is increased. This behavior is of
course consistent with the stability condition given analytically in formula
(17).

(a)

(b)

(c)

(d)

Figure 7: (Color online) Bifurcation diagram in $\tilde{a}$, for the DKDW map
with variable barrier and fixed minima: $\mu=0$ (a), $\mu=0.2$ (b), $\mu=0.5$
(d), $\mu=1$ (d).

For this first map we also simulated the bifurcation diagram with respect to
the deformability parameter $\mu$, considering four different values of
$\tilde{a}$ chosen around the first pitchfork bifurcation. Fig. 8 shows that
the bifurcation diagram in $\mu$ is dominated by period-halving bifurcations
when $\tilde{a}=3.5$, 4, 5 and 6. These period-halving bifurcations are surely
indicative of the fact that an increase in the deformability parameter $\mu$
is expected to delay priod-doubling transitions with respect to $\tilde{a}$,
as we observed in fig. 7.

(a)

(b)

(c)

(d)

Figure 8: (Color online) Bifurcation diagram in the deformability parameter
(dp) $\mu$, for the DKDW map with variable barrier and fixed minima: (a)
$\tilde{a}=3.5$, (b) $\tilde{a}=4$, (c) $\tilde{a}=5$, (d) $\tilde{a}=6$.

The bifurcation diagrams with respect to $\tilde{a}$ for the DKDW map with
variable minima are shown in figs. 9 for the same four different values of
$\mu$ used in fig. (7. Structures of the bifurcation diagram with respect to
$\tilde{a}$ for larger values of $\mu$ (i.e. $\mu=1.5$ for left graph, and 1.8
for right graph), are shown in fig. 9.

(a)

(b)

(c)

(d)

Figure 9: (Color online) Bifurcation diagram in $\tilde{a}$, for the DKDW map
with variable minima and fixed barrier: (a) $\mu=0$, (b) $\mu=0.2$, (c)
$\mu=0.5$, (d) $\mu=1$.

Figure 10: (Color online) Bifurcation diagram in $\tilde{a}$, for the DKDW map
with variable minima and fixed barrier: $\mu=1.5$ (left panel), and $\mu=1.8$
(right panel).

The bifurcation diagrams with respect to $\mu$ for this second map are plotted
in fig. 11, for four different values of $\tilde{a}$.

(a)

(b)

(c)

(d)

Figure 11: (Color online) Bifurcation diagram in the deformability parameter
(dp) $\mu$, for the DKDW map with variable minima and fixed barrier height:
(a) $\tilde{a}=3.9$, (b) $\tilde{a}=4$, (c) $\tilde{a}=5$, (d)
$\tilde{a}=6.5$.

The bifurcation structures emerging from figs. 9, 10 and 11 are all consistent
with our previous comments regarding the texture of phase space as well as the
analytical prediction on the the first and sexcond bifurcation cascades for
the second map. In particular the figures clearly sgow that for the second
map, bifurcation diagrams with respect to the two characteristic parameters
are dominated by period-doubling cascades. Here the first bifurcation in
$\tilde{a}$ is clearly insensitive to the variation of $\mu$, however the
second bifurcations leading to period-four orbits occur for values of
$\tilde{a}$ that are shifted backward more and more as $\mu$ increases. Hence
and increase of $\mu$ shrinks the regions between successive period-doubling
bifurcation cascades, so we can conclude that as we increase the values of
$\mu$ the difference in values of $\tilde{a}$ at two consecutive bifurcation
cascades will decrease. So to say the Feigenbaum number sequence for the DKDW
map is non universal, the sequence will strongly depend on values of $\mu$.

##  5 Conclusion

Most physical systems evolve in environments where they are subjected to
multiple equilibria of distinct stabilities. Of these, bistable systems form a
special class characterized by two energetically equivalent metastable states
and eventually an unstable state where the system undergo structural changes.
Bistability is a quite common feature in biology, and there are many examples
of systems which can operate stably in two completely distinct modes of
equivalent symmetries. This is for instance the case for the well known
lactose operon in the bacteria Escherichia coli [52], or the phage
$\lambda$virus which can exist in either of two states: Under normal
conditions this virus can exist in a lysogenic state and survive indefinitely
within its host (Escherichia coli). However, under adverse conditions (such as
ultraviolet radiations), the phage can switch to a reproducible (i.e. lytic)
mode leading to bacterial lysis [53]. Yet another example is the complex
system of cross-talking pathways that regulates the decision of cells to enter
the process of programmed cell death, also known as apoptosis, as opposed to
continuing normal development [53]. These processes in biological systems have
established that most often individual cells only exist in one of two distinct
states, and upon stimulations the cells undergo a change of state (or a
transition) from one state to another.

To understand how bistable systems in biology as well as other physical
systems such as solid-state materials, biochemical systems and so on, perform
complex functions, several mathematical models have been proposed. These
models have in common the existence of an appropriate feedback describing
their bistability, and represented by a function with two degenerate minima.
This double-well function, which stands for the characteristic energy
associated with the appropriate bistable feedback process, is most often
represented by the so-called $\phi^{4}$ potential. For instance the feedback
force found in the Duffing model [3] with soft anharmonicity originates from a
$\phi^{4}$-like double-well potential, this is equally the case in the
logistic model with two mestable equilibria separated by a wall, etc. However
the rigid profil of the double-well potential representing the bistable
feedback function in these mathematical models, is a weakness for it does not
enable one to take into consideration the many possible changes in
characteristic features of systems as a result of the variability of
constraints of their complex living environments. As improvement to this
rigidity parametrized double-well potentials were proposed, they include the
double-Morse potential [29], the Schmidt potential [27], the Magyari motential
[26], the razavy potential [30] and more recently, the family of parametrized
potentials whose double-well shapes can be tuned at wish by varying a
deformability parameter [32].
In the present study we explored the route to chaos for bistable systems
described by a discrete $2D$ parametric map, with emphasis on two members of
the family of parametreized double-well potentials proposed in ref. [33]. We
examined the texture of phase space of the two parametric maps as well as the
structure of the associate bifurcation diagrams. One of the two discrete $2D$
parametric maps is characterized by a double-well potential whose barrier
height can be tuned continuously, leaving unaffected its two degenerate
minima. For this first map we obtained that the shape deformability allows
shifting to higher magnitudes the values of the ratio $a/K$ where bifurcations
are expected to occur. In particular we found that the bifurcation diagram in
the deformability parameter was dominated by period-halving cascades,
suggesting unambiguously the possible control of route to chaos as well as a
non-universal character of the Feigenbaum number sequence for this map. The
second map studied has a fixed barrier height but tunable positions of its two
degenerate minima. For this second mpa, we found that the first bifurcation
always coincides with the first pitchfork instability of the $\phi^{4}$ map
irrespective of the deformability parameter. However, an incease of the
deformability parameter seemed to contract the region between consecutive
period-douboing cascades. This behavior too, clearly indicates a
non-n=universak character of the Feigenbaum number sequences for this second
map.
As we indicated in the introduction, the deformability can be associated with
the variability of the living environment of cells and chemical species, as
for instance the change of pressures or characteristic parameters of
intermocular interactions due to isotopic substitutions (changes of reactants
in processes involving chemical reactions), or the change of cell
characteristics in the bistability feedback, in reponse to temperature
variations [54]. The present study particularly points out the fact that while
the shape deformability for the models considered here does not change
qualitatively intrinsic properties of bistable systems, quantitative changes
can however be observed and marked by non-universalities of characteristic
features of their route to chaos as for instance the number sequence
associated with the recurrence of period-doubling cascades leading to chaos.

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11institutetext: Indian Institute of Technology, Mandi, India
11email<EMAIL_ADDRESS>
# AttnGrounder: Talking to Cars with Attention

Vivek Mittal

###### Abstract

We propose Attention Grounder (AttnGrounder), a single-stage end-to-end
trainable model for the task of visual grounding. Visual grounding aims to
localize a specific object in an image based on a given natural language text
query. Unlike previous methods that use the same text representation for every
image region, we use a visual-text attention module that relates each word in
the given query with every region in the corresponding image for constructing
a region dependent text representation. Furthermore, for improving the
localization ability of our model, we use our visual-text attention module to
generate an attention mask around the referred object. The attention mask is
trained as an auxiliary task using a rectangular mask generated with the
provided ground-truth coordinates. We evaluate AttnGrounder on the Talk2Car
dataset and show an improvement of 3.26% over the existing methods. Code is
available at https://github.com/i-m-vivek/AttnGrounder.

###### Keywords:

Object Detection, Visual Grounding, Attention Mechanism

##  1 Introduction

In recent years, there have been many advances in tasks involving the joint
processing of images and text. Researchers are working on various challenging
problems like image captioning [28, 1, 7], visual question answering [1, 11,
30], text-conditioned image generation [13, 31], etc. In this work, we address
the task of visual grounding [5, 24] in which our goal is to train a model
that can localize an image region based on a given natural language text
query. The task of visual grounding can be useful in many practical
applications. In Figure 1, we provide an example, in a self-driving car, the
passenger can give a command to the car like “Do you see that lady walking on
the sidewalk, up here on the left. She is the one we need to pick up. Pull
over next to her”. Additional use cases can be found in embodied agents and
human-computer interaction. The task of visual grounding is quite challenging
as it requires joint reasoning over text and images. Consider the example
shown in Figure 1: in order to correctly identify the women, a model first
needs to understand the command properly and then locate the target object in
the image.

In this paper, we propose a novel architecture for visual grounding, namely,
the AttnGrounder. At a high-level, visual grounding consists of two sub-tasks:
object detection [21, 23] and ranking detected objects. In the first task, the
model identifies all the objects present in the image, and then it calculates
a matching score between every object and given text query to rank different
proposals. These two tasks are themselves quite challenging to solve, and the
object detection task may become a bottleneck for the performance of the whole
system. In contrast to the traditional two-stage [29, 5] approaches our
AttnGrounder directly operates on raw RGB images and text expressions. Our
main goal is to combine the image and text features, and then directly predict
bounding boxes from them. To this end, we use a YOLOv3 [22] backbone to
generate visual features (Sec. 3.1) and a Bi-LSTM to generates text features
(Sec. 3.2). For jointly reasoning over visual and text features, we use two
sub-modules.: Text Feature Matrix (Sec. 3.3.1) and Attention Map (Sec. 3.3.2).
The text feature matrix calculates text representation for every image region
by selecting important words for that particular region. Predicting a rough
segmentation mask around the referred object can also help in predicting a
bounding box. Thus, we make a rectangular mask around the target object using
the ground truth coordinates, and the goal of our attention map module is to
predict that mask. The attention map module is trained jointly with rest of
the system and yields an auxiliary loss which can improve the localization
ability of our model. Finally, we fuse all the features to obtain a multi-
modal feature representation with which we predict the bounding box (Sec.
3.4). In Section 2, we give an overview of the related work. In Section 3, we
provide a detailed description of our approach. In Section 4, we evaluate our
model on the Talk2Car dataset [6]. Finally, we conclude our paper in Section
5.

Figure 1: Example of visual grounding task. Green box indicates the referred
object.

##  2 Related Work

In terms of the object detection subtask in visual grounding, there exist two
lines of work in visual grounding: a one-stage approach [27, 24, 3, 19] and a
two-stage approach [25, 26, 17, 5]. In most of the previous work, researchers
address the task of visual grounding with two-stage approaches. In the first
stage, several object proposals are generated using an off-the-shelf object
detection algorithm like Faster-RCNN [23], YOLO [21, 22]. In the second stage,
the object proposals are ranked by calculating the matching score with the
given text query.

Several works [26, 25] explore graph convolutional networks for learning the
relationship between objects and text query. The work of [26] focuses on
improving the cross-modal relationship between objects and referring
expressions by constructing a language-guided visual relation graph. They use
a gated graph convolutional network to fuse information from different
modalities. In [25], authors also use graph networks for learning the
relationship between neighboring objects. NMTree [16] constructs a language
parsing tree with which they localize referred object by accumulating
grounding confidence per node. MAttNet [29] involves a two-stage modular
approach in which they use three modules for processing visual, language,
location information, and a relationship module to understand the relationship
between the output of these three modules. In [18], the authors use attention
to generate difficult examples by discarding the most important information
from both text and image. A-ATT [4] constructs three modules to understand the
image, objects, and query. They accumulate features from different modalities
in a circular manner to guide the reasoning process in multiple steps. MSRR
[5] also uses a multi-step reasoning procedure, in which a new matching score
is calculated between every object and the language query in each step. After
all the reasoning steps, the object that has the highest score is selected as
the target object.

In two-stage methods, the offline object detector may become a bottleneck as
it may fail to provide good object proposals. For addressing this issue,
several recent works explore a one-stage paradigm for visual grounding. One-
stage methods fuse image-text features and then directly predict the bounding
box for the referred object. Recent works [24, 27] fuse visual-text features
at every spatial location and predict adjustment in predefined anchor boxes to
align the bounding boxes with the referred object. In [3], the authors
introduce a guided attention module with which their model also predicts the
center coordinates of the referred object along with the referred region. In
our AttnGrounder, we predict a mask rather than the center coordinate which
can help in better localization of image regions. Moreover, previous one-stage
methods consider a single text representation for every location, but as we
know, different regions correspond to different words, which is why we develop
a text feature matrix where every location has its unique text representation.
Additionally, one-stage methods are faster than two-stage methods as they
don’t involve matching between various image regions and text queries.

Figure 2: Overview of AttnGrounder.

##  3 Methodology

In Figure 2, we provide an overview of our proposed visual grounding
architecture. Our goal is to locate an object in an Image $I$ based on a given
text query $T$. Our proposed model AttnGrounder consists of five sub-modules:
image encoder, text encoder, visual attention module, fusion module, and a
grounding module. The image encoder uses Darknet-53 [22] with pyramid
structure [14] to extract visual features from the image $I$ in the form of
grids $\\{G_{k}\\}_{k=0}^{2}\in\mathbb{R}^{C_{k}\times H_{k}\times W_{k}}$ at
three different spatial resolutions. To encode text features, we use a Bi-LSTM
that can easily capture the long-range dependency in given query. We introduce
a visual-text attention mechanism that attends to words $w_{t}$ in the text
query $T$ at every spatial location in the grid $G_{k}$. This attention
mechanism also generates an attention map $M$ for the referred object.
Finally, we fuse the image and text feature using $1\times 1$ convolution
layers. For training, we use two loss functions: binary cross-entropy loss for
training the attention map and YOLO’s loss [21] function for complete end-to-
end training. For our use, we modify the YOLO’s loss function by replacing the
last sigmiod unit with a softmax unit.

###  3.1 Image Encoder

We adopted Darknet-53 [22] with a pyramid network [14] structure for encoding
visual features. Darknet-53 takes an image $I$ and produces feature grids
$\\{G_{k}\\}_{k=0}^{2}\in\mathbb{R}^{C_{k}\times H_{k}\times W_{k}}$ at three
different spatial resolutions, where $C_{k},H_{k}$ and $W_{k}$ are the number
of channels, height and width of the grid at $k^{th}$ resolution.
Usually, referring expressions also contain position information about the
referred object (eg. ”park in front of the second vehicle on our right
side.”). Visual features produced by Darknet lack such location information.
Similar to [27], we explicitly add location information by concatenating a
vector $C_{ij}\in\mathbb{R}^{8}$ at every spatial location of the grid
$G_{k}$,

$C_{ij}=\left(\frac{i}{W_{k}},\frac{j}{H_{k}},\frac{i+0.5}{W_{k}},\frac{j+0.5}{H_{k}},\frac{i+1}{W_{k}},\frac{j+1}{H_{k}},\frac{1}{W_{k}},\frac{1}{H_{k}}\right)$

where $i,j$ are row and column in the grid $G_{k}$ respectively. Then, we use
a $1\times 1$ convolution layer with batch normalization and ReLU unit to map
these grid feature to a common semantic dimension $D$. Now,
$G_{k}\in\mathbb{R}^{C_{k}\times H_{k}\times D}$.

###  3.2 Text Encoder

Our text encoder consists of an embedding layer and a Bi-LSTM layer. The text
query $T$ of length $n$ is first converted to its embedding
$Q=\\{e_{i}\\}_{i=0}^{n-1}$ using the embedding layer. After that, $Q$ is fed
as an input to Bi-LSTM that generates two hidden states
$[h^{right}_{i},h^{left}_{i}]$ for every word embedding $e_{i}$. The forward
and backward hidden states are concatenated to get $h_{i}\in\mathbb{R}^{2L}$,
where $L$ is the number of hidden units in Bi-LSTM. Furthermore, we use a
linear layer to project the text embedding into the common semantic space of
dimension $D$. Embeddings for all $n$ words are stacked to get
$Q^{\prime}\in\mathbb{R}^{n\times D}$.

###  3.3 Visual-Text Attention

Figure 3: Visual-Text Attention Module

Every region in an image may correspond to different words in the given query.
To better model the dependency between a word and a region, we present a
visual-text attention module. Fig. 3 illustrates this module. This module gets
a grid $G_{k}$ from the image encoder and text embeddings $Q^{\prime}$ from
the text encoder. The grid $G_{k}\in\mathbb{R}^{H_{k}\times W_{k}\times D}$
has $H_{k}\times W_{k}$ spatial locations each of dimension $D$ and text
embeddings $Q^{\prime}\in\mathbb{R}^{n\times D}$ have $n$ word features each
of dimension $D$. To measure the matching score between a location
$g_{i}\in\mathbb{R}^{D}$ and word $w_{j}\in\mathbb{R}^{D}$ in the query, we
calculate the dot product between them. We multiply $G_{k}$ with $Q^{\prime}$,
which generates a matrix $M^{k}=G_{k}\cdot(Q^{\prime})^{T}$, where
$M_{ij}^{k}$ represents the matching score between grid location $i$ and word
$j$. The matrix $M^{k}\in\mathbb{R}^{(H_{k}\times W_{k})\times n}$, where
$H_{k}$ and $W_{k}$ are the height and width of the grid at $k^{th}$
resolution generated by our image encoder. This matrix $M^{k}$ is used in two
ways described next.

####  3.3.1 Text Feature Matrix

For every location in the grid $G_{k}$, we can represent it’s text features by
summing up word features from the matrix $Q^{\prime}\in\mathbb{R}^{n\times
D}$. We normalize the matrix $M^{k}$ using the softmax function to generate a
word-level attention matrix $\alpha^{k}\in\mathbb{R}^{(H_{k}\times
W_{k})\times n}$, where $\alpha_{ij}^{k}$ denotes the correlation between
region $i$ in the grid $G_{k}$ and word $j$ in the text query $T$.

$\alpha_{ij}^{k}=\frac{\exp(M_{ij}^{k})}{\sum_{j=0}^{n-1}\exp(M_{ij}^{k})}$
(1)

To obtain text features at every location, we multiply $\alpha^{k}$ and
$Q^{\prime}$, which generates text features matrix
$T^{\prime}_{k}\in\mathbb{R}^{(H_{k}\times W_{k})\times D}$, denoted as
$T^{\prime}_{k}=\alpha^{k}\cdot Q^{\prime}$. Every row in the matrix
$T^{\prime}_{k}$ represents text features, which are effectively derived from
the word features weighted by the correlation between words and corresponding
image region. Thus, every region has its unique text representation.

####  3.3.2 Attention Map

In visual grounding, a rectangular mask around the referred object can be
predicted by just using the provided ground truth bounding boxes. Our
attention map module aims to make a rectangular mask around the referred
object, which is an auxiliary task that helps improve the overall performance.
By learning to make a mask around the object, the model can indirectly learn
to locate objects based on the given query and image. We make use of the
matrix $M^{k}\in\mathbb{R}^{(H_{k}\times W_{k})\times n}$, which already
contains the matching score between every image region and every word in the
text query. We sum the values in every row of $M^{k}$ to get a column matrix
$\beta^{k}\in\mathbb{R}^{(H_{k}\times W_{k})}$

$\beta^{k}_{i}=\sum_{j=0}^{n-1}M_{ij}^{k}$ (2)

$\beta^{k}_{i}$ provides the aggregated matching score for an image region $i$
and the given text query $T$. Now, the image regions matching with the given
query will have a high matching score and vice-versa. To create a mask, we
feed $\beta^{k}$ through a sigmoid unit, which maps the values of $\beta^{k}$
between 0 and 1. Therefore, $\beta^{k}_{i}$ can be interpreted as the
probability of having the referred object in the region $i$.

For training, we generate a rectangular mask using the ground truth
coordinates. The ground truth mask has the value of 1 inside the rectangular
region and 0 elsewhere. Masks are created for all three resolutions and
training is done using binary cross-entropy loss (See Figure 4). At the
$k^{th}$ resolution, given the ground truth mask
$\beta^{k}_{true}\in\mathbb{R}^{H_{k}\times W_{k}}$ and the predicted mask
$\beta^{k}\in\mathbb{R}^{H_{k}\times W_{k}}$; the mask loss is calculated as

$L^{k}_{mask}=-\frac{1}{H_{k}\times W_{k}}\sum_{i=0}^{(H_{k}\times
W_{k})-1}(\beta^{k}_{i,true}\log(\beta^{k}_{i})+(1-\beta^{k}_{i,true})\log(1-\beta^{k}_{i}))$
(3)

The $L_{mask}$ is obtained by summing up the $L_{mask}^{k}$ terms calculated
for all the three resolutions.

$L_{mask}=\sum_{k=0}^{2}L_{mask}^{k}$ (4)

Using $L_{mask}$ as an auxiliary loss can help in embedding visual and text
features in the common semantic space $D$, by encouraging similar visual and
text features to be nearby and vice-versa. Furthermore, it can also help
discriminate between the referred region and the others.

We also generate attended visual features $V_{k}\in\mathbb{R}^{H_{k}\times
W_{k}\times D}$ by an element-wise product between the attention map
$\beta^{k}$ and grid $G_{k}$, $V_{k}=\beta^{k}\odot G_{k}$ ($\odot$ denotes
element-wise product). In $V_{k}$, every spatial location is scaled by its
importance determined by the attention mask $\beta^{k}$. Thus, visual features
of regions where target object may be present are enhanced and visual features
of other redundant regions are reduced.

Figure 4: The mask loss is calculated for every spatial resolution using
binary cross-entropy loss and summed to get the final mask loss. First row
shows attention map produced by our AttnGrounder at three different
resolutions and second row shows ground truth mask used for training.

###  3.4 Fusion Module

For fusing image and text features, we concatenate visual feature grid
$G_{k}$, text feature matrix $T^{\prime}_{k}$ and attended visual features
$V_{k}$ along the channel dimension. This generates a matrix
$F_{k}\in\mathbb{R}^{H_{k}\times W_{k}\times 3D}$. All the features are
$l_{2}$ normalized before concatenation and fusion is done for all the three
spatial resolutions generated by our image encoder. After that, we use a
$1\times 1$ convolution layer to fuse these visual and text features. This
$1\times 1$ convolution layer maps these fused features in a new semantic
space of dimension $D^{\prime}$. After fusion, we have three feature matrices
$\\{F_{k}\\}_{k=0}^{2}\in\mathbb{R}^{H_{k}\times W_{k}\times D^{\prime}}$.

###  3.5 Grounding Module

The grounding module aims to ground the text query onto an image region. This
module takes the fused feature vector $F_{k}$ as input and generates a
bounding box prediction. We follow [27] for designing this module and similar
to [27] we replace YOLO’s [21] output sigmoid layer with softmax layer.

For every spatial location, YOLOv3 centres three different anchor boxes and
for every spatial resolution, YOLOv3 uses different set of anchor boxes. In
total, for three different spatial resolutions we have $3\times 3=9$ different
predefined anchor boxes. We denote the total number of anchor box predictions
made by our model with $m$, where $m=\sum_{k=0}^{2}H_{k}\times W_{k}\times 3$.
For every anchor box, YOLOv3 predicts changes required in location and size to
fit that particular anchor box around the target object. Specifically, YOLOv3
uses two branches: the first branch predicts the shift in centre, height,
width of the predefined anchor box and the second one uses a sigmoid layer to
predict the confidence (on the shifted box) of being the target box. As we
need only one bounding box prediction for grounding, we replace the last
sigmoid layer with softmax layer which forces to select one box out of the $m$
boxes as prediction. Accordingly, the loss function for confidence is also
changed to a cross-entropy loss between the predicted confidence (softmax
version) and a one-hot vector with a 1 entry corresponding to the anchor box
that has the highest intersection over union with the ground truth box. We
refer the readers to [21, 22] for more details and abstract away the YOLO’s
loss function as $L_{yolo}$. Thus, total loss becomes

$L=L_{yolo}+\lambda L_{mask}$ (5)

where $\lambda$ is a hyper-parameter which scales the loss $L_{mask}$.

Figure 5: Examples of results obtained using our AttnGrounder. The predicted
bounding boxes are shown in red and ground truth bounding boxes are shown in
green. The second row shows the attention maps generated by our model.

##  4 Experiments and Results

In this section, we provide details of our experiments. We evaluate our
approach on Talk2Car dataset [6] and compare with five baselines.

###  4.1 Dataset

The Talk2Car dataset is based on nuScenes dataset [2] and contains 11,959
referring expressions for 9,217 images. This dataset contains images of a
city-like environment where a self-driving car is driving. Talk2Car dataset is
a multi-modal dataset that consists of various sensor modalities such as
semantic maps, LIDAR, 360-degree RGB images, etc. However, we only use the RGB
images for training our AttnGrounder. RGB images contained in Talk2Car dataset
are taken in different weather conditions (sunny, rainy) and time of day (day,
night) which makes it even more challenging. Additional challenges that the
Talk2Car dataset presents are the ambiguity between the objects belonging to
the same class, far away object localization, long text queries, etc. In this
dataset, the size of images is $900\times 1600$ and on average, the referring
expressions contain 11 words. The dataset contains 8349, 1163 and 2447 images
for training, validation and testing respectively. Figure 1 shows an example
of Talk2Car dataset.

###  4.2 Training Details

Our image encoder backbone i.e. Darknet-53[22] is pre-trained on COCO [15]
object objection task. We use 300 dimensional GloVe [20] embeddings for
initializing word vectors. Words for which the GloVe embeddings were not
available were initialized randomly. We resize the images to $416\times 416$
and preserve the original aspect ratio. We resize the longer edge to 416 and
pad the shorter edge with the mean pixel value. We also add some data
augmentations i.e., horizontal flips, changing saturation and intensity,
random affine transformation. The anchors used in our grounding module are
(10,13), (16,30), (33,23), (30,61), (62,45), (59,119), (116,90), (156,198),
(373,326). We use $\lambda=0.1$ (defined in Sec. 3.5) for training. We train
our model using Adam [12] optimizer with an initial learning rate of $10^{-4}$
and polynomial learning rate scheduler is used with a power of 1. For the pre-
trained Darknet-53 portion, we keep a lower initial learning rate of
$10^{-3}$. We keep a batch size of 14 for training our AttnGrounder.

Table 1: Comparison with state-of-the-art methods on Talk2Car testset Method |  $AP_{50}$ Score |  Time (ms) |  Params (M)
---|---|---|---
STACK [8] |  33.71 |  52 |  35.2
SCRC [9] |  43.80 |  208 |  52.47
A-ATT [4] |  45.12 |  180 |  160.31
MAC [10] |  50.51 |  51 |  41.59
MSRR [5] |  60.04 |  270.5 |  62.25
AttnGrounder |  63.30 |  25.5 |  75.84

###  4.3 Comparison Metrics

We evaluate our approach on three different metrics. First one is $AP_{50}$
score, which is defined as the percentage of predicted bounding boxes that
have an Intersection Over Union of more than 0.5 with the ground truth
bounding boxes. The second one is inference time and third is number of
parameters in a method.

###  4.4 Results

We compare our AttnGrounder with four baselines and state-of-the-art model
MSRR [5], results on the test set of Talk2Car dataset are shown in Table 1.
AttnGrounder outperforms all the baselines and improves state-of-the-art by
3.26% in terms of $AP_{50}$ score. Thanks to our one-stage approach, we also
achieve lowest inference time among all the baseline methods. We provide
examples of prediction made by our method in Figure. 5 along with the
visualization of attention map generated by visual-text attention module (Sec.
3.3).

##  5 Conclusions

We propose AttnGrounder, a single-stage end-to-end trainable visual grounding
model. Our AttnGrounder is fast compared to two-stage approaches as it does
not involve matching various region proposals and text queries or multi-step
reasoning. We combine visual features extracted from YOLOv3 and text features
extracted from Bi-LSTM to obtain a multi-modal feature representation with
which we directly predict the bounding box for the target object. Our
AttnGrounder also generates an attention map that localizes the potential
spatial locations where the referred object may be present. This attention map
is trained as an auxiliary task which helps improve the overall performance of
our model. Finally, we evaluate our proposed method on the Talk2car dataset
and show that it outperforms all the baseline methods.

## Acknowledgements

The author wants to thank Thierry Deruyttere, Dusan Grujicic, and Monika
Jyotiyana for their feedback on an early draft of this paper. The author is
grateful to Thierry Deruyttere for his guidance and constant support.

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