# Border Figure Detection Using a Phase Oscillator Network with Dynamical Coupling

Authors: L. H. A. Monteiro; I. Gonzalez; J. R. C. Piqueira
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

Oscillator networks have been developed in order to perform specific tasks
related to image processing. Here we analytically investigate the existence of synchronism in a pair of phase oscillators that are short-range dynamically coupled. Then, we use these analytical results to design a network able of detecting border
of black-and-white figures. Each unit composing this network is a pair of such
phase oscillators and is assigned to a pixel in the image. The couplings among
the units forming the network are also dynamical. Border detection emerges from
the network activity.


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## Body

## 1. Introduction

The synchronous firing of neurons seems to be
important for accomplishing cognitive tasks such as attention (e.g., [1]), comprehension (e.g.,
[2]), coordination
(e.g., [3]),
perception (e.g., [4]), and sensory segmentation (e.g., [5]). Such experimental findings have inspired works about image processing (e.g.,
recognition, segmentation, symmetry detection) based on the synchronism of
coupled oscillators (e.g., [6–11]). In addition, several neural systems, such as spinal
cord (e.g., [12, 13]), hippocampus (e.g., [14, 15]), and visual cortex (e.g., [16, 17]), have been modelled by
phase oscillators—one of the simplest oscillator models. In spite of its simplicity,
this approach is suitable because more complex models for neurons like
pulse-coupled and Hodgkin-Huxley-type models can be transformed into a phase
oscillator model through coordinate changes [18]. Networks of phase oscillators can be electronically
built using phase-locked loops (PLLs) (e.g., [19–22]).Here we analytically study a phase oscillator model
with dynamical coupling, and then use it in order to form a network capable of
successfully detecting border of black-and-white figures. Each oscillator of
our network corresponds to a first-order PLL (e.g., [8, 11, 17, 23]), which is equivalent to an
overdamped pendulum.The aim of any border detection process is to capture
the main structural properties of an image, which can be useful, for instance,
for image compression or diagnosis in echocardiography (e.g., [24]). Usually, this kind of
image processing involves the use of partial derivatives (gradient, Laplacian;
e.g., [25]), which are
discretized in the space domain and in the time domain. Here we propose a
scheme based on a network where each unit is formed by a pair of dynamically
coupled PLLs corresponding to a pixel of the image. Border detection is obtained
from the network activity. Notice that such a model can be transformed into a
dedicated hardware for executing this image processing task. Also it naturally
presents space discretization, because it is composed of a finite number of
PLLs. This feature plus the fact that the time domain does need to be
discretized can reduce the stability problems related to the algorithms
employed for numerically calculating partial derivatives.


## 2. Model of a Single Unit

Consider that
the temporal activities of two first-order PLLs with dynamical coupling are
described by the following equations:(2.1)dθ1dt=ω1+k12sin(θ2−θ1),dθ2dt=ω2+k21sin(θ1−θ2),dk12dt=μα1cos(θ1−θ2)−μk12,dk21dt=μα2cos(θ2−θ1)−μk21,where θj and ωj(j=1,2) are the phase and the natural
frequency of the oscillator j, respectively; kij (i,j=1,2 with i≠j) is the connection strength from the
oscillator j to the oscillator i; αj (j=1,2) is the coefficient related to the Hebbian
connection modification; and μ represents a natural (exponential) decay. The
parameters ωj, αj,
and μ are positive numbers. Thus, the connection
strengths are enhanced when the oscillators are in phase, and they are weakened
when they are out of phase. This kind of (synaptic) modification between two
oscillators (neurons) was suggested by Hebb [26].By definingq≡θ2−θ1; ω≡ω2−ω1; α0≡(α1+α2)/2; k≡(k12+k21)/2,
system (2.1) can be rewritten as(2.2)dqdt=f1(q,k)=ω−2ksinq,dkdt=f2(q,k)=μα0cosq−μk.The variables q and k represent the phase difference and the average
connection strength, respectively. Notice that fj(q,k)=fj(q+2π,k) (j=1,2). The formation of synchronized clusters in
networks of phase oscillators described by system (2.2) was investigated by
Seliger et al. [27].
In this section we study the asymptotic behaviors of this model.Synchronism occurs when both nodes oscillate in a
common frequency, which means thatdθ2/dt=dθ1/dt or θ2−θ1=q=constant.
Thus, synchronism can happen if there are constants (q*,k*) so that if (q(t),k(t))=(q*,k*) then dq/dt=0 and dk/dt=0 for all time t.
Notice that a synchronous solution is an equilibrium point (q*,k*) of the dynamical system (2.2). Such a solution
corresponds to an intersection point of the nullclines f1=0 (k=ω/(2sinq)) and f2=0 (k=α0cosq).The stability of an equilibrium point(q*,k*) can usually be determined by calculating the
eigenvalues λ of the Jacobian matrix corresponding to system
(2.2) linearized around this point. It is asymptotically stable when all
eigenvalues have negative real parts (e.g., [28]). For system (2.2), the eigenvalues λ are the roots of the
polynomial(2.3)λ2+(μ+2k*cosq*)λ+2μk*cosq*−2μα0(sinq*)2=0.According to
the Routh-Hurwitz criterion, both roots of the polynomial λ2+a1λ+a2=0 have negative real parts if a1>0 and a2>0 (e.g., [29]).Whenω=0,
the equilibrium points are P1=(0,α0), P2=(π/2,0), P3=(π,−α0), P4=(3π/2,0),
where P1 and P3 are asymptotically stable, and P2 and P4 are unstable.Whenω≠0,
there are also four equilibrium solutions if ω<α0.
The points with(2.4)k*=±α021+1−(ωα0)2are
asymptotically stable; the ones with(2.5)k*=±α021−1−(ωα0)2are unstable.
When ω=α0,
there are only two equilibrium points because two saddle-node bifurcations
(e.g., [28]) occur.
When ω>α0,
there is not synchronism; however, there is a limit cycle, which corresponds to
a closed and isolated trajectory in the state space q×k.For an autonomous two-dimensional system,
Poincaré-Bendixson theorem (e.g., [28]) ensures that there is an asymptotically stable limit
cycle in a region of the state space if the vector field (f1,f2) points inward everywhere on the boundary of
this region, which must not contain any equilibrium point. For system (2.2), it
is easy to verify that if ω>α0,
then there is such an attracting trajectory inside the region R,
where R is the rectangle given by R={(q,k):0≤q<2π,α0+(ϵ/μ)<k<−α0−(ϵ/μ)} for ϵ→0+.
In this case, k oscillates and q grows as the time goes by, as shown in Figure 2. Figure 1 illustrates the case with synchronous (equilibrium) asymptotic solution.Figure 1
Temporal evolution ofq(t) and k(t) obtained by numerically integrating system
(2.2). Parameter values: ω=1, α=2, μ=1.
Initial conditions: q(0)=0, k(0)=5.
In this case, the asymptotic solution corresponds to the equilibrium point (q*,k*)=(0.26,1.93).Figure 2
Temporal evolution ofq(t) and k(t) obtained by numerically integrating system
(2.2). Parameter values: ω=10, α=2, μ=1.
Initial conditions: q(0)=0, k(0)=5.
In this case, the asymptotic solution corresponds to the limit cycle, where k(t) oscillates with frequency B≃9.8 and amplitude A≃0.20,
and q(t) linearly grows with slope A.In order to analytically characterize the limit cycle,
suppose that the asymptotic behavior ofq(t) in this case is given by qasymp(t)≃Bt.
By inserting this approximated (linear) solution in the equation for dk/dt and integrating it, the asymptotic solution of k(t) is(2.6)kasymp(t)≃Acos(Bt−φ),where A=μα0/μ2+B2 and sinφ=B/μ2+B2.
By substituting this expression in the equation for dq/dt in (2.2), an equation for calculating B is found. In fact, B is the root of the polynomial(2.7)B3−ωB2+(μ2+μα0)B−ωμ2=0.Observe that
when ω≫1,
then the angular velocity B can be written as B≃ω+Δ,
with |Δ|≪1.
The value of Δ is estimated by(2.8)Δ≃−μα0ωω2+μ2+μα0.Consequently,
the oscillation amplitude A of kasymp(t) when ω≫1 is(2.9)A≃μα0ω.For instance, forω=10, α0=2,
and μ=1,
these approximated expressions give B≃9.8 and A≃0.20,
which are in good agreement with the numerical solutions of system (2.2) shown in
Figure 2.


## 3. Network for Detecting Figure Border

Our two-dimensional network for detecting figure
border is composed of units consisting of a pair of phase oscillators forced by
an external inputI.
In this network, the coefficients α1 and α2 can vary with the time. Thus, system (2.2) is
rewritten as(3.1)dqdt=I+ω−2ksinq,dkdt=μα(t)cosq−μk.Assume that a black-and-white figure will be presented
to this network, as illustrated in Figure3(a). The variable I represents the input for each unit according
to the following rules: I=0 corresponds to a white part of the figure; I=9 corresponds to a black part. Notice that the
input can be translated into a new natural frequency Ω by defining I+ω≡Ω.
Thus, system (3.1) is reduced to system (2.2) if α(t) is a constant.(a) Att=0,
this is the image presented to the network with 9×9 units. (b) At t=40,
the border of the image is dynamically determined.
(a)(b)Each unit is coupled with its four closest neighbors
(left, right, up, and down), which is usually known as four-neighborhood in image
processing literature (e.g., [30]) or von Neumann neighborhood in cellular automaton
literature (e.g., [31]). As a consequence of this coupling, the value of α(t) for each cell is given by(3.2)α(t)={α0if0≤t≤T,pα0ift>T.The parameter T is a settling time; p≡1+qx where q is a positive number (here q=14); x=G(y) where G(y)=0 if y=0;
and G(y)=1 if y≠0.
The value of y is obtained by(3.3)y=∏i=14[ki(T)−ki(T−δT)],where δT is the time step (here δT=0.01) of the integration method (here fourth-order
Runge-Kutta method) used for numerically solving the dynamical system. The
variable ki(t) corresponds to the average connection strength
of the closest neighbor i(i=1,2,3,4).
If the neighbor j does not exist, then kj(t)≡0.An image is presented to the network att=0.
For 0≤t≤T,
all the units behave as if they were isolated, because the value of α(=α0) for each unit is independent of the neighbor
activity. By taking for all units ω=1, α0=2, μ=1, T=20, q(0)=0,
and k(0)=5,
the units with I=0 will tend to a stationary solution and the
units with I=9 will tend to a limit cycle, as explained in
the last section. For these parameter values, the permanent regime was already
reached when t=T,
as shown in Figures 1 and 2.Att=T,
the value of α for each unit can be changed. Expressions (3.2)
and (3.3) imply what follows. If all of its four neighbors are in a limit cycle
(y≠0), then the value of α is increased from 2 to 30;
if at least one neighbor is in a stationary state or if the unit does not have
a complete von Neumann neighborhood (y=0), then the value of α is kept equal to 2.
The limit cycle is characterized by k(t) oscillating with frequency B and amplitude A,
and q(t) linearly growing with slope A,
as presented in the last section.Forα=30 and Ω=10, q(t) and k(t) tend to a stationary activity, which in our
model corresponds to the white color. Thus, at t=2T=40,
the unique units remaining in oscillatory activity (the black color) are the
ones in the figure border. Figure 3(b) shows the contour of Figure 3(a), after
the units have reached the (new) permanent regime.The higher the value ofμ is, the shorter the transitory phase will be.
Hence, border detection can be made faster by increasing the value of this
parameter.


## 4. Conclusion

Networks
governed by differential (e.g., [6–8, 10, 11]) or difference equations
(e.g., [9, 32]) have been employed for
image processing. Here we used a network with local and dynamical coupling for
identifying contour of black-and-white images. The unit of such a network is
the variant of the Kuramoto model [33] for two oscillators (neurons), which was proposed by
Seliger et al. [27].
In our scheme, after applying the input corresponding to the figure, the value
of α for each unit is switched or not, depending on
the neighborhood activity. This scheme could be implemented using first-order
PLLs. This investigation could be extended for higher-order PLLs, which could
be designed in order to shorten the transient.



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*Source: 127827-2008-06-23.xml*