I. INTRODUCTION

In recent years, mobile robotics has garnered significant

attention, particularly in developing control strategies for au-
tonomous navigation. A nonholonomic robot’s circumnaviga-
tion of a target is a challenging task, especially when the robot

has limited sensory capabilities [1]–[4]. Numerous possible

applications for robotic circumnavigation exist, including mil-
itary surveillance, search and rescue, object inspection, entrap-
ment and pursuit, and orbit maintenance. Recent literature has

proposed several approaches to tackle the challenge of guiding
mobile robots to unknown target between two boundaries.
Among these, [2] stands out as a significant contribution,
presenting a control scheme for guiding a mobile robot using
range-only measurements to surround an unknown target. The
paper introduces a novel state space formulation to convert the
circumnavigation problem into a stabilizing error system. The
literature review reveals various other approaches addressing
similar problems, each with its own unique contribution. For
example, [5] focuses on networked mobile agents localizing
and enclosing control of a moving target, taking into account
unknown and bounded target velocities. [6] introduces an
estimator-controller framework for a single robot to entrap
a static target using only bearing measurements, without

prior knowledge of their positions. [7] investigates multi-
agent circumnavigation control utilizing bearing rigidity the-
ory, addressing issues such as estimating the target’s distance

and velocity. Meanwhile, [8] proposes a control algorithm
for UAVs to circumnavigate unknown targets in GPS-denied

environments based on range-only measurements, suitable for
scenarios with limited data availability. Other papers explore
specialized scenarios and methodologies. For example, in [1],
a sliding-mode control law for Dubins-car-like mobile robots

navigating irregularly moving targets is discussed, emphasiz-
ing rigorous justification and performance through simulations.

[3] introduces an estimator-controller framework for circum-
navigation problems, with plans for future extensions to con-
sider non-holonomic motion constraints and three-dimensional

space. [9] proposed controller efficiently drives the robot to
the target, ensuring that it maintains the required distance and
speed while circling the target. In [10], a way is given to
make a group of quadratic strict Lyapunov functions that can

be used for both smooth and shaky sliding mode control algo-
rithms. This approach allows for optimizing the reaching time

estimation within the class of quadratic functions, offering
flexibility in algorithm design. The reaching time estimation
relations are useful for designing algorithm gains, ensuring
convergence within a pre-fixed time for a compact set of initial
conditions. [9] proposes an algorithm enabling a single agent
to circumnavigate a target at a predefined distance using only
distance measurements to the target. We establish stability for
both stationary and slowly drifting targets. [11] presents a
distributed control law for a team of AUVs, allowing them to
cooperatively enclose a target based on local information. The
AUVs achieve asymptotic convergence to a circumnavigation
formation if the target is stationary, and bounded convergence
if the target moves with a bounded, time-varying velocity.
This approach operates without a global coordinate frame
or common reference direction. [12] introduces a distributed
control law for nonholonomic agents to achieve collective
circular motions around a target. We can shape the collective

behavior and circumnavigation patterns of the multi-agent sys-
tem by adjusting control parameters. We provide a sufficient

condition for trajectory boundedness, enabling the design of
various circumnavigation patterns. [4] addresses the problem

of enclosing a target using a single robot and multiple coop-
erative robots, relying solely on local bearing measurements

commonly available through monocular vision systems. The
proposed control schemes require no communication among
robots and guarantee asymptotic convergence to the desired
stand-off orbit using Lyapunov stability criteria. Each paper
contributes valuable insights and methodologies to the broader
field of target circumnavigation, offering diverse perspectives
and addressing various challenges. However, [2] serves as a

focal point due to its comprehensive approach. This paper
tackles this challenge by introducing a unique control scheme
that solely guides a robot around an unknown target through
range measurements. The use of range-based measurements
to tackle the boundary traversing problem for nonholonomic
robotic systems has not been investigated in the literature.
We introduce the idea of the barrier Lyapunov function in
order to tackle this problem [13]–[17]. The logarithmic BLF
is a variation of BLF that has been used to satisfy output

constraints that can be either time-invariant [13] or time-
varying [16]. In this research, we achieve our formation control

goals by utilizing the concept of time-invariant logarithmic
BLF form [13]. This approach was previously tried in [14],

[15], [18], although the analysis there was bound by fixed-
distance constraints. However, in this paper, we consider the

time-invariant radius-based constraints which allow the robot
to move in a bounded radius formation.
Main Contribution: These are the main aspects of this work:
(i) Mobile Robot within Concentric Circular Boundaries Using
ABLF: A mobile robot’s movement is constrained between
two circular boundaries—an outer boundary and an inner
boundary. To smoothly and effectively control the robot’s

angular velocity ω(t), we employ a asymmetric barrier Lya-
punov function (ABLF). This function allows us to design a

control law that ensures the robot stays within these concentric
boundaries while maintaining a specific distance (ρ) from
a target point. The control law limits the robot’s angular
velocity based on initial conditions, enabling the robot to
follow a desired circular trajectory between the inner and outer
limits. As a result, the robot’s motion is well-controlled and
predictable within the predefined boundaries.
(ii) Control Design Strategy: Using the ABLF, we design a

stabilizing control law under certain initial conditions, ensur-
ing the robot achieves its desired trajectory between the inner

and outer boundaries. The system is shown to asymptotically
stabilize, following the prescribed circular trajectory using
the proposed controller. We further analyze the selection of
these inner and outer boundaries, providing a methodology
for determining them based on the ABLF.
(iii) Evaluating Control Input (ω) for Mobile Robot: This study
evaluates the angular velocity control values required for the
robot’s movement within the defined boundaries, comparing
the results with existing research [2]. In our approach, the
control input ω has minimal significance, meaning it has
a small impact on the overall control process. In practice,
such small control values enhance operational safety and
efficiency for mobile robots. They simplify implementation,
improve safety measures, and ensure consistent performance
while maintaining navigational accuracy. Experimental results
confirm the effectiveness of these angular velocity control
strategies for achieving precise navigation within the defined
boundaries, making them ideal for real-world applications.
The paper will be addressed as follows: Section II presents
the system and problem description. Section III introducing a
Lyapunov function.Section IV introducing a controller design

using composite Lyapunov function. Section V presents a sim-
X

φ
α
ρ
u

K

(x, y)

p ̄x
p ̄y

r

Y

T
(xT , yT ) ⋆

Fig. 1: Geometry representation of Mobile Robot.
ulation results in presence or absence of range measurements.
Finally, Section VI concludes the paper and presents some
future directions of work.
II. SYSTEM AND PROBLEM DESCRIPTION
Notation
• Let σ : R → R denote a group of smooth saturation
functions adhering to certain criteria: (i) σ(0) = 0
and sσ(s) > 0 for s ̸= 0; (ii) |σ(s)| ≤ M1 and
|dσ(s)/ds| ≤ M2, where M1 and M2 are positive
constants; (iii) R s
0
σ(τ )dτ → ∞ as s → ∞. Examples

include tanh(s), arctan(s), and s/√
1 + s
2.

System Dynamic
Consider the kinematics of a mobile robot defined by the
following equations:

x ̇(t) = u cos φ(t),
y ̇(t) = u sin φ(t),
φ ̇(t) = ω(t), (1)
where, (x(t), y(t)) is represent the robot’s position, φ(t) is
represent orientation, and u is represent the linear velocity in
a stationary reference frame. The control inputs to the system
is angular velocity ω(t). The problem with using nonlinear
techniques to construct control laws for a robot’s angular
velocity is that, as the robot gets closer to its destination, the
dynamic equation governing this connection becomes single,
which makes it challenging to design a controller that works
for every robot. Furthermore, stability analysis requires an
understanding of the bearing angle α. To address these issues,
we propose an alternative control scheme. We aim to overcome
singularities and simplify stability analysis by introducing a
virtual point K on the robot’s left side, maintaining a constant
distance from it, and ensuring a direction perpendicular to its

velocity. This will enhance the overall effectiveness of robotic
control systems. Using a geometric relationship, define the
position of the virtual point.
xk = x − ρ sin φ,
yk = y + ρ cos φ, (2)
where ρ is a specified positive constant representing the
desired distance. We define the position error between the
virtual point (K) and the target (T ) is
px = xk − xT ,
py = yk − yT . (3)
The time derivative of the above position error is
p ̇x =  ̇xk − x ̇ T = (u − ρω) cos φ,
p ̇y =  ̇yk − y ̇T = (u − ρω) sin φ. (4)
As illustrated in the figure, we introduce the transformed
coordinate errors p ̄x and p ̄y. To facilitate analysis, we apply

a globally invertible coordinate transformation. This transfor-
mation rotates the original coordinate system by an angle φ,

aligning the new coordinate axes with the desired directions.
The transformation is expressed mathematically as follows:


p ̄x
p ̄y

=

cos φ sin φ
− sin φ cos φ
 px
py

. (5)
Then, the dynamic equations for the transformed errors can
be calculated as

p ̄ ̇x = u − ρω +  ̄pyω,
p ̄ ̇y = −p ̄xω. (6)
Now the dynamic equation for the norm of the transformed
errors can be calculated as
p ̄
2
e =  ̄p
2
x +  ̄p
2
y
, (7)

The time derivative of above equation

p ̄ep ̄ ̇
e =  ̄pxp ̄ ̇x +  ̄pyp ̄ ̇y,
=  ̄px(u − ρω). (8)

Given the aforementioned equations,
r
2 = (x − xT )

2 + (y − yT )
2
= (x − xk + xk − xT )

2 + (y − yk + yk − yT )
2

= p
2
x + p
2
y + ρ
2 + 2ρ(px sin θ − py cos θ)

=  ̄p
2
x + ( ̄py − ρ)
2
,

and

rr ̇ =
d
dt 
1
2
r
2

= up ̄x. (9)
We define a kinematics (1) in a polar coordinate system.
That is,

r ̇ = u cos α,
α ̇ = ω −
u
r
sin α, (10)

where α is the bearing angle between the direction of the
robot’s linear velocity and the direction from the target to the
robot. The above equations is used to establish some of the
theoretical results in this paper.
Problem Statement

Problem. We address the problem of directing a nonholo-
nomic robot to orbit an unknown target. The robot initially

starts between an outer boundary and an inner boundary. Our
goal is to design a control law for the angular velocity ω(t)
using an ABLF that guarantees main objective:
• The robot’s trajectory is confined within two circular
boundaries: the first is inner boundary has radius rmin,
and the second is outer boundary has radius rmax. The
control objective is for the robot’s trajectory to satisfy
the constraint rmin < r(t) < rmax at all times, where r(t)
is the distance of the robot from the origin. The robot
should converge to a desired circular path with radius ρ,
r(t) → ρ as t → ∞.
By achieving these objectives, we ensure that the robot
can effectively and reliably orbit the unknown target while
adhering to the desired distance and velocity constraints.
Assumption 1. We assume that the initial position of the
mobile robot is located between the defined inner boundary
rmin and outer boundary rmax.
Remark 1. In the problem statement, the control objective

is achieved by manipulating the angular velocity ω(t). Mean-
while, the linear velocity u can be chosen freely.

Barrier Lyapunov Function
Definition 1. (Barrier Lyapunov Function (BLF) [13]): A
Barrier Lyapunov Function (BLF) is a scalar function V (x)
defined for the state vector x ∈ D of the system x ̇ = f(x),on
an open region D that includes the origin. This function is
continuous, positive definite, and has continuous first-order
partial derivatives everywhere in D. It has the property that
V (x) approaches infinity as x nears the boundary of D.
Additionally, it satisfies V (x(t)) ≤ β for all t ≥ 0 along
the solution of x ̇ = f(x) for x(0) ∈ D, with some positive
constant β.
Lemma 1. (see [13]): Consider any positive constants ka and
kb, and define the sets Z := {ξ ∈ R : −kb < ξ < ka} ⊂ R
and N := R
l × Z ⊂ R
l+1, which are open. Now, consider

the system η ̇ = h(t, η), where η := [w; ξ]

T ∈ N, and

h : R
+ × N → R

l+1 is piecewise continuous in t and

locally Lipschitz in η, uniformly in t, on R

+ × N. Assume

there exist functions Y : R
l → R
+ and Vc : Z → R
+,
which are continuously differentiable and positive definite in
their respective domains, with Vc(ξ) → ∞ as ξ approaches
either −kb or ka. Additionally, suppose there exist class K∞
functions γ1 and γ2 such that γ1(∥w∥) ≤ Y (w) ≤ γ2(∥w∥).
Define V (η) := Vc(ξ) + Y (w) and assume ξ(0) ∈ Z. If
V ̇ = (∇V )
T h ≤ 0 holds in the set ξ ∈ Z, then ξ(t) ∈ Z
for all t ∈ [0, ∞). In the following sections, we use Lemma 1
to establish some theoretical results presented in this paper.

III. LYAPUNOV FUNCTION
A. Construction of Quadratic Lyapunov Function
In controller design, establishing a robotic system’s stability
typically involves choosing a quadratic Lyapunov function
[2]. This approach is later useful in demonstrating that the
composite Lyapunov function is negative semi-definite. The
quadratic Lyapunov function could take the form:

V1 =
1
2
p ̄
2
e
, (11)
We first compute the derivative of V1 with respect to time,

V ̇
1 = p ̄ ̇
ep ̄e,

By utilizing the system equations outlined in (8) along with
the relationship provided in (11), the expression for V ̇
1 can be

derived as follows:
V ̇
1 =  ̄px(u − ρω). (12)
The quadratic Lyapunov function mentioned above has been
commonly used in prior research [2]. However, the problem
in which the robot’s trajectory is confined within two circular
boundaries: an outer boundary and an inner boundary. is not
taken into consideration in the prior research [2]. In next
section we introduce the barrier Lyapunov function, which is
used to overcome this problem.
B. Construction of Barrier Lyapunov Function
The asymmetric logarithmic barrier Lyapunov function
(ABLF) [5], [8] is the basis for the control law that is proposed
in this section. Define the distance error ze as

ze = r − ρ, (13)

after differentiate the above ze, we get

z ̇e =  ̇r. (14)
For the minimization of the distance error ze, we exploit the
concept of the ABLF, defined as follows:
V2 =
Q(ze)
2ρk1
log 
k
2
a
k
2
a − z
2
e

+
(1 − Q(ze))
2ρk1
log 
k
2
b
k
2
b − z
2
e

,
(15)
where log denotes the natural logarithm and the value of logic
parameter Q(ze) is a defined as:
Q(ze) = (
1, if ze > 0,
0, if ze ≤ 0.

Here, k1, ka and kb are positive constants. The values of ka >
0 and kb > 0 can be chosen based on the specific nature of
the problem. To simplify the expression, the function V2 is
rewritten as:
V2 =



1
2ρk1
log 
k
2
a
k2
a−z
2
e

, if 0 < ze < ka

1
2ρk1
log 
k
2
a
k
2
b−z
2
e

, if − kb < ze ≤ 0.
(16)
For −ka < ze < kb, the function V2 is non-negative
and equals zero only when ze = 0, indicating that V2 is
positive definite. Additionally, V2 is piecewise smooth over
the intervals ze ∈ (−ka, 0) and ze ∈ (0, kb). Furthermore,
since limze→0+
dV2
dze
= limze→0−
dV2
dze
= 0, we can conclude
that V2 in a set ze ∈ (−ka, kb). To demonstrate the properties
of V2, we first calculate its derivative with respect to time:
V ̇
2 =
Q(ze)
ρk1
zez ̇e
k
2
a − z
2
e
+
(1 − Q(ze))
ρk1
zez ̇e
k
2
b − z
2
e
.

Using the system equations described by (6) and the relation-
ship given in (9,14), we can derive the expression for V ̇

1 as

follows:
V ̇
2 =
Q(ze)
ρk1
(r − ρ)
up ̄x
r
k
2
a − z
2
e
+
(1 − Q(ze))
ρk1
(r − ρ)
up ̄x
r
k
2
b − z
2
e
. (17)
C. Construction of Composite Lyapunov Function
In this paper, we propose a composite Lyapunov function
that combines an asymmetric barrier Lyapunov function with
a traditional quadratic Lyapunov function to enhance stability
analysis in robotic system. The Composite Lyapunov Function
is given by

V = V1 + V2. (18)
The derivative of V with respect to time is initially computed.

V ̇ = V ̇
1 + V ̇
2.

We put the value of V ̇
1 and V ̇
2 in above equation,

V ̇ =
Q(ze)
ρk1
(r − ρ)
up ̄x
r
k
2
a − z
2
e
+
(1 − Q(ze))
ρk1
(r − ρ)
up ̄x
r
k
2
b − z
2
e
+  ̄px(u − ρω). (19)
This expression shows the rate of change of the Lyapunov
function V in terms of the state variable p ̄x, the linear velocity
u, and the control input ω. By designing ω appropriately, we
can ensure that V ̇

is negative definite, which implies that the
Lyapunov function V decreases over time. Consequently, the
state variables ( ̄px, p ̄y) will converge to zero, ensuring the
desired control objective is achieved.
IV. CONTROLLER DESIGN

This section proposes a control law based on the symmetric
logarithmic barrier Lyapunov function (SBLF) [5], [8]. We
considered two strategies based on the availability of the range
rate:
A. Availability of the range rate
Theorem 1. The system (6) under the consideration of
controller
ω =
Q( ̄pe)
ρ

u +
(r − ρ)
u
r
ρk1(k
2
a − z
2
e
)
+ uσ(rr ̇)

+

(1 − Q( ̄pe))
ρ

u +
(r − ρ)
u
r
ρk1(k
2
b − z
2
e
)
+ uσ(rr ̇)

,
(20)

where k1, ka and kb is any positive constant. Then origin of
the closed-loop system is asymptotically stable with constraint
ze is −kb < ze < ka.
Proof. Substitute the value of design controller ω in the system
equation(6), if transformed error term 0 < ze < ka the
behaviors of the system are:
p ̄ ̇x =
up ̄y
ρ
−

1 −
p ̄y
ρ
  (r − ρ)
u
r
ρk1(k
2
a − z
2
e
)
+ uσ(up ̄x)

,

p ̄ ̇y = −
p ̄x
ρ

u +
(r − ρ)
u
r
ρk1(k
2
a − z
2
e
)
+ uσ(up ̄x)

. (21)
Similarly, when transformed error term −kb < ze ≤ 0 ,
p ̄ ̇x =
up ̄y
ρ
−

1 −
p ̄y
ρ
  (r − ρ)
u
r
ρk1(k
2
b − z
2
e
)
+ uσ(up ̄x)

,

p ̄ ̇y = −
p ̄x
ρ

u +
(r − ρ)
u
r
ρk1(k
2
b − z
2
e
)
+ uσ(up ̄x)

. (22)
Substitute the value of design controller ω (20) in (18), while
the transformed error term 0 < ze < ka than V ̇

is obtained,

V ̇ =
1
ρk1
(r − ρ)
up ̄x
r
k
2
a − z
2
e
+  ̄px

u − ρ

1
ρ
(u

+
(r − ρ)
u
r
ρk1(k
2
a − z
2
e
)
+ uσ(up ̄x)
 .
(23)
Similarly, substitute the value of design controller ω in (18),
while the transformed error term −kb < ze ≤ 0 than V ̇
is

obtained,
V ̇ =
1
ρk1
(r − ρ)
up ̄x
r
k
2
b − z
2
e
+  ̄px

u − ρ

1
ρ
(u

+
(r − ρ)
u
r
ρk1(k
2
b − z
2
e
)
+ uσ(up ̄x)
 .
(24)
Further, simplification of the (23) and (24) equation shows
that origin of closed-loop system is stable

V ̇ = −up ̄xσ(up ̄x) ≤ 0. (25)

In the above Lyapunov function V , which is positive defi-
nite. Let’s say that V ̇

is negative semi-definite. For show the
asymptotically stable, we use the lasalle’s invariance principle,
to prove asymptotic stability, we need to find the set where
V ̇ = 0, i.e., the set where:

up ̄xσ(up ̄x) = 0

This implies that p ̄x = 0 and u ̸= 0. Using this, we
analyze the system given by equations (21, 22), leading to

the conclusion that p ̄y = 0. Then origin of the closed-
loop system is asymptotically stable with constraint ze is

−kb < ze < ka.

B. No availability of the range rate
The controller (20) achieves target circumnavigation, but
its implementation requires explicit differentiation of distance
measurements(range rate) and may increase noise. We must
consider the scenario where only distance is available. To this
purpose, we designate the output of the system (3) as y =
0.5r
2
and suggest a dynamic output feedback controller:
ω =
Q( ̄pe)
ρ

u +
(r − ρ)
u
r
ρk1(k
2
a − z
2
e
)
+ uσ(−k2z + y)

+

(1 − Q( ̄pe))
ρ

u +
(r − ρ)
u
r
ρk1(k
2
b − z
2
e
)
+ uσ(−k2z + y)

,
z ̇ = −k2z + y, (26)
Where k1 and k2 are constants with arbitrary positive values.
We let φ = −k2z + y to make analysis easier.
Now calculate the time derivative of φ
φ ̇ = −k2z ̇ + rr. ̇

Substituting the z ̇ from (26) and rr ̇ from (9) into the above
time derivative expression, finally we get:
φ ̇ = −k2φ + up ̄x.

Remark 2. The primary concept behind the control law (26)
is to employ an observer that uses only the distance and
the dynamic equation of z to estimate rr ̇. This estimate, φ,
then eliminate rr ̇ in equation (9). While the equation for z
resembles a high-gain reduced-order observer, it has notable
differences: the gain k2 does not need to be very large.
Additionally, the output feedback controller ensures the global
stability of the closed-loop system.

Now substituting the equation (26) into (6)and If trans-
formed error term 0 < ze < ka than then we get:

p ̄ ̇x = −dk1uσ(up ̄x) + 
−k1uσ(φ)
k
2
a − p ̄
2
k
−
u
ρ

p ̄y,

p ̄ ̇y =

k1uσ(φ)
k
2
a − p ̄
2
k
+
u
ρ

p ̄x,

φ ̇ = −k2φ + up ̄x. (27)
If transformed error term −kb < ze ≤ 0 than system equation
(6),
p ̄ ̇x = −dk1uσ(up ̄x) + 
−k1uσ(φ)
k
2
b − p ̄
2
k
−
u
ρ

p ̄y,

p ̄ ̇y =

k1uσ(φ)
k
2
b − p ̄
2
k
+
u
ρ

p ̄x,

φ ̇ = −k2φ + up ̄x. (28)
Theorem 2. The origin of the closed-loop system (27,28) is
asymptotically stable with constraint ze is −kb < ze < ka.
Proof: Consider a composite Lyapunov function candidate:
Vr( ̄ex, e ̄y, φ) = V +
Z φ
0
σ(τ ) dτ. (29)

The time derivative of Vr, evaluated along the trajectory of
system (27) is:
V ̇
r( ̄ex, e ̄y, φ) = V ̇ +
d(
R φ
0
σ(τ ) dτ )
dt . (30)

After the substitute the value of V ̇

in above equation

V ̇
r =
Q(ze)
ρk1
(r − ρ)
up ̄x
r
k
2
a − z
2
e
+
(1 − Q(ze))
ρk1
(r − ρ)
up ̄x
r
k
2
b − z
2
e
+  ̄px(u − ρω) +  ̇φσ(φ). (31)
Substitute the value of design controller ω (26) in (31),
while the transformed error term 0 < ze < ka than V ̇
is

obtained,
V ̇
r =
1
ρk1
(r − ρ)
up ̄x
r
k
2
a − z
2
e
+  ̄px

u − ρ

1
ρ
(u

+
(r − ρ)
u
r
ρk1(k
2
a − z
2
e
)
+ uσ(φ)
 +  ̇φσ(φ). (32)
Similarly, put the value of design controller ω (26) in (31),
while the transformed error term −kb < ze ≤ 0 than V ̇
is

obtained,
V ̇
r =
1
ρk1
(r − ρ)
up ̄x
r
k
2
b − z
2
e
+  ̄px

u − ρ

1
ρ
(u

+
(r − ρ)
u
r
ρk1(k
2
b − z
2
e
)
+ uσ(φ)
 +  ̇φσ(φ). (33)

Further, simplification of the (32) and (33)

V ̇
r = −k2φσ(φ) ≤ 0.

This confirms that the Lyapunov function is decreasing,
thereby ensuring the stability of the system.In the above
Lyapunov function V , which is positive definite. Let’s say
that V ̇
is negative semi-definite. For show the asymptotically
stable, we use the lasalle’s invariance principle, to prove
asymptotic stability, we need to find the set where V ̇ = 0,
i.e., the set where:

φσ(φ) = 0

This implies that φ = 0. Using this, we analyze the system
(27, 28), leading to the conclusion that p ̄x = 0, p ̄y = 0. Then
origin of the closed-loop system is asymptotically stable with
constraint ze is −kb < ze < ka.
Theorem 3. Under the conditions given in Theorem 2, the
following statements hold:
(a) The distance error ze = r − ρ satisfy:
−Υb ≤ ze ≤ Υa,
Υa = ka
p
1 − e−2ρk1Vr(0)
,

Υb = kb
p
1 − e−2ρk1Vr(0)

, (34)
(b) The transformed error term remains bounded by

|pe| ≤ p
2Vr(0) (35)

Proof. (a) According to Theorem 2, we have:Vr ≤ Vr(0)using
this, along with equation (16) , the ABLF term of composite
Lyapunov function (29) of the satisfies the inequality:
Vr(0) ≥



1
2ρk1
ln 
k
2
a
(t)
k2
a−z
2
e

, if 0 < ze < ka,

1
2ρk1
ln 
k
2
b
(t)
k
2
b−z
2
e

, if − kb < ze ≤ 0,
(36)

Taking the exponential of both sides gives:
z
2
e ≤
(
k
2
a
1 − e
−2ρk1Vr(0)
, if 0 < ze < ka,

k
2
b
1 − e
−2ρk1Vr(0)
, if − kb < ze ≤ 0.
(37)
Taking the square root of these inequalities yields:
ze ≤
(
ka
√
1 − e−2ρk1Vr(0)

, if 0 < ze < ka,

−kb
√
1 − e−2ρk1Vr(0)

, if − kb < ze ≤ 0.
(38)

Since it is clear that:

−Υb ≤ ze ≤ Υa, (39)

where Υa and Υb are defined in equation (34).
(b) Applying the same argument as in part (a) for the quadratic
Lyapunov function in equation (29), we deduce:

1
2
p
2
e ≤ Vr(0) ⇒ pe ≤
p
2Vr(0). (40)

thus completing the proof.
V. SIMULATION

We develop the stabilizing control law using the asymmetric

barrier Lyapunov function (ABLF), which describes the barri-
ers in terms of the time-invariant distance functions from the

target.
The example shows how to apply control law (20) using range
rate to successfully complete target circumnavigation. The
robot’s starting parameters are (1.05, 0.20, 90◦

). The follow-
ing are the parameters for the simulation: target coordinates

(xt, yt) at (0, 0), robot velocity u at 0.5, desired distance
ρ at 0.75, function σ(s) as s, control gains k1 is set to 3,
respectively. The inner circular boundary rmin is 0.5 and outer
circular boundary rmax is 1.4 taken in Figure 2(a) demonstrates
that the robot smoothly avoids the inner boundary, maintaining
a safe distance from the target at all times. The trajectory
converges to a desired path within the allowable circular
region, confirming the effectiveness of the control strategy in
maintaining bounded region. Figure 2(b) shows the distance
traveled by the robot and the target over time, as well as
the control inputs that were applied. Figure 2(c) shows that
the proposed dynamic output feedback controller is both
computationally efficient and able to keep the robot stable.
Figure 2(d) shows the distance tracking error for the first initial
condition in the presence of range measurement uncertainty
and shows that the tracking error converges to zero. This
illustrates the robustness of the control method against ongoing
variations in distance measurements.
The example demonstrates the application of control law
(26) not presence in range rate. Starting with parameters

-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5

(a) Trajectory of robot.

0 20 40 60 80 100
0.4
0.6
0.8
1
1.2
1.4

(b) Evolution of distance.

0 20 40 60 80 100
-1
-0.5
0
0.5
1
1.5

(c) Evolution of the control .

0 20 40 60 80 100
-0.2
-0.1
0
0.1
0.2
0.3
0.4

(d) Evolution of the error.
Fig. 2: Robot trajectory, evolution of distance, control input, and error over time (presence of range rate).

-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5

(a) Trajectory of robot.

0 20 40 60 80 100
0.4
0.6
0.8
1
1.2
1.4

(b) Evolution of distance .

0 20 40 60 80 100
-0.5
0
0.5
1
1.5

(c) Evolution of the control .

0 20 40 60 80 100
-0.2
-0.1
0
0.1
0.2
0.3
0.4

(d) Evolution of the error.
Fig. 3: Robot trajectory, evolution of distance, control input, and error over time (absence of range rate).
(1.05, 0.20, 90◦

), the robot aims to reach the target at (0, 0)
with a velocity u = 0.5 and a desired distance ρ = 0.75. The
function σ(s) = s is used with control gains k1 = 15 and

k2 = 2. The inner circular boundary rmin is 0.5 and outer cir-
cular boundary rmax is 1.4 taken in figure 3(a) shows the robot

maintaining a safe distance and following the desired circular
path. Figure 3(b) plots the robot’s and target’s distances over
time, along with the control inputs. Figure 3(c) highlights the
controller’s efficiency and stability, while Figure 3(d) confirms
that the tracking error converges to zero, demonstrating robust
and reliable performance.
A. Comparison
In this study, we compare our simulation results with
previous research [2] using the same initial parameters. Our
analysis focuses on the trajectory of a robot as it traverses
both inner and outer circular boundaries under identical initial
conditions. As illustrated in figure 5(a), the robot crosses the
inner circular boundary. In contrast, our simulation results,
presented in figures 2(a) and 3(a), show that the robot remains
confined within both the inner and outer circular boundaries.
Similarly, when a different starting point (1.15, 0.20, 25◦
),
the robot aims to reach the target at (0, 0) with a velocity
u = 1.0 and a desired distance ρ = 0.75. The function
σ(s) = s is used with control gains k1 = 3, k2 = 4, inner
circular boundary rmin is 0.5 and outer circular boundary rmax
is 1.4 taken, the robot in figure 5(a) crosses both boundaries.

However, the robot in figure 4(b) does not cross the bound-
aries, regardless of whether a range rate is applied, even though

the initial conditions remain the same.

Further analysis of the time-dependent distance between the
robot and the target, shown in figure 5(b), reveals that the
robot crosses the inner circular boundary, whereas in Figure
(2(b),3(b)), it does not. Similarly, taken different starting point
(1.15, 0.20, 25◦

), the robot aims to reach the target at (0, 0)
with a velocity u = 1.0 and a desired distance ρ = 0.75.
The function σ(s) = s is used with control gains k1 = 3,
k2 = 4, inner circular boundary rmin is 0.5 and outer circular
boundary rmin is 1.4 taken, in figure 5(d), robot cross the inner
and outer circular boundaries, while in figure 4(b) in presence
of range rate, 4(d) in absence of range rate, neither boundary
is breached.

This comparison illustrates the impact of the chosen Asym-
metric Barrier Lyapunov Function (ABLF) on the robot’s

trajectory. Moreover, it confirms that the mobile robot remains
within the bounded region and does not cross outer and inner
boundary, adhering to the bounded constraints imposed by the
selected ABLF.
B. Experimental Result
To assess the effectiveness of the proposed controller
(20,26) in practical world scenario, we performed experiments
using a Khepera IV1 differential-drive ground robot. The
experiment was conducted with the help of a motion capture
(MoCap) validation setup that employed overhead cameras.
The MoCap system, operated using Qualisys Track Manager

(QTM) software, recorded and processed data to provide accu-
rate feedback on the robot’s position and orientation, including

both its location and heading angle. Control commands were
relayed to the Khepera IV robot via the Robot Operating
System (ROS) based on position data measured by QTM.

-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5

(a) Trajectory of robot.

0 10 20 30 40 50
0.4
0.6
0.8
1
1.2
1.4

(b) Evolution of distance.

-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5

(c) Trajectory of robot.

0 20 40 60 80 100
0.4
0.6
0.8
1
1.2
1.4

(d) Evolution of distance.
Fig. 4: Robot trajectory, evolution of distance(presence of range rate) and robot trajectory, evolution of distance(absence of
range rate).

-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5

(a) Trajectory of robot.

0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6

(b) Evolution of distance .

-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5

(c) Trajectory of robot.

0 10 20 30 40 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8

(d) Evolution of distance.

Fig. 5: Robot trajectory, evolution of distance [2].

(a) Trajectory of robot (t = 21sec.). (b) Trajectory in QTM.

0 20 40 60 80 100
0.4
0.6
0.8
1
1.2
1.4

(c) Evolution of distance .

0 20 40 60 80 100
-1.5
-1
-0.5
0
0.5
1
1.5

(d) Evolution of the control.

Fig. 6: Experimental results for Khepera IV robot in presence of range rate.

(a) Trajectory of robot (t = 26sec.). (b) Trajectory in QTM.

0 50 100 150
0.4
0.6
0.8
1
1.2
1.4

(c) Evolution of distance .

0 50 100 150
-1.5
-1
-0.5
0
0.5
1
1.5

(d) Evolution of the control.

Fig. 7: Experimental results for Khepera IV robot in absence of range rate.

The robot started from an initial position of x(0) = 1.05 m
and y(0) = 0.20 m with an initial heading angle of θ(0) =
90◦
. Since the Khepera IV operates with a differential drive,

the linear and angular velocities of the unicycle model were
transformed into the individual wheel speeds, vr and vl, of the
Khepera IV using the following equations: vr = v+
dw
2

ω for

the right wheel and vl = v −
dw
2

ω for the left wheel. where
dw = 10.54 cm is the distance between the robot’s wheels, and
v and ω denote the linear and angular velocities, respectively.
The robot was operated at a constant linear speed v = 0.5 m/s,
consistent with the hardware constraint of a maximum wheel
speed of 0.814 m/s.
In presence of range rate, Figure 6(a) presents a top-down
view of the experimental result followed by the robot at
t = 21 seconds. The robot path was captured throughout the
experiment using QTM, as shown in Figure 6(b). Additionally,
Figure 6(c) shown the distance from the target and Figure 6(d)
illustrates the control law (ω).
Similarly in absence of range rate, the top-down view of
the experimental result shown in Figure 7(a) followed by the
robot at t = 26seconds. Figure 7(b) shown the robot path
captured throughout the experiment using QTM. Additionally,
Figure 7(c) shown the distance from the target and Figure 7(d)
illustrates the control law (ω). For a visual illustration of
the experimental results, please refer to the following video
links: https://youtu.be/xK9oiRWVsoo for presence of range

rate measurement and https://youtu.be/eJEcr8-PU4M for ab-
sence of range rate measurement.

VI. CONCLUSION

This paper addresses the challenge of guiding a nonholo-
nomic mobile robot around an unknown target in presence

or absence of range measurements. The proposed control
approach employs an asymmetric barrier Lyapunov function
to develop a dynamic output feedback controller, ensuring the
robot’s trajectory stays within two circular boundaries. The
proposed controller guarantees that the distance between the
robot and the target asymptotically reaches a desired value,
without needing information about the target’s position or
orientation. Our approach transforms the circumnavigation
task into a stabilizing error system, ensuring global uniform
stability of the closed-loop system. Numerical simulations and
real-world experiments validate the effectiveness, simplicity,
and computational efficiency of the proposed method. Future
research will focus on extending this strategy to distributed
control algorithms for multi-robot circumnavigation of moving
targets using range-only measurements.

please write above content in below format and heading way
\begin{center} 
{\huge\textbf{\ttitle}}\\
\end{center}
\section{Introduction and background}

Cardiovascular disease, primarily attributable to coronary artery disease, is a leading cause of mortality worldwide \cite{auto}. It has the attributes of significant morbidity and mortality, contributing to almost one fourth of annual global fatalities \cite{epid}. Due to the swift advancement of the contemporary economy, individuals are facing escalating pressure in various domains. Consequently, the incidence of cardiovascular disorders is continuously surging, posing a grave menace to human well-being \cite{global}. Coronary heart disease can be diagnosed by invasive techniques. The primary approach involves inserting the pressure guide wire through the brachial artery into the heart and assessing the disease by computing the fractional flow reserve (FFR) number. FFR stands for fractional flow reserve, which measures the pressure after an injury in relation to the pressure before the damage \cite{ffr}.

In cardiovascular medicine, Intravascular ultrasonography (IVUS) \cite{ivus} has become a valuable technique, providing detailed cross-sectional pictures of coronary arteries with excellent resolution. IVUS is an essential imaging technique in interventional cardiology because to its ability to provide valuable information, namely in evaluating the structure of blood artery walls and detecting atherosclerotic plaques.

Although IVUS offers intricate visual representations, the manual examination of these images is a laborious and stressful process that requires skilled interpretation. Image segmentation is a crucial step in getting valuable information from IVUS data. It involves dividing images into separate areas or components. Precise segmentation allows clinicians to measure and describe pathological characteristics, assisting in the accurate diagnosis and planning of treatment for cardiovascular disorders.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{raw_ai01400006.jpg}
        \caption{}
        \label{fig:image1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{Figures/ai02600030.jpg}
        \caption{}
        \label{fig:image2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.3\textwidth}
        \centering
        \includegraphics[width=\textwidth]{Figures/ai02701146.jpg}
        \caption{}
        \label{fig:image3}
    \end{subfigure}
    \caption{IVUS Images of different patients}
    \label{fig:both_images1}
\end{figure}

Although IVUS has great promise, the manual analysis of its images presents notable difficulties, such as subjectivity, variability across observers, and the time-consuming nature of the process. As shown in the images, there is a need for experts to accurately diagnose. Automated and precise picture segmentation techniques are necessary to assist healthcare practitioners in quickly extracting clinically important information. The deep learning-based segmentation methodology suggested in this work directly tackles these problems and actively contributes to the continuous efforts aimed at improving the usefulness of IVUS in cardiovascular medicine.

This study focuses on the urgent requirement for sophisticated picture segmentation techniques in the field of IVUS analysis. Conventional segmentation methods encounter difficulties due to the complex characteristics of vascular systems, differences in shape, and the presence of artifacts such as plaques and speckle noise. In order to address these difficulties, we rely on the auspicious domain of deep learning, particularly utilizing convolutional neural networks (CNNs) to autonomously acquire hierarchical characteristics straight from unprocessed IVUS data.



\clearpage
\section{Problem definition and Objective}
Identifying coronary artery disease (CAD) from intravascular ultrasound (IVUS) images poses significant challenges. Firstly, precisely delineating vessel boundaries and differentiating between the vessel wall and plaque regions is difficult due to noise, artifacts, and variability in the imaging process. Secondly, understanding the extent of plaque burden, i.e., the overall plaque quantity and distribution in the arteries, is crucial for disease assessment. However, accurately measuring and quantifying the total plaque burden is complex from IVUS images alone.

Plaque burden plays a pivotal role in CAD assessment as it reflects the severity and progression of the disease. High plaque burden indicates increased arterial narrowing, which correlates with higher risks of heart-related events. Image segmentation in IVUS images is critical to addressing these challenges. 

1. Boundary Detection: Accurate segmentation delineates vessel boundaries and plaque regions, enabling more precise measurements of plaque burden and vessel dimensions.

2. Plaque burden Assessment: Segmenting and quantifying the volume and distribution of plaques facilitates a comprehensive assessment of plaque burden, aiding in risk stratification and treatment planning.

 

\clearpage
% \begin{figure}[h]
%     \centering
%     \includegraphics[width=0.25\textwidth]{Figures/IITJlogo.jpg}
%     \caption{\label{fig:myfig1}Our logo}
% \end{figure}

% \begin{figure}[h]
%     \centering
%     \begin{subfigure}{0.45\textwidth}
%         \centering
%         \includegraphics[width=\textwidth]{Figures/IITJlogo.jpg}
%         \caption{Caption for the first image.}
%         \label{fig:subfig1}
%     \end{subfigure}
%     \begin{subfigure}{0.45\textwidth}
%         \centering
%         \includegraphics[width=\textwidth]{Figures/IITJlogo.jpg}
%         \caption{Caption for the second image.}
%         \label{fig:subfig2}
%     \end{subfigure}
%     \caption{Main caption for the figure.}
%     \label{fig:myfig}
% \end{figure}

\clearpage
\section{Literature survey}

There are different algorithms for image segmentation that achieve state-of-the art performance. Each algorithm contains some feature extraction process. There are different algorithms available for density estimation. These techniques are divided based on the technique they use to predict plaque regions.

\subsection{Image Segmentation }
Image segmentation is a computer vision technique used to partition an image into meaningful segments or regions based on certain characteristics, such as color, intensity, texture, or boundaries. The goal is to simplify and/or change the representation of an image into something more meaningful and easier to analyze.

\subsubsection{Region based Segmentation}
This method involves grouping pixels associated with an object, ensuring the enclosed area for segmentation is complete without any gaps due to absent edge pixels. It identifies boundaries for segmentation, considering at least one pixel related to the region at each step. After detecting color and texture changes, the edge flow is transformed into vectors, which are then used to detect further segmentation.

\subsubsection{Edge Based Segmentation}
Segmentation can utilize edge detection methods like gradient, log, canny, sobel, laplacian, and robert. This process involves identifying boundaries to segment the image by detecting edges, which pinpoint image discontinuities. For classification purposes, support vector machines leverage both fixed and adaptive features.

\subsubsection{Feature based clustering }

Feature-based clustering \cite{multifeature} for image segmentation involves grouping image pixels based on their feature similarities. These features can include color, texture, intensity, or other visual characteristics. Techniques like k-means or mean shift clustering assign pixels into clusters by iteratively optimizing centroids or grouping pixels with similar feature representations. In image segmentation, this method partitions the image into distinct regions based on shared features, enabling the identification of different structures or objects within the image.

\subsubsection{Binary Segmentation }
This categorizes each pixel in an image into one of two classes or regions, typically representing foreground and background. It's commonly used to isolate specific features, like tumors or organs, by distinguishing them from the surrounding areas. Techniques such as thresholding, contour-based methods, or deep learning models like U-Net, enable accurate delineation of these distinct regions, aiding in medical diagnosis, object detection, and various image analysis tasks where binary separation is essential.

\subsubsection{Multiclass Semantic Segmentation }

Multiclass semantic segmentation \cite{multiclass} involves categorizing pixels in medical images into multiple classes or categories, allowing precise identification of various structures or abnormalities. In medical imaging, this technique distinguishes not only between foreground and background but also between different tissues, organs, or pathologies, aiding in comprehensive diagnostics. Utilizing deep learning models like DeepLabV3 \cite{deeplab}, PSPNet \cite{pspnet}, or UNet++ \cite{unet++}, multiclass segmentation enables accurate delineation of intricate structures, facilitating detailed analysis, treatment planning, and disease assessment in medical imaging.


\subsection{Class imbalance}
Class imbalance in image segmentation poses challenges where certain pixel classes are underrepresented in training data. This skews neural networks, causing overfitting to small structures like tumors, leading to poor generalization. In semantic segmentation, this arises from natural class occurrences. Techniques like data resampling, cost-sensitive loss functions, and ensemble learning aid in tackling this issue by balancing class representation and optimizing model performance.

\subsubsection{Class imbalance using a U-Net }
Addressing class imbalance \cite{classIm}in image segmentation involves techniques like data augmentation, class weighting, oversampling, focal loss, and ensemble methods. These approaches aim to balance dataset representation, adjust loss functions, and combine models, mitigating the impact of underrepresented classes and improving segmentation accuracy.

\clearpage





\section{Methodology}
\subsection{Dataset Collection}
IVUS (Intravascular ultrasound) was conducted following standard procedures employing an automated transducer pullback at a speed of 0.5 mm/s. This was accomplished using a commercially available imaging system operated by an expert in this specific field. Initially saved in ”.iq” format, the dataset is extracted to ”.jpeg” with QIvus Research Edition 3.1. The processed data consists of cross-sectional artery images and the annotation of vessel and lumen with red and green boundaries made by experts using the cardiac imaging software Medis. This dataset is from AIIMS Rishikesh and currently consists of 3 patients (having 1114+885+274 frames). The training and validation set consists of 1608 and 210 frames, respectively, for two patients, and the test set consists of 276 frames for a single patient. The annotated data by experts consisted of boundaries denoting the plaque region from which masks were generated.

% \begin{figure}[h]
%     \centering
%     \begin{subfigure}[b]{0.45\textwidth}
%         \centering
%         \includegraphics[width=0.5\textwidth]{raw_ai01400006.jpg}
%         \caption{Raw Image}
%         \label{fig:img1}
%     \end{subfigure}
%     \hspace{0.1cm} % Adjust the space between images here
%     \begin{subfigure}[b]{0.45\textwidth}
%         \centering
%         \includegraphics[width=0.5\textwidth]{annot_ai01400006.jpg}
%         \caption{Annotated Image}
%         \label{fig:img2}
%     \end{subfigure}
%     \caption{Annotated Image and its mask}
%     \label{fig:both_images}
% \end{figure}

\subsection{Data Preprocessing}
In order to train a model, it is necessary to have both a picture and a corresponding mask. I employ an annotated image for the purpose of mask development. The annotated image displays delineations in both green and red hues. With this boundaries, I will create a mask. This mask corresponds to a specific area of interest within an image, typically indicating a plaque region.

\subsubsection{Binary Segmentation}
For binary segmentation, we require a binary mask. In a binary mask, there are two classes: one is plaque (foreground), and the other is background. In image 4.1(c), the binary mask. where white is the plaque region and black is the background region.

For binary mask generation, we use green and red boundries. Here, first, I create a mask that has the same size as an image. Mask pixel values are initialized with 0. After this, I found the green pixels scattered in an annotated image. For this, I use the Suzuki algorithm, where I pround the range of green pixels through which the algorithm scans pixels to trace continuous white (foreground) regions by following a pixel-following technique. Starting from an initial point, it navigates through adjacent pixels, efficiently determining contour boundaries by employing a non-recursive method. By iteratively following neighboring pixels and marking visited locations, it accurately identifies and traces object boundaries, generating a list of points representing contours.


Through this, I generate green contours, and a similar method is used to generate red contours. First, I place a green contour on the mask image, and then I simply subtract a red contour from the green contours, so the resultant image will be my plaque region. This will generate a binary mask for model training.


\subsubsection{Multiclass Semantic Segmentation}
For multiclass semantic segmentation, I have 3 regions other than background, which are green areas (plaque), excluding red (vessel wall), and blue (catheter). Now for the mask generation, the steps are almost similar to those for binary mask generation, but with extra work.

First, I draw a circle that is as large as the vessel wall. Now for this first, I manually annotate an image because the vessel wall size is the same in all the images. As seen, the vessel wall creates a circle. So I will find the center coordinate, and using this coordinate, I will draw a circle. This circle will be plotted on the mask image, which is initialized with 0 and is the same size as the image.

Now I create a plaque region, the same as binary mask creation. and that region first I subtract from the mask and then add to the mask. So the region under the red boundary will be used to create a blue region. So through the above mechanism, first I draw a red region, then a plaque, and then a blue region. This way, I created a mask for multiclass semantic segmentation.

But here we might face a problem in model training because the boundary region of two masks has a different color, and for this, I have to remove that pixel value. because in multiclass semantic segmentation, we have to assign a class to each pixel. So that's why I remove the border pixels from a mask. So this way, I create a multiclass mask for the multiclass semantic segmentation. which is shown in Fig. 4.1(d).

In Fig 4.1(d) Green region is plaque, Red region is vessel wall, and Blue region is catheter region


\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \includegraphics[width=0.7\textwidth]{raw_ai01400006.jpg}
        \caption{Raw Image}
        \label{fig:img1}
    \end{subfigure}
    \hspace{0.1cm} % Adjust the space between images here
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \includegraphics[width=0.7\textwidth]{annot_ai01400006.jpg}
        \caption{Annotated Image}
        \label{fig:img2}
    \end{subfigure}
    \hspace{0.1cm} % Adjust the space between images here
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \includegraphics[width=0.7\textwidth]{Binary_ai01400006.png}
        \caption{Binary Mask}
        \label{fig:img3}
    \end{subfigure}
    \hspace{0.1cm} % Adjust the space between images here
    \begin{subfigure}[b]{0.45\textwidth}
        \centering
        \includegraphics[width=0.7\textwidth]{Multi_ai01400006.jpeg}
        \caption{Generated Mask}
        \label{fig:img4}
    \end{subfigure}
    \caption{Images and their related masks}
    \label{fig:both_images2}
\end{figure}


\subsection{Implemented Models}
To perform image segmentation. I have implemented different segmentation models in PyTorch. Here, I implemented a deep learning-based segmentation model for image analysis. To implement image segmentation on the prepared dataset, some of the models were implemented from scratch, and some were implemented using the \texttt{segmentation\_models\_pytorch} library.

First, I start implementation with UNet \cite{unet}, which is a foundational model in medical image segmentation. Because of skip connections, Unet has the ability to preserve spatial information with limited data, but it also struggles with larger resolutions and uses a large amount of memory, which also leads to overfitting problems.

I also found that UNet++ \cite{unet++} also performs well in the field of image segmentation. Implementing UNet++ offers improved feature representation through nested and dense skip pathways, enabling better feature reuse. However, it might lead to increased model complexity and longer training times.

I also implemented Multi-Resolution U-Net \cite{mrunet} to combine U-Net's abilities with multi-scale features for segmentation. Its strength lies in handling diverse resolutions, but its complexity demands higher computational resources.

Other than that, I also implemented DeepLabV3 \cite{deeplab}, a model known for its powerful semantic segmentation that allows for capturing fine details due to atrous convolutions. However, it demands substantial computational resources and can be less efficient with smaller objects.

MANet \cite{manet} integrates self-attention, combining local and global features. Strong with multi-scale data, it might struggle with single-scale information. While powerful, its attention mechanism may overlook critical details. Its computational complexity could be limited in resource-constrained environments, yet it remains impactful.

Also employed FPN \cite{fpn}, which offers multi-scale feature representation, aiding in handling objects at different scales. Yet, FPN might introduce increased complexity and computation overheads in the model architecture.

I've also worked on YoloV8 \cite{paikaryolo}, Path Aggregation Network \cite{pan}, LinkNet\cite{linknet} , PSPNet \cite{pspnet}, SAM, and more models.

\subsection{Evaluation Metrics}
Evaluation metrics for image segmentation assess model performance. The function computes various metrics including Jaccard, F1-score, accuracy, recall, precision, and Dice coefficient. The Jaccard score measures the intersection over union, while the F1-score balances precision and recall. 

Precision is a measure of the model's accuracy in avoiding false positives, calculated as the ratio of true positives to the total predicted positives. Recall, on the other hand, is the model's ability to capture all relevant instances of a class, calculated as the ratio of true positives to the total real positives. The combination of these measures provides a thorough assessment of the performance of the segmentation model, yielding useful insights for its improvement and optimization.
\begin{figure}[h]
        \centering
        \includegraphics[width=5cm,height=3cm]{Figures/preciRec.png}
        % \caption{Raw Image}
        \label{fig:img1}
\end{figure}

\begin{figure}[h]
        \centering
        \includegraphics[width=10cm,height=2cm]{Figures/f1.png}
        % \caption{Raw Image}
        \label{fig:img1}
\end{figure}

The Jaccard index, commonly referred to as the Jaccard similarity coefficient, quantifies the degree of similarity between two sets. The Jaccard index evaluates the intersection between two sets, U and V, in the context of image segmentation, where U and V denote the number of classes. More precisely, it is computed by dividing the size of the common elements between the sets by the total size of all the elements in both sets. This index quantitatively measures the alignment between the segmented regions and the true classes in image segmentation. It provides vital insights into the correctness and effectiveness of the segmentation algorithms. A higher Jaccard index signifies a larger extent of overlap and, thus, a more precise segmentation result.

\begin{figure}[h]
        \centering
        \includegraphics[width=6cm,height=2cm]{Figures/iou.png}
        % \caption{Raw Image}
        \label{fig:img1}
\end{figure}



\subsection{Used Loss Function}
Optimizing model performance requires a deliberate combination of different loss functions. The combination of Dice and Cross-Entropy, two essential loss functions, is performed with meticulous attention to their respective contributions. This union incorporates a procedure of assigning weights to get the ideal combination that produces the most advantageous results. The following text provides a summary of the four loss functions used in this process:

The cross-entropy loss, often referred to as categorical cross-entropy, is a versatile loss function that may be used for both binary and multi-class classification tasks. The cross-entropy loss for each image in a multi-class segmentation situation with C classes and N pixels can be expressed as : 
\begin{figure}[!htb]
        \centering
        \includegraphics[width=5cm,height=1.5cm]{Figures/ccrentro.png}
        % \caption{Raw Image}
        \label{fig:img1}
\end{figure}

The Dice + Cross-Entropy loss is employed in medical picture segmentation to tackle the issue of class imbalances in datasets. The Dice coefficient quantifies similarity, a critical metric for imbalanced data, such as in medical imaging. This approach considers both dice and cross-entropy losses for real and predicted pixel distributions, which enhances performance in multi-class segmentation tasks for medical images.

\begin{figure}[h]
        \centering
        \includegraphics[width=9cm,height=2cm]{Figures/diceccr.png}
        % \caption{Raw Image}
        \label{fig:img1}
\end{figure}

Weighted Dice + Cross Entropy: We employed weighted dice to address the uneven distribution of pixels across different classes. The weight assigned to each class is determined by the ratio of misclassified pixels in that class to the total number of pixels in that class.
\begin{figure}[h]
        \centering
        \includegraphics[width=11cm,height=2cm]{Figures/weigthdice.png}
        % \caption{Raw Image}
        \label{fig:img1}
\end{figure}

\clearpage

\section{Experimental Findings}

This experiment delved into the intricacies of image segmentation, evaluating a range of data split methods, data sizes, and segmentation techniques. In the initial phase, binary segmentation was employed, utilizing both mixed and patient-wise data splits. The impact of hyperparameter tuning was assessed by comparing results with and without this crucial step.

Optuna \cite{optuna} was used to do hyperparameter tuning, with batch sizes ranging from 4 to 64 based on the model sizes and a learning rate between 0.1 and 0.00001. The ReduceLROnPlateau scheduler was employed. The performance of the Adam optimizer, with a beta1 value ranging from 0.8 to 0.999, and the SGD optimizer, with a momentum value ranging from 0.0 to 0.9, were evaluated to determine the optimal configuration. The models underwent training for 100 epochs, with early halting implemented if the validation did not show improvement for ten consecutive epochs. The training was conducted using an A100 GPU.

Following these preliminary investigations, the focus shifted to multiclass segmentation, where patient-wise data splits were applied for two and three patients. This exploration encompassed a variety of loss functions, including DiceLoss, Cross Entropy, and combinations of these two metrics. Additionally, the effectiveness of different weight configurations was examined, including equal weighting, weighting based on falsely predicted pixels, and weighting based on the area of pixels.


\begin{table}[h]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Models/Metrics} & \textbf{Jaccard} & \textbf{Recall} & \textbf{Precision} & \textbf{F1} \\ \hline
UNet & 0.7888 & 0.8935 & 0.8655 & 0.8793 \\ \hline
FPN & 0.7652 & 0.8839 & 0.849 & 0.8661 \\ \hline
MultiResUNet & 0.7479 & 0.8309 & 0.8733 & 0.8516 \\ \hline
MANet & 0.8009 & 0.8868 & 0.8874 & 0.8871 \\ \hline
DeepLabV3 & 0.7867 & 0.8791 & 0.8792 & 0.8791 \\ \hline
\end{tabular}
\caption{Binary Segmentation on with random split without hyperparameter tune}
\end{table}

\begin{figure}[!htb]
    \centering
    \includegraphics[width=1\textwidth]{Figures/binarymask.png}
    \caption{Column 1: input image, Column 2: Ground Truth Mask, Column 3: UNet, Column 4: UNet++, Column 5: PSPNet, Column 6: DeepLabV3+}
    \label{fig:image_label}
\end{figure}



\begin{table}[h]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Models/Metrics} & \textbf{Jaccard} & \textbf{Recall} & \textbf{Precision} & \textbf{F1} \\ \hline
UNet & 0.796 & 0.905 & 0.867 & 0.873 \\ \hline
FPN & 0.175 & 0.249 & 0.281 & 0.230 \\ \hline
MultiResUNet & 0.768 & 0.889 & 0.841 & 0.850 \\ \hline
MANet & 0.815 & 0.893 & 0.894 & 0.887 \\ \hline
DeepLabV3 & 0.797 & 0.913 & 0.872 & 0.873 \\ \hline
\end{tabular}
\caption{Multiclass Segmentation on with patient wise with hyperparameter tune}
\end{table}

\begin{table}[h]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Models/Metrics} & \textbf{Jaccard} & \textbf{Recall} & \textbf{Precision} & \textbf{F1} \\ \hline
UNet & 0.77499 & 0.83485 & 0.89193 & 0.85925 \\ \hline
FPN & 0.75208 & 0.89781 & 0.83165 & 0.85273 \\ \hline
MultiResUNet & 0.72731 & 0.77929 & 0.86392 & 0.81498 \\ \hline
MANet & 0.77100 & 0.87053 & 0.86336 & 0.85316 \\ \hline
DeepLabV3 & 0.81142 & 0.88508 & 0.90073 & 0.88539 \\ \hline
\end{tabular}
\caption{Multiclass Segmentation on with Patient Wise split with hyperparameter tune}
\end{table}



DeepLabV3+ has the highest Jaccard index, which is a measure of the overlap between the predicted segmentation and the ground truth segmentation. It also has the highest recall, which is the proportion of ground truth pixels that are correctly identified as the target class. DeepLabV3+ also has a high precision, which is the proportion of predicted pixels that are actually the target class.

UNet has the second highest Jaccard index, recall, and precision. It is a popular model for image segmentation because it is relatively easy to train and deploy.

The DeepLabV3 model performs well on the background and red classes, with high scores across all metrics as shown in Table 5.4. However, it struggles with the green class, which is my target. The lower Jaccard and F1 scores for the green class indicate that the model is not accurately segmenting this class. This could be due to class imbalance, where the green class is underrepresented in the data, or due to the green class being more challenging to segment due to its characteristics. Despite these challenges, the model shows promising results, especially with a high recall for the green class.

\begin{table}[h]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Class} & \textbf{Jaccard} & \textbf{Recall} & \textbf{Precision} & \textbf{F1} \\ \hline
Background & 0.96597 & 0.969 & 0.99678 & 0.98268 \\ \hline
Red & 0.97585 & 0.99266 & 0.98294 & 0.98777 \\ \hline
Green & 0.57169 & 0.83378 & 0.64541 & 0.72649 \\ \hline
Blue & 0.73215 & 0.7449 & 0.97778 & 0.84461 \\ \hline
Mean & 0.81142 & 0.88508 & 0.90073 & 0.88539 \\ \hline
\end{tabular}
\caption{Class-wise evaluation metric of the deepLabV3 model}
\end{table}

The table 5.5 presents model-wise evaluation metrics for target class prediction, with a focus on accurately identifying the green region in the image. UNet exhibits moderate scores across Jaccard, Recall, Precision, and F1, indicating fair performance but room for improvement. FPN showcases high recall but lower precision, possibly due to increased false positives. MultiResUNet scores lower across metrics, indicating difficulty in precisely delineating the target class. MANet demonstrates balanced performance but struggles with precision. DeepLabV3 outperforms due to its semantic segmentation approach, leveraging atrous convolutions for context aggregation, enabling better discrimination of the intricate green regions from the background, leading to higher recall and precision in identifying the target class.

\begin{table}[h]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Models/Metrics} & \textbf{Jaccard} & \textbf{Recall} & \textbf{Precision} & \textbf{F1} \\ \hline
UNet & 0.49872 & 0.66596 & 0.66549 & 0.66464 \\ \hline
FPN & 0.56599 & 0.91092 & 0.59933 & 0.72214 \\ \hline
MultiResUNet & 0.34520 & 0.46183 & 0.57699 & 0.51112 \\ \hline
MANet & 0.46950 & 0.82417 & 0.51918 & 0.63540 \\ \hline
DeepLabV3 &	0.57169 & 0.83378 & 0.64541 & 0.72649 \\ \hline
\end{tabular}
\caption{Model Wise target class evaluation matric}
\end{table}

\begin{figure}[!htb]
    \centering
    \includegraphics[width=1\textwidth]{Figures/random1.png}
    \caption{Random split of 2 patient data with binary segmentation without hyperparameter tuning, using DiceLoss and BCE}
    \label{fig:image_label1}
\end{figure}
% \vspace{-10pt} 
\begin{figure}[!htb]
    \centering
    \includegraphics[width=1\textwidth]{Figures/random2.png}
    \caption{Patient wise split of 2 patient data with binary segmentation with hyperparameter tuning, using DiceLoss + BCE}
    \label{fig:image_label2}
\end{figure}
% \vspace{-10pt} 
\begin{figure}[!htb]
    \centering
    \includegraphics[width=1\textwidth]{Figures/random3.png}
    \caption{Patient-wise split of 2 patient data with multiclass segmentation and hyperparameter tuning using DiceLoss + BCE with equal weight}
    \label{fig:image_label3}
\end{figure}
% \vspace{-10pt} 
\clearpage

\begin{figure}[!htb]
    \centering
    \includegraphics[width=1\textwidth]{Figures/random4.png}
    \caption{Patient-wise split of 3 patient data with multiclass segmentation and hyperparameter tuning using DiceLoss + BCE with Normalize weight}
    \label{fig:image_label4}
\end{figure}
% \vspace{-15pt} 
\begin{figure}[!htb]
    \centering
    \includegraphics[width=1\textwidth]{Figures/result.png}
    \caption{Patient-wise split of 3 patient data with multiclass segmentation and hyperparameter tuning using DiceLoss + BCE with Normalize weight(Inverse area)}
    \label{fig:image_label5}
\end{figure}

% \vspace{-10pt} 




\clearpage


\section{Summary}
The evaluation metrics employed encompass Jaccard, F1-score, accuracy, recall, precision, and the Dice coefficient. These metrics offer a comprehensive insight into segmentation model efficacy, assessing accuracy, completeness, and performance in image segmentation tasks. The WeightedDiceCELoss function combines Dice and Cross-Entropy losses, optimizing segmentation models by balancing accuracy and class imbalance. This analysis explores various data split methods, hyperparameter tuning impacts, and loss functions, shedding light on effective segmentation techniques and addressing class imbalance.

The assessment of DeepLabV3+ and UNet highlights their strengths and limitations. DeepLabV3+ exhibits superior performance in segmenting background and red classes but struggles with the green class. UNet, while popular and relatively easy to train, presents moderate performance. DeepLabV3's semantic segmentation approach, leveraging atrous convolutions for context aggregation, enables superior identification of intricate regions, enhancing recall and precision in identifying the target class, unlike other models examined. 

So overall, we can say that different models exhibit varied strengths and weaknesses in segmenting target classes. DeepLabV3's semantic segmentation approach shines, effectively discerning intricate regions like the green class. This highlights its robustness and potential for challenging segmentation tasks.

\subsection{Future plan of work}
1. We extend our experiment with a larger number of patient data points.\\
2. Exploring the guided study approach, like fuzzy logic, etc., for better results. \\
3. We will also continue exploring novel modifications and strategies for further enhancement. \\
4. We also further extend this project by identifying various substances of plaque, such as fibrous, necrotic, dense, etc. \\

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