Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Abstract

In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

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mathematics
Article
Rigorous Mathematical Investigation of a Nonlocal
and Nonlinear Second-Order Anisotropic Reaction-Diffusion
Model: Applications on Image Segmentation
Costic˘a Moro¸sanu 1,* and Silviu Pav˘al 2


Citation: Moro¸sanu, C.; Pav˘al, S.
Rigorous Mathematical Investigation
of a Nonlocal and Nonlinear
Second-Order Anisotropic
Reaction-Diffusion Model:
Applications on Image Segmentation.
Mathematics 2021,9, 91. https://
doi.org/10.3390/math9010091
Received: 17 September 2020
Accepted: 29 December 2020
Published: 4 January 2021
Publisher’s Note: MDPI stays neu-
tral with regard to jurisdictional clai-
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nal affiliations.
Copyright: © 2021 by the authors. Li-
censee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and con-
ditions of the Creative Commons At-
tribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Department of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, 11, 700506 Ia¸si, Romania
2Faculty of Automatic Control and Computer Engineering, Technical University “Gheorghe Asachi” of Ia¸si,
Dimitrie Mangeron, nr. 27, 700050 Ia¸si, Romania; silviu.paval@tuiasi.ro
*Correspondence: costica.morosanu@uaic.ro
Abstract:
In this paper we are addressing two main topics, as follows. First, a rigorous qualitative
study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic
diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary
conditions. Under certain assumptions on the input data:
f(t
,
x)
,
w(t
,
x)
and
v0(x)
, we prove the
well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev
space
W1,2
p(Q)
, facilitating for the present model to be a more complete description of certain classes
of physical phenomena. The second topic refers to the construction of two numerical schemes
in order to approximate the solution of a particular mathematical model (local and nonlocal case).
To illustrate the effectiveness of the new mathematical model, we present some numerical experiments
by applying the model to image segmentation tasks.
Keywords:
nonlinear anisotropic reaction-diffusion; well-posedness of solutions; Leray-Schauder de-
gree theory; finite difference method; explicit numerical approximation scheme; image segmentation
1. Introduction
For the unknown function
v(t
,
x)
(hereafter,
v
), consider the following nonlinear
second-order boundary value problem in
Q= (
0,
T]×Ω
, with
T>
0 and a bounded
domain Ω⊂IR2of Lebesgue measures |Ω|, whose boundary ∂Ωis sufficiently smooth:

























p1
∂
∂tv(t,x) = Φvx(t,x)hΨvx(t,x)∆v(t,x) + ∇Ψvx(t,x)· ∇v(t,x)i
+p3v(t,x)−v3(t,x)+p4f(t,x)in Q
¯
q(t,x)∂
∂nv(t,x) + p5v(t,x) = w(t,x)on Σ
v(0, x) = v0(x)on Ω,
(1)
where:
•t∈(0, T],x= (x1,x2)varies in Ω,Σ= (0, T]×∂Ω;
•∇v(t
,
x) = vx(t
,
x)
(
∇v=vx
) the gradient of
v(t
,
x)
in
x
, that is
∇v=∂
∂x1
v,∂
∂x2
v
.
Setting ∂
∂xi
v=vxi,i=1, 2, then ∇v=vx1,vx2=vx;
•∆v(t
,
x)
is the Laplace operator—a second-order differential operator, defined as the
divergence (∇·) of the gradient of v(t,x)in x;
•∂
∂tv(t,x)is the partial derivative of v(t,x)with respect to t;
•p1,p3,p4,p5are positive values.
Mathematics 2021,9, 91. https://doi.org/10.3390/math9010091 https://www.mdpi.com/journal/mathematics

Mathematics 2021,9, 91 2 of 23
•Φvx(t
,
x))
is a positive and bounded nonlinear real function of class
C1(Q)
with
bounded derivatives (see [
1
]), having the role of controlling the speed of the diffusion
process and enhances the edges (e.g., in the evolving image);
•Ψ(vx(t,x)) is the mobility;
•¯
q(t,x)is a positive and bounded real function;
•f(t,x)∈Lp(Q)is the distributed control (a given function), where
p≥2; (2)
•w(t,x)∈W1−1
2p,2−1
p
p(Σ)is the boundary control (a given function);
•n
=
n
(x) is the outward unit normal vector to
Ω
at a point
x∈∂Ω
.
∂
∂n
denotes
differentiation along n;
•v0(x)∈W2−2
p
p(Ω), verifying
¯
q(t,x)∂
∂nv0(x) + p5v0(x) = w(0, x). (3)
Let us note
ai(t,x,v(t,x),vx(t,x)) = Φvx(t,x)Ψvx(t,x)vxi(t,x),i=1, 2. (4)
Then, it is easy to recognize Equation (1)1as being quasi-linear with
aij t,x,v(t,x),vx(t,x)=∂
∂vxj
ait,x,v(t,x),vx(t,x)
=∂
∂vxj
Φvx(t,x)Ψvx(t,x)vxi(t,x),i=1, 2,
at,x,v(t,x),vx(t,x)=−∂
∂vΦvx(t,x)Ψ(vx(t,x))vxi(t,x)vxi(t,x)
−∂
∂xi
Φvx(t,x)Ψvx(t,x)vxi(t,x)
−p3hv(t,x)−v3(t,x)i−p4f(t,x),
while the boundary conditions (1)2are of second type:
haij t,x,v(t,x),vx(t,x)vxj(t,x)cos αi+p5v(t,x)−w(t,x)iΣ
=0,
(see [1] and reference therein).
For the reader’s benefit, we write problem (1) in the equivalent form

























p1
∂
∂tv(t,x)−Φvx(t,x)divΨvx(t,x)∇v(t,x)
=p3v(t,x)−v3(t,x)+p4f(t,x)in Q
¯
q(t,x)∂
∂nv(t,x) + p5v(t,x) = w(t,x)on Σ
v(0, x) = v0(x)on Ω.
(5)
Concerning Equation
(5)1
, we recall that it is of quasi-linear type with principal part
in divergence form (see [1]), with ai,i=1, 2, given by (4) and

Mathematics 2021,9, 91 3 of 23
a(t,x,v(t,x),vx(t,x)) = −p3v(t,x)−v3(t,x)−p4f(t,x).
In addition, we assume that Equations (1)1[or (5)1] are uniformly parabolic, i.e.,
ν1(|u|)ζ2≤aijt,x,u,zζiζj≤ν2(|u|)ζ2(6)
for arbitrary
u(t
,
x)
and
z(t
,
x)
,
(t
,
x)∈Q
, and
ζ= (ζ1
,
ζ2)
an arbitrary real vector,
where
ν1(s)
,
ν2(s)
are positive continuous functions of
s≥
0,
ν1(s)
is nonincreasing and
ν2(s)is nondecreasing.
The nonlinear problem (1) (or (5)) is important for modeling a variety of phenomena
of life sciences, including in biology, biochemistry, economics, medicine and physics.
Particular cases of the nonlinear second-order boundary value problem
(1)
, supplied with
different boundary conditions, have been successfully applied to many complex moving
interface problems, e.g., the motion of anti-phase boundaries in crystalline solids [
2
], the
mixture of two incompressible fluids, the nucleation of solids, and vesicle membranes (see [
3
–
5
]
and the references therein). In addition, the nonlinear problems of type
(1)1
, occur in
the phase-field transition system (e.g., [
6
]) where the phase function
v(t
,
x)
describes the
transition between the solid and liquid phases in the solidification process of a material
occupying a region
Ω
. For more general assumptions and with various types of boundary
conditions, Equation
(5)
has been numerically investigated (e.g., [
6
–
17
]). The error analysis
for the implicit backward Euler approximation is presented in [
16
], and computations
with several different higher-order time-stepping schemes are used in [
11
]. For the well-
posedness (existence, estimate, uniqueness and regularity) of a solution in Sobolev spaces
we refer to [12,18–21].
Another important novelty in our paper concerns the non-homogeneous Cauchy-
Neumann boundary conditions, which can be seen as boundary control in industry. Thus, as
applications of problem
(1)
, we indicate the moving interface problems, e.g., phase separation
and transition (see [
3
,
8
,
12
,
17
,
18
,
22
–
27
]), anisotropy effects (see [
15
,
28
–
30
]), image denoising
and segmentation (see [15,24,26,30–39] and references therein), etc.
Definition 1.
The function
v(t
,
x)
is called a classical solution of the problem
(1)
if it is continuous
in
¯
Q
, has continuous derivatives
vt
,
vx
,
vxx
in
Q
, verifies
(1)1
in every
(t
,
x)∈Q
and verifies
(1)2
and (1)3for (t,x)∈Σand t =0, respectively.
In our paper, we study the solvability of the problems
(1)
in the class
W1,2
p(Q)
, char-
acterized by the presence of some new physical parameters (
p1
,
p3
,
p4
,
p5
,
Φvx(t
,
x)
,
Ψ(vx(t
,
x))
), the principal part being in divergence form and by considering the cubic nonlin-
earity p3v(t,x)−v3(t,x), satisfying for n∈ {1, 2, 3}the assumption H0in [21], that is:
H0:(v−v3)|v|3p−4v≤1+|v|3p−1− |v|3p.
In Theorem 1, we prove the existence, regularity and uniqueness of solution for
(1)
.
(see [
15
] for a numerical study of Equation
(1)
corresponding to a linear reaction term
v(t,x)−v0(x), with homogeneous Neumann boundary condition).
In the following we will denote by Cseveral positive constants.
2. Well-Posedness of the Solution of (5)
Theorem 1 of this section presents the dependence of the solution
v(t
,
x)
of
(5)
on
f(t,x)and w(t,x). In our study, we rely on the following:
• The Leray-Schauder principle (see [1,4,11–15,19–21] and reference therein);
• The Lp-theory of linear and quasi-linear parabolic equations;
• Green’s first identity
−Z
Ω
ydivz dx =Z
Ω
∇y·z dx −Z
∂Ω
y∂
∂nz dγ,

Mathematics 2021,9, 91 4 of 23
−Z
Ω
y∆z dx =Z
Ω
∇y· ∇z dx −Z
∂Ω
y∂
∂nz dγ,
for any scalar-valued function
y
and
z
, a continuously differentiable vector field in
n
dimensional space;
•
The Lions and Peetre embedding theorem (see [
1
] and references therein) to ensure
the existence of a continuous embedding
W1,2
p(Q)⊂Lµ(Q)
, where the number
µ
is
defined as follows (see (3))
µ=






any positive number ≥3pif p≥2,
1
p−1
2−1, if p<2.
(7)
and, for k∈ {1, 2, ...}and 1 ≤p≤∞,Wk,2k
p(Q)denotes the Sobolev space on Q:
Wk,2k
p(Q) = y∈Lp(Q):∂r
∂tr
∂q
∂xqy∈Lp(Q), for 2r+q≤2k, (8)
(see [1] for more details).
In addition, we use the set
C1,2(¯
Q)
(
C1,2(Q)
) of all continuous functions in
¯
Q
(in
Q
)
having continuous derivatives
ut
,
ux
and
uxx
in
¯
Q
(in
Q
), as well as the Sobolev spaces
Wl
p(Ω)
,
Wl,l/2
p(Σ)
with non-integral
l
for the initial and boundary conditions, respectively
(see [1]).
The main result for the study of the existence, a priori estimates, uniqueness and
regularity for the solution of (1) (or (5)) is the next theorem.
Theorem 1.
For any classical solution
v(t
,
x)∈C1,2(Q)
of
(5)
, suppose there are
M
,
M0
,
m1
,
M1, M2, M3and M4∈(0, ∞)such that the fpllowing hypotheses are satisfied:
I1
.
|v(t
,
x)|<M
for any
(t
,
x)∈Q
and for any
z(t
,
x)
, the map
Ψ(z(t
,
x))
is continuous,
differentiable in x, its x-derivatives are measurable bounded, satisfies (6)and
0<Ψmin ≤Ψ(vx(t,x)) <Ψmax ,f or (t,x)∈Q, (9)
|Ψ(z)vxi|(1+|z|) + 
∂
∂x1
(Ψ(z)vx1)
+
∂
∂x2
(Ψ(z)vx1)
+
∂
∂x1
(Ψ(z)vx2)
+
∂
∂x2
(Ψ(z)vx2)
+|v(t,x1,x2)| ≤ M0(1+|z|)2.
(10)
I2
.
Φvx(t
,
x)
is a positive and bounded nonlinear real function of class
C1(Q)
with bounded
derivatives and
0 < m1≤Φvx(t,x)≤M1.
In addition, for every ε>0, the functions v(t,x)and Ψ(vx(t,x)) satisfy the relations
kvkLs(Q)≤M2,kΨ(vx)vxikLr(Q)<M3,i=1, 2,
where
r=max{p, 4}p6=4
4+εp=4, s=max{p, 2}p6=2
2+εp=2.

Mathematics 2021,9, 91 5 of 23
Then,
∀f∈Lp(Q)
and
∀v0∈W2−2
p
p(Ω)
, with
p6=3
2
, the problem
(5)
has a solution
v∈W1,2
p(Q)and the next estimate holds:
kvkW1,2
p(Q)≤C"1+kv0k
W2−2
p
p(Ω)
+kv0k3−2
p
L3p−2(Ω)
+kfkLp(Q)+kwk3−2
p
L3p−2(Σ)+kwk
W1−1
2p,2−1
p
p(Σ)#,
(11)
where the constant C >0does not depend on v,f and w.
If
v1
,
v2∈W1,2
p(Q)
are two solutions to
(5)
, corresponding to
{f1
,
w1
,
v1
0}
and
{f2
,
w2
,
v2
0}
,
respectively, such that kv1kW1,2
p(Q)≤M4,kv2kW1,2
p(Q)≤M4and
0<qmin ≤¯
q(t,x)<qmax ,f or (t,x)∈Σ, (12)
then the following estimate holds:
max
(t,x)∈Q|v1−v2| ≤ C1eCT maxmax
(t,x)∈Q|f1−f2|,max
(t,x)∈Σ|w1−w2|,max
(t,x)∈Ω|v1
0−v2
0|,(13)
where the constant
C
,
C1>
0does not depend on
{v1
,
f1
,
w1
,
v1
0}
and
{v2
,
f2
,
w2
,
v2
0}
. In
particular, the solution of problem (5)is unique.
2.1. The Proof of Theorem 1
To prove this theorem, we use the Leray-Schauder principle. Thus, we consider the
Banach space
B=W0,1
p(Q)∩L3p(Q),
endowed with the norm
kukB=kukLp(Q)+kuxkLp(Q),
and a nonlinear operator H:B×[0, 1]→Bdefined by
v=v(u,λ) = H(u,λ)for all (u,λ)∈W0,1
p(Q)∩L3p(Q)×[0, 1], (14)
where v(u,λ)is the unique solution to the next problem































p1
∂
∂tv(t,x)−"λΦux(t,x)∂
∂uxj
(Ψ(ux)uxi) + (1−λ)δj
i#vxixj
=λnA(t,x,u,ux) + p3u(t,x)−u3(t,x)+p4f(t,x)oin Q
¯
q(t,x)∂
∂ν v(t,x) + p5v(t,x) = λw(t,x)on Σ
v(0, x) = λv0(x), on Ω.
(15)
with A(t,x,u,ux) = Φux∇Ψ(ux)· ∇u,∀(t,x)∈Q.
We shall prove now the following technical lemma
Lemma 1. We assume Hypotheses I1and I2to be valid. Then
A(t,x,u,uxi) + p3(u−u3) + p4f(t,x)∈Lp(Q),∀u∈W0,1
p(Q)∩L3p(Q). (16)

Mathematics 2021,9, 91 6 of 23
Proof. Indeed, since u∈L3p(Q), then kukL3p(Q)≤Konst and thus
ku3kLp(Q)= R
Q
|u3|pdxdt!1
p
=
 R
Q
|u|3pdxdt!1
3p

3p1
p
=kuk3
L3p(Q)≤(Konst)3,
i.e., the nonlinear term in (16) belongs to Lp(Q),∀u∈W0,1
p(Q)∩L3p(Q)(see also [1]).
Next, from (10) it is easy to conclude that
∂
∂xi[Ψ(uxi)uxi]≤M1(1+|uxi|)2.
Thus, to prove that
A(t,x,u,uxi)) = Φux(t,x)∇Ψ(ux(t,x)) · ∇u(t,x)∈Lp(Q),
∀u∈W0,1
p(Q)∩L3p(Q),
we have to prove that
u2
xi∈Lp(Q)
,
∀u∈W0,1
p(Q)∩L3p(Q)
. For any
u∈W0,1
p(Q)∩L3p(Q)
it follows that
kukLp(Q)+kuxkLp(Q)≤konst
, i.e.,
kuxkLp(Q)≤konst
. Making use of the
boundedness of
Φux(t
,
x)
(see I
2
), as well as the properties of
Ψux(t
,
x)
(see I
1
), and
since uxi∈Lp(Q), it results that A(t,x,u,uxi)∈Lp(Q),∀u∈W0,1
p(Q)∩L3p(Q).
Finally, we recall that
f(t
,
x)∈Lp(Q)
and, owing to the above, we easy derive that
the statement expressed by (16) is true.
2.2. The Proof of Theorem 1 (Continued)
Let us show that the nonlinear operator
H(u
,
λ)
defined by
(14)
satisfies the following
Properties Aand B.
A.
If
(15)
has a unique solution, then
H
is well-defined. By the right hand of
(
15
)1
, using
Lemma 1, it follows that,
∀u∈W0,1
p(Q)∩L3p(Q)
, then
A(t
,
x
,
u
,
ux) + p3(u−u3) +
p4f(t
,
x)∈Lp(Q)
and thus, the same reasoning as in [
1
] allows us to conclude that
for
w(t
,
x)∈W2−1
p,1−1
2p
p(Σ)
, the linear parabolic boundary value problem formulated
in
(15)
has a unique solution, that is (see
(14)
)
v=H(u
,
λ)∈W1,2
p(Q)
,
∀u∈B
and
∀λ∈[
0, 1
]
. Next, the embedding
W1,2
p(Q)⊂Lµ(Q)⊂L3p(Q)
,
p≥
2 (see
(3)
and
(7)
),
allows us to conclude that
H(u,λ) = v∈B,∀u∈Band ∀λ∈[0, 1].
Thus, the operator His well-defined.
B.
Let us now show that
H
is continuous and compact. The sketch of the proof is the
same as in [1,15]. However, for reader convenience, we present details in the sequel.
Let un→uin W0,1
p(Q)∩L3p(Q)and λn→λin [0, 1]. Making the notation
vn,λn=H(un,λn),vn,λ=H(un,λ)and vλ=H(u,λ)
and then considering the difference
H(un
,
λn)−H(un
,
λ)
, we obtain from relations
(14) and (15) that







































p1
∂
∂tVn,λn,λ−"λΦ(un
x)∂
∂un
xjΨ(un
x)un
xi+ (1−λ)δj
i#Vn,λn,λ
xixj
= (λn−λ)
Φ(un
x)∂
∂un
xj
(Ψ(un
x)un
xi)−δj
ivn,λn
xixj
+A(t,x,un,un
xi) + p3un−(un)3+p4f(t,x)oin Q
¯
q(t,x)∂
∂nVn,λn,λ+p5Vn,λn,λ= (λn−λ)w(t,x)on Σ
v(0, x) = (λn−λ)v0(x)on Ω,
(17)

Mathematics 2021,9, 91 7 of 23
where Vn,λn,λ=vn,λn−vn,λ.
The right-hand side in
(17)
belongs to
Lp(Q)
, since
vn,λn∈W1,2
p(Q)
. Therefore, the
Lp-theory of PDE gives the estimate
kVn,λn,λkW1,2
p(Q)≤C|λn−λ| × (


hΦ(un
x)∂
∂un
xjΨ(un
x)un
xi−δj
iivn,λn
xixj


Lp(Q)
+kA(t,x,un,un
xi)kLp(Q)+kun−(un)3kLp(Q)
+kv0k
W2−2
p
p(Ω)
+kfkLp(Q)+kwk
W1−1
2p,2−1
p
p(Σ)),
with a constant C(|Ω|,p1,p3,p4,M,M1,M2,M3).
Owing to Lemma 1 we can derive that
(un)3
is bounded in
Lp(Q)
,
∀un∈W0,1
p(Q)∩
L3p(Q)
. In addition, the inequality
(10)
, the working Hypothesis I
2
and the inclusion
un,λn
xixj∈Lp(Q), imply the boundedness in Lp(Q)of the terms A(t,x,un,un
xi)and
Φ(un
x)∂
∂un
xj
(Ψ(un
x)un
xi)−δj
ivn,λn
xixj
. Moreover, since
W2−2
p
p(Ω)⊂Lp(Ω)
, it results that the
remaining terms on the right-hand side from the above inequality are also bounded in
Lp(Q). Thus, making λn→λ, we obtain (Vn,λn,λ=vn,λn−vn,λ)
kvn,λn−vn,λkW1,2
p(Q)→0 for n→∞. (18)
To evaluate the difference
H(vn
,
λ)−H(v
,
λ)
, we use again the relations
(14)
,
(15)
,
and we obtain







































p1
∂
∂tVn,1,λ−"λΦ(un
x)∂
∂un
xjΨ(un
x)un
xi+ (1−λ)δj
i#Vn,1,λ
xixj
=λΦ(un
x)∂
∂un
xj
(Ψ(un
x)un
xi)−Φ(ux)∂
∂uxj(Ψ(ux)uxi)vλ
xixj
+A(t,x,un,un
xi)−A(t,x,u,uxi) + p3(un−u)−(un)3−u3oin Q
¯
q(t,x)∂
∂nVn,1,λ+p5Vn,1,λ=0 on Σ
Vn,1,λ(0, x) = 0 on Ω,
(19)
where Vn,1,λ=vn,λ−vλ.
The Lp-theory applied to (19), gives us the estimate
kVn,1,λkW1,2
p(Q)≤Cλ"


Φ(un
x)∂
∂un
xj
(Ψ(un
x)un
xi)−Φ(ux)∂
∂uxj(Ψ(ux)uxi)
vλ
xixj


Lp(Q)
+kA(t,x,un,un
xi)−A(t,x,u,uxi)kLp(Q)
+k(un−u)−((un)3−u3)kLp(Q)#,

Mathematics 2021,9, 91 8 of 23
with a new constant
C
. From the convergence
un→u
in
W0,1
p(Q)∩L3p(Q)
and the
continuity of the Nemytskij operator (see [
19
] and references therein), as well as the
continuity of Φ(un
x),∂
∂un
xj
(Ψ(un
xi)un
xi)and A(t,x,un,un
xi), it follows that
kvn,λ−vλkW1,2
p(Q)→0 as n→∞. (20)
Making use of the relations
(18)
and
(20)
, we show the continuity of the nonlinear
operator
H
defined by
(14)
. Moreover,
H
is compact. Indeed, since
µ>
3
p
, the inclusion
W1,2
p(Q),→W0,1
p(Q)∩L3p(Q)
is compact (see [
12
] and reference therein). Furthermore,
writing Has the composition
B×[0, 1]→W1,2
p(Q),→W0,1
p(Q)∩L3p(Q) = B,
the compactness of Himmediately follows.
2.2.1. The Proof of the First Part in Theorem 1: The Regularity of v(t,x)
We establish now the existence of a number δ>0 such that
(v,λ)∈W0,1
p(Q)∩L3p(Q)×[0, 1]with v=H(v,λ) =⇒ kvkB<δ. (21)
The equality v=H(v,λ)in (21) is equivalent to



























p1
∂
∂tv(t,x)−λΦ(vx)divΨ(vx)∇v−(1−λ)∆v
=λhp3v(t,x)−v3(t,x)+p4f(t,x)iin Q
¯
q(t,x)∂
∂nv(t,x) + p5v(t,x) = λw(t,x)on Σ
v(0, x) = λv0(x)on Ω.
(22)
(see (4), (6) and (15)).
Multiplying the first equation in
(22)
by
|v|3p−4v
, integrating over
Qt:= (
0,
t)×Ω
,
t∈(0, T], we get
p1Z
Qt
∂
∂t|v(τ,x)|3p−2dτdx −λZ
Qt
Φ(vx)divΨ(vx)∇v|v|3p−4v dτdx
−(1−λ)Z
Qt
∆v|v|3p−4v dτdx
=λp3Z
Qt
(v−v3)|v|3p−4v dτdx +λp4Z
Qt
f|v|3p−4v dτdx.

Mathematics 2021,9, 91 9 of 23
Owing to Green’s first identity, the left inequality in
(9)
and
(12)
, Assumption I
2
and
the boundary conditions (22)2, the previous equality leads us to
p1
3p−2Z
Ω
|v(t,x)|3p−2dx +λZ
Qt
Ψ(vx)∇v· ∇Φ(vx)|v|3p−4vdτdx
+(1−λ)(3p−3)Z
Qt
|∇v|2|v|3p−4dτdx
+λp5m1R
Σt
|v|3p−2dτdγ+(1−λ)
qmax p5R
Σt
|v|3p−2dτdγ
≤λp1
3p−2Z
Ω
|v0(x)|3p−2dx
+λp3R
Qt
(v−v3)|v|3p−4v dτdx +λp4R
Qt
f|v|3p−4v dτdx
+λM1Z
Σt
w|v|3p−4v dτdγ+(1−λ)
qmin Z
Σt
w|v|3p−4v dτdγ.
(23)
for all
t∈(
0,
T]
. The Hölder and Cauchy inequalities, applied to the last terms in
(23)
,
give us
i1λp4Z
Qt
f|v|3p−4v dτdx ≤p−1
pε
p
p−1Z
Qt
|v|3pdτdx +λp4
1
pε−pkfkp
Lp(Q)
i2λM1Z
Σt
w|v|3p−4v dτdγ
≤λp5m11−1
3p−2Z
Σt
|v|3p−2dτdγ+M1
m1
1
p5
1
3p−2Z
Σt
|w|pdτdγ,
i3
(1−λ)
qmin Z
Σt
w|v|3p−4v dτdγ
≤p51−1
3p−2(1−λ)
qmax Z
Σt
|v|3p−2dτdγ+qmax
qmin
1
p5
1
3p−2Z
Σt
|w|3p−2dτdγ.
By H0, relation (3) and Young’s inequality, we obtain
λp3Z
Qt
(v−v3)|v|3p−4v dτdx
≤λp3|Ω|T+λp3|Ω|T1
3pε−3p+3p−1
3pε
3p
3p−1Z
Qt
|v|3pdτdx
−λp3Z
Qt
|v|3pdτdx.

Mathematics 2021,9, 91 10 of 23
Owing to the above inequality as well as (
i1- i3
) and, taking into account the continu-
ous embedding L3p−2(Σt)⊂Lp(Σt), from (23), we derive the following estimate
p1
3p−2Z
Ω
|v(t,x)|3p−2dx +λZ
Qt
Ψ(vx)∇v· ∇Φ(vx)|v|3p−4vdτdx
+(1−λ)(3p−3)Z
Qt
|∇v|2|v|3p−4dτdx +λp3Z
Qt
|v|3pdτdx
+p5
1
3p−2λm1+(1−λ)
qmax Z
Σt
|v|3p−2dτdγ
≤λp1
3p−2Z
Ω
|v0(x)|3p−2dx
+3p−1
3pε
3p
3p−1+p−1
pε
p
p−1Z
Qt
|v|3pdτdx
+λp3|Ω|T+p3|Ω|T1
3pε−3p+p4
1
pε−pkfkp
Lp(Q)
+1
p5
1
3p−2M1
m1
+qmax
qmin Z
Σt
|w|3p−2dτdγ.
Taking εsmall enough, the previous inequality yields
λk|v|3kp
Lp(Q)≤C11+kv0k3p−2
L3p−2(Ω)+kfkp
Lp(Q)+kwk3p−2
L3p−2(Σt), (24)
for a positive constant C1=C(|Ω|,T,n,p,p1,p3,p4,p5,qmin,qmax ,m1,M1).
Applying Lp-theory to problem (15) (see [1] and references therein), we get
kvkW1,2
p(Q)≤C2kv0k
W2−2
p
p(Ω)
+p3k(v−v3)kLp(Q)
+kfkLp(Q)+kwk
W1−1
2p,2−1
p
p(Σ),
(25)
for a constant C2=C(|Ω|,T,n,p,p1,p3,p4)>0.
By Lemma 1.1 in [21] and (24), we get
kv−v3kLp(Q)≤C1 1+kv0k
3p−2
p
L3p−2(Ω)+kfkLp(Q)+kwk
3p−2
p
L3p−2(Σ)!
and then (25) becomes
kvkW1,2
p(Q)≤C2 1+kv0k
W2−2
p
∞(Ω)
+kv0k
3p−2
p
L3p−2(Ω)
+kfkLp(Q)+kwk
W1−1
2p,2−1
p
p(Σ)
+kwk
3p−2
p
L3p−2(Σt)!.
(26)
The continuous embedding W1,2
p(Q)⊂B=W0,1
p(Q)∩L3p(Q)ensures that
kvkB≤CkvkW1,2
p(Q)

Mathematics 2021,9, 91 11 of 23
which, owing to
(26)
, ensures that a constant
δ>
0 can be found such that the property
expressed in (21) is true.
Denoting
Bδ:=nv∈B:kvkB<δo,
relation (21) implies that
H(v,λ)6=v∀v∈∂Bδ,∀λ∈[0, 1],
provided that
δ>
0 is sufficiently large. Furthermore, following the same reasoning as
in [
1
,
4
,
11
,
15
,
19
], we conclude that problem
(6)
has a solution
v∈W1,2
p(Q)
(see also [
21
],
p. 195). The estimate
(11)
results from
(26)
, and the proof of the first part in Theorem 1
is finished.
2.2.2. The Uniqueness of the Solution v(t,x)
Now, we prove
(13)
, which implies the uniqueness of the solution of
(1)
or
(5)
. By hy-
pothesis,
v1
,
v2∈W1,2
p(Q)
solve problem
(1)
, corresponding to
{f1
,
w1
,
v1
0}
and
{f2
,
w2
,
v2
0}
,
respectively. Thus, v1−v2∈W1,2
p(Q).
Let us recall that
aij (t,x,v1,v1
x) = ∂
∂v1
xj
Φ(v1
x)Ψ(v1
x)v1
xi,
aij (t,x,v2,v2
x) = ∂
∂v2
xj
Φ(v2
x)Ψ(v2
x)v2
xi,
i=1, 2, and (following [1]) we write the increments of aij in the form
aij (t,x,v1,v1
x)−aij (t,x,v2,v2
x) =
1
Z
0
d
dλaij t,x,vλ,vλ
xdλ,
where
vλ(t,x) = λv1(t,x) + (1−λ)v2(t,x)and vλ
x(t,x) = λv1
x(t,x) + (1−λ)v2
x(t,x).
Consequently, we get
aij (t,x,v1,v1
x)v1
xixj−aij (t,x,v2,v2
x)v2
xixj
=aij (t,x,v1,v1
x)Vxixj+v2
xixjVxi
1
R
0
∂
∂vλ
xj
aij t,x,vλ,vλ
xdλ
+V
1
R
0
∂
∂vλaij t,x,vλ,vλ
xdλ
+v2
xixjΦ(v1
x)−Φ(v2
x),
(27)
where V(t,x) = v1(t,x)−v2(t,x).
Regarding A(t,x,v,vx) = Φvx∇Ψ(vx)· ∇v, we have

Mathematics 2021,9, 91 12 of 23
A(t,x,v1,v1
x)−A(t,x,v2,v2
x)
=
1
Z
0
d
dλAt,x,vλ,vλ
xdλ
=Vxi
1
Z
0
∂
∂vλ
xj
At,x,vλ,vλ
xdλ+V
1
Z
0
∂
∂vλAt,x,vλ,vλ
xdλ.
(28)
Now, we subtract Equation
(1)1
for
v2(t
,
x)
from Equation
(1)1
for
v1(t
,
x)
, and making
use of (27), (28), we obtain the following linear equation















p1
∂
∂tV−ˆ
aij (t,x)Vxixj+ˆ
ai(t,x)Vxi+ˆ
a(t,x)V=f1−f2in Q
¯
q(t,x)∂
∂nV+p5V=w1−w2on Σ
V(0, x) = v1
0(x)−v2
0(x)on Ω,
(29)
where
ˆ
aij (t,x) = ai j(t,x,v1,v1
x),
ˆ
ai(t,x) = −v2
xixj
1
Z
0
∂
∂vλ
xj
ai,jt,x,vλ,vλ
xdλ+
1
Z
0
∂
∂vλ
xj
At,x,vλ,vλ
xdλ,
ˆ
a(t,x) = −v2
xixj
1
Z
0
∂
∂vλai,jt,x,vλ,vλ
xdλ+
1
Z
0
∂
∂vλAt,x,vλ,vλ
xdλ
−p2h1−(v1)2+v1v2+ (v2)2i.
Due to (9) and the working hypotheses on v1and v2, i.e.,
kv1kW1,2
p(Q),kv2kW1,2
p(Q)≤M4,
the conditions on linear equations are fulfilled and, given this, it follows from
(29)
that
estimate (13) is valid for V, which finishes the proof of Theorem 1.
As a consequence, it results the uniqueness for the solution of (5).
Corollary 1.
For the same initial conditions, the problem
(5)
possesses a unique solution
v(t
,
x)∈
W1,2
p(Q).
Proof.
Let
f1=f2=f
and
w1=w2=w
in Theorem 1. Then
(13)
demonstrates the
corollary (see [1] and references therein).
Remark 1.
The nonlinear operator
H
in
(14)
depends on
λ∈[
0, 1
]
and its fixed point for
λ=
1
are solutions of (15).

Mathematics 2021,9, 91 13 of 23
3. A Novel Nonlinear Second-Order Anisotropic Reaction-Diffusion Model in
Image Segmentation
The nonlinear parabolic second-order PDE problem
(5)
can be applied for image de-
noising, enhancement, restoration and segmentation. Here we consider a particularization
of this mathematical model by setting the functions Φvx(t,x)and Ψvx(t,x)as follow
Φ:[0, ∞)→(0, ∞),Φ(s) =
3
pϕs2+η
α,
Ψ:[0, ∞)→(0, 1],Ψ(s) = 1
1+s
c2
(30)
where
ϕ
,
η
,
α∈(
0, 4
]
, while the parameter
c
is the conductance (see [
15
], p. 177 and [
14
],
p. 633). Therefore, the following PDE scheme with non-homogeneous Cauchy-Neumann
boundary conditions is acquired:

























p1
∂
∂tv(t,x)−Φkvx(t,x)kdivΨkvx(t,x)kvx(t,x)
=p3v(t,x)−v3(t,x)+p4f(t,x)in Q
∂
∂nv(t,x) + p5v(t,x) = w(t,x)on Σ
v(0, x) = v0(x)on Ω,
(31)
vx(t,x) = ∇v(t,x) = vx1(t,x),vx2(t,x).
The edge-stopping (diffusivity) function in
(30)2
is positive, monotonically decreasing
and converging to zero (see [
28
,
30
]) thus satisfying the conditions imposed by a proper
diffusion. Moreover, it is easy to check that
Ψ
and
Φ
in
(30)
satisfy Assumptions I
1
and
I
2
in Theorem 1 and thus the nonlinear anisotropic reaction-diffusion model
(31)
is well-
posed, as proved in the previous section. Consequently, it admits an unique classical
solution
v(t
,
x)∈W1,2
p(Q)
, that represents the evolving image of the observed image
v(0, x) = v0(x).
The corresponding nonlocal anisotropic reaction-diffusion model to
(31)
can be written
as follows:
p1
∂
∂tv(t,x) = Φkvx(t,x)kΨkvx(t,x)k(Z
Ω
K(x−y)hv(t,y)−v(t,x)idy
+R
∂Ω
K(x−ys)hw(t,ys)−p5v(t,ys)idys)
+Φkvx(t,x)k∇Ψkvx(t,x)k·vx(t,x) + p3v(t,x)−v3(t,x)+p4f(t,x),
(32)
with initial condition
v(0, x) = v0(x), (33)
where
•K:IR →IR
is a real function, symmetric, continuous, nonnegative and it’s compactly
supported in the unit sphere, such that R
IR
K(z)dz =1.

Mathematics 2021,9, 91 14 of 23
Details on certain interpretations of the terms
K(x−y)
,
Z
Ω
K(x−y)v(t
,
y)dy
and
−v(t
,
x)Z
Ω
K(x−y)dy
in the mathematical model
(32)
, can be found in the works of P. W.
Bates, S. Brown and J. Han [
3
] and J. Rubinstein and P. Sternberg [
27
] and references therein.
The solution behavior for the nonlocal model
(32)
on rescaling the kernel
K
considering
K(z) = 1
ε3Kz
ε
are studied in [
33
] and for the numerical solutions we refer to [
3
,
40
] and
references therein.
In what follows, we will approximate the solution
v(t
,
x)
in
(31)
and
(32)
using the
finite-difference method (of second-order in time, see (36)).
3.1. Numerical Approximation
In this subsection we propose two numerical schemes (see
(47)
and
(48)
) to approxi-
mate the solution of the novel nonlinear reaction-diffusion model
(31)
,
(32)
, based on the
finite difference method (see also [
3
,
4
,
7
,
9
,
16
,
23
,
28
,
40
,
41
]). By using a grid of space size
h
,
one quantizes the space coordinates x= (x1, x2)as:
x1i=ih,x2j=jh, for all i=1, 2 . . . , I,j=1, 2, . . . , J,
where [Ih ×Jh]represents the dimension of the support image.
We consider a positive value
T
as the time interval upper limit and
M
the number of
nodes which are dividing the time interval [0, T], then we can set
tm= (m−1)ε,m=1, 2, . . . , M,ε=T/(M−1).
We also denote by
vm
i,j
the approximating values in
(tm
,
x
1
i
,
x
2
j)
for the unknown
function v(t,x)used in (31) (or (32)), i.e.,
vm
i,j=v(tm,x1i,x2j),m=1, 2, . . . , M,i=1, 2 . . . , I,j=1, 2, . . . , J,
or, for later use
vm=vm
1,1,vm
2,1, . . . , vm
Ih,J h Tm=1, 2, . . . , M. (34)
From the initial condition (33), we have
v(0, x)≈v1=v(t1,x1i,x2j) = v0(x1i,x2j),i=1, 2 . . . , I,j=1, 2, . . . , J. (35)
To approximate
∂
∂tv(t
,
x)
, we employ a second-order scheme (see [
16
,
41
] and references
therein):
∂
∂tv(tm+1,x1i,x2j)≈3vm+1
i,j−4vm
i,j+vm−1
i,j
2ε,
m=1, 2, . . . , M−1, i=1, 2 . . . , I,j=1, 2, . . . , J.
(36)
We write Equation in (32) as:
p1
∂
∂tv(t,x) + p3hv3(t,x)−v(t,x)i=NlD(t,x,v,vx) + R(t,x,v,vx) + p4f(t,x)(37)
where we denote the nonlocal diffusion term by:

Mathematics 2021,9, 91 15 of 23
Nl D(t,x,v,vx) = Φkvx(t,x)kΨkvx(t,x)k(Z
Ω
K(x−y)hv(t,y)−v(t,x)idy
+Z
∂Ω
K(x−ys)hw(t,ys)−p5v(t,ys)idys)
(38)
and the reaction term by:
R(t,x,v,vx) = Φkvx(t,x)k∇Ψkvx(t,x)k·vx(t,x). (39)
The left-side term in (37) is approximated by
p1
3vm+1
i,j−4vm
i,j+vm−1
i,j
2ε+p3h(vm
i,j)3−vm
i,ji
and the right side terms are discretized using central differences (see [
16
] and references
therein).
We also denote Φi,j=Φ(k∇vi,jk)and Ψi,j=Ψ(k∇vi,jk), where
k∇vi,jk=kvx(t,xi,j)k ≈ v
u
u
t vm
i+1,j−vm
i−1,j
2h!2
+ vm
i,j+1−vm
i,j−1
2h!2
,
for all
i=
2
. . .
,
I−
1,
j=
2,
. . .
,
J−
1. To complete the discretization schema we need to
approximate NlD(t,x,v,vx)and R(t,x,v,vx)terms as follows:
Nl Dm+1(tm,xi,j,vm
i,j,vx(tm,xi,j)) =
Φm
i,jΨm
i,j(Z
Ω
K(xi,j−y)hv(tm,y)−v(tm,xi,j)idy
+R
∂Ω
K(xi,j−ys)hw(tm,ys)−p5v(tm,xi,j)idys)
(40)
Continuing the discretization by using the Riemann sums to approximate the integral
terms, we have:

Mathematics 2021,9, 91 16 of 23
Z
Ω
K(xi,j−y)hvm(tm,y)−vm(tm,xi,j)idy =
h2(I−1
∑
d1=2
J−1
∑
d2=2
K(xi,j−yd1,d2)vm
d1,d2−vm
i,j
+1
2
I−1
∑
d1=2hK(xi,j−yd1,1)vm
d1,1 −vm
i,j+K(xi,j−yd1,J)vm
d1,J−vm
i,ji
+1
2
J−1
∑
d2=2hK(xi,j−y1,d2)vm
1,d2−vm
i,j+K(xi,j−yI,d2)vm
I,d2−vm
i,ji
+1
4hK(xi,j−y1,1)vm
1,1 −vm
i,j+K(xi,j−yI,1)vm
I,1 −vm
i,j
+K(xi,j−y1,J)vm
1,J−vm
i,j+K(xi,j−yI,J)vm
I,J−vm
i,ji)
(41)
For the second integral on ∂Ω, we have:
R
∂Ω
K(xi,j−ys)hw(tm,ys)−p5v(tm,xi,j)idys
=h(I−1
∑
d1=2hK(xi,j−yd1,1)(wm
d1,1 −p5vm
i,j) + K(xi,j−yd1,J)(wm
d1,J−p5vm
i,j)i
+
J−1
∑
d2=2hK(xi,j−y1,d2)(wm
1,d2−p5vm
i,j) + K(xi,j−yI,d2)(wm
I,d2−p5vm
i,j)i
+K(xi,j−y1,1)(wm
1,1 −p5vm
i,j) + K(xi,j−yI,1)(wm
I,1 −p5vm
i,j)
+K(xi,j−y1,J)(wm
1,J−p5vm
i,j) + K(xi,j−yI,J)(wm
I,J−p5vm
i,j))
(42)
For the reaction term discretization,
Rm(tm,xi,j,vm
i,j,vx(tm,xi,j)) = Φm
i,j∇Ψkvx(tm,xi,j)k·vx(tm,xi,j)(43)
we use the following scalar product approximation:
∇Ψ(kvx(., x1, x2)k)·vx(., x1, x2)
= ∂
∂x1Ψ r∂v
∂x12+∂v
∂x22!,∂
∂x2Ψq(∂v
∂x1)2+ ( ∂v
∂x2)2!·∂v
∂x1,∂v
∂x2(44)
which leads to
∇Ψ(kvx(., x1, x2)k)·vx(., x1, x2)
=∂Ψ
∂s(kvx(., x1, x2)k)(∂v
∂x1)2∂2v
∂x12+∂v
∂x1∂v
∂x2∂2v
∂x1∂x2+ ( ∂v
∂x2)2∂2v
∂x22+∂v
∂x1∂v
∂x2∂2v
∂x1∂x2
q(∂v
∂x1)2+ ( ∂v
∂x2)2
.(45)

Mathematics 2021,9, 91 17 of 23
Further, since the second-order derivatives do not vary too much, we can use
∂2v
∂x12≈∂2v
∂x1∂x2
∂2v
∂x22≈∂2v
∂x1∂x2
to approximate
∇Ψ(kvx(., x1, x2)k)·vx(., x1, x2)
≈∂Ψ
∂s(s)
∂2v
∂x1∂x2∂v
∂x1+∂v
∂x22
q(∂v
∂x1)2+ ( ∂v
∂x2)2
≈Ψ0qv2
x1+v2
x2vx1x2(vx1+vx2),
where
vx1=∂v/∂x
1,
vx2=∂v/∂x
2 and
vx1x2=∂2v/∂x
1
∂x
2 are discretized by applying
the finite difference method (see [15,28]).
To conclude we obtain the following explicit numerical approximation for reaction
term:
Rm(tm,xi,j,vm
i,j,vx(tm,xi,j)) = Φi,jΨ0
s(vm
i+1,j−vm
i−1,j)2
4h2+(vm
i,j+1−vm
i,j−1)2
4h2
(46)
·(vm
i+1,j+1−vm
i+1,j−1−vm
i−1,j+1+vm
i−1,j−1)(vm
i+1,j−vm
i−1,j+vm
i,j+1−vm
i,j−1)
8h3
and thus we get the following explicit numerical approximation scheme for (32):
3p1
2εvm+1
i,j=2p1
ε+p3vm
i,j−p3(vm
i,j)3−p1
2εvm−1
i,j+NIDm
i,j+Rm
i,j+p4fm
i,j. (47)
In a similar manner one obtains the following explicit numerical approximation
scheme for (31):
3p1
2εvm+1
i,j=2p1
ε+p3vm
i,j−p3(vm
i,j)3−p1
2εvm−1
i,j
+Ψi,jhKi+1
2,j(vm
i+1,j−vm
i,j)−Ki−1
2,j(vm
i,j−vm
i−1,j)
+Ki,j+1
2(vm
i,j+1−vm
i,j)−Ki,j−1
2(vm
i,j−vm
i,j−1)i.
(48)
3.2. Experimental Results
The iterative numerical approximation scheme provided by
(47)
was successfully
applied in our image segmentation experiments, for each
m=
1, 2,
. . .
,
M−
1, starting with
v1=v0(x)(see (33)), which represents the [Ih ×Jh]image to be segmented.
The explicit numerical approximation scheme developed in
(47)
is consistent to the
nonlinear second-order anisotropic reaction–diffusion model given by (32).
In summary, the computations follow the procedure in Algorithm 1. For our tests,
we used the following parameter values:
h=
1,
e=
0.1,
p1=
0.6,
p3=
50,
p4=
1, p5=0.3, α=1, η=3 and φ=1.
Some image segmentation results provided by our proposed model are displayed in
Figures 1–4. All the results presented in this section are compared to standard K-means
image segmentation model with two clusters [
24
] and the Chan–Vese image segmentation
model presented in [5].
Our model successfully extracts the objects after up to three iterations. One may see
multiple objects as well as objects with boundary concavities and blurry boundaries are
accurately extracted from the background.

Mathematics 2021,9, 91 18 of 23
Algorithm 1: Reaction-diffusion based image segmentation algorithm
1Set m=1
2Initialize the unknown function v1with the input image to be segmented
3while vmdid not reach stable state do
4Compute diffusion and reaction terms according to (41), (42) and
respectively (46)
5Evolve function vmin (47) to obtain vm+1
i,j
6Increase mby 1
7Segmented image is given by vm
Figure 1shows the segmentation results of our model for a brain CT scan image.
The results are satisfactory even after only one iteration. We also see the model reaching
stability after two iterations in this case. Compared to K-means segmentation results, we
observe the extracted objects edges (brain tissue and cranium bone) are better delimited
from the background. Compared to Chan–Vese segmentation results, our model produces
more accurate results too. In this example, Chan–Vese model seems to not follow the real
object boundaries, especially at the border between cranium bone and brain tissue.
Figure 2shows the segmentation comparison between three cases: first the input
image is segmented ‘as is’, second the input image is contaminated with noise before
segmentation and third we double the noise added to the input image. For all three cases,
we can also see the results of applying K-means and Chan–Vese segmentation. We see
our model successfully removes most part of the noise in Figure 2h,l while still preserving
a good approximation for the edges on the leaf object (better than both K-means and
Chan–Vese).
In Figure 3, we see the segmentation results for a blurry boundary object as galaxy
boundaries are slowly fading. Even after one iteration, our segmentation is superior to
K-means and Chan–Vese as the real galaxy boundaries are correctly identified in Figure 3d.
Figure 4(virus microscopy) brings together noise, blur and irregular boundaries.
Again, after two iterations, the model successfully identifies all objects of interest and the
results, starting with the first iteration, are better than the compared K-means method.
The Chan–Vese segmentation does not separate the virus blobs successfully, although it
provides a good outer boundary approximation.
Regarding time complexity, due to the integral formulation of
Nl D
term in
(41)
and
(42)
, the proposed algorithm is slower than the compared K-means or Chan–Vese counter-
parts. To obtain better performance results, regarding running time, we had to implement
the program on parallel architectures such as CUDA [
42
]. Table 1shows the time taken by
a CUDA implementation for different input image sizes (total number of pixels being
I∗J
).
Using the local scheme in
(48)
, we obtained promising results for image restoration
tasks. Future work will show if we can succeed in mixing the local and nonlocal models
for better noise removal before applying segmentation tasks.
Table 1.
Running durations for the reaction-diffusion algorithm implemented on CUDA. The
durations are for only one vmiteration.
Input Area Size
(Pixels) 65,536 262,144 1,048,576
Time Taken
(Seconds) 0.3 2.0 30.0

Mathematics 2021,9, 91 19 of 23
Figure 1.
(
a
) Original input image to be segmented, (
b
) K-means segmentation results, (
c
) Chan–Vese
segmentation results; and (d–f) our model segmentation results after 1–3 iterations, respectively.
Figure 2.
(
a
) Original input image to be segmented; (
b
) K-means segmentation results; (
c
) Chan–
Vese segmentation results; (
d
) Our model segmentation results after 2 iterations; (
e
) Input image
to be segmented with Gaussian noise added; (
f
) K-means segmentation results for noisy input in
(
e
); (
g
) Chan–Vese segmentation results for noisy input in (
e
); (
h
) Our model segmentation results
for noisy input in (
e
) after 2 iterations; (
i
) Input image to be segmented with more noise added;
(
j
) K-means segmentation results for image in (
g
); (
k
) Chan–Vese segmentation results for noisy input
in (g); and (l) Our model segmentation results for noisy image in (g).

Mathematics 2021,9, 91 20 of 23
Figure 3.
(
a
) Input image to be segmented; (
b
) K-means segmentation results; (
c
) Chan–Vese segmen-
tation results; and (d–f) Our model segmentation results after 1–3 iterations. respectively.
Figure 4.
(
a
) Original input image to be segmented; (
b
) K-means segmentation results; (
c
) Chan–Vese
segmentation results; (d–f) Our model segmentation results after 1–3 iterations. respectively.
4. Conclusions
The starting point in the elaboration of the present work is the paper by Miranville, A.
and Moro¸sanu, C. [
1
], which is a major challenge for both theory and applications, fo-
cused on finding concrete cases of functions for the general case
Φ(t
,
x
,
v(t
,
x)
,
vx(t
,
x))
and
Ψ(t
,
x
,
v(t
,
x)
,
vx(t
,
x))
introduced in [
1
]. In this respect, a rigorous mathematical
investigation is performed to analyze the well-posedness of the nonlinear anisotropic
reaction–diffusion model
(1)
(in particular,
(31)
). The Leray–Schauder principle is applied

Mathematics 2021,9, 91 21 of 23
to prove the existence and uniqueness of a unique classical solution
v(t
,
x)∈W1,2
p(Q)
,
while the
Lp
theory is used to derive the regularity properties for the solutions, consider-
ing that the initial data and the boundary constraints are compatible with the regularity
and compatibility conditions (see
(3)
). In addition, the a priori estimates are made in
Lp(Q)
, which means the approximation for unknown functions
v(t
,
x)
are more precise
(see [1,11–13,15,19–21,35]).
Using the finite-difference method (of second-order in time), two numerical schemes
are constructed see
(47)
and
(48)
to approximate the solution
v(t
,
x)
of the new mathematical
model. Numerical experiments show the model can be successfully applied to image
segmentation tasks. We tested on images with multiple objects as well as objects with
complex concavities or blurry boundaries and proved our model can accurately extract
them, most of the time showing better results than the compared K-means model.
Summarizing, the main contributions in the present work are as follows:
•
We use novel techniques, such as Leray-Schauder principle, a priori estimates,
Lp
-
theory, to elaborate a rigorous qualitative study of the nonlocal and nonlinear second-
order anisotropic reaction–diffusion parabolic problem, endowed with a nonlinearity
of cubic type as well as non-homogeneous Cauchy–Neumann boundary conditions,
expressed by
(1)
and
(31)
. We note that, due to the presence of the nonlinear coeffi-
cient
Φ(kvx(t
,
x))k
(see
(30)
), the proposed second-order nonlinear reaction–diffusion
scheme
(31)
represents a non-variational PDE model. Therefore, it cannot be obtained
from a minimization of any energy cost functional, thus this scheme is not a variational
PDE model.
•
Two two numerical schemes
(47)
and
(48)
are constructed to approximate the solution
of the mathematical models (31) and (32) (local and nonlocal case).
Regarding the second theme, we aim to improve the scheme in
(47)
and
(48)
, as part
of our future research on the topic, by introducing new edge-stopping functions (see [
28
])
and by taking advantage of non-local image information which will allow us to apply the
model to images with inhomogeneity (see [33] and reference therein).
The qualitative results obtained in this current work can be used in quantitative
studies of the mathematical models in
(1)
or
(5)
as well as in the study of optimal control
problems involving such nonlinear problems. We look forward to exploiting all these in
our future works.
Author Contributions:
Conceptualization, C.M.; Formal analysis, S.P.; Project administration, C.M.;
Software, C.M. and S.P.; Supervision, C.M.; Validation, S.P.; Writing—original draft, C.M. and S.P. All
authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data sharing not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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