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mathematics
Article
Rigorous Mathematical Investigation of a Nonlocal
and Nonlinear SecondOrder Anisotropic ReactionDiffusion
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
SecondOrder Anisotropic
ReactionDiffusion 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
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tral with regard to jurisdictional clai
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tribution (CC BY) license (https://
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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 secondorder parabolic problem, equipped with nonlinear anisotropic
diffusion and cubic nonlinear reaction, as well as nonhomogeneous CauchyNeumann boundary
conditions. Under certain assumptions on the input data:
f(t
,
x)
,
w(t
,
x)
and
v0(x)
, we prove the
wellposedness (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 reactiondiffusion; wellposedness of solutions; LeraySchauder 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
secondorder 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) = Φvx(t,x)hΨvx(t,x)∆v(t,x) + ∇Ψvx(t,x)· ∇v(t,x)i
+p3v(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 secondorder 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
•Φvx(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)) = Φvx(t,x)Ψvx(t,x)vxi(t,x),i=1, 2. (4)
Then, it is easy to recognize Equation (1)1as being quasilinear with
aij t,x,v(t,x),vx(t,x)=∂
∂vxj
ait,x,v(t,x),vx(t,x)
=∂
∂vxj
Φvx(t,x)Ψvx(t,x)vxi(t,x),i=1, 2,
at,x,v(t,x),vx(t,x)=−∂
∂vΦvx(t,x)Ψ(vx(t,x))vxi(t,x)vxi(t,x)
−∂
∂xi
Φvx(t,x)Ψvx(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)−Φvx(t,x)divΨvx(t,x)∇v(t,x)
=p3v(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 quasilinear 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)) = −p3v(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≤aijt,x,u,zζiζj≤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 secondorder boundary value problem
(1)
, supplied with
different boundary conditions, have been successfully applied to many complex moving
interface problems, e.g., the motion of antiphase 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 phaseﬁeld 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 higherorder timestepping 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 nonhomogeneous 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
,
Φvx(t
,
x)
,
Ψ(vx(t
,
x))
), the principal part being in divergence form and by considering the cubic nonlin
earity p3v(t,x)−v3(t,x), satisfying for n∈ {1, 2, 3}the assumption H0in [21], that is:
H0:(v−v3)v3p−4v≤1+v3p−1− v3p.
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. WellPosedness 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 LeraySchauder principle (see [1,4,11–15,19–21] and reference therein);
• The Lptheory of linear and quasilinear 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 scalarvalued 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 nonintegral
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 xderivatives 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
.
Φvx(t
,
x)
is a positive and bounded nonlinear real function of class
C1(Q)
with bounded
derivatives and
0 < m1≤Φvx(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)∈Qv1−v2 ≤ C1eCT maxmax
(t,x)∈Qf1−f2,max
(t,x)∈Σw1−w2,max
(t,x)∈Ωv1
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 LeraySchauder 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)−"λΦux(t,x)∂
∂uxj
(Ψ(ux)uxi) + (1−λ)δj
i#vxixj
=λnA(t,x,u,ux) + p3u(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) = Φux∇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
u3pdxdt!1
p
=
R
Q
u3pdxdt!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)) = Φux(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
Φux(t
,
x)
(see I
2
), as well as the properties of
Ψux(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 welldeﬁned. 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 welldeﬁned.
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
xjΨ(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
ivn,λn
xixj
+A(t,x,un,un
xi) + p3un−(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 righthand side in
(17)
belongs to
Lp(Q)
, since
vn,λn∈W1,2
p(Q)
. Therefore, the
Lptheory of PDE gives the estimate
kVn,λn,λkW1,2
p(Q)≤Cλn−λ × (
hΦ(un
x)∂
∂un
xjΨ(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
ivn,λn
xixj
. Moreover, since
W2−2
p
p(Ω)⊂Lp(Ω)
, it results that the
remaining terms on the righthand 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
xjΨ(un
x)un
xi+ (1−λ)δj
i#Vn,1,λ
xixj
=λΦ(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−u3oin 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 Lptheory 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)divΨ(vx)∇v−(1−λ)∆v
=λhp3v(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
v3p−4v
, integrating over
Qt:= (
0,
t)×Ω
,
t∈(0, T], we get
p1Z
Qt
∂
∂tv(τ,x)3p−2dτdx −λZ
Qt
Φ(vx)divΨ(vx)∇vv3p−4v dτdx
−(1−λ)Z
Qt
∆vv3p−4v dτdx
=λp3Z
Qt
(v−v3)v3p−4v dτdx +λp4Z
Qt
fv3p−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)v3p−4vdτdx
+(1−λ)(3p−3)Z
Qt
∇v2v3p−4dτdx
+λp5m1R
Σt
v3p−2dτdγ+(1−λ)
qmax p5R
Σt
v3p−2dτdγ
≤λp1
3p−2Z
Ω
v0(x)3p−2dx
+λp3R
Qt
(v−v3)v3p−4v dτdx +λp4R
Qt
fv3p−4v dτdx
+λM1Z
Σt
wv3p−4v dτdγ+(1−λ)
qmin Z
Σt
wv3p−4v dτdγ.
(23)
for all
t∈(
0,
T]
. The Hölder and Cauchy inequalities, applied to the last terms in
(23)
,
give us
i1λp4Z
Qt
fv3p−4v dτdx ≤p−1
pε
p
p−1Z
Qt
v3pdτdx +λp4
1
pε−pkfkp
Lp(Q)
i2λM1Z
Σt
wv3p−4v dτdγ
≤λp5m11−1
3p−2Z
Σt
v3p−2dτdγ+M1
m1
1
p5
1
3p−2Z
Σt
wpdτdγ,
i3
(1−λ)
qmin Z
Σt
wv3p−4v dτdγ
≤p51−1
3p−2(1−λ)
qmax Z
Σt
v3p−2dτdγ+qmax
qmin
1
p5
1
3p−2Z
Σt
w3p−2dτdγ.
By H0, relation (3) and Young’s inequality, we obtain
λp3Z
Qt
(v−v3)v3p−4v dτdx
≤λp3ΩT+λp3ΩT1
3pε−3p+3p−1
3pε
3p
3p−1Z
Qt
v3pdτdx
−λp3Z
Qt
v3pdτdx.
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)v3p−4vdτdx
+(1−λ)(3p−3)Z
Qt
∇v2v3p−4dτdx +λp3Z
Qt
v3pdτdx
+p5
1
3p−2λm1+(1−λ)
qmax Z
Σt
v3p−2dτdγ
≤λp1
3p−2Z
Ω
v0(x)3p−2dx
+3p−1
3pε
3p
3p−1+p−1
pε
p
p−1Z
Qt
v3pdτdx
+λp3ΩT+p3ΩT1
3pε−3p+p4
1
pε−pkfkp
Lp(Q)
+1
p5
1
3p−2M1
m1
+qmax
qmin Z
Σt
w3p−2dτdγ.
Taking εsmall enough, the previous inequality yields
λkv3kp
Lp(Q)≤C11+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 Lptheory to problem (15) (see [1] and references therein), we get
kvkW1,2
p(Q)≤C2kv0k
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
xdλ,
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
xixjVxi
1
R
0
∂
∂vλ
xj
aij t,x,vλ,vλ
xdλ
+V
1
R
0
∂
∂vλaij t,x,vλ,vλ
xdλ
+v2
xixjΦ(v1
x)−Φ(v2
x),
(27)
where V(t,x) = v1(t,x)−v2(t,x).
Regarding A(t,x,v,vx) = Φvx∇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
dλAt,x,vλ,vλ
xdλ
=Vxi
1
Z
0
∂
∂vλ
xj
At,x,vλ,vλ
xdλ+V
1
Z
0
∂
∂vλAt,x,vλ,vλ
xdλ.
(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,jt,x,vλ,vλ
xdλ+
1
Z
0
∂
∂vλ
xj
At,x,vλ,vλ
xdλ,
ˆ
a(t,x) = −v2
xixj
1
Z
0
∂
∂vλai,jt,x,vλ,vλ
xdλ+
1
Z
0
∂
∂vλAt,x,vλ,vλ
xdλ
−p2h1−(v1)2+v1v2+ (v2)2i.
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 SecondOrder Anisotropic ReactionDiffusion Model in
Image Segmentation
The nonlinear parabolic secondorder 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 Φvx(t,x)and Ψvx(t,x)as follow
Φ:[0, ∞)→(0, ∞),Φ(s) =
3
pϕs2+η
α,
Ψ:[0, ∞)→(0, 1],Ψ(s) = 1
1+s
c2
(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 nonhomogeneous CauchyNeumann
boundary conditions is acquired:
p1
∂
∂tv(t,x)−Φkvx(t,x)kdivΨkvx(t,x)kvx(t,x)
=p3v(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 edgestopping (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 reactiondiffusion 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 reactiondiffusion model to
(31)
can be written
as follows:
p1
∂
∂tv(t,x) = Φkvx(t,x)kΨkvx(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)
+Φkvx(t,x)k∇Ψkvx(t,x)k·vx(t,x) + p3v(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
ε3Kz
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
ﬁnitedifference method (of secondorder 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 reactiondiffusion 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 secondorder 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) = Φkvx(t,x)kΨkvx(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) = Φkvx(t,x)k∇Ψkvx(t,x)k·vx(t,x). (39)
The leftside 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,ji
+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,ji
+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,ji)
(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∇Ψkvx(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
∂x12+∂v
∂x22!,∂
∂x2Ψq(∂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 secondorder 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
∂x22
q(∂v
∂x1)2+ ( ∂v
∂x2)2
≈Ψ0qv2
x1+v2
x2vx1x2(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+p3vm
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+p3vm
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 secondorder 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 Kmeans
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: Reactiondiffusion 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 Kmeans 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 Kmeans 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 Kmeans 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
Kmeans 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 Kmeans 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 Kmeans 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 reactiondiffusion 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
) Kmeans 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
) Kmeans 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
) Kmeans 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
) Kmeans 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
) Kmeans 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
) Kmeans 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 wellposedness 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 ﬁnitedifference method (of secondorder 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 Kmeans model.
Summarizing, the main contributions in the present work are as follows:
•
We use novel techniques, such as LeraySchauder 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 nonhomogeneous 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 secondorder nonlinear reaction–diffusion
scheme
(31)
represents a nonvariational 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 edgestopping functions (see [
28
])
and by taking advantage of nonlocal 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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