Mathematical and numerical analysis of a nonlinear diffusion model for image restoration

© Research India Publications

http://www.ripublication.com/gjpam.htm

Mathematical and numerical analysis of a nonlinear diffusion model for image restoration

R. Aboulaich¹

LIRIMA-LERMA Laboratories, Mohammadia School of Engineering,

Mohammed V University in Rabat, Ibn Sina Str., POB 765 Agdal, Rabat, Morocco.

S. Boujena

MACS, Mathematics and computing Department, Ain Chock Sciences Faculty,

Km 8 Route El Jadida POB 5366 Maârif, Casablanca, Morocco.

E. EL Guarmah Royal Air School,

Informatics and Mathematics Department, LIRIMA-LERMA Laboratories, DFST, BEFRA, POB 40002, Marrakech, Morocco.

M. Ziani

Numerical Analysis Group, Applied Mathematical Laboratory,

Dept. of Mathematics, Faculty of Sciences,

Mohammed V University in Rabat, 4, Ibn Battouta Avenue,

PO Box 1014, PC 10090, Rabat, Morocco.

1Corresponding author

diffusion model of partial differential equation in image restoration based on that of Perona Malik. The originality of this work is to prove the existence and uniqueness, in an appropriate Hilbert space, of a solution to the PDE problem proposed with a nonhomogeneous Dirichlet boundary condition.

A first numerical approach based on an explicit scheme is given. The stability of this scheme is established by means of a CFL condition. In order to overcome this shortcoming a semi-implicit scheme based on so-called additive operator splitting (AOS) method is used. The results obtained for large ranges of the parameters, show the efficiency of this AOS method.

AMS subject classification:22E46, 53C35, 57S20.

Keywords:Nonlinear diffusion, non homogeneous Dirichlet boundary condition, image restoration, Hilbert space, AOS scheme.

1. Introduction

In last decades, image restoration has been the subject of several researches. The contours preservation imposes the introduction of nonlinear models, see [2], [4], [5], [7], [8], [9], [10], [11], [12], [14] and [15]. We present in this work a modified Perona-Malik model using a nonlinear PDE. The idea is to maintain, for this model, the efficiency of Perona Malik’s one to preserve the contours while having existence and uniqueness of a solution in a suitable Hilbert space. We recall that several models using regularization by convolution have been studied before in order to prove the existence and uniqueness of regularized problems for image restoration see for more details [9], [14] and [15].

In [3], authors prove existence and uniqueness for Perona-Malik model in Orlicz space. In [2] and [4], existence and uniqueness of solution for the modified Perona Malik model with homogeneous Dirichlet boundary condition is proved in a Hilbert space. In this work, the same model is considered with non homogeneous Dirichlet boundary conditions. For the numerical resolution, we purpose at first, an explicit scheme and we prove, for its stability, a CFL condition. In order to improve the numerical resolution and to avoid the stability condition, we remplace the above explicit sheme by a semi-implicit approximation based on the AOS algorithm proposed in [16]. We end this work by some numerical simulations and a comparison between the proposed explicit and semi-implicit schemes.

2. Nonlinear diffusion model

We are interested to restore a noisy imageu₀ onusing the following nonlinear PDE model with non homogeneous Dirichlet condition on∂ .











Findφ ∈ L²(0, T , H¹())such that:

∂φ

∂t − div[µ(| ∇φ |)∇φ] =0 in Q,

φ (x, t )=u₀(x,0) ∀x ∈ ∂, ∀t ∈ [0, T], φ (x,0)=u0(x,0) ∀x ∈.

(1)

is an open bounded subset of IRⁿ,n=2 orn=3, with boundary∂andQ=x[0, T] for some givenT > 0. We setu(x, t )=φ (x, t )−u0(x,0)for all(x, t )∈ Q, then the problem (1) is equivalent to the following problem:











findu∈ L²(0, T , H¹())such that:

∂u

∂t − div[µ(| ∇(u+u0)|)∇(u+u0)] =0 in Q, u(x, t )=0 ∀x ∈∂, ∀t ∈ [0, T],

u(x,0)=0 ∀x ∈ .

(2)

A weak formulation of the problem (2) is to









Findu∈L²(0, T , V )such that:

∂u(t )

∂t vdx+

µ[| ∇u(t )+ ∇u₀|](∇u(t )+ ∇u₀).∇vdx=0 ∀v ∈V , u(0)=0.

(3)

V = H₀¹() will be provided with the inner product ((u, v)) =

∇u.∇v dx. . is its associeted norm. We indicate by H the adherence of V in L²(). The space H is provided with the scalar product of L²() defined by: (u, v) =

uv dx. The associated norm is noted by|.|.

The existence and uniqueness of a solution of the considered PDE problem with homogeneous boundary conditions are establish see [2] and [1].

In the following we will establish the existence and uniqueness of the weak solution of (1) inL²(0, T , H¹())under the same hypothesis onµ:

1. µ:IR₊ →IR₊,

2. µis a continuously differentiable function, 3. lim

s→+∞[µ(s)] =µ₀withµ₀ >0, 4. s |µ(s) |≤µ(s)∀s ∈IR₊.

The hypothesis (1. - 4.) involve thatµis bounded.

3. Existence and Uniqueness theorems

We set sup

s∈IR₊

µ(s)=aand inf

s∈IR₊µ(s)=b, witha≥0 andb≥0.

We denote byAthe operator defined by (Av, w)=

µ(| ∇v+ ∇u0 |)(∇v+ ∇u0).∇wdx for v, w∈V . (4) Whereu₀∈ H¹()is given. According to the assumptions onµ, we haveAv ∈H⁻¹() for allv∈ V. To prove the existence of a solution for the problem (3), we need the next lemma, see [1] and [6].

Lemma 3.1. Ais an operator monotone hemicontinuous, satisfying for allv, w ∈ V

(Av−Aw, v−w)≥bv−w² . (5)

Proof. Let bev, w∈H₀¹() (Av−Aw, v−w) =

µ(| ∇v+ ∇u0|)

× [∇v+ ∇u₀].∇(v−w)dx

−

µ(| ∇w+ ∇u₀|)[∇w+ ∇u₀].∇(v−w)dx.

Forα ∈ [0,1], we take (α)=

µ(s[w+u₀+α(v−w)])∇[w+u₀+α(v−w)].∇(v−w)dx, withs(u)=| ∇u|. For each fixedx ∈, we consider the functionf defined by:

f (x, α)=µ(s[(w+u₀+α(v−w))(x)])∇

× [(w+u0+α(v−w))(x)].∇(v−w)(x)on[0,1].

We need to prove thatα −→ f (x, α)is differentiable function on ]0,1[for each fixed x ∈.

Letx˜ fixed inandα fixed in]0,1[. The following situations are then discussed:

(a) 0< s((w+u₀+α(v−w))(x)) <˜ +∞. (b) s((w+u₀+α(v−w))(x))˜ =0.

In the situation (a)

∂f

∂α(x, α)˜ =µ{s[w+u₀+α(v−w)]}(∇(v−w))²

+µ{s[w+u₀+α(v−w)]}∂s(w+u0+α(v−w))

∂α

× ∇[w+u₀+α(v−w)].∇(v−w).

We have:

s(w+u₀+α(v −w))=| ∇[w+u₀+α(v−w)] |

=

∂[w+u0+α(v−w)]

∂x₁

2

+

∂[w+u0+α(v −w)]

∂x₂

2

. Then

∂s[w+u0+α(v−w)]

∂α = ∂√

C

∂C .∂C

∂α, with

C = ∂

∂x₁[w+u₀+α(v−w)] 2

+

∂[w+u0+α(v−w)]

∂x₂

2

, and

∂f

∂α(x, α)˜ =µ{s(w+u₀+α(v−w))}(∇(v−w))² +µ{s(w+u0+α(v−w))} 1

s(w+u₀+α(v−w))

× {∇(w+u0+α(v−w)).∇(v−w)}². (6) In the situation (b)

s(w+u0+α(v−w))=0

⇔ ∇w+ ∇u₀+α∇(v−w)=0

⇔



























∂

∂x1

(w+u₀)= ∂

∂x1

(v−w)

= ∂

∂x2

(w+u₀)

= ∂

∂x2

(v−w)=0 (b1) or

∂

∂x1

(w+u0)= −α ∂

∂x1

(v−w) and ∂

∂x2

(w+u0)= −α ∂

∂x2

(v−w) (b2) If we’re in the situation (b1) then using the definition off, we conclude thatf (x, δ)˜ =0,

∀δ∈]0,1[and ∂f

∂δ(x, δ)˜ =0,∀δ ∈]0,1[. If we’re in the situation (b2) then

∂

∂x₁(w+u0)

∂

∂x₁(v −w)

= −α,

and ∂

∂x2

(w+u0)

∂

∂x2

(v−w)

= −α,

and for allt ∈]0,1[such thatt =αwe haves((w+u₀+t (v−w))(x)) >˜ 0, then the functiont −→ f (x, t )˜ is differentiable. Show that t →f (x, t )˜ is also diffrentiable in t =α.

Let(t_k)_k∈IN a sequence of elements of the ]0,1[so that t_k = α for all k ∈ IN and

k→+∞lim t_k =α.

For allk ∈ IN the functiont →f (x, t )˜ is derived intk and its derivative is given by (6), then

k→+∞lim

∂f (x, t˜ _k)

∂tk

= lim

k→+∞[µ{s(w+u0+t_k(v−w))}(∇(v−w))² + µ{s(w+u0+tk(v−w))}

s(w+u0+t_k(v−w)) {∇(w+u₀+t_k(v−w)).(∇(v−w)}²]. Notice that, sinces andµare continuous then

k→+∞lim [µ{s(w+u₀+t_k(v−w))}(∇(v −w))²] =µ(0)(∇(v−w))², and

k→+∞lim

∂f (x, t˜ _k)

∂tk

=µ(0)[∇(v−w)]² + lim

k→+∞

µ{s(w+u0+tk(v −w))}

s(w+u0+t_k(v−w)) {∇(w+u₀+t_k(v−w)).(∇(v −w)}²

. We have

µ{s(w+u₀+t_k(v−w))}

s(w+u₀+t_k(v−w)) × {∇(w+u0+tk(v−w)).(∇(v−w)}²

≤|µ{s(w+u0+t_k(v−w))} |s(w+u0+t_k(v−w))| ∇(v−w)|², andµis continuous, we deduce that

µ{s(w+u0+tk(v−w))} →µ(0), and

k→+∞lim

∂f (x, t˜ k)

∂t_k

=µ(0)| ∇(v−w) |².

Sot →f (x, t )˜ is differentiable inαand we have

∂f (x, α)˜

∂α =µ(0)(∇(v−w))².

It’s now easy to show that

∂f (x, t )˜

∂t

≤K(∇(v−w))²,

withK =µ(0)in the case (b) andK =ain the case (a). Thenis differential on]0,1[ and

(α) =

∂f (x, α)

∂α dx, ∀α ∈]0,1[.

Applying the mean value theorem to on [0,1], we deduce that there exists λ∈]0,1[ such that

(Av−Aw, v−w) =(1)−(0)=(λ)=

∂f (x, λ)

∂λ dx.

We take

1 = {x ∈ /0< s[w+u0+λ(v−w)]<+∞}, and

2 = {x ∈/s[w+u0+λ(v−w)] =0}. Then=1∪2and1∩2= ∅. Furthermore,

∂f (x, λ)

∂λ =





















µ{s(w+u₀+λ(v−w))}(∇(v−w))² +µ{s(w+u₀+λ(v−w))}

s(w+u0+λ(v−w)) × {∇(w+u₀+λ(v−w)).∇(v−w)}²

if x ∈1

µ(0)(∇(v−w))² if x ∈ ₂

(Av−Aw, v −w)=

₁

∂f (x, λ)

∂λ dx

+

₂

∂f (x, λ)

∂λ dx

≥

₁

µ[s(w+u₀+α(v−w))](∇(v−w))²dx

−

1

|µ[s(w+u0+α(v−w))] | s(w+u₀+α(v−w))

× |s(w+u0+α(v−w))|²| ∇(v−w)|² dx +

₂

µ(0)| ∇(v −w)|² dx

≥bv−w ²≥0.

Theorem 3.2. (Existence theorem) Let µ satisfying (1. −4.) and u₀ a function defined onQsuch thatu₀(.,0) ∈ H¹(). Then there exists at least one weak solution φof problem (1) such thatφ ∈ L²(0, T , H¹())∩L^∞(0, T , H ).

Proof. To show the existence, we use Faedo-Galerkin method. We consider the spectral problem

((w, v))=λ(w, v) ∀v ∈V . (7)

Since the injection ofV inHis compact, the problem (7) admits a sequence of eigenvalues λ_j associated to eigenvectorsw_j such that

((wj, v))=λj(wj, v) ∀v ∈V , (8) and{w_j}_j∈IN is orthonormal in H and orthogonal inV. We denote by u_N(t )the ap- proximate solution of (3) defined by











u_N(x, t )=u_N(t )(x)∈ [w1, . . . , w_N] u_N(x, t )=

N j=1

C_j^N(t )w_j(x) u_N(.,0)=0∈ [w1, . . . , w_N].

(9)

We have then









(u_N(t ), w_j)+(µ(| ∇u_N(t )+ ∇u₀|)(∇u_N(t )+ ∇u₀),∇w_j)=0, 1≤j ≤N, t ∈ [0, T],

u_N(.,0)=u_0N ∈ [w₁, . . . , w_N]. u_0N −→u₀ in H.

(10)

EachC_j^N(t )verifies

dC_j^N(t )

dt =Gj(t, C₁^N(t ), . . . , C_N^N(t )).

Knowing that Gj is a continuous function, then by using the Cauchy–Péano theorem we deduce that there exists a local solution u_N(t ) of (10) on [0, T_N] ⊂ [0, T]. By multiplying (10) byC_j^N(t )and by adding, we deduct that:

∂u_N(t )

∂t u_N(t )dx+

µ[| ∇u_N(t )+ ∇u₀ |](∇u_N(t )+ ∇u₀).∇u_N(t )dx =0.

Then 1 2

d

dt |uN(t )|²+

µ[| ∇uN(t )+ ∇u0|](∇uN(t )+ ∇u0).∇uN(t )dx =0, (11)

and

1 2

d

dt |u_N(t )|²+

µ[| ∇u_N(t )+ ∇u₀ |](∇u_N(t ))²dx.

+

µ[| ∇u_N(t )+ ∇u₀ |]∇u₀.∇u_N(t )dx =0. (12) Case 1: b > 0.

As

µ[| ∇u_N(t )+ ∇u₀|](∇u_N(t ))²dx ≥b

(∇u_N(t ))²dx, (13) from (12) we have

1 2

d

dt |u_N(t )|² +b

(∇u_N(t ))²dx ≤ −

µ[| ∇u_N(t )+ ∇u₀ |]∇u₀.∇u_N(t )dx.

(14) Then

1 2

d

dt |u_N(t )|² +bu_N(t )²≤ a²

2bu0²+ b

2u_N(t )², (15) and

| u_N(t )|²≤ a²

b Tu₀²+ |u₀|², (16) from (14) we deduce that

|u_N(t )|² +2b _t

0

u_N(t )²dt ≤ a²

b Tu₀²+ |u₀ |², (17)

and _t

0

u_N(t )²dt ≤ a²T

2b²u₀²+ 1

2b |u0|², (18)

for allt ∈ [0, T_N]. There exists thus a constant C1=

a²

b Tu0²+ |u0 |^{2 1}₂

>0 and a constant C2=

a²

2b²Tu0²+ 1

2b |u0|^{2 1}₂

>0 depending ofa,b,T andu₀such that

|u_N(t )|≤C₁and _t

0

u_N(τ )² dτ ≤C₂∀ t ∈ [0, T_N]. (19) Case 2: b =0.

In (12), we will rewrite the second term in the form

µ{| ∇uN(t )+ ∇u0|}(∇uN(t ))²dx =

{µ[| ∇uN(t )+ ∇u0|] −µ0}(∇uN(t ))²dx +µ₀

(∇u_N(t ))²dx.

Whereµ0is given by the assumption 3. onµ. Then (12) becomes 1

2 d

dt |uN(t )|² +µ0 uN(t )²= −

{µ[| ∇uN(t )+ ∇u0 |]∇u0.∇uN(t )dx.

−

(µ[| ∇uN(t )+ ∇u0 |] −µ0)| ∇uN(t )|²dx, (20) and

1 2

d

dt | uN(t )|²+µ0uN(t )²≤au0 uN(t ) +

|µ(| ∇u_N(t )+ ∇u₀ |)−µ₀|(∇u_N(t ))²dx. (21) Let us setµ1(s) =µ(s)−µ0, then lim

s→+∞µ1(s)=0. Takeε >0, therefore there exists B >0 so that for all(x, t )∈× [0, T_N]

|µ1{| ∇uN(x, t )+ ∇u0 |} |< ε if | ∇uN(x, t )+ ∇u0|> B.

Fort ∈ [0, T_N]fixed, we consider the following sets

^t₁= {x ∈/| ∇uN(x, t )+ ∇u0 |≤B}, ^t₂= {x ∈/| ∇u_N(x, t )+ ∇u₀ |> B}, ^t₁∩^t₂= ∅ ^t₁∪^t₂=.

We have then:

µ1{| ∇u_N(x, t )+ ∇u0|}(∇u_N(x, t ))²dx

=

^t₁

µ₁{| ∇(u_N(x, t )+u₀)|}(∇u_N(x, t ))²dx +

^t₂

µ₁{| ∇(u_N(x, t )+u₀)|}(∇u_N(x, t ))²dx

≤2(B²+(∇u₀)²)

^t₁

µ₁(| ∇u_N(t )+ ∇u₀|)dx +

^t₂

| ∇u_N(t )|² dx.

From (21) we have d

dt |u_N(t )|²+2µ₀ u_N(t )²≤2a u₀u_N(t )

+4(B²+ u0²)(a+µ0)mes()+2uN(t )². We takeε = µ₀

4 and we set 2a²

µ₀ +4(µ₀+a)

mes()u₀ ²+4B²mes()(a+µ₀)=ξ(, µ₀, a, u₀),

then d

dt |u_N(t )|² +µ0

2 u_N(t )²≤ξ(, µ₀, a, u₀),

hence, by the same reasoning as in the first case, we deduce that there exists a constant C₁=

ξ(, µ₀, a, u₀)+ |u₀ |²>0 and a constant C₂= 2

µ0

(ξ(, µ₀, a, u₀)T+ |u₀|²) >0 depending of, µ₀, a, u₀ andT such that

|u_N(t )|≤C1 and _t

0

u_N(t )²dt ≤C2 ∀t ∈ [0, T_N].

Thus in both cases there exists two constantsC₁>0 andC₂ >0 independent ofN such

that 





| u_N(t )|≤C1 ∀t ∈ [0, T_N], _t

0

u_N(τ )² dτ ≤C2. (22) Moreover the problem (10) admits a global unique solutionu_Non[0, T]and according to the monotony and the hemicontinuity of the operatorAwe deduce that the approximate solutionsu_N of the problem (10) converge towards a weak solutionuof the problem (3), see [4], [6], [13] andφ =u+u₀ is a weak solution of (1).

Theorem 3.3. (Uniqueness theorem) Under Hypothesis of the existence theorem 3.2, the weak solutionφof the problem (1) is unique and

φ= dφ

dt ∈ L²(0, T , H⁻¹()).

Proof. Letφ1 andφ2 two weak solutions of the problem (1). Thenu1 = φ1−u0 and u2=φ2−u0are two solutions of the problem (2). We have for allv∈V

∂u₁

∂t (t )−∂u₂

∂t (t ), v

+(Au1(t )−Au2(t ), v)=0. (23)

Takingu₁−u₂ =wandv=w(t )in (23), we have then from (2):

1 2

d

dt|w(t )|² = −(Au₁(t )−Au₂(t ), w(t )) ≤0 Then

|w(t )|² ≤ |w(0)|²=0

Thus w(t ) = u₁(t )−u2(t ) = 0. In conclusion, uniqueness of weak solution for the problem (1) is obtained. Besides, letu= φ−u₀thenu(t ) = −Au(t )inV, from (2) and (4), and for allv ∈V, we have

(Au(t ), v)≤a u(t )v. It follows that

Au(t )V≤au(t ) for all t ∈ [0, T].

Thus Au ∈ L²(0, T , V)and u ∈ L²(0, T , V). Which allows to conclude that φ ∈

L²(0, T , V).

4. Numerical approach

In order to restore a noisy imageu0, we use a model (1), with µ(x)= 1

1+x² +α, which verifies the asymptions (1.-4.):











∂u

∂t −div

1

1+(| ∇u|)² +α

∇u

=0 in Q=x]0, T[, u(x,0)=u0(x,0) ∀x ∈ ,

u(x, t )=u0(x,0) ∀x ∈∂ ∀t ∈ [0, T],

(24)

whereurepresents the restored image.

4.1. An explicit scheme 4.1.1 Algorithm

We denote respectively byhandτ the spatial and time steps sizes. In the following, we takeh=1. The discretization of the time derivative in (24) is given by:

∂u

∂t(i, j )= uⁿ⁺¹(i, j )−uⁿ(i, j )

τ ,

the superscriptnandn+1 denote the time levelstnandtn+1, respectively.

A first-order explicit scheme employed for the spatial derivative approximation:

∇uⁿ(i, j )=











∂u

∂x

n

(i, j ) = uⁿ(i+1, j )−uⁿ(i, j )

h ,

∂u

∂y

n

(i, j ) = uⁿ(i, j +1)−uⁿ(i, j )

h .

(25)

Then

|∇uⁿ(i, j )| =

uⁿ(i +1, j )−uⁿ(i, j ) h

2

+

uⁿ(i, j +1)−uⁿ(i, j ) h

2

. We define for every fieldp=(p₁, p₂) ∈IR², the discrete divergence approximation:

(div(p))_i,j =





p₁ⁿ(i, j )−p₁ⁿ(i−1, j )=0 if 1< i < N1

p₁ⁿ(i, j ) if i=1

−p₁ⁿ(i−1, j ) if i=N1

+





p₂ⁿ(i, j )−pⁿ₂(i, j −1)=0 if 1< j < N₁

p₂ⁿ(i, j ) if j =1

−p₂ⁿ(i, j −1) if j =N1

whereN₁is an integer greater than 2 and

pⁿ(i, j )=













pⁿ₁(i, j )=

1

1+ |∇uⁿ(i, j )|² +α

∂u

∂x

n

(i, j ), pⁿ₂(i, j )=

1

1+ |∇uⁿ(i, j )|² +α

∂u

∂y

n

(i, j ),

(26)

A classical explicit finite difference scheme for(24)is then given by uⁿ⁺¹(i, j )=uⁿ(i, j )+τdiv(pⁿ(i, j )), 1≤n≤M, where

pⁿ(i, j )=(p₁ⁿ(i, j ), p₂ⁿ(i, j ))

uⁿ(i, j )=u(x_i, y_j, t_n), x_i =ih, y_j =j h, t_n =nτ and

τ = T M is the time step size.

This scheme can be explicitly solved for the unknownuⁿ⁺¹: uⁿ⁺¹(i, j )=uⁿ(i, j )+τ

1

1+ |∇uⁿ(i, j )|² +α

×(uⁿ(i, j +1)+uⁿ(i+1, j )−2uⁿ(i, j )) +τ

1

1+ |∇uⁿ(i−1, j )|² +α

(uⁿ(i−1, j )−uⁿ(i, j )) +τ

1

1+ |∇uⁿ(i, j −1)|² +α

(uⁿ(i, j −1)−uⁿ(i, j )). (27) We suppose thath=1.

4.1.2 Stability of the numerical scheme

In the following, we will study the stability of the proposed explicit scheme using the infinity norm _∞.

After reorganizing the terms, we obtain from equation (27) uⁿ⁺¹(i, j )=uⁿ(i, j )

1−2τ

1

1+ |∇uⁿ(i, j )|² +α

−τ

1

1+ |∇uⁿ(i−1, j )|² +α

−τ

1

1+ |∇uⁿ(i, j −1)|² +α

+τ

1

1+ |∇uⁿ(i, j )|² +α

(uⁿ(i, j +1)+uⁿ(i+1, j )) +τ

1

1+ |∇uⁿ(i−1, j )|² +α

(uⁿ(i−1, j )) +τ

1

1+ |∇uⁿ(i, j −1)|² +α

(uⁿ(i, j −1)).

We have

α ≤ 1

1+ |∇uⁿ(i, j )|² +α ≤1+α ∀ i, j.

Then

−(1+α)≤ − 1

1+ |∇uⁿ(i, j )|² −α ≤ −α.

Finally, we have 1−4τ (1+α)≤1−2τ

1

1+ |∇uⁿ(i, j )|² +α

−τ

1

1+ |∇uⁿ(i −1, j )|² +α

−τ

1

1+ |∇uⁿ(i, j−1)|² +α

Notice that ifτ ≤ 1

4(1+α), then

|uⁿ⁺¹(i, j )| ≤(1−4τ (1+α))|uⁿ(i, j )| +τ (1+α)(|uⁿ(i, j +1)| + |uⁿ(i+1, j )|) +τ (1+α)(|uⁿ(i−1, j )| + |uⁿ(i, j −1)|).

Then

|uⁿ⁺¹(i, j )| ≤(1−4τ (1+α))uⁿ∞+4τ (1+α)uⁿ∞. Which implies that

uⁿ⁺¹ ≤ uⁿ∞. Therefore

uⁿ⁺¹∞ ≤ u⁰∞.

And we can conclude that the CFL-like condition of the stability is given by:

τ ≤ 1

4(1+α). (28)

In order to avoid this condition, we use in the next section a semi-implicit scheme proposed in [16].

4.2. Semi-implicit scheme

In the following, we consider the discretization defined by:

u^k+1−u^k

τ =

2 l=1

Al(u^k)u^k+1. (29)

Where

A₁(u^k)= ∂

∂x

1

1+(| ∇u^k |)² +α

∂u^k

∂x

, and

A₂(u^k)= ∂

∂y

1

1+(| ∇u^k |)² +α

∂u^k

∂y

.

Figure 1: Restored image withα =1 andτ =0.2.

This scheme requires to solve a linear system at each time step. For this reason it is called a linear semi-implicit scheme. The solutionu^k+1is given by

u^k+1=

I −τ 2

l=1

A_l(u^k) −1

u^k. (30)

This semi-implicit scheme still has a major drawback at each iteration one needs to solve a large linear system whose matrix is not tridiagonal.

Let us now consider a modification of Equation (30), namely the additive operator splitting (AOS) scheme, see [16].

u^k+1= 1 2

2 l=1

(I −2τ A_l(u^k))⁻¹u^k. (31)

The operatorsB_l = I −2τ A_l(u^k)describe one-dimensional diffusion processes along the spatial axes. The above splitting scheme is efficient because at each time step a single tridiagonal matrix inversion is performed for each spatial dimension. Under a consecutive number of pixels along the direction they are reduced to tridiagonal matrices strictly diagonally dominant that can be effectively reversed by the Thomas algorithm.

Figure 2: Restored image withα =0.5 andτ =0.1.

Figure 3: Restored image by explicit scheme withα =1.

Figure 4: Restored image withα =1.

4.3. Numerical results

We recall that the SNR is the Signal-to-noise ratio used to estimate the quality of an imageI₂with respect to a reference imageI₁, defined by the expression:

SN R(I₁/I₂)=10 log₁₀ σ²(I₁) σ²(I1−I2)

, whereσ is the variance.

Figures 1 and 2 are used as a test for processing images corrupted by a Gaussian Noise.

We remark in figure 1 that the Additive Oprator Splitting scheme provides better restoration results than the explicit scheme forα = 1 andτ = 0.2. On the other hand, we observe that both the explicit and the AOS schemes give good results forα = 0.5 andτ =0.1 becuase in this case, the stability condition is satisfied.

In Figure 3 we present the value of the coefficient SN R, for the explicit scheme, according to the variation of the additional Gaussian noise. We notice that the explicit scheme is only stable and provides a better results for very small time steps. This leads to poor efficiency. Next we consider Figure 4 which depicts the filtering by a Gaussain noise. The situation is similar as in Figure 1: the Additive Oprator Splitting (AOS) scheme provides a better restoration results than an explicit scheme for α = 1 with τ =0.3 andτ =0.2 when the stability condition is not satisfied.

5. Conclusion

We proposed in this work a nonlinear diffusion model with non homogeneous boundary conditions in image restoration. After establishing the existence and uniqueness of a solution for PDE problem, we tested the model numerically by both an explicit and a semi-implicit schemes using an additive operator splitting (AOS) method. The obtained results show that an instability occurs for particular values of the parametersτ andαfor the explicit scheme if the stability condition is not verified. The semi-implicit scheme with an additive operator splitting (AOS) method permits to avoid this difficulty.

Acknowledgment

This work was supported by Euromediterranean Scomu project, the LIRIMA Laboratory and by the Franco-Moroccan Henri Curien project governed by EGIDE(MA/11/246).

References

[1] R. Aboulaich, S. Boujena, E. El Guarmah, A nonlinear diffusion model with non homogeneous boundary conditions in image restoration, ECS10 Milan Italie, June (2009) 22–26.

[2] R. Aboulaich, S. Boujena, E. El Guarmah, A nonlinear parabolic model in process- ing of medical image, Mathematical Modelling of Natural Phenomena, 3(1), 1–15 (2008).

[3] R. Aboulaich,D. Meskine, A. Souissi, New diffusion models in image processing, Journal of Computers and Mathematics with Applications, 56(4), 874–882 (2008).

[4] R. Aboulaich, S. Boujena, E. El Guarmah, Sur un modèle non-linéaire pour le débtruitage de l’image, C.R. Acad. Sci. Paris. Ser., 345, 425–429 (2007).

[5] L. Alvarez, P-L. Lions, J-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. ii, SIAM Journal on Numerical Analysis, 29(3), 845–866 (1992).

[6] S. Boujena, Etude d’une classe de fluides non newtoniens, les fluides newtoniens généralisés, thèse de 3ème cycle, Université Pierre et Marie Curie–Paris 6 (1986).

[7] S. Boujena, K. Bellaj, E. El Guarmah, O. Gouasnouane, An Improved Nonlinear Model for Image Inpainting, Applied Mathematical Sciences, 9 (124), 6189–6205 (2015).

[8] F. Catté, P.L. Lions, J. M. Morel, T. Coll, Image selective smoothing and edge detection by nonlinear, SIAM Journal on Numerical Analysis, 29(1), 182–193 (1992).

[9] P. Destuynder, Analyse et traitement des images numériques, Hermès, Paris (2004).

[10] S. German, D. German, Stochastic relaxation, gibbs distribution, and tha bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine intelli- gence, 6(6), 721–741 (1984).

[11] P. Guidotti, A family of nonlinear diffusions connecting Perona-Malik to standard diffusion, Discrete and Continuous Dynamical Systems - Series S, 5(3), 581–590 (2012).

[12] S. Levins, Y. Chen, J. Stanich, Image restoration via nonstandard diffusion, Technical-Report 04-01, Dept of Mathematics and Computer Science, Duquesne University (2004).

[13] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier Villard, Paris (1969).

[14] D. Munford, J. Shah, Optimal approximations by piecewise smooth functions and variational problems, Communication on Pure and Applied Mathematics, 42(5), 577–685 (1989).

[15] P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transaction on Pattern analysis and machine intelligence, 12(7), 629–639 (1990).

[16] J. Weickert, M. Bart, H. Romeny ter, A. Viergever Max, Efficient and reliable Schemes for nonlinear Diffusion Filtering, IEEE Trans., Image Processing, 7(3), 398–410 (1998).