2. Definition of the cost function and variational model

DOI:10.1051/cocv/2014061 www.esaim-cocv.org

TOPOLOGICAL GRADIENT FOR A FOURTH ORDER OPERATOR USED IN IMAGE ANALYSIS

Gilles Aubert

and Audric Drogoul

Abstract. This paper is concerned with the computation of the topological gradient associated to a fourth order Kirchhoff type partial differential equation and to a second order cost function. This computation is motivated by fine structure detection in image analysis. The study of the topological sensitivity is performed both in the cases of a circular inclusion and a crack.

Mathematics Subject Classification. 35J30, 49Q10, 49Q12, 94A08, 94A13.

Received December 9, 2013. Revised November 24, 2014 Published online June 30, 2015.

1. Introduction

The notion of topological gradient which has been rigorously formalized in [13,17] for shape optimization problems has a wide range of applications: structural mechanics, optimal design, inverse analysis and more recently image processing [6–8]. Roughly speaking the topological gradient approach performs as follows: letΩ be an open bounded set of R² and j(Ω) =J(Ω, u_Ω) be a cost function where u_Ω is the solution of a given PDE onΩ. For small≥0, letΩ=Ω\x₀+ω, wherex₀∈Ω andω is a given subset ofR². The topological analysis provides an asymptotic expansion ofj(Ω) as→0. In most cases it takes the form:

j(Ω) =j(Ω) +²I(x0) +o(²) (1.1) I(x₀) is called the topological gradient atx₀.

Thus, in optimal design for example, if we want to minimizej(Ω) it would be preferable to create holes at pointsx₀ whereI(x0) is “the most negative”. In practice, we keep points x₀ where the topological gradient is less than a given negative threshold. In image processing the choice of the cost function is guided by the aimed application. For example for detection or segmentation problems, we have to choose a cost function which blows up in a neighbourhood of the structure we want to detect. Thus removing from the initial domain such a neighbourhood implies a large variation of the cost function and so a large topological gradient. In [8] the method was applied for edge detection by studying the topological sensitivity ofj(Ω) =

Ω|∇u_Ω|²dxwhereu_Ω is the solution of a Laplace equation. For filament (or point) detection, the problematic is different. Indeed there is no jump of the image intensity across this type of structure which is of zero Lebesgue measure. Typically the intensity takes the value 1 on the fine structure and 0 elsewhere. It is known (see for example Steger [18]) that

Keywords and phrases.Topological gradient, fourth order PDE, fine structures, 2D imaging.

1 Universit´e Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France.gaubert@unice.fr; drogoula@unice.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2015

TOPOLOGICAL GRADIENT FOR A FOURTH ORDER OPERATOR USED IN IMAGE ANALYSIS

Figure 1. (a) Perforated domain, (b) cracked domain.

the gradient operator is not adapted since “it does not see” these structures. In this case we have to use a cost function defined from the Hessian matrix of a regularized version of the initial image. Inspired from the theory of thin plates ([9], Chap. 8), the main goal of this paper is to compute the topological gradient associated to the following model: ifΩdenotes the image domain and iff is the initial grey values image (the data, possibly degraded) we search foru_Ω as the solution of the fourth order PDE:

(P)

⎧⎪

⎨

⎪⎩

Δ²u+u=f, inΩ B₁(u) = 0, on∂Ω B₂(u) = 0, on∂Ω

(1.2)

whereB₁(u) andB₂(u) are natural boundary conditions to be specified in the next section (in fact we will study a more general model). The cost function is then defined as:

J_Ω(u) =

Ω

(Δu)²+ 2(1−ν) ∂²u

∂x₁∂x_{2 2}

−∂²u

∂x²₁

∂²u

∂x²₂

, (1.3)

where 1< ν <0 is a parameter (the Poisson ratio). The link betweenu_Ω andJ_Ω(u) is u_Ω = argmin

u∈H²(Ω)

J_Ω(u) +u−f²_L2(Ω). Let us notice that for ν = 0 the cost function simplifies as J_Ω(u) =

Ω∇²u²₂. This latter cost function has been used in the numerical companion paper [10] for detecting fine structures in 2D or 3D images (see also [11]).

In this work we compute the topological gradient associated to (1.2) and (1.3) when Ω =Ω\x₀+B(0,1) and Ω =Ω\x₀+σ(n), whereB(0,1) is the unit ball of R² andσ(n) is a straight segment with normaln (a crack, see Fig.1).

We warn the reader that the proofs are very technical and we only give the main steps. For the complete proofs and some results in 3D we refer the reader to [11]. The numerical analysis of the model as well numerous examples will be given in [10]. Most of the results of our work have been announced in [5].

Remark 1.1. The study of topological sensitivity for fourth order operators is not new. In [4], the authors in a different context, compute the topological gradient for the Kirchhoff plate bending in the case of a circular inclusion. Our model is simpler and we are able to give explicit expressions of the topological gradient both in the cases of circular inclusions and of cracks.

Remark 1.2. The link between the PDE and the cost function simplifies the topological gradient computation but it is easy to adapt the method for some other cost functions.

In the following, we denote by um,Ω =u_H^m_(Ω)(respectively |u|m,Ω =|u|_H^m_(Ω)) the norm (respectively the seminorm) on the Sobolev spacesH^m(Ω) andus,Γ the norm on the fractional Sobolev spaceH^s(Γ) with Γ =∂Ω.

G. AUBERT AND A. DROGOUL

The outline of the paper is as follows. In Section 2we define precisely the cost function and the variational problem. In Section3(respectively Sect.4) we give the main steps of the computation of the topological gradient in the case of a circular inclusion (respectively of a crack). The paper ends with three appendices in which are developed details not given in Sections3 and4.

2. Definition of the cost function and variational model

In this section we specify the general cost function and the variational model we want to study. Before, we give a lemma explaining why the natural operator we have to use in the cost function for detecting points and curves in 2D must be of second order.

2.1. Detecting fine structures: what is the good operator?

We denote by D(R²) the space of C^∞- functions with compact support in R² and D(R²) the space of distributions onR². We refer the reader to [11] for the proof of the following lemma.

Lemma 2.1. Let ϕ:R²−→Rbe a Lipschitz continuous function, and let(g_h)_h>0 be a sequence of functions defined by

g_h(x) = 1

θ₁(h)e^{−ϕ2(x)θ2(h)} where θ_i:R⁺−→R⁺ and lim

h→0θ_i(h) = 0.

(i)Let a∈R², by settingϕ(x) =x−a,θ₁(h) =πhandθ₂(h) =hthen g_h−→

h→0δ_a, inD(R²).

Besides we have

∇g_h(a) = [0,0]^T, ∇²g_h(a) =− 2 πh²I, whereI denotes the identity in R².

(ii)Let Γ be a smooth closed curve or a smooth infinite curve ofR² delimiting two sub-domainsR²_Γ⁻ andR²_Γ⁺ forming a partition of R². Let ϕbe the signed distance toΓ defined by:

ϕ(x) = dist

x,R²_Γ⁻

−dist

x,R²_Γ⁺ . We denote by R²_Γ⁺ (resp.R²_Γ⁻), the sub-domain {ϕ >0}(resp.{ϕ <0}).

Taking the following scalings:θ₁(h) =√

πhandθ₂(h) =h, we have g_h−→

h→0δ_Γ, inD(R²) Besides for allx∈Γ we have

∇g_h(x) = [0,0]^T, spec

∇²g_h(x)

=

− 2 h^3/2,0

,

where spec(M) denotes the eigenvalues of the matrix M. The associated eigenvectors to ∇²g_h on Γ are (∇ϕ(x),∇ϕ(x)^⊥), where∇ϕ(x) =n(x).

So this lemma shows heuristically that the gradient “does not see” fines structures inR²(points and filaments).

On the other hand second derivatives are singular on these structures.

TOPOLOGICAL GRADIENT FOR A FOURTH ORDER OPERATOR USED IN IMAGE ANALYSIS

2.2. Definition of the cost function and the fourth order PDE

We give now the general cost function we are going to study. The cost function is more general than (1.3). It is defined by:

J_Ω(u) =

Ω

(Δu)²+ 2(1−ν) ∂²u

∂x₁∂x₂ 2

−∂²u

∂x²₁

∂²u

∂x²₂

+γ|∇u|²dx, 0< ν <1, γ≥0 (2.1)

Remark 2.2.

(i) Whenγ= 0 we retrieve (1.3).

(ii) We check easily thatJ_Ω:

J_Ω(u)≥(1−ν)|u|²_H2(Ω)+γ|u|²_H1(Ω), ∀u∈H²(Ω). (2.2) For small >0, let (a)Ω=Ω\{x₀+ω}or (b)Ω=Ω\{x₀+σ(n)}, where x₀∈Ω, ω=B(O,1) is the unit ball ofR²andσ(n) is a straight segment centered at the origin and with normaln(a crack). We introduce the bilinear and linear forms:

a(u, v) =

Ω

ΔuΔv+ (1−ν) 2 ∂²u

∂x₁∂x₂

∂²v

∂x₁∂x₂ −∂²u

∂x²₁

∂²v

∂x²₂ −∂²u

∂x²₂

∂²v

∂x²₁

+γ∇u.∇v+uv, l(v) =

Ω

fv.

(2.3)

Thanks to Lax–Milgram lemma, it is easy to prove that for ≥0 fixed there exists a unique u∈ H²(Ω) such that

a(u, v) =l(v), ∀v ∈H²(Ω). (2.4) This solutionu necessarily satisfies the Euler equation:

(P)

⎧⎪

⎨

⎪⎩

Δ²u−γΔu+u=f, onΩ, B₁(u)−γ∂_nu= 0, on∂Ω,

B₂(u) = 0, on∂Ω,

(2.5)

where

B₁(u) =∂_n(Δu)−(1−ν)∂_σ n₁n₂ ∂²u

∂x²₁ −∂²u

∂x²₂

−(n²₁−n²₂) ∂²u

∂x₁∂x₂

and

B₂(u) =νΔu+ (1−ν) n²₁∂²u

∂x²₁ +n²₂∂²u

∂x²₂ + 2n₁n₂ ∂²u

∂x₁∂x₂

,

where f ∈L²(Ω);n= (n₁, n₂) is the outward normal to the domain, and σ = (σ₁, σ₂) is the tangent vector such that (n,σ) forms an orthonormal basis.

Remark 2.3.

(i) Whenγ= 0, (2.4) is well-posed by using Gagliardo–Nirenberg inequalities [1,15]. The computation of the topological gradient is the same as in the caseγ = 0.

(ii) We have the following relation:a(u, u) =J_Ω(u) +u²_0,Ω.

We denote byP₁ the set of polynomial of degree less or equal than 1, and byC all constants not depending on . Finally, we will use the quotient spaceH^m(Ω)/P1 which is the set ofH^m(Ω) functions defined up to a polynomial of degree less or equal than 1. In the paper, we study independently the case of a domain perforated by a ball and the case of a cracked domain.

G. AUBERT AND A. DROGOUL

3. Computation of the topological gradient in the case of the ball

3.1. Notations and statement of the problem

Letx₀∈Ω andB(O,1) the unit ball. We defineΩ=Ω\x₀+B(O,1) withx₀ and chosen such that the perturbation does not touch the border. By denotingB =B(x₀,1) andB=B(x₀, ) we have∂Ω=∂B∪Γ (Γ =∂Ω).

Problem (P) rewrites as

(P^b)

⎧⎪

⎨

⎪⎩

Δ²u−γΔu+u=f, onΩ, B₁(u)−γ∂_nu= 0, on∂B∪Γ,

B₂(u) = 0, on∂B∪Γ.

(3.1)

Let v∈ H²(Ω), andu the solution of P^b

. By a classical regularity result,u ∈H⁴(Ω). Then by using integration by parts onΩ, and ([9], p. 376), we get from (2.4) and (2.5):

Ω

fv=

Ω

Δ²u−γΔu+u v

=a(u, v)−

B

((B₁(u)−γ∂_nu)v−B₂(u)∂_nv) +

Γ

((B₁(u)−γ∂_nu)v−B₂(u)∂_nv). To simplify notations we suppose thatx₀≡0.

Now to compute the topological gradient, we have to estimate the leading term when→0 in the difference J(u)−J₀(u₀). Using equations satisfied by u andu₀ we have

J(u)−J₀(u₀) =

Ω

(f−2u₀)(u−u₀)−

Ω

(u−u₀)²−

B

(f−u₀)u₀. (3.2) Let us denoteL:H²(Ω)−→Rthe linear map

L(u) =

Ω

(f−2u₀)u, ∀u∈H²(Ω), (3.3)

and

J=−

Ω

(u−u₀)²−

B

(f−u₀)u₀. (3.4)

The first step for evaluating (3.2) is to introduce v the unique solution of the adjoint problem (see [2]

and [17])

a(u, v) =−L(u), ∀u∈H²(Ω). (3.5) From (3.5) and (3.4) we rewrite (3.2)

J(u)−J₀(u₀) =−a(u−u₀, v) +J=−l(v) +a(u₀, v) +J

=−

Ω

fv+

Ω

Δ²u₀−γΔu₀+u₀ v+

∂B

(B₁(u₀)−γ∂_nu₀)v−B₂(u₀)∂_nv+J

=

∂B

(B₁(u₀)−γ∂_nu₀)v−B₂(u₀)∂_nv+J.

Then we setw=v−v₀, wherev₀is the solution (3.5) with= 0, (Ω₀=Ω), thus we rewrite J(u)−J₀(u₀) =

∂B

(B₁(u₀)−γ∂_nu₀)v₀−B₂(u₀)∂_nv₀+

∂B

(B₁(u₀)−γ∂_nu₀)w−B₂(u₀)∂_nw+J. Now we express the differenceJ(u)−J₀(u₀) as a sum of more simple terms.

TOPOLOGICAL GRADIENT FOR A FOURTH ORDER OPERATOR USED IN IMAGE ANALYSIS

Forϕ₁∈H^3/2(∂B),ϕ₂∈H^1/2(∂B) let l^ϕ1,ϕ2 ∈H²(B) the solution of the problem

⎧⎪

⎨

⎪⎩

Δ²l^ϕ1,ϕ2 = 0, onB, l^ϕ1,ϕ2 =ϕ₁, on∂B,

∂_nl^ϕ1,ϕ2 =ϕ₂, on∂B.

(3.6)

Foru∈H²(Ω) we denote byl^u the functionl^{u,∂ nu}, and for= 1 byl^ϕ1,ϕ2 the function l₁^ϕ1,ϕ2. The difference becomes:

J(u)−J₀(u₀) =

∂B

(B₁(u₀)−γ∂_nu₀)v₀−B₂(u₀)∂_nv₀+

∂B

(B₁(u₀)−γ∂_nu₀)l^w−B₂(u₀)∂_nl^w+J

=L+K+J, (3.7)

where K=

∂B

(B₁(u₀)−γ∂_nu₀)l^w−B₂(u₀)∂_nl^w, L =

∂B

(B₁(u₀)−γ∂_nu₀)v₀−B₂(u₀)∂_nv₀. Letu₀(x) =u₀(x)−u₀(0)− ∇u₀(0).x, it is straightforward that

K=

∂B

(B₁(u₀)−γ∂_nu₀)l^w−B₂(u₀)∂_nl^w. Then integration by parts (see [9], p. 376) gives

K=

B

Δ²u₀l^w−b(u₀, l^w)−

B

γ(Δu₀l^w− ∇u₀.∇l^w) whereb(u, v) is the bilinear form associated to (3.6) and defined by

b(u, v) =

B

ΔuΔv+ (1−ν) 2 ∂²u

∂x₁∂x₂

∂²v

∂x₁∂x₂ −∂²u

∂x²₁

∂²v

∂x²₂ −∂²u

∂x²₂

∂²v

∂x²₁

· Integration by parts again gives:

K=

B

Δ²u₀−γΔu₀

l^w+γ∇u₀.∇l^w−

B

Δ²l^w =0

u₀+

∂B

B₁(l^w)u₀−B₂(l^w)∂_nu₀. (3.8)

In a similar manner L=

B

Δ²u₀−γΔu₀

v₀+γ∇u₀.∇v₀−

B

Δ²v₀u₀+

∂B

B₁(v₀)u₀−B₂(v₀)∂_nu₀. (3.9) We setF = (f−2u₀), thus we have in the distributional sense for→0:

Δ²v₀−γΔv₀+v₀=−F inD(B). (3.10) From (3.7)–(3.10) we get

L+K=

B

(f−u₀)v₀+γ∇u₀.∇v₀+

B

(F−γΔv₀+v₀)u₀+

B

Δ²u₀−γΔu₀ l^w +γ∇u₀.∇l^w+

∂B

(B₁(l^w) +B₁(v₀))u₀−

∂B

(B₂(l^w) +B₂(v₀))∂_nu₀

=

B

(f−u₀)v₀+γ∇u₀.∇v₀+J_A−J_B+E1+E2+E3, (3.11)

G. AUBERT AND A. DROGOUL

where

J_A=

∂B

(B₁(l^w) +B₁(v₀))u₀, J_B =

∂B

(B₂(l^w) +B₂(v₀))∂_nu₀, E₁=

B

(F−γΔv₀+v₀)u₀, E₂=

B

Δ²u₀−γΔu₀

l^w, E₃=

B

γ∇u₀.∇l^w. (3.12)

In the next section, we will show that E1, E2 and E3 areo(²) and that J_A andJ_B are O(²). Before we establish the asymptotic expansion ofB₁(v₀),B₂(v₀) andw.

3.2. Estimates of B

( v

)( x ) and B

( v

)( x ) for x ∈ ∂B

Proposition 3.1. Supposev₀ regular, then when →0 we have the following boundary expansions:

B₁(v₀)(X) =−g₁(X)

+O(1), B₂(v₀)(X) =−g₂(X) +O(), where settingX = (cos(θ),sin(θ))∈∂B

g₁(X) =g₁(θ) = (1−ν) ∂²v₀

∂x²₁(0)−∂²v₀

∂x²₂(0)

cos(2θ) + 2(1−ν) ∂²v₀

∂x₁∂x₂(0) sin(2θ), g₂(X) =g₂(θ) =−(1 +ν)

2 Δv₀(0)−(1−ν) 2

∂²v₀

∂x²₁(0)−∂²v₀

∂x²₂(0)

cos(2θ)−(1−ν) ∂²v₀

∂x₁∂x₂(0) sin(2θ).

Proof. It suffices to expand B₁(v₀)(X) and B₂(v₀)(X) around = 0 by using Taylor formula (see [11] for

details).

3.3. Asymptotic expansion of w

(Q^b)

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

Δ²w−γΔw+w= 0, onΩ,

B₁(w)−γ∂_nw=−B₁(v₀) +γ∂_nv₀, on∂B, B₂(w) =−B₂(v₀), on∂B,

B₁(w)−γ∂_nw= 0, onΓ, B₂(w) = 0, onΓ.

(3.13)

We denote byB the exterior domainR²\B, and we introduce the weighted Sobolev space [14]:

W²(B) =

u, u

(1 +r²) log(2 +r²) ∈L²(B), ∇u

(1 +r²)^1/2log(2 +r²) ∈L²(B),∇²u∈L²(B)

, (3.14) wherer=|x|. We denote byW²(B)/P1 the set of functionsW²(B) defined up to a polynomial of degree less or equal than 1. To estimatew, we introduce (see [3,17]) the following exterior problem

(Pext)

⎧⎪

⎨

⎪⎩

Δ²P = 0, onB, B₁(P) =g₁, on∂B, B₂(P) =g₂, on∂B,

(3.15)

TOPOLOGICAL GRADIENT FOR A FOURTH ORDER OPERATOR USED IN IMAGE ANALYSIS

where g₁ ∈H^−3/2(∂B) andg₂ ∈H^−1/2(∂B) are given in Proposition3.1. From Theorem B.3(Appendix B), we deduce that problem (P_ext) admits a unique solutionP ∈W²(B)/P₁ andP can be written as the sum of simple and double layers potential:

P(x) =

∂B

λ₁(y)E(x−y)dσ(y) +

∂B

λ₂(y)∂_ny(E(x−y))dσ(y)

where λ₁ and λ₂ are densities that we can determine in function of the boundary data; E(x) denotes the bilaplacian fundamental solution:

E(x) =− 1

8π|x|²log(|x|) (3.16)

From TheoremB.3(Appendix B) and Proposition3.1we get

λ₁=αcos(2θ) +βsin(2θ), λ₂=c+acos(2θ) +bsin(2θ) (3.17) where

a=−21−ν 3 +ν

∂²v₀

∂x²₁(0)−∂²v₀

∂x²₁ (0)

, b=−41−ν 3 +ν

∂²v₀

∂x₁∂x₂(0), c=−1 +ν

1−νΔv₀(0), α= 2a, β = 2b.

(3.18)

Then from LemmaB.6(Appendix B), we prove thatw=²P_x

+e withe2,Ω=O(−²log()).

In the next subsection we estimateJ_A, J_B,E₁,E₂ andE₃.

3.4. Estimates of J

, J

, E

, E

and E

Lemma 3.2. Let J_A,J_B,E1,E2 andE3 given in (3.12), we have the following estimates J_A=²1

2

∂B

λ₁(y)∇²u₀(0)y.ydσ(y) +o(²), J_B =−²

∂B

λ₂(y)∇²u₀(0)y.ydσ(y) +o(²), E1=O

³

, E2=O

−³log()

, E3=O

−³log() .

Proof. From the linearity of the solution of (3.6), by using the jump relations (B.7) given in Theorem B.3 (Appendix B) and the asymptotic expansion ofwwe get

J_A=

∂B

B₁(v₀) +B₁(l^2P(^x)+e

)

u₀=

∂B

B₁(v₀)(X) +1

B₁(l^P)(X)

u₀(X)dσ(X) +F₁

=

∂B

−g₁(X) +B₁(l^P)(X)

u₀(X)dσ(X) +F₁+F₂

=

∂B

−g₁(X) +B₁(l^P)(X)

²

2∇²u₀(0)X.Xdσ(X) +F1+F2+F3

=²1 2

∂B

λ₁(y)∇²u₀(0)y.ydσ(y) +F₁+F₂+F₃ where

F1=

∂B

B₁(l^e)u₀, F2=

∂B

B₁(v₀)(X) +g₁(X)

u₀(X)dσ(X), F₃=

∂B

−g₁(X) +B₁(l^P)(X)

u₀(X)−²1

2∇²u₀(0)X.X

dσ(X).

G. AUBERT AND A. DROGOUL

LetB_rsuch asB B_r⊂¹Ω. By a change of variable, by using LemmaB.2(Appendix B), a Taylor expansion ofu₀(X), the trace theorem applied onB_r\B, a change of variable and LemmaB.6(Appendix B), the following estimate holds

F1=

∂B

B₁(l^e)(X)u₀(X)dσ(X) =

∂B

1

³B₁(l^e(X))u₀(X)dσ(X)

≤C|l^e(X)|_2,B=C|l^e(X)|_2,B≤Ce(X)_H2(Br\B)/P1≤C|e|_2,Ω ≤C³log().

From estimates given in Proposition3.1, and by a Taylor expansion ofu₀(X) at 0 we easily see that F₂=O(³), F₃=O

³ . Similarly

J_B=

∂B

B₂(v₀) +B₂ l^2P(^x)+e

∂_nu₀=

∂B

B₂(v₀)(X) +B₂(l^P)(X)

∂_nu₀(X)dσ(X) +F4,

=

∂B

−g₂(X) +B₂(l^P)(X)

∂_nu₀(X)dσ(X) +F₄+F₅,

=²

∂B

−g₂(X) +B₂(l^P)(X)

∇²u₀(0)X.ndσ(X) +F4+F5+F6,

=−²

∂B

λ₂(y)∇²u₀(0)y.ydσ(y) +F4+F5+F6, where

F4=

∂B

B₂(l^e)∂_nu₀, F5=

∂B

(B₂(v₀)(X) +g₂(X))∂_nu₀(X)dσ(X), F₆=

∂B

−g₂(X) +B₂(l^P)(X) ∂_nu₀(X)−²∇²u₀(0)X.n dσ(X).

Similarly to theF1 computation, and from a Taylor expansion of∂_nu₀(X) we can prove that F4=O

−³log() .

Then from estimates given in Proposition3.1and a Taylor expansion of∂_nu₀(X) we obtain F₅=O

³

, F₆=O ³

. Thus, we deduce the estimates of J_Aand J_B given in the lemma.

ForE₁, by using a change of variable, a Taylor expansion ofu₀(X) and the definition ofF given in (3.10), it is straightforward that

E1=O ³

.

ForE2, a change of variable, Lemma B.2(see Appendix B) with= 1, the trace theorem applied on B_r\B, a change of variable again, and finally LemmaB.6 (see Appendix B) lead to

E₂=²

B

Δ²u₀(X)l^w(X)≤C²l^w(X)_0,Br\B ≤C²w(X)_2,B,

≤C²

w(X)_0,Br\B+|w(X)|_1,Br\B+|w(X)|_2,Br\B ,

≤C² 1

w_0,Ω+|w|_1,Ω+|w|_2,Ω

≤C³log().

whereB B_r⊂ ¹Ω. Similarly forE3we have

E3≤C(|w|1,Ω+|w|2,Ω)≤ −C³log()

TOPOLOGICAL GRADIENT FOR A FOURTH ORDER OPERATOR USED IN IMAGE ANALYSIS

3.5. Computation of the topological gradient in the case of the ball

J(u)−J₀(u₀)

² =π(f(0)−u₀(0))v₀(0) +γπ∇u₀(0).∇v₀(0) +1

2

∂B

λ₁(y)∇²u₀(0)y.ydσ(y) +

∂B

λ₂(y)∇²u₀(0)y.ydσ(y) +J

² +o(1).

Using polar coordinates and from the expressions ofλ₁ andλ₂ given in (3.17) and (3.18), we obtain J(u)−J₀(u₀)

² =π(f(0)−u₀(0))v₀(0) +γπ∇u₀(0).∇v₀(0) + ∂²u₀

∂x²₁ (0)−∂²u₀

∂x²₂ (0)

∂²v₀

∂x²₁(0)−∂²v₀

∂x²₂(0)

πa + ∂²u₀

∂x₂∂x₁(0) ∂²v₀

∂x₂∂x₁(0)π4a+πcΔu₀(0)Δv₀(0) +J

² +o(1), where

a=−21−ν

3 +ν, c =−1 +ν 1−ν.

From (3.4) and by using LemmaB.6applied tou−u₀ and a Taylor expansion off andu₀ at 0, we have J=−π²(f(0)−u₀(0))u₀(0) +o

² .

3.6. Conclusion: general expression for all x

∈ Ω

The topological gradient of the cost functionJ in the case of the ball associated with P^b

given in (3.1) is for all pointx₀∈Ω:

I(x₀) =π(f(x₀)−u₀(x₀)) (v₀(x₀)−u₀(x₀)) +γπ∇u₀(x₀).∇v₀(x₀)

−2π(1−ν) 3 +ν

∂²u₀

∂x²₁ (x₀)−∂²u₀

∂x²₂ (x₀)

∂²v₀

∂x²₁ (x₀)−∂²v₀

∂x²₂(x₀)

+ 4 ∂²u₀

∂x₂∂x₁(x₀) ∂²v₀

∂x₂∂x₁(x₀)

−π(1 +ν)

1−ν Δu₀(x₀)Δv₀(x₀). (3.19)

4. Study in the case of the crack

4.1. Notations and statement of the problem

For each smooth manifoldΣ⊂Ω, we define the following spaces:

H₀₀^1/2(Σ) ={u_|Σ, u∈H^1/2(Σ), u _|Σ\Σ = 0}, H₀₀^3/2(Σ) ={u_|Σ, u∈H^3/2(Σ), u _|Σ\Σ = 0}, whereΣ is a smooth closed manifold containingΣand of the same dimension.

We define on these spaces the following norms u_|ΣH1/2

00 (Σ)=u_H1/2(Σ) , u_|ΣH3/2

00 (Σ)=u_H3/2(Σ) .

Now, let σ⊂Ω a C¹-manifold of dimension 1, with normal nand containing the origin. We denote by τ the tangent vector to the crackσsuch as (n,τ) forms an orthonormal basis.∂τdenotes the differentiation along the