The Deblurring-Denoising case

This case is motivated by image restoration. More precisely, in the special case N = 2, letΩbe a bounded domain in❘² andf ∈L²(Ω)be an observed image, which is the result of a compact transformation R (blur) and corrupted by an additive zero-mean Gaussian noise η ∈ L²(Ω) both performed on the ideal image uâ; that is, f = Ruâ+η. It is well-known since the seminal work of Tikhonov and Arsenin [39], that a good approximation of uâ is provided by the minimizer of a certain (strictly) convex energy

T(u) := λ 2

Ú

Ω(f −Ru)²dx+

Ú

ΩΨ(|∇u|)dx,

on an adequate Sobolev space, where the functionΨhas to be chosen to realize some desired smoothing effects. The parameter λ > 0 can be interpreted as the Lagrange multiplier with respect to the constraint on the variance of η or as a regularizing weight.

In what follows, consider the functional J(u) := λ

2

Ú

Ω(f −Ru)²dx+

Ú

ΩΦ(|Du|) (2.3.1) defined on the more realistic functional spaceBV(Ω), whereλ >0is a given parameter and the regularization term^s_ΩΦ(|Du|)is defined as in Definition2.2.1.

It is clear that we can assume, without less of generality, Φ(0) = 0.

The sublinearity at infinity of the function Φ will be assumed, that is, we set the hypothesis :

(H₁) lim

s→+∞

Φ(s) s = 0.

We quote here that in [10], the authors consider the special case Φ is bounded on ❘ andR=Id_L²_(Ω), the identity operator L²(Ω).

In this first subsection, the operator R is assumed to be compact from L²(Ω) to itself, it is not necessarily a Hilbert-Schmidt operator.

2.3. SUBLINEAR REGULARIZATION IN IMAGE RESTORATION 33

Now, we introduce the following subspace of L^∞(Ω)

X :={u : Ω−→❘ s. t.uis piecewise constant}. (2.3.2) The algebra X ⊂ BV(Ω) is dense in L²(Ω), with respect to the norm ë.ëL²(Ω). In particular,X is dense in ¹BV(Ω),ë.ëL²(Ω)

2. Then the closure ofR(X) with respect to theL²(Ω)norm is not other than the closure ofR(L²(Ω))with respect to theL²(Ω) norm, since R(X)⊂R(X). That is

R(X) = Im(R), where Im(R) is the range ofR.

It is clear thatIm(R)is a (convex) and closed subspace ofL²(Ω), then the orthogonal projection operator on Im(R),i.e., Π_Im(R) : L²(Ω)−→L²(Ω), is uniquely defined.

Before stating our results, we introduce some technical tools and lemmas.

For every positive integern, leth_n = 2⁻ⁿ and the sequence of squaresC_i,jⁿ := [i h_n,(i+

1)h_n]×[j h_n,(j+ 1)h_n],(i, j)∈❩×❩. Let the subsets of ❘²

Ωⁿ₊:= ^Û

{(i,j)∈❩×❩:C_i,jⁿ∩ΩÓ=∅}

C_i,jⁿ and Ωⁿ₋:= ^Û

{(i,j)∈❩×❩:C_i,jⁿ⊂ΩÓ=∅}

C_i,jⁿ . and the sets of indices

■ⁿ₊:=^î(k, l)∈❩×❩s.t.C_k,lⁿ ∩ Ω Ó= ∅^ï and ■ⁿ₋ :=^î(k, l)∈❩×❩s.t.C_k,lⁿ ⊂ Ω^ï. (2.3.3) In is clear thatΩⁿ₋⊂Ω⊂Ωⁿ₊, for every integer n and that

+∞Û

n=0

Ωⁿ₋= Ω and

+∞Ü

n=0

Ωⁿ₊= Ω.

We introduce the subspace

X_n⁺:={u∈L²(Ωⁿ₊) : uis constant on every C_i,jⁿ ⊂Ωⁿ₊} and

X_n :={u_|Ω : u∈X_n⁺}, whereu_|Ω denotes the restriction ofuto Ω.

It is obvious thatX_n is a strictly increasing sequence of finite dimension subspaces ofX ⊂L^∞(Ω)and that

Û

n∈◆

X_n =L²(Ω).

Moreover, sinceR is continuous from L²(Ω) to itself then

Û

n∈◆

R(X_n) = Im(R).

Hence, we get the following lemma :

Lemma 2.3.1 For every g ∈Im(R), there is a sequence (g_n)_n such that g_n ∈ X_n ⊂ L^∞(Ω), for every n, and (g_n)_n converges to g in L²(Ω).

Lemma 2.3.2 The hypothesis (H₁) is equivalent to : for every n ∈ ◆^∗, there exists a_n >0such that 0≤Φ(s)≤a_n+ _n^s2.

Proof. Assume that (H₁) holds true and let n ∈ ◆^∗. Then there is A_n > 0 such that for every s ≥A_n we have ^Φ(s)_s ≤ _n¹2. If we set a_n := max

s∈[0,An]Φ(s), it follows that 0 ≤ Φ(s) ≤ a_n + _n^s2, for every s ≥ 0. Conversely, for every n ∈ ◆^∗ (fixed), one has

s→lim+∞

Φ(s) s ≤ 1

n². Hence lim

s→+∞

Φ(s)

s ≤0, which implies (H₁) since Φ≥0.

Remark 2.3.3 In the previous lemma, the sequence (A_n)_n can be choosed an increa- sing one, in this situation the sequence (a_n)_n is non decreasing. If the sequence (a_n)_n is bounded, this means that the functionΦis bounded and this case was studied in [10].

Here, we are concerned with the more complex situation when (an)n is not bounded.

Then, without loss of generality, we may assume in the sequel that a_n −→ +∞ as n→+∞, (if not, we can replacea_n by a_n+ 2ⁿ, for example).

Theorem 2.3.4 Assume that the operator R is compact from L²(Ω) to itself and the hypothesis (H) is satisfied. Then, for every u ∈ BV(Ω), there exists a sequence (w_n)∈BV(Ω) such that











w_n −→ u in L¹(Ω)

Ú

ΩΦ(|Dw_n|) −→ 0

asn goes to +∞. Consequently, if the infimum of J on BV(Ω) is achieved at uâ then

Ú

ΩΦ(|Duâ|) = 0.

In particular, we get uâ is constant.

2.3. SUBLINEAR REGULARIZATION IN IMAGE RESTORATION 35

Proof. Step 1. Let u ∈ BV(Ω), then applying Theorem 3.9, p. 122 of [5], we can find a sequence (u_n)_n ⊂ C^∞(Ω)∩BV(Ω) such that

n→lim+∞ëu_n−uëL¹(Ω)= 0 and lim

n→+∞

-- -- Ú

Ω|∇u_n|dx−

Ú

Ω|Du|

-- --= 0.

For any integerp and any (i, j)∈■^p₋, defined by (2.3.3), let x^p_i,j be the center of the squareC_i,j^p ⊂Ω^p₊.

For every fixed integer n, consider the sequence (v^p_n)_p⊂X_p defined by v_n^p := ^Ø

(i,j)∈■^p−

u_n(x^p_i,j)1C^p_i,j,

where1C_i,j^p denotes the characteristic function ofC_i,j^p . It is clear that

p→+∞lim ëv_n^p−u_nëL¹(Ω)= 0.

Then for every integer n, there isθ(n)> θ(n−1), such that ëv_n^θ(n)−u_nëL¹(Ω)≤ 1

2ⁿ.

By construction, the mapθ is strictly increasing. If we set v_n :=v^θ(n)_n , we get

n→lim+∞ëvn−unëL¹(Ω)= 0.

At this stage, let introduce the parameter δ_k := min

3

h_k, 1 a_k

4

, (2.3.4)

wherea_k is defined in Lemma 2.3.2.

Step 2. For any integerk and(i, j)∈■^k₋, let D_i,j^k ⊂C_i,j^k defined by D_i,j^k := [i hk+δ^β_k,(i+ 1)hk−δ_k^β]×[j hk+δ^β_k,(j+ 1)hk−δ_k^β],

where β >2. Using elementary geometrical properties, we can construct a sequence ofcontinuousfunctions(w_n)_n such thatw_n =v_n on each squareD^θ(n)_i,j andw_n is affine on C_i,j^θ(n)\D^θ(n)_i,j . Therefore, the slope of these affine functions are controlled by the inverse of δ_θ(n)^β multiplied by the jump of v_n from a rectangle to its neighbors (see Figure 1).

Figure 2.3.1.1 – Continuous approximations w_n of v_n s.t. D^sw_n ≡ 0 (Dw_n has no singular part)

.

We show that the the sequence (∇w_n)_n is bounded in L¹(Ω). Indeed, for every g∈BV(Ω) and(i, j)∈■ⁿ₋, set

∆₁g(xⁿ_i,j) :=g((i+1)h_n, jh_n)−g(ih_n, jh_n) and ∆₂g(xⁿ_i,j) :=g(ih_n,(j+1)h_n)−g(ih_n, jh_n), where g((i+ 1)h_n, jh_n) := g(ih_n, jh_n) if (i + 1, j) Ó∈ ■ⁿ₋ and g(ih_n,(j + 1)h_n) :=

g(ih_n, jh_n)if (i, j+ 1)Ó∈■ⁿ₋.

Ú

Ω|∇w_n|dx =

Ú

Ωu5t

(i,j)∈■θ(n)

−

(C_i,j^θ(n)\D^θ(n)i,j )

6|∇w_n|dx

≤ ^Ø

(i,j)∈■^θ(n)−

1|∆₁w_n(x^θ(n)_i,j )|h_θ(n)+|∆₂w_n(x^θ(n)_i,j )|h_θ(n)²

= ^Ø

(i,j)∈■^θ(n)−

1|∆₁u_n(x^θ(n)_i,j )|h_θ(n)+|∆₂u_n(x^θ(n)_i,j )|h_θ(n)²

= ^Ø

(i,j)∈■^θ(n)−



|∆₁u_n(x^θ(n)_i,j )|

h_θ(n) + |∆₂u_n(x^θ(n)_i,j )| h_θ(n)



h²_θ(n).

2.3. SUBLINEAR REGULARIZATION IN IMAGE RESTORATION 37

Moreover, since

p→lim+∞

Ø

(i,j)∈■^p−

A|∆₁u_n(x^p_i,j)|

h_p + |∆₂u_n(x^p_i,j)| h_p

B

h²_p =

Ú

Ω

A-----

∂u_n

∂x₁

-- -- -+

-- -- -

∂u_n

∂x₂

-- -- - B

dx(2.3.5)

≤ √ 2

Ú

Ω|∇u_n|dx (2.3.6)

≤ c₀ (2.3.7)

it follows that

Ú

Ω|∇w_n|dx≤ ^Ø

(i,j)∈■^θ(n)−



|∆₁u_n(x^θ(n)_i,j )|

h_θ(n) +|∆₂u_n(x^θ(n)_i,j )| h_θ(n)



h²_θ(n)≤c₁,

wherec₀, c₁ are positives constants independent of n.

We claim now that lim

n→+∞ëw_n−v_nëL¹(Ω)= 0. Indeed, ëw_n−v_në²L¹(Ω) =

Ú

Ωu t

(i,j)∈■θ(n)

−

(C_i,j^θ(n)\D_i,j^θ(n))|w_n −v_n|

≤ ^Ø

(i,j)∈■^θ(n)−

1|∆₁v_n(x^θ(n)_i,j )|h²_θ(n)+|∆₂v_n(x^θ(n)_i,j )|h²_θ(n)²

= h_θ(n) ^Ø

(i,j)∈■^θ(n)−



|∆1un(x^θ(n)_i,j )|

h_θ(n) +|∆2un(x^θ(n)_i,j )| h_θ(n)



h²_θ(n).

Using (2.3.5) we obtain that

Ø

(i,j)∈■^θ(n)−



|∆₁u_n(x^θ(n)_i,j )|

h_θ(n) + |∆₂u_n(x^θ(n)_i,j )| h_θ(n)



h²_θ(n)≤c₂

and we getlim_n→+∞ëw_n−v_nëL¹(Ω)= 0and obviously

n→lim+∞ëw_n−uëL¹(Ω)= 0.

Therefore, the sequence(w_n)_n is uniformly bounded in BV(Ω). Using the continuous embedding BV(Ω) ⊂ L²(Ω), (w_n)_n is uniformly bounded in L²(Ω). Then, up to a subsequence, (w_n)_n converges weakly to u in L²(Ω). Moreover, since the operator R is compact fromL²(Ω)to itself, it follows that

n→+∞lim ëR w_n−R uëL²(Ω)= 0. (2.3.8)

Concerning the sublinear term :

Ú

ΩΦ(|Dw_n|) =

Ú

ΩΦ(|∇w_n|)dx =

Ú

Ωu t

(i,j)∈■θ(n)

−

(C^θ(n)_i,j \D^θ(n)_i,j )Φ(|∇w_n|)dx

≤ ^Ú

Ωu t

(i,j)∈■θ(n)

−

(C^θ(n)_i,j \D^θ(n)_i,j )

A

a_θ(n)+ |∇w_n| n²

B

dx.

Using the facts that the cardinal of ■k is proportional toh⁻_k² andδ_k = min(a⁻_k¹, h_k), for any positive integer k, we get for the first term

Ú

Ωu t

(i,j)∈■θ(n)

−

(C_i,j^θ(n)\D_i,j^θ(n))a_θ(n)dx = 2a_θ(n) ^Ø

(i,j)∈■^θ(n)−

h_θ(n)δ^β_θ(n)

≤ a_θ(n)h_θ(n)δ^β_θ(n)C(Ω) h²_θ(n)

≤ C(Ω)δ_θ(n)^β−2 −→0asn→+∞sinceβ >2.

Finally, the second term gives

Ú

Ωu t

(i,j)∈■θ(n)

−

(C_i,j^θ(n)\D_i,j^θ(n))

|∇w_n|

n² dx ≤ c₃

n² −→0asn→ +∞. We conclude that

n→lim+∞

Ú

ΩΦ(|Dw_n|) = 0.

Now, ifuâ∈BV(Ω)is a minimizer of the functional J onBV(Ω). Consider a sequence wâ_n ∈BV(Ω)such that











wân −→ uâ in L¹(Ω)

Ú

ΩΦ(|Dwâ_n|) −→ 0 as n tends to+∞. It follows that

n→lim+∞J(wâ_n) =ëf −RuâëL²(Ω)≥J(u).â

Whence ^s_ΩΦ(|Duâ|) = 0, which achieves the proof.

At this stage, we show that the minimization problem Find u∈BV(Ω) such that J(u) = inf

v∈BV(Ω)J(v) (2.3.9)

2.3. SUBLINEAR REGULARIZATION IN IMAGE RESTORATION 39

is ill-posed. For this, we will assume that the operatorRdoes not annihilate constant functions. More precisely, we assume that :

(H₂) R·1Ω Ó= 0 (2.3.10)

Notice that hypothesis (H₂) is very natural in applications and was introduced first in [?]. We show in the following that Problem (2.3.9) is ill-posed in general, unless the data f and the compact operatorR are linked by a binding relation on the support of the functionR·1Ω.

Theorem 2.3.5 Assume that the hypotheses of Theorem4.1.4and (H₂)are satisfied.

Then Problem (2.3.9) has a solution if the data f satisfies

f(x) =t^∗_R(Ω, f)×(R·1Ω)(x)on the set {x∈Ω : R·1Ω(x)Ó= 0}, where t^∗_R(Ω, f) is defined by (2.3.11) and 1Ω is the indicatrice function of Ω.

Proof. If Problem (2.3.9) admits a solution u. From Theoremâ 4.1.4, the function uâ = t^∗_R(Ω, f) ·1Ω, where the constant t^∗_R(Ω, f) realizes the minimum of the real function tÔ→^sΩ(f −t R·1Ω)²dx. Then

t^∗_R(Ω, f) =

s

Ωf ×(R·1Ω)dx

s

Ω(R·1Ω)²dx . (2.3.11) Suppose that there existsω⊂ {x∈Ω : R·1Ω(x)Ó= 0}such thatt^∗_R(ω, f)Ó= t^∗_R(Ω, f), wheret^∗_R(ω, f)is defined by

t^∗_R(ω, f) =

s

ωf ×(R·1Ω)dx

s

ω(R·1Ω)²dx .

Then^s_ω(f −t^∗_R(ω, f)R·1Ω)²dx <^s_ω(f −t^∗_R(Ω, f)R·1Ω)²dx. Let uå =t^∗_R(ω, f)1ω+ t^∗_R(Ω, f)1Ω\ω. Let (w_n)_n ∈BV(Ω)such that











w_n −→ uå in L¹(Ω)

Ú

ΩΦ(|Dw_n|) −→ 0 Then

n→lim+∞J(w_n) =

Ú

Ω(f −Ru)å ²dx < J(u),â

which leads to a contradiction. It follows that for everyω⊂ {x∈Ω : R·1Ω(x)Ó= 0} we get t^∗_R(ω) =t^∗_R(Ω, f). Now, let a∈ {x∈Ω : R·1Ω(x)Ó= 0}, forω =B(a, ε), the ball centered at aand of radius ε, we get

limε→0t^∗_R(B(a, ε), f) = f(a)×(R·1Ω)(a)

(R·1Ω)²(a) =t^∗_R(Ω, f).

Therefore, for every a∈ {x∈Ω : R·1Ω(x)Ó= 0}, f(a) =t^∗_R(Ω, f)(R·1Ω)(a),

which ends the proof.

Remark 2.3.6 In the case where the operator R is the identity, the constant t^∗_Id(Ω, f) =

s

ωf dx

s

ω1Ωdx,

which is no other than the mean of f on Ω. Therefore, t^∗_R(Ω, f) gives a natural gene- ralization, in our context, taking into account of the presence of the compact operator R.

At this stage, we can state the following result which generalizes the results ob- tained in [24] for every Φ sublinear, for f ∈L²(Ω) and in the presence of a compact operatorR, in the framework of Sobolev spaces.

Consider the energy K : W^1,1(Ω)−→[0,+∞[ defined by K(u) := λ

2

Ú

Ω(f −Ru)²dx+

Ú

ΩΦ(|∇u|)dx.

Then we have the following

Corollary 2.3.7 For every u ∈ W^1,1(Ω), there exists a sequence (w_n) ∈ W^1,1(Ω)

such that _









w_n −→ u in L¹(Ω)

Ú

ΩΦ(|∇w_n|) −→ 0

asn goes to +∞. Consequently, the infimum of K on W^1,1(Ω)is achieved if the data f satisfies

f(x) =t^∗_R(Ω, f)×(R·1Ω)(x)on the set {x∈Ω : R·1Ω(x)Ó= 0}, where t^∗(Ω, f) is defined by (2.3.11).

2.3. SUBLINEAR REGULARIZATION IN IMAGE RESTORATION 41

Proof.Using similar arguments as in Theorem4.1.4, it follows that^s_ΩΦ(|∇u^â|)dx= 0 and thenuâ is constant. We end the proof in the same way as in the previous corollary.

No documento Méthodes variationnelles : Applications à l’analyse d’image et au modèle de Frenkel-Kontorova (páginas 33-42)