# The Problem of Image Recovery by the Metric Projections in Banach Spaces

Authors: Yasunori Kimura; Kazuhide Nakajo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the problem of image recovery by the metric projections in a real Banach space. For a countable family of nonempty closed convex subsets, we generate an iterative sequence converging weakly to a point in the intersection of these subsets. Our convergence theorems extend the results proved by Bregman and Crombez.


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## Body

## 1. Introduction

LetC1,C2,…,Cr be nonempty closed convex subsets of a real Hilbert space H such that ⋂i=1rCi≠∅. Then, the problem of image recovery may be stated as follows: the original unknown image z is known a priori to belong to the intersection of {Ci}i=1r; given only the metric projections PCi of H onto Ci for i=1,2,…,r, recover z by an iterative scheme. Such a problem is connected with the convex feasibility problem and has been investigated by a large number of researchers.Bregman [1] considered a sequence {xn} generated by cyclic projections, that is, x0=x∈H,x1=PC1x,x2=PC2x1,x3=PC3x2,…,xr=PCrxr-1,xr+1=PC1xr,xr+2=PC2xr+1,…. It was proved that {xn} converges weakly to an element of ⋂i=1rCi for an arbitrary initial point x∈H.Crombez [2] proposed a sequence {yn} generated by y0=y∈H, yn+1=α0yn+∑i=1rαi(yn+λi(PCiyn-yn)) for n=0,1,2,…, where 0<αi<1 for all i=0,1,2,…,r with ∑i=0rαi=1 and 0<λi<2 for every i=1,2,…,r. It was proved that {yn} converges weakly to an element of ⋂i=1rCi for an arbitrary initial point y∈H.Later, Kitahara and Takahashi [3] and Takahashi and Tamura [4] dealt with the problem of image recovery by convex combinations of nonexpansive retractions in a uniformly convex Banach space E. Alber [5] took up the problem of image recovery by the products of generalized projections in a uniformly convex and uniformly smooth Banach space E whose duality mapping is weakly sequentially continuous (see also [6, 7]).On the other hand, using the hybrid projection method proposed by Haugazeau [8] (see also [9–11] and references therein) and the shrinking projection method proposed by Takahashi et al. [12] (see also [13]), Nakajo et al. [14] and Kimura et al. [15] considered this problem by the metric projections and proved convergence of the iterative sequence to a common point of countable nonempty closed convex subsets in a uniformly convex and smooth Banach space E and in a strictly convex, smooth, and reflexive Banach space E having the Kadec-Klee property, respectively. Kohsaka and Takahashi [16] took up this problem by the generalized projections and obtained the strong convergence to a common point of a countable family of nonempty closed convex subsets in a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable (see also [17, 18]). Although these results guarantee the strong convergence, they need to use metric or generalized projections onto different subsets for each step, which are not given in advance.In this paper, we consider this problem by the metric projections, which are one of the most familiar projections to deal with. The advantage of our results is that we use projections onto the given family of subsets only, to generate the iterative scheme. Our convergence theorems extend the results of [1, 2] to a Banach space E, and they deduce the weak convergence to a common point of a countable family of nonempty closed convex subsets of E.There are a number of results dealing with the image recovery problem from the aspects of engineering using nonlinear functional analysis (see, e.g., [19]). Comparing with these researches, we may say that our approach is more abstract and theoretical; we adopt a general Banach space with several conditions for an underlying space, and therefore, the technique of the proofs can be applied to various mathematical results related to nonlinear problems defined on Banach spaces.


## 2. Preliminaries

Throughout this paper, letℕ be the set of all positive integers, ℝ the set of all real numbers, E a real Banach space with norm ∥·∥, and E* the dual of E. For x∈E and x*∈E*, we denote by 〈x,x*〉 the value of x* at x. We write xn→x to indicate that a sequence {xn} converges strongly to x. Similarly, xn⇀x and xn⇀*x will symbolize weak and weak* convergence, respectively. We define the modulus δE of convexity of E as follows: δE is a function of [0,2] into [0,1] such that
(1)δE(ϵ)=inf{1-∥x+y∥2:x,y∈E,∥x∥=1,=infkk∥y∥=1,∥x-y∥≥ϵ∥x+y∥2}
for every ϵ∈[0,2]. E is called uniformly convex if δE(ϵ)>0 for each ϵ>0. Let p>1. E is said to be p-uniformly convex if there exists a constant c>0 such that δE(ϵ)≥cϵp for every ϵ∈[0,2]. It is obvious that a p-uniformly convex Banach space is uniformly convex. E is said to be strictly convex if ∥x+y∥/2<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. We know that a uniformly convex Banach space is strictly convex and reflexive. For every p>1, the (generalized) duality mapping Jp:E→2E* of E is defined by
(2)Jpx={y*∈E*:〈x,y*〉=∥x∥p,∥y*∥=∥x∥p-1}
for all x∈E. When p=2, J2 is called the normalized duality mapping. We have that for p,q>1, ∥x∥pJqx=∥x∥qJpx for all x∈E. E is said to be smooth if the limit
(3)limt→0∥x+ty∥-∥x∥t
exists for every x,y∈E with ∥x∥=∥y∥=1. We know that the duality mapping Jp of E is single valued for each p>1 if E is smooth. It is also known that if E is strictly convex, then the duality mapping Jp of E is one to one in the sense that x≠y implies that Jpx∩Jpy=∅ for all p>1. If E is reflexive, then Jp is surjective, and Jp-1 is identical to the duality mapping Jq*:E*→2E defined by
(4)Jq*y*={x∈E:〈x,y*〉=∥y*∥q,∥x∥=∥y*∥q-1}
for every y*∈E*, where q∈ℝ satisfies 1/p+1/q=1. For p>1, the duality mapping Jp of a smooth Banach space E is said to be weakly sequentially continuous if xn⇀x implies that Jpxn⇀*Jpx (see [20, 21] for details). The following is proved by Xu [22] (see also [23]).Theorem 1 (Xu [22]).
LetE be a smooth Banach space and p>1. Then, E is p-uniformly convex if and only if there exists a constant c>0 such that ∥x+y∥p≥∥x∥p+p〈y,Jpx〉+c∥y∥p holds for every x,y∈E.Remark 2.
For ap-uniformly convex and smooth Banach space E, we have that the constant c in the theorem above satisfies c≤1. Let
(5)c0=sup{c>0:∥x+y∥p≥∥x∥p+p〈y,Jpx〉=sump+c∥y∥p∀x,y∈E∥x∥pp〈y,Jpx〉}.
Then, there exists a positive real sequence {cn} such that limn→∞cn=c0 and ∥x+y∥p≥∥x∥p+p〈y,Jpx〉+cn∥y∥p for all x,y∈E and n∈ℕ. So, we get ∥x+y∥p≥∥x∥p+p〈y,Jpx〉+c0∥y∥p for every x,y∈E. Therefore, c0 is the maximum of constants. In the case of Hilbert spaces, the normalized duality mapping J2 is the identity mapping and c0=1.LetE be a smooth Banach space and p>1. The function ϕp:E×E→ℝ is defined by
(6)ϕp(y,x)=∥y∥p-p〈y,Jpx〉+(p-1)∥x∥p
for every x,y∈E. We have ϕp(x,y)≥0 for all x,y∈E and ϕp(z,x)+ϕp(x,y)=ϕp(z,y)+p〈x-z,Jpx-Jpy〉 for every x,y,z∈E. It is known that if E is strictly convex and smooth, then, for x,y∈E,ϕp(y,x)=ϕp(x,y)=0 if and only if x=y. Indeed, suppose that ϕp(y,x)=ϕp(x,y)=0. Then, since
(7)0=ϕp(y,x)+ϕp(x,y)=p(∥x∥p+∥y∥p-〈x,Jpy〉-〈y,Jpx〉)≥p(∥x∥p+∥y∥p-∥x∥∥y∥p-1-∥y∥∥x∥p-1)=p(∥x∥-∥y∥)(∥x∥p-1-∥y∥p-1)≥0,
we have ∥x∥=∥y∥. It follows that 〈y,Jpx〉=p-1(∥y∥p+(p-1)∥x∥p-ϕp(y,x))=∥y∥p and ∥Jpx∥=∥x∥p-1=∥y∥p-1, which implies that Jpy=Jpx. Since Jp is one to one, we have x=y (see also [17]). We have the following result from Theorem 1.Lemma 3.
Letp>1 and E be a p-uniformly convex and smooth Banach space. Then, for each x,y∈E,
(8)ϕp(x,y)≥c0∥x-y∥p
holds, where c0 is maximum in Remark 2.Proof.
Letx,y∈E. By Theorem 1, we have
(9)∥x∥p≥∥y∥p+p〈x-y,Jpy〉+c0∥x-y∥p,
where c0 is maximum in Remark 2. Hence, we get
(10)ϕp(x,y)=∥x∥p-∥y∥p-p〈x-y,Jpy〉≥c0∥x-y∥p,
which is the desired result.LetC be a nonempty closed convex subset of a strictly convex and reflexive Banach space E, and let x∈E. Then, there exists a unique element x0∈C such that ∥x0-x∥=infy∈C∥y-x∥. Putting x0=PCx, we call PC the metric projection onto C (see [24]). We have the following result [25, p. 196] for the metric projection.Lemma 4.
LetC be a nonempty closed convex subset of a strictly convex, reflexive, and smooth Banach space E, and let x∈E. Then, y=PCx if and only if 〈y-z,J2(x-y)〉≥0 for all z∈C, where PC is the metric projection onto C.Remark 5.
Forp>1, it holds that ∥x∥Jpx=∥x∥p-1J2x for every x∈E. Therefore, under the same assumption as Lemma 4, we have that y=PCx if and only if 〈y-z,Jp(x-y)〉≥0 for all z∈C.


## 3. Main Results

Firstly, we consider the iteration of Crombez’s type and get the following result.Theorem 6.
Letp,q>1 be such that 1/p+1/q=1. Let {Cn}n∈ℕ be a family of nonempty closed convex subsets of a p-uniformly convex and smooth Banach space E whose duality mapping Jp is weakly sequentially continuous. Suppose that ⋂n∈ℕCn≠∅. Let λn,k∈]0,(1+1/(p-1))p-1c0[ and αn,k∈[0,1] for all n∈ℕ and k=1,2,…,n with ∑k=1nαn,k=1 for every n∈ℕ, where c0 is maximum in Remark 2. Let {xn} be a sequence generated by x1=x∈E and
(11)xn+1=Jq*(∑k=1nαn,k(Jpxn-λn,kJp(xn-PCkxn)))
for every n∈ℕ. If 0<liminfn→∞λn,k≤limsupn→∞λn,k<(1+1/(p-1))p-1c0 and liminfn→∞αn,k>0 for each k∈ℕ, then {xn} converges weakly to a point in ⋂n=1∞Cn.Proof.
Letyn,k=Jq*(Jpxn-λn,kJp(xn-PCkxn)) for n∈ℕ and k=1,2,…,n. Then, for z∈⋂n∈ℕCn, we obtain
(12)ϕp(z,yn,k)-ϕp(z,xn)=-ϕp(yn,k,xn)+p〈yn,k-z,Jpyn,k-Jpxn〉=-ϕp(yn,k,xn)-pλn,k〈yn,k-z,Jp(xn-PCkxn)〉=-ϕp(yn,k,xn)-pλn,k〈yn,k-xn,Jp(xn-PCkxn)〉-pλn,k〈xn-z,Jp(xn-PCkxn)〉
for all n∈ℕ and k=1,2,…,n. Using Remark 5 with that z∈Ck, we get
(13)〈xn-z,Jp(xn-PCkxn)〉=〈xn-PCkxn,Jp(xn-PCkxn)〉+〈PCkxn-z,Jp(xn-PCkxn)〉≥∥xn-PCkxn∥p
for every n∈ℕ and k=1,2…,n. Thus, by Lemma 3 we have
(14)ϕp(z,yn,k)-ϕp(z,xn)≤-c0∥yn,k-xn∥p-pλn,k〈yn,k-xn,Jp(xn-PCkxn)〉-pλn,k∥xn-PCkxn∥p≤-c0∥yn,k-xn∥p+pλn,k∥yn,k-xn∥∥xn-PCkxn∥p-1-pλn,k∥xn-PCkxn∥p
for each n∈ℕ and k=1,2,…,n. Since it holds that
(15)st≤1βspp+βq-1tqq
for s,t≥0, p,q>1 with 1/p+1/q=1, and β>0, we have
(16)∥yn,k-xn∥∥xn-PCkxn∥p-1≤1βkp∥yn,k-xn∥p+βk1/(p-1)p-1p∥xn-PCkxn∥p
for every k∈ℕ, βk>0 and n≥k. Therefore, it follows that
(17)ϕp(z,yn,k)-ϕp(z,xn)≤(λn,kβk-c0)∥yn,k-xn∥p+λn,k((p-1)βk1/(p-1)-p)∥xn-PCkxn∥p
for every n∈ℕ, k=1,2,…,n, and βk>0. Since
(18)ϕp(z,xn+1)=∥z∥p-p〈z,∑k=1nαn,kJpyn,k〉+(p-1)∥∑k=1nαn,kJpyn,k∥p/(p-1)≤∥z∥p-p∑k=1nαn,k〈z,Jpyn,k〉+(p-1)∑k=1nαn,k∥yn,k∥p=∑k=1nαn,kϕp(z,yn,k)
for every n∈ℕ, we have
(19)ϕp(z,xn+1)-ϕp(z,xn)≤∑k=1nαn,k(λn,kβk-c0)∥yn,k-xn∥p+∑k=1nαn,kλn,k((p-1)βk1/(p-1)-p)∥xn-PCkxn∥p
for all n∈ℕ and β1,β2,…,βn>0. Since λn,k∈]0,(1+1/(p-1))p-1c0[, αn,k∈[0,1] for all n∈ℕ and k=1,2,…,n,
(20)0<liminfn→∞λn,k≤limsupn→∞λn,k<(1+1(p-1))p-1c0,liminfn→∞αn,k>0
for each k∈ℕ, we can choose βk>0 for every k∈ℕ such that αn,k(λn,k/βk-c0)≤0, αn,kλn,k((p-1)βk1/(p-1)-p)≤0 for all n≥k and
(21)limsupn→∞αn,k(λn,kβk-c0)<0,limsupn→∞αn,kλn,k((p-1)βk1/(p-1)-p)<0
for each k∈ℕ. Hence, there exists limn→∞ϕp(z,xn) for every z∈⋂n∈ℕCn and
(22)limn→∞∥yn,k-xn∥=limn→∞∥xn-PCkxn∥=0
for all k∈ℕ. It follows from Lemma 3 that {xn} is bounded. Let {xni} and {xmj} be subsequences of {xn} such that xni⇀u1 and xmj⇀u2. Then, we get ∥xni-PCkxni∥→0 which implies that u1∈Ck for every k∈ℕ. In the same way, we also have u2∈Ck for every k∈ℕ. Let limn→∞ϕp(u1,xn)=μ1 and limn→∞ϕp(u2,xn)=μ2. Since
(23)μ1-μ2=limi→∞(ϕp(u1,xni)-ϕp(u2,xni))=∥u1∥p-∥u2∥p+plimi→∞〈u2-u1,Jpxni〉
and Jp is weakly sequentially continuous, we have
(24)μ1-μ2=∥u1∥p-∥u2∥p+p〈u2-u1,Jpu1〉=-ϕp(u2,u1).
Similarly, we obtain μ2-μ1=-ϕp(u1,u2). So, we get ϕp(u1,u2)+ϕp(u2,u1)=0, that is, u1=u2. Therefore, {xn} converges weakly to a point in ⋂n∈ℕCn.Using the idea of [9, p. 256], we also have the following result by the iteration of Bregman’s type.Theorem 7.
Letp,q>1 be such that 1/p+1/q=1. Let I be a countable set and {Cj}j∈I a family of nonempty closed convex subsets of a p-uniformly convex and smooth Banach space E whose duality mapping Jp is weakly sequentially continuous. Suppose that ⋂j∈ICj≠∅. Let λn∈]0,(1+1/(p-1))p-1c0[ for all n∈ℕ, where c0 is maximum in Remark 2, and let {xn} be a sequence generated by x1=x∈E and
(25)xn+1=Jq*(Jpxn-λnJp(xn-PCi(n)xn))
for every n∈ℕ, where the index mapping i:ℕ→I satisfies that, for every j∈I, there exists Mj∈ℕ such that j∈{i(n),…,i(n+Mj-1)} for each n∈ℕ. If 0<liminfn→∞λn≤limsupn→∞λn<(1+1/(p-1))p-1c0, then, {xn} converges weakly to a point in ⋂j∈ICj.Proof.
Letz∈⋂j∈ICj. As in the proof of Theorem 6, we have
(26)ϕp(z,xn+1)-ϕp(z,xn)≤(λnβ-c0)∥xn+1-xn∥p+λn((p-1)β1/(p-1)-p)∥xn-PCi(n)xn∥p
for all n∈ℕ and β>0. Since λn∈]0,(1+1/(p-1))p-1c0[ for all n∈ℕ and 0<liminfn→∞λn≤limsupn→∞λn<(1+1/(p-1))p-1c0, we can find that β>0 such that
(27)limsupn→∞(λnβ-c0)<0,limsupn→∞λn((p-1)β1/(p-1)-p)<0.
Then, there exists limn→∞ϕp(z,xn) for every z∈⋂i∈ICi and
(28)limn→∞∥xn+1-xn∥=limn→∞∥xn-PCi(n)xn∥=0.
So, we have that {xn} is bounded from Lemma 3. Let {xnk} be a subsequence of {xn} such that xnk⇀u. For fixed j∈I, there exists a strictly increasing sequence {mk}⊂ℕ such that nk≤mk≤nk+Mj-1 and i(mk)=j for every k∈ℕ. It follows that
(29)∥xmk-xnk∥≤∑l=nknk+Mj-1∥xl+1-xl∥
for all k∈ℕ which implies that xmk⇀u. Since limk→∞∥xmk-PCjxmk∥=0, u∈Cj for every j∈I. So, we get u∈⋂j∈ICj. As in the proof of Theorem 6, using that Jp is weakly sequentially continuous, we get that {xn} converges weakly to a point in ⋂j∈ICj.Suppose that the index setI is a finite set {0,1,2,…,N-1}. For the cyclic iteration, the index mapping i is defined by i(j)=jmodN for each j∈I. Clearly it satisfies the assumption in Theorem 7. In the case where the index set I is countably infinite, that is, I=ℕ, one of the simplest examples of i:ℕ→ℕ can be defined as follows:
(30)i(n)={1(n=2m-1forsomem∈ℕ),2(n=2(2m-1)forsomem∈ℕ),3(n=4(2m-1)forsomem∈ℕ),…,k(n=2k-1(2m-1)forsomem∈ℕ),….
Then, the assumption in Theorem 7 is satisfied by letting Mj=2j for each j∈I=ℕ.


## 4. Deduced Results

Since a real Hilbert spaceH is 2-uniformly convex and the maximum c0 in Remark 2 is equal to 1, we get the following results. At first, we have the following theorem which generalizes the results of [2] by Theorem 6.Theorem 8.
Let{Cn}n∈ℕ be a family of nonempty closed convex subsets of H such that ⋂n∈ℕCn≠∅. Let λn,k∈]0,2[ and αn,k∈[0,1] for all n∈ℕ and k=1,2,…,n with ∑k=1nαn,k=1 for every n∈ℕ. Let {xn} be a sequence generated by x1=x∈H and
(31)xn+1=∑k=1nαn,k(xn-λn,k(xn-PCkxn))
for every n∈ℕ. If it holds that 0<liminfn→∞λn,k≤limsupn→∞λn,k<2 and liminfn→∞αn,k>0 for each k∈ℕ, then, {xn} converges weakly to a point in ⋂n=1∞Cn.Next, we have the following theorem which extends the result of [1] by Theorem 7.Theorem 9.
LetI be a countable set and {Cj}j∈I a family of nonempty closed convex subsets of H such that ⋂j∈ICj≠∅. Let λn∈]0,2[ for all n∈ℕ, and let {xn} be a sequence generated by x1=x∈H and
(32)xn+1=xn-λn(xn-PCi(n)xn)
for every n∈ℕ, where the index mapping i:ℕ→I satisfies that, for every j∈I, there exists Mj∈ℕ such that j∈{i(n),…,i(n+Mj-1)} for each n∈ℕ. If 0<liminfn→∞λn≤limsupn→∞λn<2, then, {xn} converges weakly to a point in ⋂j∈ICj.



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