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

Yasunori Kimura,Kazuhide Nakajo

doi:10.1155/2013/817392

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.

Highlights

Let C1, C2, . . . , Cr be a real Hilbert space nonempty closed convex H such that ⋂ri=1 Ci ≠ 0.subsets of 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}ri=1; given only the metric projections PCi of H onto Ci for i = 1, 2, . . . , r, recover z by an iterative scheme
For each x, y ∈ E, φp (x, y) ≥ c0󵄩󵄩󵄩󵄩x − y󵄩󵄩󵄩󵄩p holds, where c0 is maximum in Remark 2
[0, 1] 1 for every n ∈ N, where c0 is maximum in Remark 2

Summary

Introduction

Let C1, C2, . . . , Cr be a real Hilbert space nonempty closed convex H such that ⋂ri=1 Ci ≠ 0. 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 Gateaux differentiable (see [17, 18]). 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. 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 the technique of the proofs can be applied to various mathematical results related to nonlinear problems defined on Banach spaces

Preliminaries

Main Results

Deduced Results