Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated . Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 65-02, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and phrases. Angle between two subspaces, averaged mapping, Cimmino's method, computerized tomography, convex feasibility problem, convex function, convex inequalities, convex programming, convex set, Fej'er monotone sequence, firmly nonexpansive mapping, H...

We consider a wide class of iterative methods arising in numerical mathematics and optimization which are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods which makes them strongly convergent without additional assumptions. Several applications are discussed. AMS 1991 subject classification. Primary: 65J15, 47N10; secondary 41A29, 47H05, 47H09, 65K10, 90C25. Key words. Convex feasibility, Fej'er-monotonicity, firmly nonexpansive mapping, fixed point, Haugazeau, maximal monotone operator, projection, proximal point algorithm, resolvent, subgradient algorithm. 1 Introduction Let H be a real Hilbert space with scalar product h\Delta j \Deltai, norm k \Delta k, and distance d. In 1965, Bregman [5] proposed a simple iterative method for finding a common point of m intersecting closed convex sets (S i ) 1im in H. He showed that, given an arbitrary starting point x 0 2 H, the sequence (x n ) n0 gene...

The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the error reduction algorithm was identified as a nonconvex alternating projection algorithm. The purpose of this paper is to formulate the phase retrieval problem with mathematical care and to establish new connections between well established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup’s basic inputoutput algorithm corresponds to Dykstra’s algorithm, and that Fienup’s hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. This work provides a theoretical framework to better understand and, potentially, improve existing phase recovery algorithms. 1 1

Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the Douglas-Rachford algorithm for monotone operator-splitting, is obtained under general conditions. Applications to non-Gaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, Douglas-Rachford, frame, nondifferentiable optimization, Poisson noise,

The formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals. In recent years, nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges. They also arise naturally in minimax inconsistent set theoretic recovery problems. At the same time, the issue of developing reliable numerical algorithms to solve such convex programs in the context of image recovery applications has received little attention. In this paper, we address this issue and propose an adaptive level set method for nondifferentiable constrained image recovery. The asymptotic properties of the method are analyzed and its implementation is discussed. Numerical experiments illustrate applications to total variation and minimax set theoretic image restoration and denoising problems.

A quasi-Fejér sequence is a sequence which satisfies the standard Fejér monotonicity property to within an additional error term. This notion is studied in detail in a Hilbert space setting and shown to provide a powerful framework to analyze the convergence of a wide range of optimization algorithms in a systematic fashion. A number of convergence theorems covering and extending existing results are thus established. Special emphasis is placed on the design and the analysis of parallel algorithms.

Abstract — Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex programming framework. This approach places no limitation on the incorporation of additional constraints in the restoration process and the resulting optimization problem can be solved efficiently via block-iterative methods. Image denoising and deconvolution applications are demonstrated. I. Problem statement The classical linear restoration problem is to find the original form of an image x in a real Hilbert space (H, � · �) from the observation of a degraded image y = Lx + u, (1) where L: H → H is a bounded linear operator modeling the blurring process and u ∈ H models an additive noise component. Numerous approaches have been developed over the past three decades to solve this problem; see for instance [2], [9], [16], [33], [35], [39] and the references therein. Roughly speaking, restoration problems are typically posed as optimization problems in which an appropriate objective function is minimized under certain constraints. Restricting ourselves to convex problems, a general formulation is therefore m� Find x ∈ S = Si such that J(x) = inf J(S), (2) i=1 where the objective J: H →]−∞, +∞] is a convex function and the constraint sets (Si)1≤i≤m are closed convex subsets of H. These constraints arise from a priori knowledge about the model (1) and the original image x. For instance, the classical formulation of [21] concerns problems with smooth images, in which the energy δ of the noise u is known. The goal is then to find the smoothest image in terms of some high-pass filtering operator C: H → H which is consistent with (1) and the noise information, whence J: x ↦ → �Cx � 2 and S = � x ∈ H | �Lx − y � 2 ≤ δ � in (2).

Abstract — We consider the problem of synthesizing feasible signals in a Hilbert space in the presence of inconsistent convex constraints, some of which must imperatively be satisfied. This problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones. The resulting convex optimization problem is analyzed, and numerical solution schemes are presented along with convergence results. The proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems. An application to signal synthesis is demonstrated. Index Terms—Convex feasibility problem, fixed point, Hilbert space, inconsistent constraints, monotone operator, optimization,

A block-iterative parallel decomposition method is proposed to solve general quadratic signal recovery problems under convex constraints. The proposed method proceeds by local linearizations of blocks of constraints and it is therefore not sensitive to their analytical complexity. In addition, it naturally lends itself to implementation on parallel computing architectures due to its flexible block-iterative structure. Comparisons with existing methods are carried out and the case of inconsistent constraints is also discussed. Numerical results are presented.

In set theoretic image recovery, the constraints which do not yield convex sets in the chosen Hilbert solution space cannot be enforced. In some cases, however, such constraints may yield convex sets in other Hilbert spaces. In this paper we introduce a generalized product space formalism, through which constraints that are convex in different Hilbert spaces can be combined. A nonconvex problem with several sets is reduced to a convex problem with two sets in the product space, where it is solved via an alternating projection method. Applications are discussed. INTRODUCTION Let h be an image in a Hilbert space \Xi. The set theoretic image recovery problem is to produce an estimate of h that satisfies constraints (\Psi i ) 1im . Upon defining S i = fa 2 \Xi j a satisfies \Psi i g, the recovery problem takes the form of the feasibility problem Find a 2 S = m " i=1 S i : (1) There exists no numerical method that can solve this program in its full generality. However, the case whe...