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

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.

We report on progress in algorithms for iterative phase retrieval. The theory of convex optimization is used to develop and to gain insight into counterparts for the nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fixed point set of the related operator in the convex case. A numerical study supports our theoretical observations and demonstrates the effectiveness of the algorithm compared to the current state of the art. 1

We survey some key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging.

The phase retrieval problem, fundamental in applied physics and engineering, asks to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently, we identified two important methods for phase retrieval, namely Fienup’s Basic Input-Output (BIO) and Hybrid Input-Output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the Hybrid Projection Reflection (HPR) algorithm, for solving phase retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than twenty years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments, we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust. 1 1

We apply nonsmooth analysis to a well known optical inverse problem, phase retrieval. The phase retrieval problem arises in many di#erent modalities of electromagnetic imaging and has been studied in the optics literature for over forty years. The state of the art for this problem in two dimensions involves iterated projections for solving a nonconvex feasibility problem. Despite widespread use of these algorithms, current mathematical theory cannot explain their success. At the heart of projection algorithms is a nonconvex, nonsmooth optimization problem. We obtain some insight into these algorithms by applying techniques from nonsmooth analysis. In particular, we show that the weak closure of the set of directions toward the projection generate the subdi#erential of the corresponding squared set distance function. Following a pattern of proof described in F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer (1998), this result is generalized to provide conditions under which the subdi#erential of an integral function equals the integral of the subdi#erential. Key words. phase retrieval, least squares, nonsmooth analysis, variational analysis AMS subject classifications. 78A45, 93E24, 49J52, 49J53 1.

The phase retrieval problem, fundamental in applied physics and engineering, asks to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently, we identified two important methods for phase retrieval, namely Fienup's Basic Input-Output (BIO) and Hybrid Input-Output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the Hybrid Projection Reflection (HPR) algorithm, for solving phase retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than twenty years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments, we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust.

is 5 m with a microscope-based system and 100 m for an embedded photodiode system. The photodiode system also provides a detection limit of 2.4 m for ATP/luciferase bioluminescence. Introduction The past decade has seen a rapid growth in the use of microtechnology to produce ever-smaller instrumentation systems for use in medical diagnostics, studies in cell biology, biochemical process control, and the detection of contaminants and pathogens in the environment. The general goal of these systems is to provide high-speed, low-cost, reliable measurements of various biochemical molecules that occur either naturally or as markers. The objective of our research over the past few years has been to design and build an intelligent analytical system that measures different molecular analytes simultaneously using specific molecular interactions in wells on specially constructed chips. The technology we propose is generic and could be useful in a variety of applications, such as quality manag

Abstract. We propose strongly consistent algorithms for reconstructing the characteristic function 1K of an unknown convex body K in Rn from possibly noisy measurements of the modulus of its Fourier transform ̂1K. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k2) noisy covariogram measurements to construct a convex polytope Pk that approximates K or its reflection −K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all threedimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between Pk and ±K tends to zero as k tends to infinity. 1.

1 A new recursive augmented Lagrangian (AL) algorithm is presented for reconstruction of a 3D wave field for intensityonly measurements obtained from two or more sensor planes parallel to the object plane. This reconstruction is framed as a maximum likelihood constrained nonlinear optimization problem for Gaussian additive noise observations. A contribution of this paper concerns a development of a novel recursive algorithm and demonstration that this algorithm enables a better accuracy and better imaging comparing with the successive iterative method