Hard-constrained inconsistent signal feasibility problems

Combettes, Patrick L.; Bondon, Pascal

doi:10.1109/78.782189

Cited by 78 publications

(60 citation statements)

References 14 publications

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“…21,[46][47][48][49] Several investigations have been devoted to analyzing and coping with this situation in convex problems. [50][51][52][53][54][55] While the infinite-dimensional space L appears to be the most appropriate signal space to model the physics of the problem and to describe the subtle properties of the algorithms in their full generality, we shall also call attention to finite-dimensional versions of the results whenever these happen to differ from their infinitedimensional counterparts. The reason for this is that in most numerical applications, the signals are sampled on a finite grid and the algorithms are implemented on a digital computer.…”

Section: Phase Retrieval and Feasibilitymentioning

confidence: 99%

Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

2002

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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. Our purpose 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 input-output 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. We provide a theoretical framework to better understand and, potentially, to improve existing phase recovery algorithms.

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Section: Phase Retrieval and Feasibilitymentioning

confidence: 99%

Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

2002

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“…Several approaches can be used to solve (5), of which a convex programming formulation for inconsistent problems [25] was chosen because it can incorporate the Bmode image information. Appendix B provides details of the convex programming formulation.…”

Section: Reconstruction Of the Sound Velocity Distributionmentioning

confidence: 99%

Computed Tomography Sound Velocity Reconstruction Using Incomplete Data

Huang

2004

IEEE Trans. Ultrason., Ferroelect., Freq. Contr.

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An approach based on limited-angle transmission tomography for reconstruction of the sound velocity distribution in the breast is proposed. The imaging setup is similar to that of x-ray mammography. With this setup, the time-of-flight data are acquired by a linear array positioned at the top of the compressed breast that both transmits and receives, and a metal plate is placed at the bottom as a reflector. The setup allows acoustic data acquisition for simultaneous B-mode image formation and the tomographic sound velocity reconstruction. In order to improve the sound velocity estimation accuracy, a new reconstruction algorithm based on a convex programming formulation has been developed. Extensive simulations for both imaging and time-of-flight data based on a 5-MHz linear array were performed on tissues with different geometries and acoustic parameters. Results show that the sound velocity error was generally 1-3 m/s, with a maximum of 5.8 m/s. The radii of the objects under investigation varied from 2 to 6 mm, and all of them were detected successfully. Thus, the proposed approach has been shown to be both feasible and accurate. The approach can be used to complement conventional Bmode imaging to further enhance the detection of breast cancer.

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“…The main objective of this paper is to develop a novel distributed computing approach for resolving the following minimization problem (see [6,8,9,11,12,13,14,16,18,19,23,24] and references therein for applications of problem (1.1)) on a Hilbert space H:…”

Section: Introductionmentioning

confidence: 99%

Acceleration Method Combining Broadcast and Incremental Distributed Optimization Algorithms

Iiduka¹,

Hishinuma²

2014

SIAM J. Optim.

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This paper considers a networked system consisting of an operator, which manages the system, and a finite number of subnetworks with all users, and studies the problem of minimizing the sum of the operator's and all users' objective functions over the intersection of the operator's and all users' constraint sets. When users in each subnetwork can communicate with each other, they can implement an incremental subgradient method that uses the transmitted information from their neighbor users. Since the operator can communicate with users in the subnetworks, it can implement a broadcast distributed algorithm that uses all available information in the subnetworks. We present an iterative method combining broadcast and incremental distributed optimization algorithms. Our method has faster convergence and a wider range of application compared with conventional distributed algorithms. We also prove that under certain assumptions our method converges to the solution to the problem in the sense of the strong topology of a Hilbert space. Moreover, we numerically compare our method with the conventional distributed algorithms in the case of a data storage system. The numerical results demonstrate the effectiveness and fast convergence of our method.

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Hard-constrained inconsistent signal feasibility problems

Cited by 78 publications

References 14 publications

Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

Computed Tomography Sound Velocity Reconstruction Using Incomplete Data

Acceleration Method Combining Broadcast and Incremental Distributed Optimization Algorithms

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