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Journal ArticleDOI

Adaptive Reconstruction of Discrete-Valued Objects from few Projections

TL;DR: This paper describes how the binary reconstruction problem to multi-valued objects can be reconstructed just by combining binary decisions, and shows how approximately known absorption levels can be adaptively estimated within the reconstruction process.
About: This article is published in Electronic Notes in Discrete Mathematics.The article was published on 2005-07-01. It has received 17 citations till now. The article focuses on the topics: Discrete tomography & Combinatorial optimization.
Citations
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Journal ArticleDOI
TL;DR: A short survey on thirty years of developments of DC (Difference of Convex functions) programming and DCA (DC Algorithms) which constitute the backbone of nonconvex programming and global optimization.
Abstract: The year 2015 marks the 30th birthday of DC (Difference of Convex functions) programming and DCA (DC Algorithms) which constitute the backbone of nonconvex programming and global optimization. In this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools. The survey is comprised of three parts. In the first part we present a brief history of the field, while in the second we summarize the state-of-the-art results and recent advances. We focus on main theoretical results and DCA solvers for important classes of difficult nonconvex optimization problems, and then give an overview of real-world applications whose solution methods are based on DCA. The third part is devoted to new trends and important open issues, as well as suggestions for future developments.

257 citations

Book ChapterDOI
23 May 2011
TL;DR: A new discrete tomography reconstruction algorithm developed for reconstruction of images that consist of a small number of gray levels based on the minimization of the objective function which combines the regularized squared projection error with the multi-well potential function.
Abstract: In this paper we present a new discrete tomography reconstruction algorithm developed for reconstruction of images that consist of a small number of gray levels. The proposed algorithm, called DTMWP is based on the minimization of the objective function which combines the regularized squared projection error with the multi-well potential function. The minimization is done by a gradient based method. We present experimental results obtained by application of the proposed algorithm for reconstruction of images that consist from three gray levels using small number of projections.

21 citations

Journal ArticleDOI
TL;DR: A binary optimization approach to the transmembrane voltage (TMV)-based problem and a hybrid metaheuristic approach and the difference of convex functions (DC) algorithm were tested, showing their potential for application in ECGI.
Abstract: The goal of ECG-imaging (ECGI) is to reconstruct heart electrical activity from body surface potential maps. The problem is ill-posed, which means that it is extremely sensitive to measurement and modeling errors. The most commonly used method to tackle this obstacle is Tikhonov regularization, which consists in converting the original problem into a well-posed one by adding a penalty term. The method, despite all its practical advantages, has however a serious drawback: The obtained solution is often over-smoothed, which can hinder precise clinical diagnosis and treatment planning. In this paper, we apply a binary optimization approach to the transmembrane voltage (TMV)-based problem. For this, we assume the TMV to take two possible values according to a heart abnormality under consideration. In this work, we investigate the localization of simulated ischemic areas and ectopic foci and one clinical infarction case. This affects only the choice of the binary values, while the core of the algorithms remains the same, making the approximation easily adjustable to the application needs. Two methods, a hybrid metaheuristic approach and the difference of convex functions (DC), algorithm were tested. For this purpose, we performed realistic heart simulations for a complex thorax model and applied the proposed techniques to the obtained ECG signals. Both methods enabled localization of the areas of interest, hence showing their potential for application in ECGI. For the metaheuristic algorithm, it was necessary to subdivide the heart into regions in order to obtain a stable solution unsusceptible to the errors, while the analytical DC scheme can be efficiently applied for higher dimensional problems. With the DC method, we also successfully reconstructed the activation pattern and origin of a simulated extrasystole. In addition, the DC algorithm enables iterative adjustment of binary values ensuring robust performance.

17 citations


Cites background or methods from "Adaptive Reconstruction of Discrete..."

  • ...and using (19) [34] ,we find the subgradient of the dual function g...

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  • ...So in this work, we adopted a hybrid metaheuristic algorithm [23] and the difference of convex functions (DC) method [33, 34] for solving the inverse problem of ECg....

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Book ChapterDOI
24 Aug 2012
TL;DR: A new algorithm for multivalued discrete tomogra phy, that reconstructs images from few projections by approximating the minimum of a suitably constructed energy function with a deterministic optimization method is proposed.
Abstract: There is a wide range of algorithms for binary and non-binary (called multivalued) discrete tomography. For example, the DART, Discrete Algebraic Reconstruction Technique (4) is capable of producing highly accurate reconstructions by thresholding a continuous reconstruction and then adjusting the object boundaries. Also, there are reconstruction algorithms based on minimizing an energy function by deterministic (13; 15; 16; 18) or randomized (1; 2; 8; 14) optimization strategies.

13 citations

References
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Book
15 Nov 1996
TL;DR: The EM Algorithm and Extensions describes the formulation of the EM algorithm, details its methodology, discusses its implementation, and illustrates applications in many statistical contexts, opening the door to the tremendous potential of this remarkably versatile statistical tool.
Abstract: The first unified account of the theory, methodology, and applications of the EM algorithm and its extensionsSince its inception in 1977, the Expectation-Maximization (EM) algorithm has been the subject of intense scrutiny, dozens of applications, numerous extensions, and thousands of publications. The algorithm and its extensions are now standard tools applied to incomplete data problems in virtually every field in which statistical methods are used. Until now, however, no single source offered a complete and unified treatment of the subject.The EM Algorithm and Extensions describes the formulation of the EM algorithm, details its methodology, discusses its implementation, and illustrates applications in many statistical contexts. Employing numerous examples, Geoffrey McLachlan and Thriyambakam Krishnan examine applications both in evidently incomplete data situations-where data are missing, distributions are truncated, or observations are censored or grouped-and in a broad variety of situations in which incompleteness is neither natural nor evident. They point out the algorithm's shortcomings and explain how these are addressed in the various extensions.Areas of application discussed include: Regression Medical imaging Categorical data analysis Finite mixture analysis Factor analysis Robust statistical modeling Variance-components estimation Survival analysis Repeated-measures designs For theoreticians, practitioners, and graduate students in statistics as well as researchers in the social and physical sciences, The EM Algorithm and Extensions opens the door to the tremendous potential of this remarkably versatile statistical tool.

5,998 citations


"Adaptive Reconstruction of Discrete..." refers methods in this paper

  • ...Using the expectation-maximization (EM) algorithm, we describe how these absorption coefficients can also be estimated as hidden variables and adapted to increase the reconstruction quality....

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01 Jan 2001
TL;DR: This chapter discusses reconstruction algorithms, stability and resolution in tomography, and problems that have peculiarities in relation to nonlinear tomography.
Abstract: 1. Introduction 2. Integral geometry 3. Tomography 4. Stability and resolution 5. Reconstruction algorithms 6. Problems that have peculiarities 7. Nonlinear tomography.

848 citations

Book
01 Jan 2001
TL;DR: In this article, the authors present a reconstruction algorithm for nonlinear tomography problems that have peculiarities, based on integral geometry and structural and resolution properties of the tomography images.
Abstract: 1. Introduction 2. Integral geometry 3. Tomography 4. Stability and resolution 5. Reconstruction algorithms 6. Problems that have peculiarities 7. Nonlinear tomography.

763 citations

Journal ArticleDOI
TL;DR: The problem of image reconstruction and restoration is first formulated, and some of the current regularization approaches used to solve the problem are described, and a Bayesian interpretation of the regularization techniques is given.
Abstract: Developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined. Particular attention is paid to common estimation structures and to practical problems not properly solved yet. The problem of image reconstruction and restoration is first formulated. Some of the current regularization approaches used to solve the problem are then described. The concepts of a priori information and compound criterion are introduced. A Bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated. The practical aspects of computing the solution, first when the hyperparameters are known and second when they must be estimated, are then considered. Conclusions are drawn, and points that still need to be investigated are outlined. >

716 citations