arXiv subjects

Elias S. Helou

Publications and source records attributed to Elias S. Helou.

10 recordsLinked to original sources

arXiv2021

Unbiased Bregman-Risk Estimators: Application to Regularization Parameter Selection in Tomographic Image Reconstruction

Unbiased estimators are introduced for averaged Bregman divergences which generalize Stein's Unbiased (Predictive) Risk Estimator, and the minimization of these estimators is proposed as a regularization parameter selection method for regularization of inverse problems. Numerical experiments are presented in order to show the performance of the proposed technique. Experimental results indicate a useful occurence of a concentration of measure phenomena and some implications of this hypothesis are analyzed.

math.NA↗

arXiv2021

A Markovian Incremental Stochastic Subgradient Algorithm

A stochastic incremental subgradient algorithm for the minimization of a sum of convex functions is introduced. The method sequentially uses partial subgradient information and the sequence of partial subgradients is determined by a general Markov chain. This makes it suitable to be used in networks where the path of information flow is stochastically selected. We prove convergence of the algorithm to a weighted objective function where the weights are given by the Cesàro limiting probability distribution of the Markov chain. Unlike previous works in the literature, the Cesàro limiting distribution is general (not necessarily uniform), allowing for general weighted objective functions and flexibility in the method.

math.OC↗

arXiv2020

Derivative-Free Superiorization: Principle and Algorithm

The superiorization methodology is intended to work with input data of constrained minimization problems, that is, a target function and a set of constraints. However, it is based on an antipodal way of thinking to what leads to constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce (not necessarily minimize) target function values. This is done by inserting target-function-reducing perturbations into a feasibility-seeking algorithm while retaining its feasibility-seeking ability and without paying a high computational price. A superiorized algorithm that employs component-wise target function reduction steps is presented. This enables derivative-free superiorization (DFS), meaning that superiorization can be applied to target functions that have no calculable partial derivatives or subgradients. The numerical behavior of our derivative-free superiorization algorithm is illustrated on a data set generated by simulating a problem of image reconstruction from projections. We present a tool (we call it a proximity-target curve) for deciding which of two iterative methods is "better" for solving a particular problem. The plots of proximity-target curves of our experiments demonstrate the advantage of the proposed derivative-free superiorization algorithm.

math.OC↗

arXiv2020

Fast Proximal Gradient Methods for Nonsmooth Convex Optimization for Tomographic Image Reconstruction

The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a $O\left(1/k^{2}\right)$ non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in practical scenarios. Experimental comparison with Algebraic Reconstruction Techniques are performed uncovering certain behaviors of accelerated Proximal Gradient algorithms that apparently have not yet been noticed when these are applied to tomographic image reconstruction.

math.OC↗

arXiv2019

The Discrete Fourier Transform for Golden Angle Linogram Sampling

Estimation of the Discrete-Time Fourier Transform (DTFT) at points of a finite domain arises in many imaging applications. A new approach to this task, the Golden Angle Linogram Fourier Domain (GALFD), is presented, together with a computationally fast and accurate tool, named Golden Angle Linogram Evaluation (GALE), for approximating the DTFT at points of a GALFD. A GALFD resembles a Linogram Fourier Domain (LFD), which is efficient and accurate. A limitation of linograms is that embedding an LFD into a larger one requires many extra points, at least doubling the domain's cardinality. The GALFD, on the other hand, allows for incremental inclusion of relatively few data points. Approximation error bounds and floating point operations counts are presented to show that GALE computes accurately and efficiently the DTFT at the points of a GALFD. The ability to extend the data collection in small increments is beneficial in applications such as Magnetic Resonance Imaging. Experiments for simulated and for real-world data are presented to substantiate the theoretical claims. The mathematical analysis, algorithms, and software developed in the paper are equally suitable to other angular distributions of rays and therefore we bring the benefits of linograms to arbitrary radial patterns.

cs.GR↗

arXiv2018

Superiorization of Preconditioned Conjugate Gradient Algorithms for Tomographic Image Reconstruction

Properties of Superiorized Preconditioned Conjugate Gradient (SupPCG) algorithms in image reconstruction from projections are examined. Least squares (LS) is usually chosen for measuring data-inconsistency in these inverse problems. Preconditioned Conjugate Gradient algorithms are fast methods for finding an LS solution. However, for ill-posed problems, such as image reconstruction, an LS solution may not provide good image quality. This can be taken care of by superiorization. A superiorized algorithm leads to images with the value of a secondary criterion (a merit function such as the total variation) improved as compared to images with similar data-inconsistency obtained by the algorithm without superiorization. Numerical experimentation shows that SupPCG can lead to high-quality reconstructions within a remarkably short time. A theoretical analysis is also provided.

math.NA↗

arXiv2017

A Fast Gradient and Function Sampling Method for Finite Max-Functions

This paper tackles the unconstrained minimization of a class of nonsmooth and nonconvex functions that can be written as finite max-functions. A gradient and function-based sampling method is proposed which, under special circumstances, either moves superlinearly to a minimizer of the problem of interest or superlinearly improves the optimality certificate. Global and local convergence analysis are presented, as well as illustrative examples that corroborate and elucidate the obtained theoretical results.

math.OC↗

arXiv2017

Phase-Retrieval as a Regularization Problem

It was recently shown that the phase retrieval imaging of a sample can be modeled as a simple convolution process. Sometimes, such a convolution depends on physical parameters of the sample which are difficult to estimate a priori. In this case, a blind choice for those parameters usually lead to wrong results, e.g., in posterior image segmentation processing. In this manuscript, we propose a simple connection between phase-retrieval algorithms and optimization strategies, which lead us to ways of numerically determining the physical parameters

math.NA↗

arXiv2016

Superiorization of Incremental Optimization Algorithms for Statistical Tomographic Image Reconstruction

We propose the superiorization of incremental algorithms for tomographic image reconstruction. The resulting methods follow a better path in its way to finding the optimal solution for the maximum likelihood problem in the sense that they are closer to the Pareto optimal curve than the non-superiorized techniques. A new scaled gradient iteration is proposed and three superiorization schemes are evaluated. Theoretical analysis of the methods as well as computational experiments with both synthetic and real data are provided.

math.OC↗

arXiv2016

Fast Backprojection Techniques for High Resolution Tomography

Fast image reconstruction techniques are becoming important with the increasing number of scientific cases in high resolution micro and nano tomography. The processing of the large scale three-dimensional data demands new mathematical tools for the tomographic reconstruction task because of the big computational complexity of most current algorithms as the sizes of tomographic data grow with the development of more powerful acquisition hardware and more refined scientific needs. In the present paper we propose a new fast back-projection operator for the processing of tomographic data and compare it against other fast reconstruction techniques.

math.NA↗