Elias S. Helou

String-averaging expectation-maximization for maximum likelihood estimation in emission tomography

We study the maximum likelihood model in emission tomography and propose a new family of algorithms for its solution, called string-averaging expectationmaximization (SAEM). In the string-averaging algorithmic regime, the index set of all underlying equations is split into subsets, called ‘strings’, and the algorithm separately proceeds along each string, possibly in parallel. Then, the end-points of all strings are averaged to form the next iterate. SAEM algorithms with several strings present better practical merits than the classical row-action maximum-likelihood algorithm. We present numerical experiments showing the effectiveness of the algorithmic scheme, using data of image reconstruction problems. Performance is evaluated from the computational

Mesh-Free Discrete Laplace-Beltrami Operator

In this work we propose a new discretization method for the Laplace–Beltrami operator defined on point‐based surfaces. In contrast to the existing point‐based discretization techniques, our approach does not rely on any triangle mesh structure, turning out truly mesh‐free. Based on a combination of Smoothed Particle Hydrodynamics and an optimization procedure to estimate area elements, our discretization method results in accurate solutions while still being robust when facing abrupt changes in the density of points. Moreover, the proposed scheme results in numerically stable discrete operators. The effectiveness of the proposed technique is brought to bear in many practical applications. In particular, we use the eigenstructure of the discrete operator for filtering and shape segmentation. Point‐based surface deformation is another application that can be easily carried out from the proposed discretization method.

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.

Accelerating Overrelaxed and Monotone Fast Iterative Shrinkage-Thresholding Algorithms With Line Search for Sparse Reconstructions

Recently, specially crafted unidimensional optimization has been successfully used as line search to accelerate the overrelaxed and monotone fast iterative shrinkage-threshold algorithm (OMFISTA) for computed tomography. In this paper, we extend the use of fast line search to the monotone fast iterative shrinkage-threshold algorithm (MFISTA) and some of its variants. Line search can accelerate the FISTA family considering typical synthesis priors, such as the

On the differentiability check in gradient sampling methods

\ell _{1}

String-averaging incremental subgradients for constrained convex optimization with applications to reconstruction of tomographic images

-norm of wavelet coefficients, as well as analysis priors, such as anisotropic total variation. This paper describes these new MFISTA and OMFISTA with line search, and also shows through numerical results that line search improves their performance for tomographic high-resolution image reconstruction.

Generalized Backprojection Operator: Fast Calculation

The present study aims to carefully discuss the importance of differentiability checks during the execution of methods based on gradient sampling. We stress the significance of this procedure not only from the theoretical perspective, but also in the practical implementation. We support our claims exhibiting illustrative examples where the absence of the differentiability check in the method prevents the achievement of the minimization problem solution. As possible alternatives, this manuscript presents two procedures that suppress the differentiability check without affecting the convergence of the method (both in theory and in practice). Lastly, by solving a difficult control problem, we show that besides the theoretical appeal our changes may also be useful to address real problems.

Fast Backprojection Operator for Synchrotron Tomographic Data

We present a method for non-smooth convex minimization which is based on subgradient directions and string-averaging techniques. In this approach, the set of available data is split into sequences (strings) and a given iterate is processed independently along each string, possibly in parallel, by an incremental subgradient method (ISM). The end-points of all strings are averaged to form the next iterate. The method is useful to solve sparse and large-scale non-smooth convex optimization problems, such as those arising in tomographic imaging. A convergence analysis is provided under realistic, standard conditions. Numerical tests are performed in a tomographic image reconstruction application, showing good performance for the convergence speed when measured as the decrease ratio of the objective function, in comparison to classical ISM.

On the Local Convergence Analysis of the Gradient Sampling Method for Finite Max-Functions

The inverse Radon transform and his straightforward implementation, known as filtered backprojection (also known as FBP), has become a powerful algorithm for solving a tomographic inverse problem. It has a wide range of applications, including geophysics, medicine and synchrotrons, and from kilo to centi to micro scale respectively. Such a classical inversion has a major computational disadvantage: increasing slowness proportionally to the data size. An ordinary implementation of this algorithm relies on a simple integral that has to be done pixelwise. Many accelerating techniques were proposed in the literature so as to make this part of the inversion as fast as possible. One the most promising strategies is converting the backprojection as a convolution operator (at log-polar coordinates). The generalized backprojector has many applications, for instance in the analytical inversion of single-photon emission tomography or x-ray fluorescence tomography. Our aim in this paper is to show how these ideas can be used for other inversion methods, the iterative ones; which deal much better with noise.

Accelerating the over-relaxed iterative shrinkage-thresholding algorithms with fast and exact line search for high resolution tomographic image reconstruction

Reduction of computational time in high resolution image reconstruction is essential in basic research and applications as well. This reduction is important for different types of traditional non diffractive tomography in medical diagnosis as well as for applications in nanomaterials research, related to modern technologies. Alternatives to alleviate the computationally intense part of each iteration of iterative methods in tomographic reconstruction have all been based on interpolation over a regular grid in the Fourier domain or in fast nonuniform Fourier transforms. Both approaches speed up substantially the computation of each iteration of classical algorithms, but are not suitable for being used in a large class of more advanced faster algorithms: incremental methods such as OS-EM, BRAMLA or BSREM, among others, cannot benefit from these techniques. The backprojection is a stacking operator, known to be the adjoint of the Radon transform. As a mapping \(\mathcal{B}\), the backprojection can be recast as a convolution operator, in a different coordinate system, which is an improvement in accelerating the computation of \(\mathcal{B}\). In this work, we propose several analytical representations for the operator \(\mathcal{B}\), in order to find a fast algorithm.