The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space.

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.

Total variation superiorization schemes in proton computed tomography image reconstruction.

PURPOSE: Iterative projection reconstruction algorithms are currently the preferred reconstruction method in proton computed tomography (pCT). However, due to inconsistencies in the measured data arising from proton energy straggling and multiple Coulomb scattering, the noise in the reconstructed image increases with successive iterations. In the current work, the authors investigated the use of total variation superiorization (TVS) schemes that can be applied as an algorithmic add-on to perturbation-resilient iterative projection algorithms for pCT image reconstruction. METHODS: The block-iterative diagonally relaxed orthogonal projections (DROP) algorithm was used for reconstructing GEANT4 Monte Carlo simulated pCT data sets. Two TVS schemes added on to DROP were investigated; the first carried out the superiorization steps once per cycle and the second once per block. Simplifications of these schemes, involving the elimination of the computationally expensive feasibility proximity checking step of the TVS framework, were also investigated. The modulation transfer function and contrast discrimination function were used to quantify spatial and density resolution, respectively. RESULTS: With both TVS schemes, superior spatial and density resolution was achieved compared to the standard DROP algorithm. Eliminating the feasibility proximity check improved the image quality, in particular image noise, in the once-per-block superiorization, while also halving image reconstruction time. Overall, the greatest image quality was observed when carrying out the superiorization once per block and eliminating the feasibility proximity check. CONCLUSIONS: The low-contrast imaging made possible with TVS holds a promise for its incorporation into future pCT studies.

Perturbation Resilience and Superiorization of Iterative Algorithms.

Iterative algorithms aimed at solving some problems are discussed. For certain problems, such as finding a common point in the intersection of a finite number of convex sets, there often exist iterative algorithms that impose very little demand on computer resources. For other problems, such as finding that point in the intersection at which the value of a given function is optimal, algorithms tend to need more computer memory and longer execution time. A methodology is presented whose aim is to produce automatically for an iterative algorithm of the first kind a "superiorized version" of it that retains its computational efficiency but nevertheless goes a long way towards solving an optimization problem. This is possible to do if the original algorithm is "perturbation resilient," which is shown to be the case for various projection algorithms for solving the consistent convex feasibility problem. The superiorized versions of such algorithms use perturbations that steer the process in the direction of a superior feasible point, which is not necessarily optimal, with respect to the given function. After presenting these intuitive ideas in a precise mathematical form, they are illustrated in image reconstruction from projections for two different projection algorithms superiorized for the function whose value is the total variation of the image.

Perturbation-resilient block-iterative projection methods with application to image reconstruction from projections.

A block-iterative projection algorithm for solving the consistent convex feasibility problem in a finite-dimensional Euclidean space that is resilient to bounded and summable perturbations (in the sense that convergence to a feasible point is retained even if such perturbations are introduced in each iterative step of the algorithm) is proposed. This resilience can be used to steer the iterative process towards a feasible point that is superior in the sense of some functional on the points in the Euclidean space having a small value. The potential usefulness of this is illustrated in image reconstruction from projections, using both total variation and negative entropy as the functional.

Transfer function restoration in 3D electron microscopy via iterative data refinement.

Three-dimensional electron microscopy (3D-EM) is a powerful tool for visualizing complex biological systems. As with any other imaging device, the electron microscope introduces a transfer function (called in this field the contrast transfer function, CTF) into the image acquisition process that modulates the various frequencies of the signal. Thus, the 3D reconstructions performed with these CTF-affected projections are also affected by an implicit 3D transfer function. For high-resolution electron microscopy, the effect of the CTF is quite dramatic and limits severely the achievable resolution. In this work we make use of the iterative data refinement (IDR) technique to ameliorate the effect of the CTF. It is demonstrated that the approach can be successfully applied to noisy data.

BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting.

Component averaging (CAV) was recently introduced by Censor, Gordon, and Gordon as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations. Based on earlier work of Byrne and Censor, it uses diagonal weighting matrices, with pixel-related weights determined by the sparsity of the system matrix. CAV is inherently parallel (similar to the very slowly converging Cimmino method) but its practical convergence on problems of image reconstruction from projections is similar to that of the algebraic reconstruction technique (ART). Parallel techniques are becoming more important for practical image reconstruction since they are relevant not only for supercomputers but also for the increasingly prevalent multiprocessor workstations. This paper reports on experimental results with a block-iterative version of component averraging (BICAV). When BICAV is optimized for block size and relaxation parameters, its very first iterates are far superior to those of and more or less on a par with ART. Similar to CAV, BICAV is also inherently parallel. The fast convergence is demonstrated on problems of image reconstruction from projections, using the SNARK93 image reconstruction software package. Detailed plots of various measures of convergence, and reconstructed images are presented.

An iterative approach to plan combination in radiotherapy.

Radiation therapy concerns the delivery of a proper dose of radiation to a tumor volume without causing irreparable damage to surrounding healthy tissue and critical organs. The problem of plan combination in radiation therapy treatment planning (RTTP) proposed, formulated and studied here, addresses a situation when for a specific clinical case, a set of several treatment plans is proposed, but each one of them violates the prescribed dose in at least one significant region of the volume that has to be treated. We represent treatment plans as vectors in the Euclidean space, and define their equivalence, acceptability and realizability. A simple linear algebraic model for combining them is then used in order to derive, from the given set of approximate plans, a combined treatment plan, which will be both acceptable, and technically realizable. In the event that such a combined plan dose not exist, the alternatives for relaxing the treatment requirements can be systematically considered.

Semi-automated radiotherapy treatment planning with a mathematical model to satisfy treatment goals.

Iterative algorithms can provide a feasible solution, if any exists, to specified treatment goals. Our model subdivides both the patient's cross section into a fine grid of points and the radiation beam into a set of "pencil" rays. The anatomy, treatment machine parameters, dose limits and homogeneity, are all defined. This process of subdivision leads to a large system of linear inequalities with a solution that provides a radiation intensity distribution that will deliver a prescribed dose distribution. The clinical results from two different algorithms will be presented and contrasted. Once the anatomy, treatment, and machine parameters have been entered, the computerized algorithms yield an answer in several minutes. The Cimmino algorithm also allows "weights" or priority assignments of the treatment goals. The resulting solution is biased towards fulfilling the specified doses for the anatomic regions which were given greater weight. It is desirable to have a systematic search of possible treatment alternatives in complex clinical situations, including 3-dimensional radiation therapy treatment planning (RTTP). Our method has been applied to 2-D RTTP, but is equally applicable to 3-D RTTP with minor modifications.

Demonstration of a software package for the reconstruction of the dynamically changing structure of the human heart from cone beam x-ray projections.

The Dynamic Spatial Reconstructor (DSR) is a device constructed at the Biodynamics Research Unit of the Mayo Clinic for (among other things) the visualization of the beating heart inside the intact thorax. The device consists of 28 rotating X-ray sources arranged on a circular arc at 6 degrees intervals (total span 162 degrees) and a matching set of 28 imaging systems. The whole thorax of the patient is projected onto the two-dimensional screen of the imaging systems by cone beams of X rays from the sources. All of the X-ray sources are switched on and off within a total period of 10 milliseconds. The Medical Image Processing Group at the State University of New York at Buffalo has developed a software package for the design and evaluation of algorithms to be used by the DSR. In this paper we illustrate the operation of the package and a particular algorithm for the reconstruction of the dynamically changing structure of the heart from data collected by the DSR.