Ran Davidi

On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints

The effectiveness of projection methods for solving systems of linear inequalities is investigated. It is shown that they often have a computational advantage over alternatives that have been proposed for solving the same problem and that this makes them successful in many real-world applications. This is supported by experimental evidence provided in this paper on problems of various sizes (up to tens of thousands of unknowns satisfying up to hundreds of thousands of constraints) and by a discussion of the demonstrated efficacy of projection methods in numerous scientific publications and commercial patents (dealing with problems that can have over a billion unknowns and a similar number of constraints).

Stable Convergence Behavior Under Summable Perturbations of a Class of Projection Methods for Convex Feasibility and Optimization Problems

We study the convergence behavior of a class of projection methods for solving convex feasibility and optimization problems. We prove that the algorithms in this class converge to solutions of the consistent convex feasibility problem, and that their convergence is stable under summable perturbations. Our class is a subset of the class of string-averaging projection methods, large enough to contain, among many other procedures, a version of the Cimmino algorithm, as well as the cyclic projection method. A variant of our approach is proposed to approximate the minimum of a convex functional subject to convex constraints. This variant is illustrated on a problem in image processing: namely, for optimization in tomography.

Improved compressed sensing-based cone-beam CT reconstruction using adaptive prior image constraints

Volumetric cone-beam CT (CBCT) images are acquired repeatedly during a course of radiation therapy and a natural question to ask is whether CBCT images obtained earlier in the process can be utilized as prior knowledge to reduce patient imaging dose in subsequent scans. The purpose of this work is to develop an adaptive prior image constrained compressed sensing (APICCS) method to solve this problem. Reconstructed images using full projections are taken on the first day of radiation therapy treatment and are used as prior images. The subsequent scans are acquired using a protocol of sparse projections. In the proposed APICCS algorithm, the prior images are utilized as an initial guess and are incorporated into the objective function in the compressed sensing (CS)-based iterative reconstruction process. Furthermore, the prior information is employed to detect any possible mismatched regions between the prior and current images for improved reconstruction. For this purpose, the prior images and the reconstructed images are classified into three anatomical regions: air, soft tissue and bone. Mismatched regions are identified by local differences of the corresponding groups in the two classified sets of images. A distance transformation is then introduced to convert the information into an adaptive voxel-dependent relaxation map. In constructing the relaxation map, the matched regions (unchanged anatomy) between the prior and current images are assigned with smaller weight values, which are translated into less influence on the CS iterative reconstruction process. On the other hand, the mismatched regions (changed anatomy) are associated with larger values and the regions are updated more by the new projection data, thus avoiding any possible adverse effects of prior images. The APICCS approach was systematically assessed by using patient data acquired under standard and low-dose protocols for qualitative and quantitative comparisons. The APICCS method provides an effective way for us to enhance the image quality at the matched regions between the prior and current images compared to the existing PICCS algorithm. Compared to the current CBCT imaging protocols, the APICCS algorithm allows an imaging dose reduction of 10-40 times due to the greatly reduced number of projections and lower x-ray tube current level coming from the low-dose protocol.

Superiorization: An optimization heuristic for medical physics

PURPOSE To describe and mathematically validate the superiorization methodology, which is a recently developed heuristic approach to optimization, and to discuss its applicability to medical physics problem formulations that specify the desired solution (of physically given or otherwise obtained constraints) by an optimization criterion. METHODS The superiorization methodology is presented as a heuristic solver for a large class of constrained optimization problems. The constraints come from the desire to produce a solution that is constraints-compatible, in the sense of meeting requirements provided by physically or otherwise obtained constraints. The underlying idea is that many iterative algorithms for finding such a solution are perturbation resilient in the sense that, even if certain kinds of changes are made at the end of each iterative step, the algorithm still produces a constraints-compatible solution. This property is exploited by using permitted changes to steer the algorithm to a solution that is not only constraints-compatible, but is also desirable according to a specified optimization criterion. The approach is very general, it is applicable to many iterative procedures and optimization criteria used in medical physics. RESULTS The main practical contribution is a procedure for automatically producing from any given iterative algorithm its superiorized version, which will supply solutions that are superior according to a given optimization criterion. It is shown that if the original iterative algorithm satisfies certain mathematical conditions, then the output of its superiorized version is guaranteed to be as constraints-compatible as the output of the original algorithm, but it is superior to the latter according to the optimization criterion. This intuitive description is made precise in the paper and the stated claims are rigorously proved. Superiorization is illustrated on simulated computerized tomography data of a head cross section and, in spite of its generality, superiorization is shown to be competitive to an optimization algorithm that is specifically designed to minimize total variation. CONCLUSIONS The range of applicability of superiorization to constrained optimization problems is very large. Its major utility is in the automatic nature of producing a superiorization algorithm from an algorithm aimed at only constraints-compatibility; while nonheuristic (exact) approaches need to be redesigned for a new optimization criterion. Thus superiorization provides a quick route to algorithms for the practical solution of constrained optimization problems.

Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction

We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency of the linear system. For a consistent system, the limit point is a solution of the system. In the inconsistent case, the symmetric version of our method converges to a weighted least squares solution. Perturbation resilience is utilized to approximate the minimum of a convex functional subject to the equations. A main contribution, as compared to previously published approaches to achieving similar aims, is a more than an order of magnitude speed-up, as demonstrated by applying the methods to problems of image reconstruction from projections. In addition, the accelerated algorithms are illustrated to be better, in a strict sense provided by the method of statistical hypothesis testing, than their unaccelerated versions for the task of detecting small tumors in the brain from X-ray CT projection data.

Projected Subgradient Minimization Versus Superiorization

The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to regain feasibility. The latter poses a computational difficulty, and, therefore, the projected subgradient method is applicable only when the feasible region is “simple to project onto.” In contrast to this, in the superiorization methodology a feasibility-seeking algorithm leads the overall process, and objective function steps are interlaced into it. This makes a difference because the feasibility-seeking algorithm employs projections onto the individual constraints sets and not onto the entire feasible region.We present the two approaches side-by-side and demonstrate their performance on a problem of computerized tomography image reconstruction, posed as a constrained minimization problem aiming at finding a constraint-compatible solution that has a reduced value of the total variation of the reconstructed image.

Reconstruction from a few projections by ℓ1-minimization of the Haar transform

Much recent activity is aimed at reconstructing images from a few projections. Images in any application area are not random samples of all possible images, but have some common attributes. If these attributes are reflected in the smallness of an objective function, then the aim of satisfying the projections can be complemented with the aim of having a small objective value. One widely investigated objective function is total variation (TV), it leads to quite good reconstructions from a few mathematically ideal projections. However, when applied to measured projections that only approximate the mathematical ideal, TV-based reconstructions from a few projections may fail to recover important features in the original images. It has been suggested that this may be due to TV not being the appropriate objective function and that one should use the ℓ(1)-norm of the Haar transform instead. The investigation reported in this paper contradicts this. In experiments simulating computerized tomography (CT) data collection of the head, reconstructions whose Haar transform has a small ℓ(1)-norm are not more efficacious than reconstructions that have a small TV value. The search for an objective function that provides diagnostically efficacious reconstructions from a few CT projections remains open.

SNARK09 - A software package for reconstruction of 2D images from 1D projections

The problem of reconstruction of slices and volumes from 1D and 2D projections has arisen in a large number of scientific fields (including computerized tomography, electron microscopy, X-ray microscopy, radiology, radio astronomy and holography). Many different methods (algorithms) have been suggested for its solution. In this paper we present a software package, SNARK09, for reconstruction of 2D images from their 1D projections. In the area of image reconstruction, researchers often desire to compare two or more reconstruction techniques and assess their relative merits. SNARK09 provides a uniform framework to implement algorithms and evaluate their performance. It has been designed to treat both parallel and divergent projection geometries and can either create test data (with or without noise) for use by reconstruction algorithms or use data collected by another software or a physical device. A number of frequently-used classical reconstruction algorithms are incorporated. The package provides a means for easy incorporation of new algorithms for their testing, comparison and evaluation. It comes with tools for statistical analysis of the results and ten worked examples.

TH‐C‐BRA‐12: Tomographic Measurement of X‐Ray Beam Spot Profiles Using a Rotating Edge

Purpose: X‐ray beam spot size and shape are critical performance determinants of an imaging or treatment system. However, quantitative assessment of such spot profiles can prove difficult, particularly at MV energies. We have developed a novel and convenient tomographic spot measurement technique that uses a rotating edge phantom. The method can be applied to x‐ray systems equipped with flat panel imagers.Methods: Data were acquired at 10MV and 6MV on a Varian TrueBeam system. A 0.5 mm thick tantalum sheet (attenuation ∼5%) was placed on the systems rotatable treatment head with an edge abutting the axis of rotation. A total of 144 projections, each 10MU, were acquired at 2.5 degree steps. For each projection, a line‐spread function (LSF) is generated by differentiating the measured edge spread function. For sufficiently high source magnifications, the LSF, taken at a given rotation angle, is a tomographic projection of the x‐ray beam spot at that angle. The LSFs were then assembled into a sinogram and the corresponding spot profile was reconstructed using a parallel‐beam CT algorithm. The reconstructed profiles were compared to those measured using a film‐based ‘spot camera’ made from a 15 cm thick tungsten cylinder pierced with an array of small holes. Monte Carlo simulations of the edge and the spot camera experiments were also performed. Results: The edge technique produced profiles similar to those from the spot camera yet with higher resolution (0.17mm vs. 0.25mm). These results were confirmed by Monte Carlo simulations. The measured FWHM of the 10 MV spot was 1.6 mm. The 6 MV spot was slightly asymmetric with an average FWHM of 1.5 mm. Conclusions: A thin rotating edge with low attenuation can be used to accurately and conveniently measure x‐ray beam spot profiles. NIH NIH 1R01CA138426 Employees of Varian Medical Systems