[Show abstract][Hide abstract]ABSTRACT: We apply the recently proposed superiorization methodology (SM) to the
inverse planning problem in radiation therapy. The inverse planning problem is
represented here as a constrained minimization problem of the total variation
(TV) of the intensity vector over a large system of linear two-sided
inequalities. The SM can be viewed conceptually as lying between
feasibility-seeking for the constraints and full-fledged constrained
minimization of the objective function subject to these constraints. It is
based on the discovery that many feasibility-seeking algorithms (of the
projection methods variety) are perturbation-resilient, and can be proactively
steered toward a feasible solution of the constraints with a reduced, thus
superiorized, but not necessarily minimal, objective function value.
[Show abstract][Hide abstract]ABSTRACT: 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
Full-text · Article · Aug 2013 · Journal of Optimization Theory and Applications
[Show abstract][Hide abstract]ABSTRACT: 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.
No preview · Article · Feb 2013 · Computer methods and programs in biomedicine
[Show abstract][Hide abstract]ABSTRACT: 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.
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.
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.
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.
[Show abstract][Hide abstract]ABSTRACT: 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.
No preview · Article · Apr 2012 · Physics in Medicine and Biology
[Show abstract][Hide abstract]ABSTRACT: 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.
[Show abstract][Hide abstract]ABSTRACT: 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.
[Show abstract][Hide abstract]ABSTRACT: 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.
[Show abstract][Hide abstract]ABSTRACT: 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).
KeywordsProjection methods–Convex feasibility problems–Numerical evaluation–Optimization–Linear inequalities–Sparse matrices
Preview · Article · Dec 2009 · Computational Optimization and Applications
[Show abstract][Hide abstract]ABSTRACT: 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.
Preview · Article · Jun 2009 · International Transactions in Operational Research
[Show abstract][Hide abstract]ABSTRACT: Image reconstruction from projections suffers from an inherent difficulty: there are different images that have identical projections in any finite number of directions. However, by identifying the type of image that is likely to occur in an application area, one can design algorithms that may be efficacious in that area even when the number of projections is small. One such approach uses total variation minimization. We report on an algorithm based on this approach, and show that sometimes it produces medically-desirable reconstructions in computerized tomography (CT) even from a small number of projections. However, we also demonstrate that such a reconstruction is not guaranteed to provide the medically-relevant information: when data are collected by an actually CT scanner for a small number projections, the noise in such data may very well result in a tumor in the brain not being visible in the reconstruction.
[Show abstract][Hide abstract]ABSTRACT: 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.
Preview · Article · Jan 2008 · IEEE Journal of Selected Topics in Signal Processing