bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

convergence of em image reconstruction algorithms with gibbs smoothing p j green has defined an osl one step late algorithm that retains the e step of the em algorithm for image reconstruction in emission tomography but provides an approximate solution to the m step further modifications of the osl algorithm guarantee convergence to the unique maximum of the log posterior function convergence is proved under a specific set of sufficient conditions several of these conditions concern the potential function of the gibb s prior and a number of candidate potential functions are identified generalization of the osl algorithm to transmission tomography is also considered

three dimensional underwater sound propagation using a split step pade parabolic equation solution the majority of current three dimensional 3 d parabolic equation propagation model development work has been focused on implementations recent attention has been on the modeling of internal wave fields and propagation in the presence of strong internal tides and eddies state of the art solutions are experimental water tank comparisons against azimuthal schemes sturm etal jasa 113 and split step fourier based propagation models lin etal jasa 126 a current thrust is to establish 3 d benchmarks for propagation simulations in range dependent environments where effects due to generic bottom features are present in this paper a 3 d extension to current 2 d split step pade solutions is developed and benchmarked transverse and depth operator discretizations are performed using a galerkin method cartesian versus cylindrical coordinate systems are discussed as is the nature of the acoustic source in either geometry

grouped coordinate ascent algorithms for penalized likelihood transmission image reconstruction presents a new class of algorithms for penalized likelihood reconstruction of attenuation maps from low count transmission scans we derive the algorithms by applying to the transmission log likelihood a version of the convexity technique developed by de pierro for emission tomography the new class includes the single coordinate ascent sca algorithm and lange s convex algorithm for transmission tomography as special cases the new grouped coordinate ascent gca algorithms in the class overcome several limitations associated with previous algorithms 1 fewer exponentiations are required than in the transmission maximum likelihood expectation maximization ml em algorithm or in the sca algorithm 2 the algorithms intrinsically accommodate nonnegativity constraints unlike many gradient based methods 3 the algorithms are easily parallelizable unlike the sca algorithm and perhaps line search algorithms we show that the gca algorithms converge faster than the sca algorithm even on conventional workstations an example from a low count positron emission tomography pet transmission scan illustrates the method

an iterative algorithm for x ray ct fluoroscopy x ray computed tomography fluoroscopy ctf enables image guidance of interventions synchronization of scanning with contrast bolus arrival and motion analysis however filtered backprojection fb the current method for ctf image reconstruction is subject to motion and metal artifacts from implants needles or other surgical instruments reduced target lesion conspicuity may result from increased image noise associated with reduced tube current in this report the authors adapt the row action expectation maximization em algorithm for ctf because time dependent variation in images is localized during ctf the row action em like algorithm allows rapid convergence more importantly this iterative ctf algorithm has fewer metal artifacts and better low contrast performance than fb

choice of initial conditions in the ml reconstruction of fan beam transmission with truncated projection data the authors investigate the effects of initial conditions in the iterative maximum likelihood ml reconstruction of fan beam transmission projection data with truncation in an iterative ml reconstruction the estimate of the transmission reconstructed image in the previous iteration is multiplied by some factors to obtain the current estimate normally a flat initial condition fic or an image with equal positive pixel values is used as initial condition for an ml reconstruction usage of fic has also been perceived as a way of preventing any bias on the reconstruction which may have come from the initial condition when projection data have truncation the authors show that using are fic in an ml iterative reconstruction can introduce a bias to the reconstruction inside the densely sampled region dsr whose projection data have no truncation at any angle to reduce this bias the authors propose to use the largest right singular vector lrsv of the system matrix as the initial condition and demonstrate that the bias can be reduced with the lrsv when data truncation is reduced the lrsv approaches the fic this result does not contradict to the use of fic when projection data are not truncated the authors also demonstrate that the reconstructed transmission image using lrsv as initial condition provides a more accurate attenuation coefficient distribution than that using fic however the improvement is mostly in the area outside the dsr

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm iterative maximum likelihood ml transmission computed tomography algorithms have distinct advantages over fourier based reconstruction but unfortunately require increased computation time the convex algorithm is a relatively fast iterative ml algorithm but it is nevertheless too slow for many applications therefore an acceleration of this algorithm by using ordered subsets of projections is proposed ordered subsets convex algorithm osc osc applies the convex algorithm sequentially to subsets of projections osc was compared with the convex algorithm using simulated and physical thorax phantom data reconstructions were performed for osc using eight and 16 subsets eight and four projections subset respectively global errors image noise contrast recovery and likelihood increase were calculated results show that osc is faster than the convex algorithm the amount of acceleration being approximately proportional to the number of subsets in osc and it causes only a slight increase of noise and global errors in the reconstructions images and image profiles of the reconstructions were in good agreement in conclusion osc and the convex algorithm result in similar image quality but osc is more than an order of magnitude faster

pet regularization by envelope guided conjugate gradients the authors propose a new way to iteratively solve large scale ill posed problems and in particular the image reconstruction problem in positron emission tomography by exploiting the relation between tikhonov regularization and multiobjective optimization to obtain iteratively approximations to the tikhonov l curve and its corner monitoring the change of the approximate l curves allows the authors to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration so that the desired solution can be reconstructed with a small number of iterations

an interior point iterative maximum likelihood reconstruction algorithm incorporating upper and lower bounds with application to spect transmission imaging the algorithm we consider here is a block iterative or ordered subset version of the inferior point algorithm for transmission reconstruction our algorithm is an interior point method because each vector of the iterative sequence x sup k k 0 1 2 satisfies the constraints a sub j x sub j sup k b sub j j 1 j because it is a block iterative algorithm that reconstructs the transmission attenuation map and places constraints above and below the pixel values of the reconstructed image we call it the bitab method computer simulations using the three dimensional mathematical cardiac and torso phantom reveal that the bitab algorithm in conjunction with reasonably selected prior upper and lower bounds has the potential to improve the accuracy of the reconstructed attenuation coefficients from truncated fan beam transmission projections by suitably selecting the bounds it is possible to restrict the over estimation of coefficients outside the fully sampled region that results from reconstructing truncated fan beam projections with iterative transmission algorithms such as the maximum likelihood gradient type algorithm

a theoretical study of some maximum likelihood algorithms for emission and transmission tomography this paper has the dual purpose of introducing some new algorithms for emission and transmission tomography and proving mathematically that these algorithms and related antecedent algorithms converge like the em algorithms for positron single photon and transmission tomography the algorithms provide maximum likelihood estimates of pixel concentration or linear attenuation parameters one particular innovation we discuss is a computationally practical scheme for modifying the em algorithms to include a bayesian prior the bayesian versions of the em algorithms are shown to have superior convergence properties in a vicinity of the maximum we anticipate that some of the other algorithms will also converge faster than the em algorithms

on the convergence of an em type algorithm for penalized likelihood estimation in emission tomography recently we proposed an extension of the expectation maximization em algorithm that was able to handle regularization terms in a natural way although very general convergence proofs were not valid for many possibly useful regularizations we present here a simple convergence result that is valid assuming only continuous differentiability of the penalty term and can be also extended to other methods for penalized likelihood estimation in tomography

hybrid poisson polynomial objective functions for tomographic image reconstruction from transmission scans this paper describes rapidly converging algorithms for computing attenuation maps from poisson transmission measurements using penalized likelihood objective functions we demonstrate that an under relaxed cyclic coordinate ascent algorithm converges faster than the convex algorithm of lange see ibid vol 4 no 10 p 1430 1438 1995 which in turn converges faster than the expectation maximization em algorithm for transmission tomography to further reduce computation one could replace the log likelihood objective with a quadratic approximation however we show with simulations and analysis that the quadratic objective function leads to biased estimates for low count measurements therefore we introduce hybrid poisson polynomial objective functions that use the exact poisson log likelihood for detector measurements with low counts but use computationally efficient quadratic or cubic approximations for the high count detector measurements we demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias

a framework for bayesian and likelihood approximations in statistics summary regular bayesian and frequentist approximations in statistics are studied within a unified framework in particular it is shown how some higher order likelihood based approximations arise from their bayesian counterparts via an unsmoothing argument this approach serves to shed new light on these formulae and to clarify relationships between bayesian and frequentist inferences for example bayesian interpretations of higher order approximations can give insights into the acceptability or otherwise of these approximations from the point of view of relevance to the actual data observed furthermore this approach is a very natural one for the study of more general nonregular problems models for dependent data and approximate conditional inference for ease of exposition the development is in terms of a single real parameter the main analytic development proceeds in terms of signed roots of log density ratios

monotonic algorithms for transmission tomography we present a framework for designing fast and monotonic algorithms for transmission tomography penalized likelihood image reconstruction the new algorithms are based on paraboloidal surrogate functions for the log likelihood due to the form of the log likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity unlike previous methods the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events such as scatter and random coincidences the gradient and the curvature of the likelihood terms are evaluated only once per iteration since the problem is simplified at each iteration the cpu time is less than that of current algorithms which directly minimize the objective yet the convergence rate is comparable the simplicity monotonicity and speed of the new algorithms are quite attractive the convergence rates of the algorithms are demonstrated using real and simulated pet transmission scans

globally convergent algorithms for maximum a posteriori transmission tomography this paper reviews and compares three maximum likelihood algorithms for transmission tomography one of these algorithms is the em algorithm one is based on a convexity argument devised by de pierro see ieee trans med imaging vol 12 p 328 333 1993 in the context of emission tomography and one is an ad hoc gradient algorithm the algorithms enjoy desirable local and global convergence properties and combine gracefully with bayesian smoothing priors preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the em algorithm this superiority stems from the larger number of exponentiations required by the em algorithm the convex and gradient algorithms are well adapted to parallel computing

convergent incremental optimization transfer algorithms application to tomography no convergent ordered subsets os type image reconstruction algorithms for transmission tomography have been proposed to date in contrast in emission tomography there are two known families of convergent os algorithms methods that use relaxation parameters and methods based on the incremental expectation maximization em approach this paper generalizes the incremental em approach by introducing a general framework incremental optimization transfer the proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters the general optimization transfer framework allows the use of a very broad family of surrogate functions enabling the development of new algorithms this paper provides the first convergent os type algorithm for nonconcave penalized likelihood pl transmission image reconstruction by using separable paraboloidal surrogates sps which yield closed form maximization steps we found it is very effective to achieve fast convergence rates by starting with an os algorithm with a large number of subsets and switching to the new transmission incremental optimization transfer triot algorithm results show that triot is faster in increasing the pl objective than nonincremental ordinary sps and even os sps yet is convergent

on the relation between the isra and the em algorithm for positron emission tomography the image space reconstruction algorithm isra was proposed as a modification of the expectation maximization em algorithm based on physical considerations for application in volume emission computered tomography as a consequence of this modification isra searches for least squares solutions instead of maximizing poisson likelihoods as the em algorithm it is shown that both algorithms may be obtained from a common mathematical framework this fact is used to extend isra for penalized likelihood estimates

majorization minimization algorithms for wavelet based image restoration standard formulations of image signal deconvolution under wavelet based priors regularizers lead to very high dimensional optimization problems involving the following difficulties the non gaussian heavy tailed wavelet priors lead to objective functions which are nonquadratic usually nondifferentiable and sometimes even nonconvex the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet based denoising this paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems placing them in a common majorization minimization mm framework one of the classes of algorithms considered when using quadratic bounds on nondifferentiable log priors shares the infamous singularity issue si of iteratively re weighted least squares irls algorithms the possibility of having to handle infinite weights which may cause both numerical and convergence issues in this paper we prove several new results which strongly support the claim that the si does not compromise the usefulness of this class of algorithms exploiting the unified mm perspective we introduce a new algorithm resulting from using bounds for nonconvex regularizers the experiments confirm the superior performance of this method when compared to the one based on quadratic majorization finally an experimental comparison of the several algorithms reveals their relative merits for different standard types of scenarios

a gradient algorithm locally equivalent to the em algorithm in many problems of maximum likelihood estimation it is impossible to carry out either the e step or the m step of the em algorithm the present paper introduces a gradient algorithm that is closely related to the em algorithm this em gradient algorithm approximately solves the m step of the em algorithm by one iteration of newton s method since newton s method converges quickly the local properties of the em gradient algorithm are almost identical with those of the em algorithm any strict local maximum point of the observed likelihood locally attracts the em and em gradient algorithm at the same rate of convergence and near the maximum point the em gradient algorithm always produces an increase in the likelihood with proper modification the em gradient algorithm also exhibits global convergence properties that are similar to those of the em algorithm our proof of global convergence applies and improves existing theory for the em algorithm these theoretical points are reinforced by a discussion of three realistic examples illustrating how the em gradient algorithm can succeed where the em algorithm is intractable

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

rectification for cone beam projection and backprojection the purpose of this paper is to derive a technique for accelerating the computation of cone beam forward and backward projections that are the basic steps of tomographic reconstruction the cone beam geometry of c arm systems is commonly described with projection matrices such matrices provide a continuous framework for analyzing the flow of operations needed to compute backprojection for analytical reconstruction as well as the combination of forward and backward projections for iterative reconstruction the proposed rectification technique resamples the original data to planes that are aligned with two of the reconstructed volume main axes so that the original cone beam geometry can be replaced by a simpler geometry where succession of plane magnifications are involved only rectification generalizes previous independent results to the cone beam backprojection of preprocessed data as well as to cone beam iterative reconstruction the memory access pattern of simple magnifications provides superior predictability and is therefore easier to optimize independently of the choice of the interpolation technique rectification is also shown to provide control over interpolation errors through oversampling allowing tradeoffs between computation speed and precision to be set experimental results are provided for linear and nearest neighbor interpolations based on simulations as well as phantom and patient data acquired on a digital c arm system

a modified expectation maximization algorithm for penalized likelihood estimation in emission tomography the maximum likelihood ml expectation maximization em approach in emission tomography has been very popular in medical imaging for several years in spite of this no satisfactory convergent modifications have been proposed for the regularized approach here a modification of the em algorithm is presented the new method is a natural extension of the em for maximizing likelihood with concave priors convergence proofs are given

alternating minimization algorithms for transmission tomography a family of alternating minimization algorithms for finding maximum likelihood estimates of attenuation functions in transmission x ray tomography is described the model from which the algorithms are derived includes polyenergetic photon spectra background events and nonideal point spread functions the maximum likelihood image reconstruction problem is reformulated as a double minimization of the i divergence a novel application of the convex decomposition lemma results in an alternating minimization algorithm that monotonically decreases the objective function each step of the minimization is in closed form the family of algorithms includes variations that use ordered subset techniques for increasing the speed of convergence simulations demonstrate the ability to correct the cupping artifact due to beam hardening and the ability to reduce streaking artifacts that arise from beam hardening and background events