a primal dual method for large scale image reconstruction in emission tomography in emission tomography images can be reconstructed from a set of measured projections using a maximum likelihood ml criterion in this paper we present a primal dual algorithm for large scale three dimensional image reconstruction the primal dual method is specialized to the ml reconstruction problem the reconstruction problem is extremely large in several of our data sets the hessian of the objective function is the product of a 1 4 million by 63 million matrix and its scaled transpose as such we consider only approaches that are suitable for large scale parallel computation we apply a stabilization technique to the system of equations for computing the primal direction and demonstrate the need for stabilization when approximately solving the system using an early terminated conjugate gradient iteration we demonstrate that the primal dual method for this problem converges faster than the logarithmic barrier method and considerably faster than the expectation maximization algorithm the use of extrapolation in conjunction with the primal dual method further reduces the overall computation required to achieve convergence

a rapidly convergent iteration method and gâteaux differentiable operators abstract let fr 0 r p be a family of banach spaces satisfying if 0 r1 r2 p i fr1 fr2 ii f r1 f r2 f f r1 and iii r ln f r is a convex function let g0 be a banach space and f be a g teaux differentiate mapping and suppose that f x fp is dense in g0 under appropriate assumptions the equation f x 0 has a solution in fr for 0 r p the results extend the inverse function theorem of j moser to the class of g teaux differentiable operators

list mode likelihood em algorithm and image quality estimation demonstrated on 2 d pet using a theory of list mode maximum likelihood ml source reconstruction presented recently by barrett et al 1997 this paper formulates a corresponding expectation maximization em algorithm as well as a method for estimating noise properties at the ml estimate list mode ml is of interest in cases where the dimensionality of the measurement space impedes a binning of the measurement data it can be advantageous in cases where a better forward model can be obtained by including more measurement coordinates provided by a given detector different figures of merit for the detector performance can be computed from the fisher information matrix fim this paper uses the observed fim which requires a single data set thus avoiding costly ensemble statistics the proposed techniques are demonstrated for an idealized two dimensional 2 d positron emission tomography pet 2 d pet detector the authors compute from simulation data the improved image quality obtained by including the time of flight of the coincident quanta

function variational principles and coercivity abstract the function type extension of ekeland s variational principle j math anal appl 47 1974 324 353 due to zhong nonlinear anal 29 1997 1421 1431 is deductible in a simplified manner and in a larger functional context this is also true for his normed coercivity result based on palais smale techniques

on the distribution of inversive congruential pseudorandom numbers in parts of the period the inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation in this paper we present the first nontrivial bounds on the discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period the proof is based on a new bound for certain incomplete exponential sums

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

the gamma generalized normal distribution a descriptor of sar imagery abstract we propose a new four parameter distribution for modeling synthetic aperture radar sar imagery named the gamma generalized normal ggn by combining the gamma and generalized normal distributions a mathematical characterization of the new distribution is provided by identifying the limiting behavior and by calculating the density and moment expansions the ggn model performance is evaluated on both synthetic and actual data and for that maximum likelihood estimation and random number generation are discussed the proposed distribution is compared with the beta generalized normal bgn distribution which has already shown to appropriately represent sar imagery the performance of these two distributions are measured by means of statistics which provide evidence that the ggn can outperform the bgn distribution in some contexts

iterative image reconstruction algorithms based on cross entropy minimization the related problems of minimizing the functionals f x alpha kl y px 1 alpha kl p x and g x alpha kl px y 1 alpha kl x p respectively over the set of vectors x or 0 are considered kl a b is the cross entropy or kullback leibler distance between two nonnegative vectors a and b iterative algorithms for minimizing both functionals using the method of alternating projections are derived a simultaneous version of the multiplicative algebraic reconstruction technique mart algorithm called smart is introduced and its convergence is proved

block iterative methods for image reconstruction from projections the simultaneous mart algorithm smart and the expectation maximization method for likelihood maximization emml are extended to block iterative versions bi smart and bi emml that converge to a solution in the feasible case for any choice of subsets the bi emml reduces to the ordered subset emml of hudson et al 1992 1994 when their subset balanced property holds

penalized maximum likelihood image reconstruction using space alternating generalized em algorithms most expectation maximization em type algorithms for penalized maximum likelihood image reconstruction converge slowly particularly when one incorporates additive background effects such as scatter random coincidences dark current or cosmic radiation in addition regularizing smoothness penalties or priors introduce parameter coupling rendering intractable the m steps of most em type algorithms this paper presents space alternating generalized em sage algorithms for image reconstruction which update the parameters sequentially using a sequence of small hidden data spaces rather than simultaneously using one large complete data space the sequential update decouples the m step so the maximization can typically be performed analytically we introduce new hidden data spaces that are less informative than the conventional complete data space for poisson data and that yield significant improvements in convergence rate this acceleration is due to statistical considerations not numerical overrelaxation methods so monotonic increases in the objective function are guaranteed we provide a general global convergence proof for sage methods with nonnegativity constraints

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

on the ray walker extension of the caristi kirk fixed point theorem on demontre que le theoreme de ray walker pour des operateurs non lineaires sur des espaces de banach est equivalent au theoreme de caristi kirk

acceleration of maximum likelihood reconstruction using frequency amplification and attenuation compensation algorithms that calculate maximum likelihood ml and maximum a posteriori solutions using expectation maximization have been successfully applied to spect and pet these algorithms are appealing because of their solid theoretical basis and their guaranteed convergence a major drawback is the slow convergence which results in long processing times the authors present 2 new heuristic acceleration methods for maximum likelihood reconstruction of ect images the first method incorporates a frequency dependent amplification in the calculations to compensate for the low pass filtering of the backprojection operation in the second method an amplification factor is incorporated that suppresses the effect of attenuation on the updating factors both methods are compared to the 1 dimensional line search method proposed by lewitt all 3 methods accelerate the ml algorithm on the authors test images lewitt s method produced the strongest acceleration of the three individual methods however the combination of the frequency amplification with the line search method results in a new algorithm with still better performance under certain conditions an effective frequency amplification can be already achieved by skipping some of the calculations required for ml

a row action alternative to the em algorithm for maximizing likelihood in emission tomography the maximum likelihood ml approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography ect since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection fbp algorithm the expectation maximization em algorithm is an often used iterative approach for maximizing the poisson likelihood in ect because of its attractive theoretical and practical properties its major disadvantage is that due to its slow rate of convergence a large amount of computation is often required to achieve an acceptable image here the authors present a row action maximum likelihood algorithm ramla as an alternative to the em algorithm for maximizing the poisson likelihood in ect the authors deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of ramla increase the poisson likelihood in ect at an order of magnitude faster that the standard em algorithm specifically the authors show that from the point of view of measuring total radionuclide uptake in simulated brain phantoms iterations 1 2 3 and 4 of ramla perform at least as well as iterations 45 60 70 and 80 respectively of em moreover the authors show that iterations 1 2 3 and 4 of ramla achieve comparable likelihood values as iterations 45 60 70 and 80 respectively of em the authors also present a modified version of a recent fast ordered subsets em os em algorithm and show that ramla is a special case of this modified os em furthermore the authors show that their modification converges to a ml solution whereas the standard os em does not

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

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

sirt and cg type methods for the iterative solution of sparse linear least squares problems abstract sirt and cg type methods have been successfully employed for the approximate solution of least squares problems arising in tomography in this paper we study and compare their convergence and regularization properties it is pointed out that sirt methods apply an uncontrollable implicit rescaling which affects the statistical characteristics of the system whereas cg type methods do not for a large class of model problems it is shown that virtually the same solutions as obtained by sirt methods can be obtained by applying a cg type method to a properly rescaled system but with an amount of work proportional to the square root of the amount of work with sirt

mapping theorems for gateaux differentiable and accretive operators let p be a nonlinear operator mapping the banach space x into the banach space y there are many approaches to studying solvability of the equation px y for y e y a considerable number of which involve local or infinitesimal assumptions on the operator p this is due in part to the fact that p frequently arises from a differential or integral operator which has monotonicity or smoothness properties for which local apriori estimates are readily obtainable in this paper we derive mapping theorems for operators p satisfying two different kinds of local assumptions differentiability assumptions and monotonicity assumptions our approach in the former case relies on an abstract newton kantorovich scheme which was pioneered by m altman see for example l and further developed in 19 in the latter case we extend some results of f e browder using a combination of both new and standard techniques in each case our basic technique involves an application of a new version of ekeland s theorem which we derive from a fundamental maximal principle of h brezis and f e browder in section 2 of this paper we derive our new version of ekeland s theorem and in section 3 apply this result to nonlinear operators using different and by now standard techniques we prove a domain invariance result in section 4 needed for our result an accretive operators

a fast bayesian reconstruction algorithm for emission tomography with entropy prior converging to feasible images the development and tests of an iterative reconstruction algorithm for emission tomography based on bayesian statistical concepts are described the algorithm uses the entropy of the generated image as a prior distribution can be accelerated by the choice of an exponent and converges uniformly to feasible images by the choice of one adjustable parameter a feasible image has been defined as one that is consistent with the initial data i e it is an image that if truly a source of radiation in a patient could have generated the initial data by the poisson process that governs radioactive disintegration the fundamental ideas of bayesian reconstruction are discussed along with the use of an entropy prior with an adjustable contrast parameter the use of likelihood with data increment parameters as conditional probability and the development of the new fast maximum a posteriori with entropy fmape algorithm by the successive substitution method it is shown that in the maximum likelihood estimator mle and fmape algorithms the only correct choice of initial image for the iterative procedure in the absence of a priori knowledge about the image configuration is a uniform field

joint map bayesian tomographic reconstruction with a gamma mixture prior we address the problem of bayesian image reconstruction with a prior that captures the notion of a clustered intensity histogram the problem is formulated in the framework of a joint map maximum a posteriori estimation with the prior pdf modeled as a mixture of gammas density this prior pdf has appealing properties including positivity enforcement the joint map optimization is carried out as an iterative alternating descent wherein a regularized likelihood estimate is followed by a mixture decomposition of the histogram of the current tomographic image estimate the mixture decomposition step estimates the hyperparameters of the prior pdf the objective functions associated with the joint map estimation are complicated and difficult to optimize but we show how they may be transformed to allow for much easier optimization while preserving the fixed point of the iterations we demonstrate the method in the context of medical emission and transmission tomography

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

on use of the em algorithm for penalized likelihood estimation summary the em algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation this paper discusses properties of the em algorithm in such contexts concentrating on rates of conver gence and presents an alternative that is usually more practical and converges at least as quickly the em algorithm is a general approach to maximum likelihood estimation rather than a specific algorithm dempster et al 1977 discussed the method and derived basic properties demonstrating that a variety of procedures previously developed rather informally could be unified the common strand to problems where the approach is applicable is a notion of incomplete data this includes the conventional sense of missing data but is much broader than that the em algorithm demon strates its strength in situations where some hypothetical experiment yields data from which estimation is particularly convenient and economical the incomplete data actually at hand are regarded as observable functions of these complete data the resulting algorithms while usually slow to converge are often extremely simple and remain practical in large problems where no other approaches may be feasible dempster et al 1977 briefly refer to the use of the same approach to the problem of finding the posterior mode maximum a posteriori estimate in a bayesian estima

a theoretical study of the contrast recovery and variance of map reconstructions from pet data the authors examine the spatial resolution and variance properties of pet images reconstructed using maximum a posteriori map or penalized likelihood methods resolution is characterized by the contrast recovery coefficient crc of the local impulse response simplified approximate expressions are derived for the local impulse response crcs and variances for each voxel using these results the authors propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local crc to noise ratio in the reconstructed image

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

operator theoretic and computational approaches to ill posed problems with applications to antenna theory a general framework for regularization and approximation methods for ill posed problems is developed three levels in the resolution processes are distinguished and emphasized philosophy of resolution regularization approximation schema and regularization algorithms dilemmas and methodologies of resolution of ill posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted norm least squares solutions of first kind integral equations and more generally of linear operator equations with nonclosed range a common problem in all these methods is delineated each method reduces the problem of resolution to a nonstandard minimization problem involving an unknown critical parameter whose optimal value is crucial to the numerical realization and amenability of the method the nonstandardness results from the fact that one does not have explicitly or a priori the function to be minimized it has to built up using additional information convergence rate estimates and robustness conditions etc several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill posed identification and pattern synthesis problems an emphasis is placed on the role of constraints function space methods the role of generalized inverses and reproducing kernels in the regularization and stable computational resolution of these problems the results will be applied specifically to problems of antenna synthesis and identification however the thrust of the paper is devoted to the interdisciplinary character of operator theoretic and numerical methods for ill posed problems

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

on bayesian image reconstruction from projections uniform and nonuniform a priori source information a method that incorporates a priori uniform or nonuniform source distribution probabilistic information and data fluctuations of a poisson nature is presented the source distributions are modeled in terms of a priori source probability density functions maximum a posteriori probability solutions as determined by a system of equations are given interactive bayesian imaging algorithms for the solutions are derived using an expectation maximization technique comparisons of the a priori uniform and nonuniform bayesian algorithms to the maximum likelihood algorithm are carried out using computer generated noise free and poisson randomized projections improvement in image reconstruction from projections with the bayesian algorithm is demonstrated superior results are obtained using the a priori nonuniform source distribution

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

estimation of discrete distributions with a class of simplex constraints abstract simplex constraints such as monotonicity and convexity or concavity on the probabilities of a set of discrete distributions are useful for modeling and analyzing discrete data this article considers both maximum likelihood estimation and bayesian estimation of discrete distribution with a class of simplex constraints using the expectation maximization em algorithm and the data augmentation da algorithm the formulation and implementation of em and da for binomial poisson hierarchical poisson binomial multinomial and hierarchical multinomial distributions are considered in detail and illustrated with examples

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

improved iterative image reconstruction with automatic noise artifact suppression a method for stabilizing iterative image reconstruction techniques has been developed for improving the image quality of position emission tomography a damping matrix is introduced which suppresses noisy correction on a pixel by pixel basis depending on the statistical precision of the iterative correction the precision is evaluated by comparing a certain number of correction submatrices each of which is formed from a subset of the projection data simulation studies showed that statistical noise is effectively suppressed while the image of the source object is reconstructed with high resolution as long as the signal level is higher than the local noise level in the application to the mle maximum likelihood estimator the minimum rms error of the image was reduced to 84 for 500 k total counts and the rms error increased more slowly with further iterations as compared with the simple mle the method was also applied to the fir filtered iterative reconstruction algorithm and the images were found to be better than those obtained by the convolution backprojection method

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 multigrid expectation maximization reconstruction algorithm for positron emission tomography the problem of reconstruction in positron emission tomography pet is basically estimating the number of photon pairs emitted from the source using the concept of the maximum likelihood ml algorithm the problem of reconstruction is reduced to determining an estimate of the emitter density that maximizes the probability of observing the actual detector count data over all possible emitter density distributions a solution using this type of expectation maximization em algorithm with a fixed grid size is severely handicapped by the slow convergence rate the large computation time and the nonuniform correction efficiency of each iteration which makes the algorithm very sensitive to the image pattern an efficient knowledge based multigrid reconstruction algorithm based on the ml approach is presented to overcome these problems

developments with maximum likelihood x ray computed tomography an approach to the maximum likelihood estimation of attenuation coefficients in transmission tomography is presented as an extension of earlier theoretical work by k lange and r carson j comput assist tomography vol 8 p 306 16 1984 the reconstruction algorithm is based on the expectation maximization em algorithm several simplifying approximations are introduced which make the maximization step of the algorithm available computer simulations are presented using noise free and poisson randomized projections the images obtained with the em type method are compared to those reconstructed with the em method of lange and carson and with filtered backprojection preliminary results show that there are potential advantages in using the maximum likelihood approaches in situations where a high contrast object such as bone is embedded in low contrast soft tissue

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

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

a statistical multiscale framework for poisson inverse problems this paper describes a statistical multiscale modeling and analysis framework for linear inverse problems involving poisson data the framework itself is founded upon a multiscale analysis associated with recursive partitioning of the underlying intensity a corresponding multiscale factorization of the likelihood induced by this analysis and a choice of prior probability distribution made to match this factorization by modeling the splits in the underlying partition the class of priors used here has the interesting feature that the noninformative member yields the traditional maximum likelihood solution other choices are made to reflect prior belief as to the smoothness of the unknown intensity adopting the expectation maximization em algorithm for use in computing the maximum a posteriori map estimate corresponding to our model we find that our model permits remarkably simple closed form expressions for the em update equations the behavior of our em algorithm is examined and it is shown that convergence to the global map estimate can be guaranteed applications in emission computed tomography and astronomical energy spectral analysis demonstrate the potential of the new approach

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

synthetic magnetic resonance imaging revisited synthetic magnetic resonance mr imaging is an approach suggested in the literature to predict mr images at different design parameter settings from at least three observed mr scans however performance is poor when no regularization is used in the estimation and otherwise computationally impractical to implement for 3 d imaging methods we propose a method which accounts for spatial context in mr images by the imposition of a gaussian markov random field mrf structure on a transformation of the spin lattice relaxation time the spin spin relaxation time and the proton density at each voxel the mrf structure is specified through a matrix normal distribution we also model the observed magnitude images using the more accurate but computationally challenging rice distribution a one step late expectation maximization approach is adopted to make our approach computationally practical we evaluate predictive performance in generating synthetic mr images in a clinical setting our results indicate that our suggested approach is not only computationally feasible to implement but also shows excellent performance

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

accelerating the emml algorithm and related iterative algorithms by rescaled block iterative methods analysis of convergence of the algebraic reconstruction technique art shows it to be predisposed to converge to a solution faster than simultaneous methods such as those of the cimmino landweber 1992 type the expectation maximization maximum likelihood method for the poisson model emml and the simultaneous multiplicative art smart which use all the data at each step although the choice of ordering of the data and of relaxation parameters are important as herman and meyer 1993 have shown they are not the full story the analogous multiplicative art mart which applies only to systems y px in which y 0 p spl ges 0 and a nonnegative solution is sought is also sequential or row action rather than simultaneous but does not generally exhibit the same accelerated convergence relative to its simultaneous version smart by dividing each equation by the maximum of the corresponding row of p we find that this rescaled mart rmart does converge faster when solutions exist significantly so in cases in which the row maxima are substantially less than one such cases arise frequently in tomography and when the columns of p have been normalized to have sum one between simultaneous methods which use all the data at each step and sequential or row action methods which use only a single data value at each step there are the block iterative or ordered subset methods in which a single block or subset of the data is processed at each step the ordered subset em osem of hudson et al see ieee trans med imag vol 13 p 601 9 1994 is significantly faster than the emml but often fails to converge the rescaled block iterative emml rbi emml is an accelerated block iterative version of emml that converges in the consistent case to a solution for any choice of subsets it reduces to osem when the restrictive subset balanced condition holds rescaled block iterative versions of smart and mart also exhibit accelerated convergence

a smoothed em approach to indirect estimation problems with particular reference to stereology and emission tomography there are many practical problems where the observed data are not drawn directly from the density g of real interest but rather from another distribution derived from g by the application of an integral operator the estimation of g then entails both statistical and numerical difficulties a natural statistical approach is by maximum likelihood conveniently implemented using the em algorithm but this provides unsatisfactory reconstructions of g in this paper we modify the maximum likelihood em approach by introducing a simple smoothing step at each em iteration in our experience this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration some theoretical background is given that relates this smoothed em algorithm to a maximum penalized likelihood approach two applications are considered in detail the first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium the second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography for this problem we also suggest several technical improvements on existing methodology

numerical study of multigrid implementations of some iterative image reconstruction algorithms the numerical behavior of multigrid implementations of the landweber generalized landweber art and mlem iterative image reconstruction algorithms is investigated comparisons between these algorithms and with their single grid implementations are made on two small scale synthetic pet systems for phantom objects exhibiting different characteristics and on one full scale synthetic system for a shepp logan phantom the authors also show analytically the effects of noise and initial condition on the generalized landweber iteration and note how to choose the shaping operator to filter out noise in the data or to enhance features of interest in the reconstructed image original contributions include 1 numerical studies of the convergence rates of single grid and multigrid implementations of the landweber generalized landweber art and mlem iterations and 2 effects of noise and initial condition on the generalized landweber iteration with procedures for filtering out noise or enhancing image features

on the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators in this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degree growth of these iterations we use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation

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

a unified approach to statistical tomography using coordinate descent optimization over the past years there has been considerable interest in statistically optimal reconstruction of cross sectional images from tomographic data in particular a variety of such algorithms have been proposed for maximum a posteriori map reconstruction from emission tomographic data while map estimation requires the solution of an optimization problem most existing reconstruction algorithms take an indirect approach based on the expectation maximization em algorithm we propose a new approach to statistically optimal image reconstruction based on direct optimization of the map criterion the key to this direct optimization approach is greedy pixel wise computations known as iterative coordinate decent icd we propose a novel method for computing the icd updates which we call icd newton raphson we show that icd newton raphson requires approximately the same amount of computation per iteration as em based approaches but the new method converges much more rapidly in our experiments typically five to ten iterations other advantages of the icd newton raphson method are that it is easily applied to map estimation of transmission tomograms and typical convex constraints such as positivity are easily incorporated

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

interior point methodology for 3 d pet reconstruction interior point methods have been successfully applied to a wide variety of linear and nonlinear programming applications this paper presents a class of algorithms based on path following interior point methodology for performing regularized maximum likelihood ml reconstructions on three dimensional 3 d emission tomography data the algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region the nonnegative orthant the authors propose two methods a primal method which updates only the primal image variables and a primal dual method which simultaneously updates the primal variables and the lagrange multipliers a parallel implementation permits the interior point methods to scale to very large reconstruction problems termination is based on well defined convergence measures namely the karush kuhn tucker first order necessary conditions for optimality the authors demonstrate the rapid convergence of the path following interior point methods using both data from a small animal scanner and monte carlo simulated data the proposed methods can readily be applied to solve the regularized weighted least squares reconstruction problem

the drop theorem the petal theorem and ekeland s variational principle on montre que le principe variationnel d ekeland ou plus precisement une forme legerement modifiee est une consequence d un resultat geometrique connu sous le terme de theoreme de la goutte

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

fixed point theorems for a broad class of multimaps abstract multimaps from a metric space into the collection of all its nonempty closed subsets are considered and fixed point theorems for such maps have been obtained under new contractive conditions not involving the extended hausdorff metric

acceleration of landweber type algorithms by suppression of projection on the maximum singular vector a procedure that speeds up convergence during the initial stage the first 100 forward and backward projections of landweber type algorithms for iterative image reconstruction for positron emission tomography pet which include the landweber generalized landweber and steepest descent algorithms is discussed the procedure first identifies the singular vector associated with the maximum singular value of the pet system matrix and then suppresses projection of the data on this singular vector after a single landweber iteration it is shown that typical pet system matrices have a significant gap between their two largest singular values hence this suppression allows larger gains in subsequent iterations speeding up convergence by roughly a factor of three

vector extrapolated fast maximum likelihood estimation algorithms for emission tomography a new class of fast maximum likelihood estimation mle algorithms for emission computed tomography ect is developed in these cyclic iterative algorithms vector extrapolation techniques are integrated with the iterations in gradient based mle algorithms with the objective of accelerating the convergence of the base iterations this results in a substantial reduction in the effective number of base iterations required for obtaining an emission density estimate of specified quality the mathematical theory behind the minimal polynomial and reduced rank vector extrapolation techniques in the context of emission tomography is presented these extrapolation techniques are implemented in a positron emission tomography system the new algorithms are evaluated using computer experiments with measurements taken from simulated phantoms it is shown that with minimal additional computations the proposed approach results in substantial improvement in reconstruction

improved image quality and computation reduction in 4 d reconstruction of cardiac gated spect images spatiotemporal reconstruction of cardiac gated spect images permits one to obtain valuable information related to cardiac function however the task of reconstructing this four dimensional 4 d data set is computation intensive typically these studies are reconstructed frame by frame a nonoptimal approach because temporal correlations in the signal are not accounted for in this work the authors show that the compression and signal decorrelation properties of the karhunen loeve kl transform may be used to greatly simplify the spatiotemporal reconstruction problem the gated projections are first kl transformed in the temporal direction this results in a sequence of kl transformed projection images for which the signal components are uncorrelated along the time axis as a result the 4 d reconstruction task is simplified to a series of three dimensional 3 d reconstructions in the kl domain the reconstructed kl components are subsequently inverse kl transformed to obtain the entire spatiotemporal reconstruction set the authors simulation and clinical results indicate that kl processing provides image sequences that are less noisy than are conventional frame by frame reconstructions additionally by discarding high order kl components that are dominated by noise one can achieve savings in computation time because fewer reconstructions are needed in comparison to conventional frame by frame reconstructions

fast gradient based methods for bayesian reconstruction of transmission and emission pet images the authors describe conjugate gradient algorithms for reconstruction of transmission and emission pet images the reconstructions are based on a bayesian formulation where the data are modeled as a collection of independent poisson random variables and the image is modeled using a markov random field a conjugate gradient algorithm is used to compute a maximum a posteriori map estimate of the image by maximizing over the posterior density to ensure nonnegativity of the solution a penalty function is used to convert the problem to one of unconstrained optimization preconditioners are used to enhance convergence rates these methods generally achieve effective convergence in 15 25 iterations reconstructions are presented of an sup 18 fdg whole body scan from data collected using a siemens cti ecat931 whole body system these results indicate significant improvements in emission image quality using the bayesian approach in comparison to filtered backprojection particularly when reprojections of the map transmission image are used in place of the standard attenuation correction factors

parallelizable bayesian tomography algorithms with rapid guaranteed convergence bayesian tomographic reconstruction algorithms generally require the efficient optimization of a functional of many variables in this setting as well as in many other optimization tasks functional substitution fs has been widely applied to simplify each step of the iterative process the function to be minimized is replaced locally by an approximation having a more easily manipulated form e g quadratic but which maintains sufficient similarity to descend the true functional while computing only the substitute we provide two new applications of fs methods in iterative coordinate descent for bayesian tomography the first is a modification of our coordinate descent algorithm with one dimensional 1 d newton raphson approximations to an alternative quadratic which allows convergence to be proven easily in simulations we find essentially no difference in convergence speed between the two techniques we also present a new algorithm which exploits the fs method to allow parallel updates of arbitrary sets of pixels using computations similar to iterative coordinate descent the theoretical potential speed up of parallel implementations is nearly linear with the number of processors if communication costs are neglected

constrained iterative reconstruction by the conjugate gradient method the conjugate gradient method incorporating the object extent constraint is applied to image reconstruction of a three dimensional object using an incomplete projection data set the missing information is recovered by constraining the solution with the knowledge of the outer boundary of the object extent which may be a priori measured or known the algorithm is derived from the least squares criterion as an advanced version of conventional iterative reconstruction algorithms such as sirt simultaneous iterative reconstruction technique and ilst iterative least squares technique in the case of reconstruction from noisy projection data a method based on the minimum mean square error criterion is also proposed computer simulated reconstruction images of a phantom using limited angle and number of views are presented the result shows that the conjugate gradient method incorporating the object extent constraining provides the fastest convergence and the least error

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

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

accelerated iterative reconstruction for positron emission tomography based on the em algorithm for maximum likelihood estimation the em method that was originally developed for maximum likelihood estimation in the context of mathematical statistics may be applied to a stochastic model of positron emission tomography pet the result is an iterative algorithm for image reconstruction that is finding increasing use in pet due to its attractive theoretical and practical properties its major disadvantage is the large amount of computation that is often required due to the algorithm s slow rate of convergence this paper presents an accelerated form of the em algorithm for pet in which the changes to the image as calculated by the standard algorithm are multiplied at each iteration by an overrelaxation parameter the accelerated algorithm retains two of the important practical properties of the standard algorithm namely the selfnormalization and nonnegativity of the reconstructed images experimental results are presented using measured data obtained from a hexagonal detector system for pet the likelihood function and the norm of the data residual were monitored during the iterative process according to both of these measures the images reconstructed at iterations 7 and 11 of the accelerated algorithm are similar to those at iterations 15 and 30 of the standard algorithm for two different sets of data important theoretical properties remain to be investigated namely the convergence of the accelerated algorithm and its performance as a maximum likelihood estimator

likelihood maximization for list mode emission tomographic image reconstruction the maximum a posteriori map bayesian iterative algorithm using priors that are gamma distributed due to lange bahn and little is extended to include parameter choices that fall outside the gamma distribution model special cases of the resulting iterative method include the expectation maximization maximum likelihood emml method based on the poisson model in emission tomography as well as algorithms obtained by parra and barrett and by huesman et al that converge to maximum likelihood and maximum conditional likelihood estimates of radionuclide intensities for list mode emission tomography the approach taken here is optimization theoretic and does not rely on the usual expectation maximization em formalism block iterative variants of the algorithms are presented a self contained elementary proof of convergence of the algorithm is included