2 5 d simultaneous multislice reconstruction by series expansion methods from fourier rebinned pet data true three dimensional 3 d volume reconstruction from fully 3 d data in positron emission tomography pet has only a limited clinical use because of its large computational burden fourier rebinning fore of the fully 3 d data into a set of 2 d sinogram data decomposes the 3 d reconstruction process into multiple 2 d reconstructions of decoupled 2 d image slices thus substantially decreasing the computational burden even in the case when the 2 d reconstructions are performed by an iterative reconstruction algorithm on the other hand the approximations involved in the rebinning combined with the decoupling of the image slices cause a certain reduction of image quality especially when the signal to noise ratio of the data is low the authors propose a 2 5 d simultaneous multislice reconstruction approach based on the series expansion principle where the volume is represented by the superposition of 3 d spherically symmetric bell shaped basis functions it takes advantage of the time reduction due to the use of the fore 2 d data instead of the original fully 3 d data but at the same time uses a 3 d iterative reconstruction approach with 3 d basis functions the same general approach can be applied to any reconstruction algorithm belonging to the class of series expansion methods iterative or noniterative using 3 d basis functions that span multiple slices and can be used for any multislice sinogram or list mode data whether obtained by a special rebinning scheme or acquired directly by a pet scanner in the 2 d mode using septa the authors studies confirm that the proposed 2 5 d approach provides a considerable improvement in reconstruction quality as compared to the standard 2 d reconstruction approach while the reconstruction time is of the same order as that of the 2 d approach and is clinically practical even on a general purpose computer

some new multiplicative algorithms for image reconstruction from projections abstract new iterative algorithms qmart 1 and qmart 2 are proposed to solve large systems of linear equations the general step of these algorithms results from a quadratic approximation to bregman s method of taking successive entropy projections onto the linear constraints the algorithms qmart 1 and qmart 2 are shown to be linear approximations to the multiplicative algebraic reconstruction techniques mart a geometric interpretation of the algorithms is provided to compare their behavior experimental results show that qmart 1 and qmart 2 have better image processing performance than mart but they require more iterations

fast maximum entropy approximation in spect using the rbi map algorithm presents a method for approximating constrained maximum entropy me reconstructions of spect data with modifications to a block iterative maximum a posteriori map algorithm maximum likelihood ml based reconstruction algorithms require some form of noise smoothing constrained me provides a more formal method of noise smoothing without requiring the user to select parameters in the context of spect constrained me seeks the minimum information image estimate among those whose projections are a given distance from the noisy measured data with that distance determined by the magnitude of the poisson noise images that meet the distance criterion are referred to as feasible images the authors find that modeling of all principal degrading factors attenuation detector response and scatter in the reconstruction is critical because feasibility is not meaningful unless the projection model is as accurate as possible because the constrained me solution is the same as a map solution for a particular value of the map weighting parameter spl beta the constrained me solution can be found with a map algorithm if the correct value of spl beta is found the authors show that the rbi map algorithm if used with a dynamic scheme for estimating spl beta can approximate constrained me solutions in 20 or fewer iterations the authors compare results for various methods of achieving feasible images on a simulation of tl 201 cardiac spect data results show that the rbi map me approximation provides images and quantitative estimates close to those from a slower algorithm that gives the true me solution also the authors find that the me results have higher spatial resolution and greater high frequency noise content than a feasibility based stopping rule feasibility based low pass filtering and a quadratic gibbs prior with spl beta selected according to the feasibility criterion the authors conclude that fast me approximation is possible using either rbi map with the dynamic procedure or a feasibility based stopping rule and that such reconstructions may be particularly useful in applications where resolution is critical

augmented is lm model based on particular functions abstract in this paper we create our own static and dynamic augmented is lm model based on concrete particular functions these functions represent a mathematical and analytical description of economical quantities on which the is lm model is based and whose properties follow from the corresponding economical theories we prove several theorems about parameters of this model we describe dynamical behaviour of this system and prove un stability of the singular points including their classification then we demonstrate this model on some illustrative examples and we suggest our own econometric non linear regression models of the particular functions courses

convergence studies on iterative algorithms for image reconstruction we introduce a general iterative scheme for image reconstruction based on landweber s method in our configuration a sequential block iterative seqbi version can be readily formulated from a simultaneous block iterative simbi version and vice versa this provides a mechanism to derive new algorithms from known ones it is shown that some widely used iterative algorithms such as the algebraic reconstruction technique art simultaneous art sart cimmino s and the recently designed diagonal weighting and component averaging algorithms are special examples of the general scheme we prove convergence of the general scheme under conditions more general than assumed in earlier studies for its seqbi and simbi versions in the consistent and inconsistent cases respectively our results suggest automatic relaxation strategies for the seqbi and simbi versions and characterize the dependence of the limit image on the initial guess it is found that in all cases the limit is the sum of the minimum norm solution of a weighted least squares problem and an oblique projection of the initial image onto the null space of the system matrix

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

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

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

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

a stochastic dynamic model to evaluate the influence of economy and well being on unemployment control abstract this paper presents a stochastic dynamic mathematical model to study the evolution of the unemployment rate and other relevant related variables in a country this model is composed by three basic interrelated subsystems demographic economic and well being ones a key aspect of this model is that it considers three un well being variables simultaneously human development index gender empowerment index and gender differentiation index these variables involve key concepts for human development as health education economy and female labor with this model the most outstanding variables found in the literature in relation with unemployment control can be used to design strategies and scenarios to reduce the unemployment rate in the future the model has been fitted for the case of spain in the 2002 2014 period the largest one with information about all the variables involved in the model finally several tentative scenarios and strategies have been tested to reduce the unemployment rate in spain in the horizon of year 2025 and the corresponding forecast evolution of the gender differentiation index national income per capita public debt and the ratio between public debt and gross domestic product are shown

globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms we present two types of globally convergent relaxed ordered subsets os algorithms for penalized likelihood image reconstruction in emission tomography modified block sequential regularized expectation maximization bsrem and relaxed os separable paraboloidal surrogates os sps the global convergence proof of the existing bsrem de pierro and yamagishi 2001 required a few a posteriori assumptions by modifying the scaling functions of bsrem we are able to prove the convergence of the modified bsrem under realistic assumptions our modification also makes stepsize selection more convenient in addition we introduce relaxation into the os sps algorithm erdogan and fessler 1999 that otherwise would converge to a limit cycle we prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed os sps is a special case main results of the proofs are from nedic and bertsekas 2001 and correa and lemarechal 1993 simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms

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

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 fast fully 4 d incremental gradient reconstruction algorithm for list mode pet data we describe a fast and globally convergent fully four dimensional incremental gradient 4dig algorithm to estimate the continuous time tracer density from list mode positron emission tomography pet data detection of 511 kev photon pairs produced by positron electron annihilation is modeled as an inhomogeneous poisson process whose rate function is parameterized using cubic b splines the rate functions are estimated by minimizing the cost function formed by the sum of the negative log likelihood of arrival times spatial and temporal roughness penalties and a negativity penalty we first derive a computable bound for the norm of the optimal temporal basis function coefficients based on this bound we then construct and prove convergence of an incremental gradient algorithm fully 4 d simulations demonstrate the substantially faster convergence behavior of the 4dig algorithm relative to preconditioned conjugate gradient four dimensional reconstructions of real data are also included to illustrate the performance of this method

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

a dynamic is lm business cycle model with two time delays in capital accumulation equation in this paper we analyze a augmented is lm business cycle model with the capital accumulation equation that two time delays are considered in investment processes according to kalecki s idea applying stability switch criteria and hopf bifurcation theory we prove that time delays cause the equilibrium to lose or gain stability and hopf bifurcation occurs

a general framework for nonlinear multigrid inversion a variety of new imaging modalities such as optical diffusion tomography require the inversion of a forward problem that is modeled by the solution to a three dimensional 3d partial differential equation for these applications image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems the multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems the method works by dynamically adjusting the cost functionals at different scales so that they are consistent with and ultimately reduce the finest scale cost functional in this way the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem importantly the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions an application of our method to bayesian optical diffusion tomography with a generalized gaussian markov random field image prior model shows the potential for very large computational savings numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem

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

fixed points imply chaos for a class of differential inclusions that arise in economic models we consider multi valued dynamical systems with continuous time of the form f x where f x is a set valued function such models have been studied recently in mathematical economics we provide a definition for chaos chaos and topological entropy for these differential inclusions that is in terms of the natural r action on the space of all solutions of the model by considering this more complicated topological space and its r action we show that chaos is the typical behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic chaotic and has infinite topological entropy

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

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

accelerated image reconstruction using ordered subsets of projection data the authors define ordered subset processing for standard algorithms such as expectation maximization em for image restoration from projections ordered subsets methods group projection data into an ordered sequence of subsets or blocks an iteration of ordered subsets em is defined as a single pass through all the subsets in each subset using the current estimate to initialize application of em with that data subset this approach is similar in concept to block kaczmarz methods introduced by eggermont et al 1981 for iterative reconstruction simultaneous iterative reconstruction sirt and multiplicative algebraic reconstruction mart techniques are well known special cases ordered subsets em os em provides a restoration imposing a natural positivity condition and with close links to the em algorithm os em is applicable in both single photon spect and positron emission tomography pet in simulation studies in spect the os em algorithm provides an order of magnitude acceleration over em with restoration quality maintained

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

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

analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay in this paper a delayed fractional order financial system is proposed and the complex dynamical behaviors of such a system are discussed by numerical simulations a great variety of interesting dynamical behaviors of such a system including single periodic multiple periodic and chaotic motions are displayed in particular the effect of time delay on the chaotic behavior is investigated it is found that an approximate time delay can enhance or suppress the emergence of chaos meanwhile corresponding to different values of delay the lowest orders for chaos to exist in the delayed fractional order financial systems are determined respectively

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

information theoretic image formation the emergent role of information theory in image formation is surveyed unlike the subject of information theoretic communication theory information theoretic imaging is far from a mature subject the possible role of information theory in problems of image formation is to provide a rigorous framework for defining the imaging problem for defining measures of optimality used to form estimates of images for addressing issues associated with the development of algorithms based on these optimality criteria and for quantifying the quality of the approximations the definition of the imaging problem consists of an appropriate model for the data and an appropriate model for the reproduction space which is the space within which image estimates take values each problem statement has an associated optimality criterion that measures the overall quality of an estimate the optimality criteria include maximizing the likelihood function and minimizing mean squared error for stochastic problems and minimizing squared error and discrimination for deterministic problems the development of algorithms is closely tied to the definition of the imaging problem and the associated optimality criterion algorithms with a strong information theoretic motivation are obtained by the method of expectation maximization related alternating minimization algorithms are discussed in quantifying the quality of approximations global and local measures are discussed global measures include the mean squared error and discrimination between an estimate and the truth and probability of error for recognition or hypothesis testing problems local measures include fisher information

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

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

feasible images and practical stopping rules for iterative algorithms in emission tomography the discussion of the causes of image deterioration in the maximum likelihood estimator mle method of tomographic image reconstruction initiated with the publication of a stopping rule for that iterative process e veklerov and j llacer 1987 is continued the concept of a feasible image is introduced which is a result of a reconstruction that if it were a radiation field could have generated the initial projection data by the poisson process that governs radioactive decay from the premise that the result of a reconstruction should be feasible the shape and characteristics of the region of feasibility in projection space are examined with a new rule reconstructions from real data can be tested for feasibility results of the tests and reconstructed images for the hoffman brain phantom are shown a comparative examination of the current methods of dealing with mle image deterioration is included

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

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

strong underrelaxation in kaczmarz s method for inconsistent systems we investigate the behavior of kaczmarz s method with relaxation for inconsistent systems we show that when the relaxation parameter goes to zero the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system this point minimizes the sum of the squares of the euclidean distances to the hyperplanes of the system if the starting point is chosen properly then the limits approach the minimum norm weighted least squares solution the proof is given for a block kaczmarz method

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

space alternating generalized expectation maximization algorithm the expectation maximization em method can facilitate maximizing likelihood functions that arise in statistical estimation problems in the classical em paradigm one iteratively maximizes the conditional log likelihood of a single unobservable complete data space rather than maximizing the intractable likelihood function for the measured or incomplete data em algorithms update all parameters simultaneously which has two drawbacks 1 slow convergence and 2 difficult maximization steps due to coupling when smoothness penalties are used the paper describes the space alternating generalized em sage method which updates the parameters sequentially by alternating between several small hidden data spaces defined by the algorithm designer the authors prove that the sequence of estimates monotonically increases the penalized likelihood objective derive asymptotic convergence rates and provide sufficient conditions for monotone convergence in norm two signal processing applications illustrate the method estimation of superimposed signals in gaussian noise and image reconstruction from poisson measurements in both applications the sage algorithms easily accommodate smoothness penalties and converge faster than the em algorithms

event by event image reconstruction from list mode pet data this paper adapts the classical list mode osem and the globally convergent list mode cosem methods to the special case of singleton subsets the image estimate is incrementally updated for each coincidence event measured by the pet scanner events are used as soon as possible to improve the current image estimate and therefore the convergence speed toward the maximum likelihood solution is accelerated an alternative online formulation of the list mode cosem algorithm is proposed first this method saves memory resources by re computing previous incremental image contributions while processing a new pass over the complete dataset this online expectation maximization principle is applied to the list mode osem method as well image reconstructions have been performed from a simulated dataset for the ncat torso phantom and from a clinical dataset results of the classical and event by event list mode algorithms are discussed in a systematic and quantitative way

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

choosing parameters in block iterative or ordered subset reconstruction algorithms viewed abstractly all the algorithms considered here are designed to provide a nonnegative solution x to the system of linear equations y px where y is a vector with positive entries and p a matrix whose entries are nonnegative and with no purely zero columns the expectation maximization maximum likelihood method as it occurs in emission tomography and the simultaneous multiplicative algebraic reconstruction technique are slow to converge on large data sets accelerating convergence through the use of block iterative or ordered subset versions of these algorithms is a topic of considerable interest these block iterative versions involve relaxation and normalization parameters the correct selection of which may not be obvious to all users the algorithms are not faster merely by virtue of being block iterative the correct choice of the parameters is crucial through a detailed discussion of the theoretical foundations of these methods we come to a better understanding of the precise roles these parameters play

a maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography the expectation maximization method for maximum likelihood image reconstruction in emission tomography based on the poisson distribution of the statistically independent components of the image and measurement vectors is extended to a maximum aposteriori image reconstruction using a multivariate gaussian a priori probability distribution of the image vector the approach is equivalent to a penalized maximum likelihood estimation with a special choice of the penalty function the expectation maximization method is applied to find the a posteriori probability maximizer a simple iterative formula is derived for a penalty function that is a weighted sum of the squared deviations of image vector components from their a priori mean values the method is demonstrated to be superior to pure likelihood maximization in that the penalty function prevents the occurrence of irregular high amplitude patterns in the image with a large number of iterations the so called checkerboard effect or noise artifact

existence of chaos in the plane r 2 and its application in macroeconomics the existence of devaney li yorke and distributional chaos in the plane r 2 the chaos in a continuous dynamical system generated by euler equation branching all combinations of hyperbolic singular points of both branches admit chaos the chaotic set of solutions associated with chaotic subset of r 2 exists new overall macroeconomic equilibrium model called is lm qy ml model the existence of chaos in an economy the devaney li yorke and distributional chaos in the plane r 2 can occur in the continuous dynamical system generated by euler equation branching euler equation branching is a type of differential inclusion x f x g x where f g x r n r n are continuous and f x g x in every point x x stockman and raines 2010 defined the so called chaotic set in the plane r 2 whose existence leads to the existence of devaney li yorke and distributional chaos in this paper we follow up on stockman and raines 2010 and we show that chaos in the plane r 2 is always admitted for hyperbolic singular points in both branches not lying in the same point in r 2 but the chaos existence is also caused by a set of solutions of euler equation branching we research this set of solutions in the second part we create the new overall macroeconomic equilibrium model called is lm qy ml model this model is based on the fundamental macroeconomic equilibrium model called is lm model we model an inflation effect an endogenous money supply an economic cycle etc in addition to the original is lm model we research the dynamical behaviour of this new is lm qy ml model and show when a chaos exists with relevant economic interpretation

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

convergent block iterative algorithms for image reconstruction from inconsistent data it has been shown that convergence to a solution can be significantly accelerated for a number of iterative image reconstruction algorithms including simultaneous cimmino type algorithms the expectation maximization method for maximizing likelihood emml and the simultaneous multiplicative algebraic reconstruction technique smart through the use of rescaled block iterative bi methods these bi methods involve partitioning the data into disjoint subsets and using only one subset at each step of the iteration one drawback of these methods is their failure to converge to an approximate solution in the inconsistent case in which no image consistent with the data exists they are always observed to produce limit cycles lcs of distinct images through which the algorithm cycles no one of these images provides a suitable solution in general the question that arises then is whether or not these lc vectors retain sufficient information to construct from them a suitable approximate solution we show that they do to demonstrate that we employ a feedback technique in which the lc vectors are used to produce a new data vector and the algorithm restarted convergence of this nested iterative scheme to an approximate solution is then proven preliminary work also suggests that this feedback method may be incorporated in a practical reconstruction method

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

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

regularized image reconstruction algorithms for dual isotope myocardial perfusion spect mps imaging using a cross tracer prior in simultaneous dual isotope myocardial perfusion spect mps imaging data are simultaneously acquired to determine the distributions of two radioactive isotopes the goal of this work was to develop penalized maximum likelihood pml algorithms for a novel cross tracer prior that exploits the fact that the two images reconstructed from simultaneous dual isotope mps projection data are perfectly registered in space we first formulated the simultaneous dual isotope mps reconstruction problem as a joint estimation problem a cross tracer prior that couples voxel values on both images was then proposed we developed an iterative algorithm to reconstruct the mps images that converges to the maximum a posteriori solution for this prior based on separable surrogate functions to accelerate the convergence we developed a fast algorithm for the cross tracer prior based on the complete data os em cosem framework the proposed algorithm was compared qualitatively and quantitatively to a single tracer version of the prior that did not include the cross tracer term quantitative evaluations included comparisons of mean and standard deviation images as well as assessment of image fidelity using the mean square error we also evaluated the cross tracer prior using a three class observer study with respect to the three class mps diagnostic task i e classifying patients as having either no defect reversible defect or fixed defects for this study a comparison with conventional ordered subsets expectation maximization os em reconstruction with postfiltering was performed the comparisons to the single tracer prior demonstrated similar resolution for areas of the image with large intensity changes and reduced noise in uniform regions the cross tracer prior was also superior to the single tracer version in terms of restoring image fidelity results of the three class observer study showed that the proposed cross tracer prior and the convergent algorithms improved the image quality of dual isotope mps images compared to os em

performance evaluation of an iterative image reconstruction algorithm for positron emission tomography an image reconstruction method motivated by positron emission tomography pet is discussed the measurements tend to be noisy and so the reconstruction method should incorporate the statistical nature of the noise the authors set up a discrete model to represent the physical situation and arrive at a nonlinear maximum a posteriori probability map formulation of the problem an iterative approach which requires the solution of simple quadratic equations is proposed the authors also present a methodology which allows them to experimentally optimize an image reconstruction method for a specific medical task and to evaluate the relative efficacy of two reconstruction methods for a particular task in a manner which meets the high standards set by the methodology of statistical hypothesis testing the new map algorithm is compared to a method which maximizes likelihood and with two variants of the filtered backprojection method

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 spatio temporal image reconstruction for dynamic pet in tomographic imaging dynamic images are typically obtained by reconstructing the frames of a time sequence independently one by one a disadvantage of this frame by frame reconstruction approach is that it fails to account for temporal correlations in the signal ideally one should treat the entire image sequence as a single spatio temporal signal however the resulting reconstruction task becomes computationally intensive fortunately as the authors show in this paper the spatio temporal reconstruction problem call be greatly simplified by first applying a temporal karhunen loeve kl transformation to the imaging equation the authors show that if the regularization operator is chosen to be separable into space and time components penalized weighted least squares reconstruction of the entire image sequence is approximately equivalent to frame by frame reconstruction in the space kl domain by this approach spatio temporal reconstruction can be achieved at reasonable computational cost one can achieve further computational savings by discarding high order kl components to avoid reconstructing them performance of the method is demonstrated through statistical evaluations of the bias variance tradeoff obtained by computer simulation reconstruction

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

fast em like methods for maximum a posteriori estimates in emission tomography the maximum likelihood ml approach in emission tomography provides images with superior noise characteristics compared to conventional filtered backprojection fbp algorithms the expectation maximization em algorithm is an iterative algorithm for maximizing the poisson likelihood in emission computed tomography that became very popular for solving the ml problem because of its attractive theoretical and practical properties recently browne and depierro 1996 and hudson and larkin 1991 block sequential versions of the em algorithm that take advantage of the scanner s geometry have been proposed in order to accelerate its convergence in hudson and larkin 1991 the ordered subsets em os em method was applied to the hit problem and a modification os gp to the maximum a posteriori map regularized approach without showing convergence in browne and depierro 1996 we presented a relaxed version of os em ramla that converges to an ml solution in this paper we present an extension of ramla for map reconstruction we show that if the sequence generated by this method converges then it must converge to the true map solution experimental evidence of this convergence is also shown to illustrate this behavior we apply the algorithm to positron emission tomography simulated data comparing its performance to os gp

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 generalized dynamic is lm model with delayed time in investment processes abstract in this paper we analyze a generalized dynamic is lm model with a government expenditure constraint and the capital accumulation equation that capital stock changes are caused by past investment decisions according to kalecki s idea of time delay in investment processes applying stability switch criteria we prove that the time delay causes the equilibrium point may lose or gain local stability so that the hopf bifurcation occurs

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

block iterative fisher scoring algorithms for maximum penalized likelihood image reconstruction in emission tomography this paper introduces and evaluates a block iterative fisher scoring bfs algorithm the algorithm provides regularized estimation in tomographic models of projection data with poisson variability regularization is achieved by penalized likelihood with a general quadratic penalty local convergence of the block iterative algorithm is proven under conditions that do not require iteration dependent relaxation we show that when the algorithm converges it converges to the unconstrained maximum penalized likelihood mpl solution simulation studies demonstrate that with suitable choice of relaxation parameter and restriction of the algorithm to respect nonnegative constraints the bfs algorithm provides convergence to the constrained mpl solution constrained bfs often attains a maximum penalized likelihood faster than other block iterative algorithms which are designed for nonnegatively constrained penalized reconstruction

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

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

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