the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

the complex lorenz equations we have undertaken a study of the complex lorenz equations x x y y r z x ay z bz 12 x y xy where x and y are complex and z is real the complex parameters r and a are defined by r r1 ir2 a 1 ie and and b are real behaviour remarkably different from the real lorenz model occurs only the origin is a fixed point except for the special case e r2 0 we have been able to determine analytically two critical values of r1 namely r1c and r1c the origin is a stable fixed point for 0 r1c a hopf bifurcation to a limit cycle occurs we have an exact analytic solution for this limit cycle which is always stable if 1 then this limit is only stable in the region r1c rlc a transition to a finite amplitude oscillation about the limit cycle occurs the nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the stuart landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency this latter bifurcation is either a sub or super critical hopf like bifurcation to a doubly periodic motion the direction of bifurcation depending on the parameter values the nature of the bifurcation is complicated by the existence of a zero eigenvalue

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

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

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

synchronization and secure communication of uncertain chaotic systems based on full order and reduced order output affine observers this paper deals with the problems of chaos synchronization and secure communication for a class of uncertain chaotic systems the uncertain chaotic systems under consideration are subject to modeling uncertainties and depend affinely on a set of measurable system outputs two types of observer are proposed to achieve chaos synchronization and secure communication first a full order output affine observer is proposed to simultaneously estimate the system states and the message signals second a reduced order output affine observer is proposed to simultaneously estimate the unmeasured system states and the message signals a numerical example is given for a chaotic system to show the design procedure and the effectiveness of the proposed observers the simulation results indicate that the proposed observers work well

regularization in tomographic reconstruction using thresholding estimators in tomographic medical devices such as single photon emission computed tomography or positron emission tomography cameras image reconstruction is an unstable inverse problem due to the presence of additive noise a new family of regularization methods for reconstruction based on a thresholding procedure in wavelet and wavelet packet wp decompositions is studied this approach is based on the fact that the decompositions provide a near diagonalization of the inverse radon transform and of prior information in medical images a wp decomposition is adaptively chosen for the specific image to be restored corresponding algorithms have been developed for both two dimensional and full three dimensional reconstruction these procedures are fast noniterative and flexible numerical results suggest that they outperform filtered back projection and iterative procedures such as ordered subset expectation maximization

bias reduction for low statistics pet maximum likelihood reconstruction with a modified poisson distribution positron emission tomography data are typically reconstructed with maximum likelihood expectation maximization mlem however mlem suffers from positive bias due to the non negativity constraint this is particularly problematic for tracer kinetic modeling two reconstruction methods with bias reduction properties that do not use strict poisson optimization are presented and compared to each other to filtered backprojection fbp and to mlem the first method is an extension of negml where the poisson distribution is replaced by a gaussian distribution for low count data points the transition point between the gaussian and the poisson regime is a parameter of the model the second method is a simplification of abml abml has a lower and upper bound for the reconstructed image whereas aml has the upper bound set to infinity aml uses a negative lower bound to obtain bias reduction properties different choices of the lower bound are studied the parameter of both algorithms determines the effectiveness of the bias reduction and should be chosen large enough to ensure bias free images this means that both algorithms become more similar to least squares algorithms which turned out to be necessary to obtain bias free reconstructions this comes at the cost of increased variance nevertheless negml and aml have lower variance than fbp furthermore randoms handling has a large influence on the bias reconstruction with smoothed randoms results in lower bias compared to reconstruction with unsmoothed randoms or randoms precorrected data however negml and aml yield both bias free images for large values of their parameter

combining ordered subsets and momentum for accelerated x ray ct image reconstruction statistical x ray computed tomography ct reconstruction can improve image quality from reduced dose scans but requires very long computation time ordered subsets os methods have been widely used for research in x ray ct statistical image reconstruction and are used in clinical pet and spect reconstruction in particular os methods based on separable quadratic surrogates os sqs are massively parallelizable and are well suited to modern computing architectures but the number of iterations required for convergence should be reduced for better practical use this paper introduces os sqs momentum algorithms that combine nesterov s momentum techniques with os sqs methods greatly improving convergence speed in early iterations if the number of subsets is too large the os sqs momentum methods can be unstable so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior experiments with simulated and real 3d ct scan data illustrate the performance of the proposed algorithms

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

recursive parameter estimation using incomplete data abstract stochastic approximation procedures are considered for the estimation of parameters using incomplete data one procedure is stated and illustrated which often leads to asymptotically efficient estimators others are developed which although possibly not optimal in the above sense will be very much easier to apply this will be particularly advantageous when quick recursive estimates are required examples are given and a link is made between one of the sub optimal methods and the em algorithm author

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

joint mr pet reconstruction using a multi channel image regularizer while current state of the art mr pet scanners enable simultaneous mr and pet measurements the acquired data sets are still usually reconstructed separately we propose a new multi modality reconstruction framework using second order total generalized variation tgv as a dedicated multi channel regularization functional that jointly reconstructs images from both modalities in this way information about the underlying anatomy is shared during the image reconstruction process while unique differences are preserved results from numerical simulations and in vivo experiments using a range of accelerated mr acquisitions and different mr image contrasts demonstrate improved pet image quality resolution and quantitative accuracy

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

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

a generalized em algorithm for 3 d bayesian reconstruction from poisson data using gibbs priors a generalized expectation maximization gem algorithm is developed for bayesian reconstruction based on locally correlated markov random field priors in the form of gibbs functions and on the poisson data model for the m step of the algorithm a form of coordinate gradient ascent is derived the algorithm reduces to the em maximum likelihood algorithm as the markov random field prior tends towards a uniform distribution three different gibbs function priors are examined reconstructions of 3 d images obtained from the poisson model of single photon emission computed tomography are presented

improved regional activity quantitation in nuclear medicine using a new approach to correct for tissue partial volume and spillover effects we have developed a ne w method of compensating for effects of partial volume and spillover in dual modality imaging the approach requires segmentation of just a few tissue types within a small volume of interest voi surrounding a lesion the algorithm estimates simultaneously from projection data the activity concentration within each segmented tissue inside the voi measured emission projections were fitted to the sum of resolution blurred projections of each such tissue scaled by its unknown activity concentration plus a global background contribution obtained by reprojection through the reconstructed image volume outside the voi the method was evaluated using multiple pinhole spect data simulated for the moby mouse phantom containing two spherical lung tumors and one liver tumor as well as using multiple bead phantom data acquired on spect and ct scanners each voi in the simulation study was 4 8 mm 12 voxels cubed and depending on location contained up to four tissues tumor liver heart lung with different values of relative 99mtc concentration all tumor activity estimates achieved 3 bias after 15 ordered subsets expectation maximization osem iterations 10 subsets with better than 8 precision 25 greater than the cramer rao low er bound the projection based fitting approach also outperformed three standardized uptake value suv like metrics one of which was corrected for count spillover in the bead phantom experiment the mean standard deviation of the bias of voi estimates of bead concentration were 0 9 9 5 comparable to those of a perturbation geometric transfer matrix pgtm approach 5 4 8 6 however voi estimates were more stable with increasing iteration number than pgtm estimates even in the presence of substantial axial misalignment between ct and spect image volumes

tomographic image reconstruction based on a content adaptive mesh model in this paper we explore the use of a content adaptive mesh model camm for tomographic image reconstruction in the proposed framework the image to be reconstructed is first represented by a mesh model an efficient image description based on nonuniform sampling in the camm image samples represented as mesh nodes are placed most densely in image regions having fine detail tomographic image reconstruction in the mesh domain is performed by maximum likelihood ml or maximum a posteriori map estimation of the nodal values from the measured data a camm greatly reduces the number of unknown parameters to be determined leading to improved image quality and reduced computation time we demonstrated the method in our experiments using simulated gated single photon emission computed tomography spect cardiac perfusion images a channelized hotelling observer cho was used to evaluate the detectability of perfusion defects in the reconstructed images a task based measure of image quality a minimum description length mdl criterion was also used to evaluate the effect of the representation size in our application both mdl and cho suggested that the optimal number of mesh nodes is roughly five to seven times smaller than the number of projection bins when compared to several commonly used methods for image reconstruction the proposed approach achieved the best performance in terms of defect detection and computation time the research described in this paper establishes a foundation for future development of a four dimensional space time reconstruction framework for image sequences in which a built in deformable mesh model is used to track the image motion

effects of attenuation correction and reconstruction method on pet activation studies the outcome of statistical parametric mapping spm analyses of pet activation studies depends among others on the quality of reconstructed data in general filtered back projection fbp is used for reconstruction in pet activation studies there is however increasing interest in iterative reconstruction algorithms such as ordered subset expectation maximization osem algorithms the aim of the present study was to investigate the effects of reconstruction techniques and attenuation correction ac on the detection of activation foci following statistical analysis with spm first a replicate study was performed to assess the effects of the reconstruction method on pixel variance second a phantom study was performed to evaluate the influence of both locations of an activated area and applied reconstruction method on spm outcome a volumetric method was used to compute the number of false positive voxels for all reconstructions in addition average t values within activation foci and for false positive voxels were calculated for the assessment of the effects of reconstruction on clinical data a group of 11 patients was studied for all reconstructions spm maps were created and compared both the clinical and the phantom data showed that use of iterative reconstruction methods reduced false positive results while showing similar spm results within activated areas as fbp reconstruction of data without attenuation correction reduced noise for fbp only but did not affect the quality of spm results for osem it is concluded that osem is a good alternative for fbp reconstructions providing spm results with less noise

unmatched projector backprojector pairs in an iterative reconstruction algorithm computational burden is a major concern when an iterative algorithm is used to reconstruct a three dimensional 3 d image with attenuation detector response and scatter corrections most of the computation time is spent executing the projector and backprojector of an iterative algorithm usually the projector and the backprojector are transposed operators of each other the projector should model the imaging geometry and physics as accurately as possible some researchers have used backprojectors that are computationally less expensive than the projectors to reduce computation time this paper points out that valid backprojectors should satisfy a condition that the projector backprojector matrix must not contain negative eigenvalues this paper also investigates the effects when unmatched projector backprojector pairs are used

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

reconstruction from uniformly attenuated spect projection data using the dbh method an algorithm was developed for the 2 d reconstruction of truncated and nontruncated uniformly attenuated data acquired from single photon emission computed tomography spect the algorithm is able to reconstruct data from half scan 180deg and short scan 180deg fan angle acquisitions for parallel and fan beam geometries respectively as well as data from full scan 360deg acquisitions the algorithm is a derivative backprojection and hilbert transform dbh method which involves the backprojection of differentiated projection data followed by an inversion of the finite weighted hilbert transform the kernel of the inverse weighted hilbert transform is solved numerically using matrix inversion numerical simulations confirm that the dbh method provides accurate reconstructions from half scan and short scan data even when there is truncation however as the attenuation increases finer data sampling is required

pet image reconstruction using kernel method image reconstruction from low count positron emission tomography pet projection data is challenging because the inverse problem is ill posed prior information can be used to improve image quality inspired by the kernel methods in machine learning this paper proposes a kernel based method that models pet image intensity in each pixel as a function of a set of features obtained from prior information the kernel based image model is incorporated into the forward model of pet projection data and the coefficients can be readily estimated by the maximum likelihood ml or penalized likelihood image reconstruction a kernelized expectation maximization algorithm is presented to obtain the ml estimate computer simulations show that the proposed approach can achieve better bias versus variance trade off and higher contrast recovery for dynamic pet image reconstruction than the conventional maximum likelihood method with and without post reconstruction denoising compared with other regularization based methods the kernel method is easier to implement and provides better image quality for low count data application of the proposed kernel method to a 4 d dynamic pet patient dataset showed promising results

passivity analysis and passification for uncertain signal processing systems the problem of passivity analysis finds important applications in many signal processing systems such as digital quantizers decision feedback equalizers and digital and analog filters equally important is the problem of passification where a compensator needs to be designed for a given system to become passive this paper considers these two problems for a large class of systems that involve uncertain parameters time delays quantization errors and unmodeled high order dynamics by characterizing these and many other types of uncertainty using a general tool called integral quadratic constraints iqcs we present solutions to the problems of robust passivity analysis and robust passification more specifically for the analysis problem we determine if a given uncertain system is passive for all admissible uncertainty satisfying the iqcs similarly for the problem of robust passification we are concerned with finding a loop transformation such that a particular part of the uncertain signal processing system becomes passive for all admissible uncertainty the solutions are given in terms of the feasibility of one or more linear matrix inequalities lmis which can be solved efficiently

noise characterization of block iterative reconstruction algorithms i theory researchers have shown increasing interest in block iterative image reconstruction algorithms due to the computational and modeling advantages they provide although their convergence properties have been well documented little is known about how they behave in the presence of noise in this work the authors fully characterize the ensemble statistical properties of the rescaled block iterative expectation maximization rbi em reconstruction algorithm and the rescaled block iterative simultaneous multiplicative algebraic reconstruction technique rbi smart also included in the analysis are the special cases of rbi em maximum likelihood em ml em and ordered subset em os em and the special case of rbi smart smart a theoretical formulation strategy similar to that previously outlined for ml em is followed for the rbi methods the theoretical formulations in this paper rely on one approximation namely that the noise in the reconstructed image is small compared to the mean image in a second paper the approximation will be justified through monte carlo simulations covering a range of noise levels iteration points and subset orderings the ensemble statistical parameters could then be used to evaluate objective measures of image quality

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

analytical calculation of volumes of intersection for iterative fully 3 d pet reconstruction use of iterative algorithms to reconstruct three dimensional 3 d positron emission tomography pet data requires the computation of the system probability matrix the pure geometrical contribution can easily be approximated by the length of intersection loi between lines of response lor and individual voxels however more accurate geometrical projectors are desirable therefore we have developed a fast method for the analytical calculation of the 3 d shape and volume of volumes of intersection voi this method provides an alternative robust projector with a uniformly continuous sampling of the image space the enhanced calculation effort is facilitated by using several speedup techniques exploiting intrinsic symmetry relations and the sparseness of the system matrix allows to create an efficiently compressed matrix which can be precomputed and completely stored in memory in addition a new voxel addressing scheme has been implemented this scheme avoids time consuming symmetry transformations of voxel addresses by using an octant wise symmetrically ordered field of voxels the above methods have been applied for a fully 3 d iterative reconstruction of 3 d sinograms recorded with a siemens cti ecat hr pet scanner a comparison of the performance of the reconstruction using loi weighting and voi weighting is presented

statistical modeling and reconstruction of randoms precorrected pet data randoms precorrected positron emission tomography pet data is formed as the difference of two poisson random variables its exact probability mass function pmf is inconvenient for use in likelihood based iterative image reconstruction as it contains an infinite summation the shifted poisson model is a tractable approximation to this pmf but requires that negative values are truncated resulting in positively biased reconstructions in low count studies here we analyze the properties of the exact pmf and propose a simple but accurate approximation that allows negative valued data we investigate the properties of this approximation and demonstrate its application to penalized maximum likelihood image reconstruction

classification based adaptive filtering for multiframe blind image restoration in this paper the blind restoration of a scene is investigated when multiple degraded blurred and noisy acquisitions are available an adaptive filtering technique is proposed where the distorted images are filtered classified and then fused based upon the classification decisions finite normal density mixture fnm models are used to model the filtered outputs at each iteration for simplicity fixed number of gaussian components classes is initially considered for each degraded frame and the selection of the optimal number of classes is performed according to the global relative entropy criterion however there exist cases where dynamically varying fnm models should be considered where the optimal number of classes is selected according to the akaike information criterion the iterative application of classification and fusion followed by optimal adaptive filtering converges to a global enhanced representation of the original scene in only a few iterations the proposed restoration method does not require knowledge of the point spread function support size or exact alignment of the acquired frames simulation results on synthetic and real data using both fixed and dynamically varying fnm models demonstrate its efficiency under both noisy and noise free conditions

implementation of a model based nonuniform scatter correction scheme for spect four scatter compensation schemes are considered the 4 schemes are all based on a previously developed two dimensional 2 d scatter model reconstruction is achieved using the iterative expectation maximization maximum likelihood em ml algorithm the schemes consist of 1 including the model in both the forward and back projector 2 just including the model in the forward projector 3 and 4 implementing the model in a subtraction and addition scheme respectively monte carlo simulated projection data are used to test the accuracy convergence properties and noise properties of the 4 scatter compensation schemes data are simulated for both uniformly and nonuniformly attenuating objects the results show that all 4 correction schemes yield images which are similar in terms of accuracy to that obtained from reconstructing scatter free data the subtraction scheme is shown to converge faster than the other compensation schemes both in terms of iterations and actual time required for reconstruction the scheme in which the model is only used in the forward projector and the scatter addition scheme both performs slightly better in terms of signal to noise ratio snr than the subtraction scheme however the forward projector scheme requires significantly more cpu time for reconstruction the correction scheme in which the scatter model was included in both the forward and backprojectors is shown to produce accurate images with snr s higher than even a perfect scatter rejection scheme while the scatter correction scheme with the model in both the forward projector and backprojector has superior noise properties to the other algorithms the results suggest that the faster subtraction addition schemes will probably prove most useful for routine clinical scatter compensation

postreconstruction nonlocal means filtering of whole body pet with an anatomical prior positron emission tomography pet images usually suffer from poor signal to noise ratio snr due to the high level of noise and low spatial resolution which adversely affect its performance for lesion detection and quantification the complementary information present in high resolution anatomical images from multi modality imaging systems could potentially be used to improve the ability to detect and or quantify lesions however previous methods that use anatomical priors usually require matched organ lesion boundaries in this study we investigated the use of anatomical information to suppress noise in pet images while preserving both quantitative accuracy and the amplitude of prominent signals that do not have corresponding boundaries on computerized tomography ct the proposed approach was realized through a postreconstruction filter based on the nonlocal means nlm filter which reduces noise by computing the weighted average of voxels based on the similarity measurement between patches of voxels within the image anatomical knowledge obtained from ct was incorporated to constrain the similarity measurement within a subset of voxels in contrast to other methods that use anatomical priors the actual number of neighboring voxels and weights used for smoothing were determined from a robust measurement on pet images within the subset thus the proposed approach can be robust to signal mismatches between pet and ct a 3 d search scheme was also investigated for the volumetric pet ct data the proposed anatomically guided median nonlocal means filter amnlm was first evaluated using a computer phantom and a physical phantom to simulate realistic but challenging situations where small lesions are located in homogeneous regions which can be detected on pet but not on ct the proposed method was further assessed with a clinical study of a patient with lung lesions the performance of the proposed method was compared to gaussian edge preserving bilateral and nlm filters as well as median nonlocal means mnlm filtering without an anatomical prior the proposed amnlm method yielded improved lesion contrast and snr compared with other methods even with imperfect anatomical knowledge such as missing lesion boundaries and mismatched organ boundaries

voxel based comparison of state of the art reconstruction algorithms for 18f fdg pet brain imaging using simulated and clinical data abstract the resolution of a pet scanner 2 5 4 5 mm for brain imaging is similar to the thickness of the cortex in the human brain 2 5 mm on average hampering accurate activity distribution reconstruction many techniques to compensate for the limited resolution during or post reconstruction have been proposed in the past and have been shown to improve the quantitative accuracy in this study state of the art reconstruction techniques are compared on a voxel basis for quantification accuracy and group analysis using both simulated and measured data of healthy volunteers and patients with epilepsy methods maximum a posteriori map reconstructions using either a segmentation based or a segmentation less anatomical prior were compared to maximum likelihood expectation maximization mlem reconstruction with resolution recovery as anatomical information a spatially aligned 3d t1 weighted magnetic resonance image was used firstly the algorithms were compared using normal brain images to detect systematic bias with respect to the true activity distribution as well as systematic differences between two methods secondly it was verified whether the algorithms yielded similar results in a group comparison study results significant differences were observed between the reconstructed and the true activity with the largest errors when using post smoothed mlem only 5 10 underestimation in cortical gray matter voxel activity was found for both map reconstructions higher errors were observed at gm edges map with the segmentation based prior also resulted in a significant bias in the subcortical regions due to segmentation inaccuracies while map with the anatomical prior which does not need segmentation did not significant differences in reconstructed activity were also found between the algorithms at similar locations mainly in gray matter edge voxels and in cerebrospinal fluid voxels in the simulated as well as in the clinical data sets nevertheless when comparing two groups very similar regions of significant hypometabolism were detected by all algorithms conclusion including anatomical a priori information during reconstruction in combination with resolution modeling yielded accurate gray matter activity estimates and a significant improvement in quantification accuracy was found when compared to post smoothed mlem reconstruction with resolution modeling asymbowsher provided the most accurate subcortical gm activity estimates it is also reassuring that the differences found between the algorithms did not hamper the detection of hypometabolic regions in the gray matter when performing a voxel based group comparison nevertheless the size of the detected clusters differed more elaborated and application specific studies are required to decide which algorithm is best for a group analysis

intensity normalization methods in brain fdg pet quantification abstract background the lack of standardization of intensity normalization methods and its unknown effect on the quantification output is recognized as a major drawback for the harmonization of brain fdg pet quantification protocols the aim of this work is the ground truth based evaluation of different intensity normalization methods on brain fdg pet quantification output methods realistic fdg pet images were generated using monte carlo simulation from activity and attenuation maps directly derived from 25 healthy subjects adding theoretical relative hypometabolisms on 6 regions of interest and for 5 hypometabolism levels single subject statistical parametric mapping spm was applied to compare each simulated fdg pet image with a healthy database after intensity normalization based on reference regions methods such as the brain stem rrbs cerebellum rrc and the temporal lobe contralateral to the lesion rrtl and data driven methods such as proportional scaling ps histogram based method hn and iterative versions of both methods ips and ihn the performance of these methods was evaluated in terms of the recovery of the introduced theoretical hypometabolic pattern and the appearance of unspecific hypometabolic and hypermetabolic findings results detected hypometabolic patterns had significantly lower volumes than the introduced hypometabolisms for all intensity normalization methods particularly for slighter reductions in metabolism among the intensity normalization methods rrc and hn provided the largest recovered hypometabolic volumes while the rrbs showed the smallest recovery in general data driven methods overcame reference regions and among them the iterative methods overcame the non iterative ones unspecific hypermetabolic volumes were similar for all methods with the exception of ps where it became a major limitation up to 250 cm3 for extended and intense hypometabolism on the other hand unspecific hypometabolism was similar far all methods and usually solved with appropriate clustering conclusions our findings showed that the inappropriate use of intensity normalization methods can provide remarkable bias in the detected hypometabolism and it represents a serious concern in terms of false positives based on our findings we recommend the use of histogram based intensity normalization methods reference region methods performance was equivalent to data driven methods only when the selected reference region is large and stable

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

single and multipinhole collimator design evaluation method for small animal spect high resolution functional imaging of small animals is often obtained by single pinhole spect with circular orbit acquisition multipinhole spect adds information due to its improved sampling and can improve the trade off between resolution and sensitivity to evaluate different pinhole collimator designs an efficient method is needed that quantifies the reconstruction image quality in this paper we propose a fast approximate method that examines the quality of individual voxels of a postsmoothed maximum likelihood expectation maximization mlem reconstruction by studying their linearized local impulse response llir and co variance for a predefined target resolution for validation the contrast to noise ratios cnrs in some voxels of a homogeneous sphere and of a realistic rat brain software phantom were calculated for many single and multipinhole designs a good agreement was observed between the cnrs obtained with the approximate method and those obtained with postsmoothed mlem reconstructions of simulated noisy projections this good agreement was quantified by a least squares fit through these results which yielded a line with slope 1 02 1 00 expected and a y intercept close to zero 0 expected 95 4 of the validation points lie within three standard deviations from that line using the approximate method the influence on the cnr of varying a parameter in realistic single and multipinhole designs was examined the investigated parameters were the aperture diameter the distance between the apertures and the axis of rotation the focal distance the acceptance angle the position of the apertures the focusing distance and the number of pinholes the results can generally be explained by the change in sensitivity the amount of postsmoothing and the amount of overlap in the projections the method was applied to multipinhole designs with apertures focusing at a single point but is also applicable to other designs

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

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

operator extensions of hua s inequality abstract we give an extension of hua s inequality in pre hilbert c modules without using convexity or the classical hua s inequality as a consequence some known and new generalizations of this inequality are deduced providing a jensen inequality in the content of hilbert c modules another extension of hua s inequality is obtained we also present an operator hua s inequality which is equivalent to operator convexity of given continuous real function

operator aczel inequality we establish several operator versions of the classical aczel inequality one of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms some applications including the unital positive linear maps on c algebras and the unitarily invariant norms on matrices are presented

a model based iterative reconstruction algorithm for high angle annular dark field scanning transmission electron microscope haadf stem tomography high angle annular dark field haadf scanning transmission electron microscope stem data is increasingly being used in the physical sciences to research materials in 3d because it reduces the effects of bragg diffraction seen in bright field tem data typically tomographic reconstructions are performed by directly applying either filtered back projection fbp or the simultaneous iterative reconstruction technique sirt to the data since haadf stem tomography is a limited angle tomography modality with low signal to noise ratio these methods can result in significant artifacts in the reconstructed volume in this paper we develop a model based iterative reconstruction algorithm for haadf stem tomography we combine a model for image formation in haadf stem tomography along with a prior model to formulate the tomographic reconstruction as a maximum a posteriori probability map estimation problem our formulation also accounts for certain missing measurements by treating them as nuisance parameters in the map estimation framework we adapt the iterative coordinate descent algorithm to develop an efficient method to minimize the corresponding map cost function reconstructions of simulated as well as experimental data sets show results that are superior to fbp and sirt reconstructions significantly suppressing artifacts and enhancing contrast

an improved secure communication scheme based passive synchronization of hyperchaotic complex nonlinear system in this paper an improved secure communication scheme is proposed based on passive synchronization of hyperchaotic complex nonlinear system the hyperchaotic complex lu system is employed to encrypt the emitted signal comparing with the existing approaches the useful information in our work can be unbounded in the transmitter the original information signal is modulated into one parameter of hyperchaotic complex lu system and it is assumed that the parameter of the receiver system is unknown in the receiver based on the passivity theory the controllers and corresponding parameter update rule are designed to make two identical hyperchaotic complex lu systems asymptotically synchronized and identify the unknown parameter simultaneously the information signal can be recovered accurately by the estimated parameter corresponding theoretical proofs and numerical simulations demonstrate the feasibility and effectiveness of the proposed method it is also shown that the presented secure communication scheme is robust against different channel noise

theoretical study of penalized likelihood image reconstruction for region of interest quantification region of interest roi quantification is an important task in emission tomography e g positron emission tomography and single photon emission computed tomography it is essential for exploring clinical factors such as tumor activity growth rate and the efficacy of therapeutic interventions statistical image reconstruction methods based on the penalized maximum likelihood pml or maximum a posteriori principle have been developed for emission tomography to deal with the low signal to noise ratio of the emission data similar to the filter cut off frequency in the filtered backprojection method the regularization parameter in pml reconstruction controls the resolution and noise tradeoff and hence affects roi quantification in this paper we theoretically analyze the performance of roi quantification in pml reconstructions building on previous work we derive simplified theoretical expressions for the bias variance and ensemble mean squared error emse of the estimated total activity in an roi that is surrounded by a uniform background when the mean and covariance matrix of the activity inside the roi are known the theoretical expressions are readily computable and allow for fast evaluation of image quality for roi quantification with different regularization parameters the optimum regularization parameter can then be selected to minimize the emse computer simulations are conducted for small rois with variable uniform uptake the results show that the theoretical predictions match the monte carlo results reasonably well

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

on artifact free projection overlaps in multi pinhole tomographic imaging in multi pinhole spect overlapping the projections from different pinholes has been used to increase sensitivity however the prevailing view is that the overall quality of the reconstructed image is not improved by the overlaps in the projections it is often stated in literatures that overlaps introduce ambiguous information which can lead to nonuniqueness of solution for the inverse problem and thus artifacts are introduced in the reconstructed image on the other hand contrary to the prevailing view a recent study on slit slat collimators shows that artifacts can be removed with the help of an extra complete nonoverlapped projection data set in this paper two types of artifact free projection overlaps are defined in general and the criteria for designing artifact free multi pinhole systems with overlaps are proposed it is shown that once the criteria are satisfied the solution of the inverse problem is unique and thus no artifact is expected in the reconstructed image via extensive simulation study various artifact free overlapping multi pinhole systems are designed and validated it is shown that overlaps in the artifact free systems can improve contrast to noise ratio cnr with a proper design the cnr for an artifact free overlapping system can be significantly higher than that for the corresponding nonoverlapping system the improved image quality is also confirmed with noisy reconstructions

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

tackling box constrained optimization via a new projected quasi newton approach numerous scientific applications across a variety of fields depend on box constrained convex optimization box constrained problems therefore continue to attract research interest we address box constrained strictly convex problems by deriving two new quasi newton algorithms our algorithms are positioned between the projected gradient j b rosen j siam 8 1960 pp 181 217 and projected newton d p bertsekas siam j control optim 20 1982 pp 221 246 methods we also prove their convergence under a simple armijo step size rule we provide experimental results for two particular box constrained problems nonnegative least squares nnls and nonnegative kullback leibler nnkl minimization for both nnls and nnkl our algorithms perform competitively as compared to well established methods on medium sized problems for larger problems our approach frequently outperforms the competition

fast and reliable estimation of multiple parametric images using an integrated method for dynamic spect dynamic single photon emission computed tomography spect has demonstrated the potential to quantitatively estimate physiological parameters in the brain and the heart the generalized linear least square glls method is a well established method for solving linear compartment models with fast computational speed however the high level of noise intrinsic in the spect data leads to reliability and instability problems of glls for generating parametric images an integrated method is proposed to restrict the noise in both the temporal and spatial domains to estimate multiple parametric images for dynamic spect this method comprises three steps which are optimum image sampling schedule in the projection space cluster analysis applied postreconstruction and parametric image generation with glls the simulation and experimental studies for the neuronal nicotine acetylcholine receptor tracer of 5 123i iodo a 85380 were employed to evaluate the performance of the proposed method the results of influx rate of k1 and volume of distribution of vd demonstrated that the integrated method was successful in generating low noise parametric images for high noise spect data without enhancing the partial volume effect furthermore the integrated method is computationally efficient for potential clinical applications

iterative algorithms for large partitioned linear systems with applications to image reconstruction abstract we present a unifying framework for a wide class of iterative methods in numerical linear algebra in particular the class of algorithms contains kaczmarz s and richardson s methods for the regularized weighted least squares problem with weighted norm the convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for kaczmarz s and richardson s methods the algorithms in the class may be characterized as being group iterative and incorporate relaxation matrices as opposed to a single relaxation parameter we show that some well known iterative methods of image reconstruction fall into the class of algorithms under consideration and are thus covered by the convergence theory we also describe a novel application to truly three dimensional image reconstruction

positron emission tomography by markov chain monte carlo with auxiliary variables we propose a new algorithm for positron emission tomography pet image reconstruction the algorithm belongs to the family of markov chain monte carlo methods with auxiliary variables the idea is to iteratively generate hidden variables at one step and use them for image restoration at another step the well known model of vardi et al j amer statist assoc 80 1985 8 for pet is combined with the bayesian model of lasota and niemiro pattern recognition 36 2003 931 for the underlying images this latter model takes advantage of the fact that medical images often consist of relatively few grey levels of unknown intensity the algorithm of lasota and niemiro pattern recognition 36 2003 931 is used in the image restoration part of the pet algorithm essentially as a noise filtering and smoothing device it is now equipped with an additional data reconstruction step we include simulation results which suggest that the method is truly reliable we also describe a version of the basic algorithm in which a random simulation step is replaced by computation of expected value similarly as in the em algorithm

pose estimation of known objects during transmission tomographic image reconstruction we address the problem of image formation in transmission tomography when metal objects of known composition and shape but unknown pose are present in the scan subject using an alternating minimization am algorithm derived from a model in which the detected data are viewed as poisson distributed photon counts we seek to eliminate the streaking artifacts commonly seen in filtered back projection images containing high contrast objects we show that this algorithm which minimizes the i divergence or equivalently maximizes the log likelihood between the measured data and model based estimates of the means of the data converges much faster when knowledge of the high density materials such as brachytherapy applicators or prosthetic implants is exploited the algorithm incorporates a steepest descent based method to find the position and orientation collectively called the pose of the known objects this pose is then used to constrain the image pixels to their known attenuation values or for example to form a mask on the missing projection data in the shadow of the objects results from two dimensional simulations are shown in this paper the extension of the model and methods used to three dimensions is outlined

the evaluation of data completeness and image quality in multiplexing multi pinhole spect multi pinhole collimators are often used in pre clinical spect systems because they have a better resolution sensitivity tradeoff than parallel hole collimators when imaging small objects most multi pinhole collimators are designed to allow no or only a limited amount of overlap between the different pinhole projections because the ambiguity introduced by multiplexing pinholes can result in artifacts the origin of these artifacts is still not fully understood but previous research has already shown that data incompleteness could be part of the explanation therefore we developed a method to investigate data completeness in multiplexing multi pinhole systems and showed that a certain activity distribution can be successfully reconstructed when the nonmultiplexed data is complete or when the overlap can be sufficiently de multiplexed we validated this method using computer simulated phantom data of different multiplexing systems we also studied contrast to noise and nonprewhitening matched filter signal to noise ratio npw snr to compare the image quality in a single pinhole system with multiplexing systems we found that our method can indeed be used to evaluate data completeness in multiplexing systems and found no artifacts in the systems that had complete data sensitivity increased significantly with multiplexing but we found only small nonsignificant differences in contrast to noise ratio however the npw snr did slightly improve in the multiplexing setups we conclude that more multiplexing does not necessarily result in more artifacts and that even a high amount of multiplexing can still result in artifact free images if the nonmultiplexed data is complete or when the overlap can be sufficiently de multiplexed

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

edge preserving pet image reconstruction using trust optimization transfer iterative image reconstruction for positron emission tomography can improve image quality by using spatial regularization the most commonly used quadratic penalty often oversmoothes sharp edges and fine features in reconstructed images while nonquadratic penalties can preserve edges and achieve higher contrast recovery existing optimization algorithms such as the expectation maximization em and preconditioned conjugate gradient pcg algorithms work well for the quadratic penalty but are less efficient for high curvature or nonsmooth edge preserving regularizations this paper proposes a new algorithm to accelerate edge preserving image reconstruction by using two strategies trust surrogate and optimization transfer descent trust surrogate approximates the original penalty by a smoother function at each iteration but guarantees the algorithm to descend monotonically optimization transfer descent accelerates a conventional optimization transfer algorithm by using conjugate gradient and line search results of computer simulations and real 3 d data show that the proposed algorithm converges much faster than the conventional em and pcg for smooth edge preserving regularization and can also be more efficient than the current state of art algorithms for the nonsmooth ell 1 regularization

direct parametric image reconstruction in reduced parameter space for rapid multi tracer pet imaging the separation of multiple pet tracers within an overlapping scan based on intrinsic differences of tracer pharmacokinetics is challenging due to limited signal to noise ratio snr of pet measurements and high complexity of fitting models in this study we developed a direct parametric image reconstruction dpir method for estimating kinetic parameters and recovering single tracer information from rapid multi tracer pet measurements this is achieved by integrating a multi tracer model in a reduced parameter space rps into dynamic image reconstruction this new rps model is reformulated from an existing multi tracer model and contains fewer parameters for kinetic fitting ordered subsets expectation maximization osem was employed to approximate log likelihood function with respect to kinetic parameters to incorporate the multi tracer model an iterative weighted nonlinear least square wnls method was employed the proposed multi tracer dpir mt dpir algorithm was evaluated on dual tracer pet simulations 18 rm f fdg and 11 rm c met as well as on preclinical pet measurements 18 rm f flt and 18 rm f fdg the performance of the proposed algorithm was compared to the indirect parameter estimation method with the original dual tracer model the respective contributions of the rps technique and the dpir method to the performance of the new algorithm were analyzed in detail for the preclinical evaluation the tracer separation results were compared with single 18 rm f fdg scans of the same subjects measured two days before the dual tracer scan the results of the simulation and preclinical studies demonstrate that the proposed mt dpir method can improve the separation of multiple tracers for pet image quantification and kinetic parameter estimations

application of task based measures of image quality to optimization and evaluation of three dimensional reconstruction based compensation methods in myocardial perfusion spect in this paper we apply the channelized hotelling observer cho using a defect detection task to the optimization and evaluation of three dimensional iterative reconstruction based compensation methods for myocardial perfusion single photon emission computed tomography spect we used a population of 24 mathematical cardiac torso phantoms that realistically model the activity and attenuation distribution in three classes of patients females and males with flat diaphragms and raised diaphragms projection data were generated and subsequently reconstructed using methods based on the ordered subsets expectation maximization osem algorithm the methods evaluated included compensation for attenuation detector response blurring and scatter in various combinations we applied the cho to optimize the number of iterations for osem and the cutoff frequency and order of a three dimensional postreconstruction butterworth filter using the optimal parameters we then compared the compensation methods the index of comparison in these studies was the area under the receiver operating characteristics curve auc for the cho we found that attenuation compensation with either detector response or scatter compensation gave statistically significant increases in the auc compared to attenuation compensation alone the greatest increase in the auc occurred when all three compensations were applied these results indicate that compensation for detector response and scatter in addition to attenuation compensation will improve defect detectability in myocardial spect images

iterative versus filtered backprojection reconstruction for statistical parametric mapping of pet activation measurements a comparative case study the significance of task induced cerebral blood flow responses assessed using statistical parametric mapping depends among other things on the signal to noise ratio snr of these responses generally positron emission tomography sinograms of h2o o 15 activation studies are reconstructed using filtered backprojection fbp alternatively the acquired data can be reconstructed using an iterative reconstruction procedure it has been demonstrated that the application of iterative reconstruction methods improves image snr as compared with fbp the aim of this study was to compare fbp with iterative reconstruction to assess the statistical power of h2o o 15 pet activation studies using statistical parametric mapping for this case study pet data originating from a bimanual motor task were reconstructed using both fbp and maximum likelihood expectation maximization ml em an iterative algorithm both resulting data sets were statistically analyzed using statistical parametric mapping it was found with this dataset that the statistical analysis of the iteratively reconstructed data confirm the a priori expected physiological response in addition increased z scores were obtained in the iteratively reconstructed data in particular for the expected task related response activation of the posterior border of the left angular gyrus the z score increased from 3 00 to 3 96 furthermore the number of statistically significant clusters doubled while their volume increased by more than 50 in conclusion iterative reconstruction has the potential to increase the statistical power in h2o o 15 pet activation studies as compared with fbp reconstruction c 2002 elsevier science

the ordered subsets mirror descent optimization method with applications to tomography we describe an optimization problem arising in reconstructing three dimensional medical images from positron emission tomography pet a mathematical model of the problem based on the maximum likelihood principle is posed as a problem of minimizing a convex function of several million variables over the standard simplex to solve a problem of these characteristics we develop and implement a new algorithm ordered subsets mirror descent and demonstrate theoretically and computationally that it is well suited for solving the pet reconstruction problem

synchronization of unified chaotic system based on passive control chaos synchronization in the unified chaotic system was discussed using passive control on the basis of the property of the passive system the passive controller is designed and synchronization of two identical unified chaotic systems with different initial conditions is realized simulation results for two lorenz two lu and two chen chaotic systems are provided to show the effectiveness of the proposed synchronization method

new pixellation scheme for ct algebraic reconstruction to exploit matrix symmetries in this article we propose a new pixellation scheme which makes it possible to speed up the time of reconstruction this proposal consists in splitting the field of view of the scanner into as many circular sectors as rotation positions of the detector the sectors are pixellated using circular pixels whose size is always smaller than the resolution needed the geometry of the pixels and the arrangement on circular sectors make it possible to compute the entire matrix from only one position of the scanner therefore the size of the matrix decreases as many times as the number of rotations this results in a significant reduction of the system matrix which allows algebraic methods to be applied within a reasonable time of reconstruction and speeds up the time of matrix generation the new model is studied by means of analytical ct simulations which are reconstructed using the maximum likelihood emission maximization algorithm for transmission tomography and is compared to the cartesian pixellation model therefore two different grids of pixels were developed for the same scanner geometry one that employs circular pixels within a cartesian grid and another that divides the field of view into a polar grid which is composed by identical sectors with circular pixels too the results of both models are that polar matrix is built in a few seconds and the cartesian one needs several hours the size of the matrix is significantly smaller than the circular one and the time of reconstruction per iteration using the same iterative method is less in the polar pixel model than in the square pixel model several figures of merit have been computed in order to compare the original phantom with the reconstructed images finally we can conclude that both reconstructions have been proved to have enough quality but the polar pixel model is more efficient than the square pixel model

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

projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the spect collimator the spatially varying geometric response of the collimator detector system in single photon emission computed tomography spect causes loss in resolution shape distortions reconstructed density nonuniformity and quantitative inaccuracies a projection space image reconstruction algorithm is used to correct these reconstruction artifacts the projectors f use strip functions to calculate pixels more natural for modeling the two dimensional 2 d geometric response of the spect collimator transaxially to the axis of rotation these projectors are defined by summing the intersection of an array of multiple strips rotated at equal angles to approximate the ideal system geometric response of the collimator two projection models were evaluated for modeling the system geometric response function for one projector each strip is of equal weight for the other projector a gaussian weighting is used parallel beam and fan beam projections of a physical three dimensional 3 d hoffman brain phantom and a jaszczak cold rod phantom were used to evaluate the geometric response correction reconstructions were obtained by using the singular value decomposition svd method and the iterative conjugate gradient algorithm to solve for q in the imaging equation fgq p where p is the projection measurement the projector f included the new models for the geometric response whereas the backprojector g did not always model the geometric response in order to increase the computational speed the final reconstruction was obtained by sampling the backprojection gq at a discrete array of points reconstructions produced by the two proposed projectors showed improved resolution when compared against a unit strip natural pixel model the conventional image pixelized model with ray tracing to calculate the geometric response and the filtered backprojection algorithm when the reconstruction is displayed on fine grid points the continuity and resolution of the image is preserved without the ring artifacts seen in the unit strip natural pixel model with present computing power the geometric response correction using the proposed projection space reconstruction approach is not yet feasible for routine clinical use

multi material decomposition using statistical image reconstruction for spectral ct spectral computed tomography ct provides information on material characterization and quantification because of its ability to separate different basis materials dual energy de ct provides two sets of measurements at two different source energies in principle two materials can be accurately decomposed from dect measurements however many clinical and industrial applications require three or more material images for triple material decomposition a third constraint such as volume conservation mass conservation or both is required to solve three sets of unknowns from two sets of measurements the recently proposed flexible image domain id multi material decomposition method assumes each pixel contains at most three materials out of several possible materials and decomposes a mixture pixel by pixel we propose a penalized likelihood pl method with edge preserving regularizers for each material to reconstruct multi material images using a similar constraint from sinogram data we develop an optimization transfer method with a series of pixel wise separable quadratic surrogate pwsqs functions to monotonically decrease the complicated pl cost function the pwsqs algorithm separates pixels to allow simultaneous update of all pixels but keeps the basis materials coupled to allow faster convergence rate than our previous proposed material and pixel wise sqs algorithms comparing with the id method using 2 d fan beam simulations the pl method greatly reduced noise streak and cross talk artifacts in the reconstructed basis component images and achieved much smaller root mean square errors

generalized em type reconstruction algorithms for emission tomography we provide a general form for many reconstruction estimators of emission tomography these estimators include shepp and vardi s maximum likelihood ml estimator the quadratic weighted least squares wls estimator anderson s wls estimator and liu and wang s multi objective estimator and others we derive a generic update rule by constructing a surrogate function this work is inspired by the ml em em expectation maximization where the latter naturally arises as a special case a regularization with a specific form can also be incorporated by de pierro s trick we provide a general and quite different convergence proof compared with the proofs of the ml em and de pierro theoretical analysis shows that the proposed algorithm monotonically decreases the cost function and automatically meets nonnegativity constraints we have introduced a mechanism to provide monotonic self constraining and convergent algorithms from which some interesting existing and new algorithms can be derived simulation results illustrate the behavior of these algorithms in term of image quality and resolution noise tradeoff

effect of anatomical variability reconstruction algorithms and scattered photons on the spm output of brain pet studies statistical parametric mapping spm has become the standard technique to statistically evaluate differences between functional images the aim of this paper was to assess the effect of anatomical variability of skull the reconstruction algorithm and the scattering of photons in the brain on the output of an spm analysis of brain pet studies to this end monte carlo simulation was used to generate suitable pet sinograms and bootstrap techniques were employed to increase the reliability of the conclusions activity distribution maps were obtained by segmenting thirty nine t1 weighted magnetic resonance images foci were placed on the posterior cingulate cortex pcc and the superior temporal cortex stc and activation factors ranging between 25 and 25 were simulated preprocessing of the reconstructed images and statistical analysis were performed using spm2 our findings show that intersubject anatomical differences can cause the minimum sample size to increase between 10 and 42 for posterior cingulate cortex and between 40 and 80 for superior temporal cortex ideal scatter correction isc allowed us to diminish the sample size up to 18 and fully 3d reconstruction reduced the minimum sample size between 8 and 33 detection sensitivity was higher for hypo activation than for hyper activation situations and higher for superior temporal cortex than for posterior cingulate cortex

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

lo keng hua inequality and dynamic programming 2 where k cx n lx h n a 3 inequality 1 is a very interesting and useful as indicated in hua s inequality it appears that this inequality has not been cited as a problem or example elsewhere in an english text however ozeki and aoyagi cited it as an example on page 157 of their inequalities 9 in japanese with a proof indeed there are at least four proofs availabe for inequality 1 moreover inequality 1 can be naturally generalized and its continuous counterpart can also be given this is a major purpose of this paper we now summarize our findings in two theorems as follows

kullback proximal algorithms for maximum likelihood estimation accelerated algorithms for maximum likelihood image reconstruction are essential for emerging applications such as three dimensional 3 d tomography dynamic tomographic imaging and other high dimensional inverse problems in this paper we introduce and analyze a class of fast and stable sequential optimization methods for computing maximum likelihood estimates and study its convergence properties these methods are based on a proximal point algorithm implemented with the kullback liebler kl divergence between posterior densities of the complete data as a proximal penalty function when the proximal relaxation parameter is set to unity one obtains the classical expectation maximization em algorithm for a decreasing sequence of relaxation parameters relaxed versions of em are obtained which can have much faster asymptotic convergence without sacrifice of monotonicity we present an implementation of the algorithm using more s 1983 trust region update strategy for illustration the method is applied to a nonquadratic inverse problem with poisson distributed data