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

exploring estimator bias variance tradeoffs using the uniform cr bound we introduce a plane which we call the delta sigma plane that is indexed by the norm of the estimator bias gradient and the variance of the estimator the norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space using a uniform cramer rao cr bound on estimator variance a delta sigma tradeoff curve is specified that defines an unachievable region of the delta sigma plane for a specified statistical model in order to place an estimator on this plane for comparison with the delta sigma tradeoff curve the estimator variance bias gradient and bias gradient norm must be evaluated we present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log likelihood we demonstrate the methods developed in this paper for linear gaussian and nonlinear poisson inverse problems

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

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

a splitting based iterative algorithm for accelerated statistical x ray ct reconstruction statistical image reconstruction using penalized weighted least squares pwls criteria can improve image quality in x ray computed tomography ct however the huge dynamic range of the statistical weights leads to a highly shift variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly we propose to alleviate the problem by using a variable splitting scheme that separates the shift variant and nearly invariant components of the statistical data model and also decouples the regularization term this leads to an equivalent constrained problem that we tackle using the classical method of multipliers framework with alternating minimization the specific form of our splitting yields an alternating direction method of multipliers admm algorithm with an inner step involving a nearly shift invariant linear system that is suitable for fft based preconditioning using cone type filters the proposed method can efficiently handle a variety of convex regularization criteria including smooth edge preserving regularizers and non smooth sparsity promoting ones based on the l1 norm and total variation numerical experiments with synthetic and real in vivo human data illustrate that cone filter preconditioners accelerate the proposed admm resulting in fast convergence of admm compared to conventional nonlinear conjugate gradient ordered subsets and state of the art mfista split bregman algorithms that are applicable for ct

edge preserving tomographic reconstruction with nonlocal regularization tomographic image reconstruction using statistical methods can provide more accurate system modeling statistical models and physical constraints than the conventional filtered backprojection fbp method because of the ill posedness of the reconstruction problem a roughness penalty is often imposed on the solution to control noise to avoid smoothing of edges which are important image attributes various edge preserving regularization methods have been proposed most of these schemes rely on information from local neighborhoods to determine the presence of edges in this paper we propose a cost function that incorporates nonlocal boundary information into the regularization method we use an alternating minimization algorithm with deterministic annealing to minimize the proposed cost function jointly estimating region boundaries and object pixel values we apply variational techniques implemented using level sets methods to update the boundary estimates then using the most recent boundary estimate we minimize a space variant quadratic cost function to update the image estimate for the positron emission tomography transmission reconstruction application we compare the bias variance tradeoff of this method with that of a conventional penalized likelihood algorithm with local huber roughness penalty

analysis of observer performance in known location tasks for tomographic image reconstruction we consider the task of detecting a statistically varying signal of known location on a statistically varying background in a reconstructed tomographic image we analyze the performance of linear observer models in this task we show that if one chooses a suitable reconstruction method a broad family of linear observers can exactly achieve the optimal detection performance attainable with any combination of a linear observer and linear reconstructor this conclusion encompasses several well known observer models from the literature including models with a frequency selective channel mechanism and certain types of internal noise interestingly the optimal reconstruction methods are unregularized and in some cases quite unconventional these results suggest that for the purposes of designing regularized reconstruction methods that optimize lesion detectability known location tasks are of limited use

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

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

sparsity constrained mixture modeling for the estimation of kinetic parameters in dynamic pet the estimation and analysis of kinetic parameters in dynamic positron emission tomography pet is frequently confounded by tissue heterogeneity and partial volume effects we propose a new constrained model of dynamic pet to address these limitations the proposed formulation incorporates an explicit mixture model in which each image voxel is represented as a mixture of different pure tissue types with distinct temporal dynamics we use cramer rao lower bounds to demonstrate that the use of prior information is important to stabilize parameter estimation with this model as a result we propose a constrained formulation of the estimation problem that we solve using a two stage algorithm in the first stage a sparse signal processing method is applied to estimate the rate parameters for the different tissue compartments from the noisy pet time series in the second stage tissue fractions and the linear parameters of different time activity curves are estimated using a combination of spatial regularity and fractional mixture constraints a block coordinate descent algorithm is combined with a manifold search to robustly estimate these parameters the method is evaluated with both simulated and experimental dynamic pet data

resolution and noise properties of map reconstruction for fully 3 d pet derives approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three dimensional 3 d positron emission tomography pet data using maximum a posteriori map estimation these expressions explicitly account for the spatially variant detector response and sensitivity of a 3 d tomograph the resulting spatially variant impulse response and covariance are computed using 3 d fourier transforms a truncated gaussian distribution is used to account for the effect on the variance of the nonnegativity constraint used in map reconstruction using monte carlo simulations and phantom data from the micropet small animal scanner the authors show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of map reconstruction for priors with quadratic energy functions they also describe how these analytical results can be used to achieve near uniform contrast recovery throughout the reconstructed volume

effect of overlapping projections on reconstruction image quality in multipinhole spect multipinhole single photon emission computed tomography spect imaging has several advantages over single pinhole spect imaging including an increased sensitivity and an improved sampling however the quest for a good design is challenging due to the large number of design parameters the effect of one of these the amount of overlap in the projection images on the reconstruction image quality is examined in this paper the evaluation of the quality is based on efficient approximations for the linearized local impulse response and the covariance in a voxel and on the bias of the reconstruction of the noiseless projection data two methods are proposed that remove the overlap in the projection image by blocking certain projection rays with the use of extra shielding between the pinhole plate and the detector also two measures to quantify the amount of overlap are suggested first the approximate method predicting the contrast to noise ratio cnr is validated using postsmoothed maximum likelihood expectation maximization mlem reconstructions with an imposed target resolution second designs with different amounts of overlap are evaluated to study the effect of multiplexing in addition the cnr of each pinhole design is also compared with that of the same design where overlap is removed third the results are interpreted with the overlap quantification measures fourth the two proposed overlap removal methods are compared from the results we can conclude that once the complete detector area has been used the extra sensitivity due to multiplexing is only able to compensate for the loss of information not to improve the cnr removing the overlap however improves the cnr the gain is most prominent in the central field of view though often at the cost of the cnr of some voxels at the edges since after overlap removal very little information is left for their reconstruction the reconstruction images provide insight in the multiplexing and truncation artifacts

regularization for uniform spatial resolution properties in penalized likelihood image reconstruction traditional space invariant regularization methods in tomographic image reconstruction using penalized likelihood estimators produce images with nonuniform spatial resolution properties the local point spread functions that quantify the smoothing properties of such estimators are space variant asymmetric and object dependent even for space invariant imaging systems the authors propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the least squares fitting of a parameterized local impulse response to a desired global response the authors have developed computationally efficient methods for pet systems with shift invariant geometric responses they demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated pet thorax scans

penalized weighted least squares image reconstruction for positron emission tomography presents an image reconstruction method for positron emission tomography pet based on a penalized weighted least squares pwls objective for pet measurements that are precorrected for accidental coincidences the author argues statistically that a least squares objective function is as appropriate if not more so than the popular poisson likelihood objective the author proposes a simple data based method for determining the weights that accounts for attenuation and detector efficiency a nonnegative successive over relaxation sor algorithm converges rapidly to the global minimum of the pwls objective quantitative simulation results demonstrate that the bias variance tradeoff of the pwls sor method is comparable to the maximum likelihood expectation maximization ml em method but with fewer iterations and is improved relative to the conventional filtered backprojection fbp method qualitative results suggest that the streak artifacts common to the fbp method are nearly eliminated by the pwls sor method and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over fbp

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

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

fast spatial resolution analysis of quadratic penalized least squares image reconstruction with separate real and imaginary roughness penalty application to fmri penalized least squares iterative image reconstruction algorithms used for spatial resolution limited imaging such as functional magnetic resonance imaging fmri commonly use a quadratic roughness penalty to regularize the reconstructed images when used for complex valued images the conventional roughness penalty regularizes the real and imaginary parts equally however these imaging methods sometimes benefit from separate penalties for each part the spatial smoothness from the roughness penalty on the reconstructed image is dictated by the regularization parameter s one method to set the parameter to a desired smoothness level is to evaluate the full width at half maximum of the reconstruction method s local impulse response previous work has shown that when using the conventional quadratic roughness penalty one can approximate the local impulse response using an fft based calculation however that acceleration method cannot be applied directly for separate real and imaginary regularization this paper proposes a fast and stable calculation for this case that also uses fft based calculations to approximate the local impulse responses of the real and imaginary parts this approach is demonstrated with a quadratic image reconstruction of fmri data that uses separate roughness penalties for the real and imaginary parts

preconditioning methods for improved convergence rates in iterative reconstructions because of the characteristics of the tomographic inversion problem iterative reconstruction techniques often suffer from poor convergence rates especially at high spatial frequencies by using preconditioning methods the convergence properties of most iterative methods can be greatly enhanced without changing their ultimate solution to increase reconstruction speed spatially invariant preconditioning filters that can be designed using the tomographic system response and implemented using 2 d frequency domain filtering techniques have been applied in a sample application reconstructions from noiseless simulated projection data were performed using preconditioned and conventional steepest descent algorithms the preconditioned methods demonstrated residuals that were up to a factor of 30 lower than the assisted algorithms at the same iteration applications of these methods to regularized reconstructions from projection data containing poisson noise showed similar although not as dramatic behavior

fast iterative image reconstruction for mri in the presence of field inhomogeneities in magnetic resonance imaging magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast fourier transform fft methods several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities but these methods assume that the field map that characterizes the off resonance frequencies is spatially smooth recently iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off resonance effects however straightforward implementations of such iterative methods suffer from inconveniently long computation times this paper describes a tool for accelerating iterative reconstruction of field corrected mr images a novel time segmented approximation to the mr signal equation we use a min max formulation to derive the temporal interpolator speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast fourier transform with normalized root mean squared approximation errors of 0 07 the proposed method provides fast accurate field corrected image reconstruction even when the field map is not smooth

evaluation of fisher information matrix based methods for fast assessment of image quality in pinhole spect the accurate determination of the local impulse response and the covariance in voxels from penalized maximum likelihood reconstructed images requires performing reconstructions from many noise realizations of the projection data as this is usually a very time consuming process efficient analytical approximations based on the fisher information matrix fim have been extensively used in pet and spect to estimate these quantities for 3d imaging however additional approximations need to be made to the fim in order to speed up the calculations the most common approach is to use the local shift invariant lsi approximation of the fim but this assumes specific conditions which are not always necessarily valid in this paper we take a single pinhole spect system and compare the accuracy of the lsi approximation against two other methods that have been more recently put forward the non uniform object space pixelation nuop and the subsampled fim these methods do not assume such restrictive conditions while still increasing the speed of the calculations considerably our results indicate that in pinhole spect the nuop and subsampled fim approaches could be more reliable than the lsi approximation especially when a high accuracy is required

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

precision and accuracy of regional radioactivity quantitation using the maximum likelihood em reconstruction algorithm the imaging characteristics of maximum likelihood ml reconstruction using the em algorithm for emission tomography have been extensively evaluated there has been less study of the precision and accuracy of ml estimates of regional radioactivity concentration the authors developed a realistic brain slice simulation by segmenting a normal subject s mri scan into gray matter white matter and csf and produced pet sinogram data with a model that included detector resolution and efficiencies attenuation scatter and randoms noisy realizations at different count levels were created and ml and filtered backprojection fbp reconstructions were performed the bias and variability of roi values were determined in addition the effects of ml pixel size image smoothing and region size reduction were assessed hit estimates at 3 000 iterations 0 6 sec per iteration on a parallel computer for 1 cm sup 2 gray matter rois showed negative biases of 6 spl plusmn 2 which can be reduced to 0 spl plusmn 3 by removing the outer 1 mm rim of each roi fbp applied to the full size rois had 15 spl plusmn 4 negative bias with 50 less noise than hit shrinking the fbp regions provided partial bias compensation with noise increases to levels similar to ml smoothing of ml images produced biases comparable to fbp with slightly less noise because of its heavy computational requirements the ml algorithm will be most useful for applications in which achieving minimum bias is important

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

theoretical study of lesion detectability of map reconstruction using computer observers the low signal to noise ratio snr in emission data has stimulated the development of statistical image reconstruction methods based on the maximum a posteriori map principle experimental examples have shown that statistical methods improve image quality compared to the conventional filtered backprojection fbp method however these results depend on isolated data sets here the authors study the lesion detectability of map reconstruction theoretically using computer observers these theoretical results can be applied to different object structures they show that for a quadratic smoothing prior the lesion detectability using the prewhitening observer is independent of the smoothing parameter and the neighborhood of the prior while the nonprewhitening observer exhibits an optimum smoothing point the authors also compare the results to those of fbp reconstruction the comparison shows that for ideal positron emission tomography pet systems where data are true line integrals of the tracer distribution the map reconstruction has a higher snr for lesion detection than fbp reconstruction due to the modeling of the poisson noise for realistic systems map reconstruction further benefits from accurately modeling the physical photon detection process in pet

optimal space varying regularization in iterative image restoration it has been shown that space variant regularization in image restoration provides better results than space invariant regularization however the optimal choice of the regularization parameter is usually unknown a priori in previous work the generalized cross validation gcv criterion was shown to provide accurate estimates of the optimal regularization parameter the author introduces a modified form of the gcv criterion that incorporates space variant regularization and data error terms furthermore he presents an efficient method for estimating the gcv criterion for the space variant case using iterative image restoration techniques this method performs nearly as well as the exact criterion for the image restoration problem in addition he proposes a wiener filter interpretation for choosing the local weighting of the regularization this interpretation suggests the use of a multistage estimation procedure to estimate the optimal choice of the local regularization weights experiments confirm the value of the modified gcv estimation criterion as well as the multistage procedure for estimating the local regularization weights

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

stopping rule for the mle algorithm based on statistical hypothesis testing it is known that when the maximum likelihood estimator mle algorithm passes a certain point it produces images that begin to deteriorate we propose a quantitative criterion with a simple probabilistic interpretation that allows the user to stop the algorithm just before this effect begins the mle algorithm searches for the image that has the maximum probability to generate the projection data the underlying assumption of the algorithm is a poisson distribution of the data therefore the best image according to the mle algorithm is the one that results in projection means which are as close to the data as possible it is shown that this goal conflicts with the assumption that the data are poisson distributed we test a statistical hypothesis whereby the projection data could have been generated by the image produced after each iteration the acceptance or rejection of the hypothesis is based on a parameter that decreases as the images improve and increases as they deteriorate we show that the best mle images which pass the test result in somewhat lower noise in regions of high activity than the filtered back projection results and much improved images in low activity regions the applicability of the proposed stopping rule to other iterative schemes is discussed

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

task driven optimization of fluence field and regularization for model based iterative reconstruction in computed tomography this paper presents a joint optimization of dynamic fluence field modulation ffm and regularization in quadratic penalized likelihood reconstruction that maximizes a task based imaging performance metric we adopted a task driven imaging framework for prospective designs of the imaging parameters a maxi min objective function was adopted to maximize the minimum detectability index d throughout the image the optimization algorithm alternates between ffm represented by low dimensional basis functions and local regularization including the regularization strength and directional penalty weights the task driven approach was compared with three ffm strategies commonly proposed for fbp reconstruction as well as a task driven tcm strategy for a discrimination task in an abdomen phantom the task driven ffm assigned more fluence to less attenuating anteroposterior views and yielded approximately constant fluence behind the object the optimal regularization was almost uniform throughout image furthermore the task driven ffm strategy redistribute fluence across detector elements in order to prescribe more fluence to the more attenuating central region of the phantom compared with all strategies the task driven ffm strategy not only improved minimum d by at least 17 8 but yielded higher d over a large area inside the object the optimal ffm was highly dependent on the amount of regularization indicating the importance of a joint optimization sample reconstructions of simulated data generally support the performance estimates based on computed d the improvements in detectability show the potential of the task driven imaging framework to improve imaging performance at a fixed dose or equivalently to provide a similar level of performance at reduced dose

fast variance prediction for iteratively reconstructed ct images with locally quadratic regularization predicting noise properties of iteratively reconstructed ct images is useful for analyzing reconstruction methods for example local noise power spectrum nps predictions may be used to quantify the detectability of an image feature to design regularization methods or to determine dynamic tube current adjustment during a ct scan this paper presents a method for fast prediction of reconstructed image variance and local nps for statistical reconstruction methods using quadratic or locally quadratic regularization previous methods either require impractical computation times to generate an approximate map of the variance of each reconstructed voxel or are restricted to specific ct geometries our method can produce a variance map of the entire image for locally shift invariant ct geometries with sufficiently fine angular sampling using a computation time comparable to a single back projection the method requires only the projection data to be used in the reconstruction not a reconstruction itself and is reasonably accurate except near image edges where edge preserving regularization behaves highly nonlinearly we evaluate the accuracy of our method using reconstructions of both simulated ct data and real ct scans of a thorax phantom

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

analysis of resolution and noise properties of nonquadratically regularized image reconstruction methods for pet we present accurate and efficient methods for estimating the spatial resolution and noise properties of nonquadratically regularized image reconstruction for positron emission tomography pet it is well known that quadratic regularization tends to over smooth sharp edges many types of edge preserving nonquadratic penalties have been proposed to overcome this problem however there has been little research on the quantitative analysis of nonquadratic regularization due to its nonlinearity in contrast quadratically regularized estimators are approximately linear and are well understood in terms of resolution and variance properties we derive new approximate expressions for the linearized local perturbation response llpr and variance using the taylor expansion with the remainder term although the expressions are implicit we can use them to accurately predict resolution and variance for nonquadratic regularization where the conventional expressions based on the first order taylor truncation fail they also motivate us to extend the use of a certainty based modified penalty to nonquadratic regularization cases in order to achieve spatially uniform perturbation responses analogous to uniform spatial resolution in quadratic regularization finally we develop computationally efficient methods for predicting resolution and variance of nonquadratically regularized reconstruction and present simulations that illustrate the validity of these methods

a new convex edge preserving median prior with applications to tomography in a bayesian tomographic maximum a posteriori map reconstruction an estimate of the object f is computed by iteratively minimizing an objective function that typically comprises the sum of a log likelihood data consistency term and prior or penalty term the prior can be used to stabilize the solution and to also impose spatial properties on the solution one such property preservation of edges and locally monotonic regions is captured by the well known median root prior mrp an empirical method that has been applied to emission and transmission tomography we propose an entirely new class of convex priors that depends on f and also on m an auxiliary field in register with f we specialize this class to our median prior mp the approximate action of the median prior is to draw at each iteration an object voxel toward its own local median this action is similar to that of mrp and results in solutions that impose the same sorts of object properties as does mrp our map method is not empirical since the problem is stated completely as the minimization of a joint on f and m objective we propose an alternating algorithm to compute the joint map solution and apply this to emission tomography showing that the reconstructions are qualitatively similar to those obtained using mrp

emission image reconstruction for randoms precorrected pet allowing negative sinogram values most positron emission tomography pet emission scans are corrected for accidental coincidence ac events by real time subtraction of delayed window coincidences leaving only the randoms precorrected data available for image reconstruction the real time randoms precorrection compensates in mean for ac events but destroys the poisson statistics the exact log likelihood for randoms precorrected data is inconvenient so practical approximations are needed for maximum likelihood or penalized likelihood image reconstruction conventional approximations involve setting negative sinogram values to zero which can induce positive systematic biases particularly for scans with low counts per ray we propose new likelihood approximations that allow negative sinogram values without requiring zero thresholding with negative sinogram values the log likelihood functions can be nonconcave complicating maximization nevertheless we develop monotonic algorithms for the new models by modifying the separable paraboloidal surrogates and the maximum likelihood expectation maximization ml em methods these algorithms ascend to local maximizers of the objective function analysis and simulation results show that the new shifted poisson sp model is nearly free of systematic bias yet keeps low variance despite its simpler implementation the new sp performs comparably to the saddle point model which has shown the best performance as to systematic bias and variance in randoms precorrected pet emission reconstruction

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

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

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

topological charge of lattice gauge fields using recently derived explicit formulae for the 2 and 3 cochains in su 2 gauge theory we are able to integrate the chern simons density analytically we arrive in su 2 at a local algebraic expression for the topological charge which is the sum of local winding numbers associated with the corners lattice points of the cells covering the manifold plus contributions from possible isolated gauge singularities which manifest themselves as vortices in the 1 2 or 3 cochains among others we consider hypercubic geometry i e covering the manifold by hypercubes which is of particular interest to lattice monte carlo applications finally we extend our results to su 3 gauge theory

fast predictions of variance images for fan beam transmission tomography with quadratic regularization accurate predictions of image variances can be useful for reconstruction algorithm analysis and for the design of regularization methods computing the predicted variance at every pixel using matrix based approximations is impractical even most recently adopted methods that are based on local discrete fourier approximations are impractical since they would require a forward and backprojection and two fast fourier transform fft calculations for every pixel particularly for shift variant systems like fan beam tomography this paper describes new analytical approaches to predicting the approximate variance maps of 2 d images that are reconstructed by penalized likelihood estimation with quadratic regularization in fan beam geometries the simplest of the proposed analytical approaches requires computation equivalent to one backprojection and some summations so it is computationally practical even for the data sizes in x ray computed tomography ct simulation results show that it gives accurate predictions of the variance maps the parallel beam geometry is a simple special case of the fan beam analysis the analysis is also applicable to 2 d positron emission tomography pet

statistic based map image reconstruction from poisson data using gibbs priors a statistical method for selecting the gibbs parameter in map image restoration from poisson data using gibbs priors is presented the gibbs parameter determines the degree to which the prior influences the restoration the presented method yields a map restored image minimally influenced by the prior for which a statistic falls within an appropriate confidence interval the method assumes that a close approximation to the blurring function is known a simple iterative feedback algorithm is presented to statistically select the parameter as the map image restoration is being performed this algorithm is heuristically based on a model reference control formulation but it requires only a minimal number of iterations for the parameter to settle to its statistically specified value the performance of the statistical method for selecting the prior parameter and that of the iterative feedback algorithm are demonstrated using both 2 d and 3 d images

a local update strategy for iterative reconstruction from projections a method for bayesian reconstruction which relies on updates of single pixel values rather than the entire image at each iteration is presented the technique is similar to gauss seidel gs iteration for the solution of differential equations on finite grids the computational cost per iteration of the gs approach is found to be approximately equal to that of gradient methods for continuously valued images gs is found to have significantly better convergence at modes representing high spatial frequencies in addition gs is well suited to segmentation when the image is constrained to be discretely valued it is shown that bayesian segmentation using gs iteration produces useful estimates at much lower signal to noise ratios than required for continuously valued reconstruction the convergence properties of gradient ascent and gs for reconstruction from integral projections are analyzed and simulations of both maximum likelihood and maximum a posteriori cases are included

efficient calculation of resolution and covariance for penalized likelihood reconstruction in fully 3 d spect resolution and covariance predictors have been derived previously for penalized likelihood estimators these predictors can provide accurate approximations to the local resolution properties and covariance functions for tomographic systems given a good estimate of the mean measurements although these predictors may be evaluated iteratively circulant approximations are often made for practical computation times however when numerous evaluations are made repeatedly as in penalty design or calculation of variance images these predictors still require large amounts of computing time in stayman and fessler 2000 we discussed methods for precomputing a large portion of the predictor for shift invariant system geometries in this paper we generalize the efficient procedure discussed in stayman and fessler 2000 to shift variant single photon emission computed tomography spect systems this generalization relies on a new attenuation approximation and several observations on the symmetries in spect systems these new general procedures apply to both two dimensional and fully three dimensional 3 d spect models that may be either precomputed and stored or written in procedural form we demonstrate the high accuracy of the predictions based on these methods using a simulated anthropomorphic phantom and fully 3 d spect system the evaluation of these predictors requires significantly less computation time than traditional prediction techniques once the system geometry specific precomputations have been made

proximal admm for multi channel image reconstruction in spectral x ray ct the development of spectral x ray computed tomography ct using binned photon counting detectors has received great attention in recent years and has enabled selective imaging of contrast agents loaded with k edge materials a practical issue in implementing this technique is the mitigation of the high noise levels often present in material decomposed sinogram data in this work the spectral x ray ct reconstruction problem is formulated within a multi channel mc framework in which statistical correlations between the decomposed material sinograms can be exploited to improve image quality specifically a mc penalized weighted least squares pwls estimator is formulated in which the data fidelity term is weighted by the mc covariance matrix and sparsity promoting penalties are employed this allows the use of any number of basis materials and is therefore applicable to photon counting systems and k edge imaging to overcome numerical challenges associated with use of the full covariance matrix as a data fidelity weight a proximal variant of the alternating direction method of multipliers is employed to minimize the mc pwls objective function computer simulation and experimental phantom studies are conducted to quantitatively evaluate the proposed reconstruction method

spatial resolution properties of penalized likelihood image reconstruction space invariant tomographs this paper examines the spatial resolution properties of penalized likelihood image reconstruction methods by analyzing the local impulse response the analysis shows that standard regularization penalties induce space variant local impulse response functions even for space invariant tomographic systems paradoxically for emission image reconstruction the local resolution is generally poorest in high count regions we show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections this analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution the modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized likelihood methods

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

fourier based forward and back projectors in iterative fan beam tomographic image reconstruction fourier based forward and back projection methods can reduce computation in iterative tomographic image reconstruction recently an optimized nonuniform fast fourier transform nufft approach was shown to yield accurate parallel beam projections in this paper we extend the nufft approach to describe an o n sup 2 log n projector backprojector pair for fan beam transmission tomography simulations and experiments with real ct data show that fan beam fourier based forward and back projection methods can reduce computation for iterative reconstruction while still providing accuracy comparable to their o n sup 3 space based counterparts

regularized field map estimation in mri in fast magnetic resonance mr imaging with long readout times such as echo planar imaging epi and spiral scans it is important to correct for the effects of field inhomogeneity to reduce image distortion and blurring such corrections require an accurate field map a map of the off resonance frequency at each voxel standard field map estimation methods yield noisy field maps particularly in image regions with low spin density this paper describes regularized methods for field map estimation from two or more mr scans having different echo times these methods exploit the fact that field maps are generally smooth functions the methods use algorithms that decrease monotonically a regularized least squares cost function even though the problem is highly nonlinear results show that the proposed regularized methods significantly improve the quality of field map estimates relative to conventional unregularized methods

iterative image reconstruction using inverse fourier rebinning for fully 3 d pet we describe a fast forward and back projector pair based on inverse fourier rebinning for use in iterative image reconstruction for fully 3 d positron emission tomography pet the projector pair is used as part of a factored system matrix that takes into account detector pair response by using shift variant sinogram blur kernels thereby combining the computational advantages of fourier rebinning with iterative reconstruction using accurate system models the forward projector consists of a 2 d projector which maps 3 d images into 2 d direct sinograms followed by exact inverse rebinning which maps the 2 d into fully 3 d sinograms the back projector is implemented as the transpose of the forward projector and differs from the true exact rebinning operator in the sense that it does not require reprojection to compute missing lines of response lors we compensate for two types of inaccuracies that arise in a cylindrical pet scanner when using inverse fourier rebinning 1 nonuniform radial sampling and 2 nonconstant oblique angles in the radial direction in a single oblique sinogram we examine the effects of these corrections on sinogram accuracy and reconstructed image quality we evaluate performance of the new projector pair for maximum a posteriori map reconstruction of simulated and in vivo data the new projector results in only a small loss in resolution towards the edge of the field of view when compared to the fully 3 d geometric projector and requires an order of magnitude less computation

penalized likelihood estimators and noise analysis for randoms precorrected pet transmission scans this paper analyzes and compares image reconstruction methods based on practical approximations to the exact log likelihood of randoms precorrected positron emission tomography pet measurements the methods apply to both emission and transmission tomography however in this paper the authors focus on transmission tomography the results of experimental pet transmission scans and variance approximations demonstrate that the shifted poisson sp method avoids the systematic bias of the conventional data weighted least squares wls method and leads to significantly lower variance than conventional statistical methods based on the log likelihood of the ordinary poisson of model the authors develop covariance approximations to analyze the propagation of noise from attenuation maps into emission images via the attenuation correction factors acf s empirical pixel and region variances from real transmission data agree closely with the analytical predictions both the approximations and the empirical results show that the performance differences between the op model and sp model are even larger when considering noise propagation from the transmission images into the final emission images than the differences in the attenuation maps themselves

diffusion and fracton dimensionality on fractals and on percolation clusters the authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions the results for the fractal support the rammal and toulouse proposition 1983 that ds n dn varies as b n s n the results for percolation are in excellent agreement with the alexander and orbach conjecture 1982 that the fracton dimensionality d 4 3

penalized likelihood sinogram restoration for computed tomography we formulate computed tomography ct sinogram preprocessing as a statistical restoration problem in which the goal is to obtain the best estimate of the line integrals needed for reconstruction from the set of noisy degraded measurements ct measurement data are degraded by a number of factors including beam hardening and off focal radiation that produce artifacts in reconstructed images unless properly corrected currently such effects are addressed by a sequence of sinogram preprocessing steps including deconvolution corrections for off focal radiation that have the potential to amplify noise noise itself is generally mitigated through apodization of the reconstruction kernel which effectively ignores the measurement statistics although in high noise situations adaptive filtering methods that loosely model data statistics are sometimes applied as an alternative we present a general imaging model relating the degraded measurements to the sinogram of ideal line integrals and propose to estimate these line integrals by iteratively optimizing a statistically based objective function we consider three different strategies for estimating the set of ideal line integrals one based on direct estimation of ideal monochromatic line integrals that have been corrected for single material beam hardening one based on estimation of ideal polychromatic line integrals that can be readily mapped to monochromatic line integrals and one based on estimation of ideal transmitted intensities from which ideal monochromatic line integrals can be readily estimated the first two approaches involve maximization of a penalized poisson likelihood objective function while the third involves minimization of a quadratic penalized weighted least squares pwls objective applied in the transmitted intensity domain we find that at low exposure levels typical of those being considered for screening ct the poisson likelihood based approaches outperform the pwls objective as well as a standard approach based on adaptive filtering followed by deconvolution at higher exposure levels the approaches all perform similarly

alternating direction method of multiplier for tomography with nonlocal regularizers the ordered subset expectation maximization osem algorithm approximates the gradient of a likelihood function using a subset of projections instead of using all projections so that fast image reconstruction is possible for emission and transmission tomography such as spect pet and ct however osem does not significantly accelerate reconstruction with computationally expensive regularizers such as patch based nonlocal nl regularizers because the regularizer gradient is evaluated for every subset we propose to use variable splitting to separate the likelihood term and the regularizer term for penalized emission tomographic image reconstruction problem and to optimize it using the alternating direction method of multiplier admm we also propose a fast algorithm to optimize the admm parameter based on convergence rate analysis this new scheme enables more sub iterations related to the likelihood term we evaluated our admm for 3 d spect image reconstruction with a patch based nl regularizer that uses the fair potential function our proposed admm improved the speed of convergence substantially compared to other existing methods such as gradient descent em and osem using de pierro s approach and the limited memory broyden fletcher goldfarb shanno algorithm

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

nonparametric regression sinogram smoothing using a roughness penalized poisson likelihood objective function the authors develop and investigate an approach to tomographic image reconstruction in which nonparametric regression using a roughness penalized poisson likelihood objective function is used to smooth each projection independently prior to reconstruction by unapodized filtered backprojection fbp as an added generalization the roughness penalty is expressed in terms of a monotonic transform known as the link function of the projections the approach is compared to shift invariant projection filtering through the use of a hanning window as well as to a related nonparametric regression approach that makes use of an objective function based on weighted least squares wls rather than the poisson likelihood the approach is found to lead to improvements in resolution noise tradeoffs over the hanning filter as well as over the wls approach the authors also investigate the resolution and noise effects of three different link functions the identity square root and logarithm links the choice of link function is found to influence the resolution uniformity and isotropy properties of the reconstructed images in particular in the case of an idealized imaging system with intrinsically uniform and isotropic resolution the choice of a square root link function yields the desirable outcome of essentially uniform and isotropic resolution in reconstructed images with noise performance still superior to that of the hanning filter as well as that of the wls approach

theoretical analysis of penalized maximum likelihood patlak parametric image reconstruction in dynamic pet for lesion detection detecting cancerous lesions is a major clinical application of emission tomography in a previous work we studied penalized maximum likelihood pml image reconstruction for lesion detection in static pet here we extend our theoretical analysis of static pet reconstruction to dynamic pet we study both the conventional indirect reconstruction and direct reconstruction for patlak parametric image estimation in indirect reconstruction patlak parametric images are generated by first reconstructing a sequence of dynamic pet images and then performing patlak analysis on the time activity curves tacs pixel by pixel in direct reconstruction patlak parametric images are estimated directly from raw sinogram data by incorporating the patlak model into the image reconstruction procedure pml reconstruction is used in both the indirect and direct reconstruction methods we use a channelized hotelling observer cho to assess lesion detectability in patlak parametric images simplified expressions for evaluating the lesion detectability have been derived and applied to the selection of the regularization parameter value to maximize detection performance the proposed method is validated using computer based monte carlo simulations good agreements between the theoretical predictions and the monte carlo results are observed both theoretical predictions and monte carlo simulation results show the benefit of the indirect and direct methods under optimized regularization parameters in dynamic pet reconstruction for lesion detection when compared with the conventional static pet reconstruction