statistical analysis of maximum likelihood estimator images of human brain fdg pet studies the work presented evaluates the statistical characteristics of regional bias and expected error in reconstructions of real positron emission tomography pet data of human brain fluoro deoxiglucose fdg studies carried out by the maximum likelihood estimator mle method with a robust stopping rule and compares them with the results of filtered backprojection fbp reconstructions and with the method of sieves the task of evaluating radioisotope uptake in regions of interest rois is investigated an assessment of bias and variance in uptake measurements is carried out with simulated data then by using three different transition matrices with different degrees of accuracy and a components of variance model for statistical analysis it is shown that the characteristics obtained from real human fdg brain data are consistent with the results of the simulation studies

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

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

a region based segmentation method for multichannel image data abstract segmentation has become a widely used tool in modern image analysis typical methods are highly nonlinear and often ad hoc and none so far have proved accessible to detailed statistical analysis even by asymptotic approximations can such methods be any good this article describes a segmentation procedure for multichannel image data and attempts to develop an understanding of its statistical performance characteristics in terms of nonparametric regression the procedure is based on a recursive merging algorithm defined via a nested sequence of discretizations of the image domain a cross validation rule with a near neighbor block replacement strategy is proposed for selecting the final segmentation model idealized numerical simulation experiments are used to evaluate the rate of convergence of the mean square error of estimation and also to study the efficiency of the cross validation technique interestingly the results show that the rate of convergence tends to decrease as the degree of smo

corrections for accidental coincidences and attenuation in maximum likelihood image reconstruction for positron emission tomography reconstruction procedures that account for attenuation in forming maximum likelihood estimates of activity distributions in positron emission tomography are extended to include regularization constraints and accidental coincidences a mathematical model is used for these effects the corrections are incorporated into the iterations of an expectation maximization algorithm for numerically producing the maximum likelihood estimate of the distribution of radioactivity within a patient the images reconstructed with this procedure are unbiased and exhibit lower variance than those reconstructed from precorrected data

preconditioners for image restoration by reblurring techniques it is well known that iterative algorithms for image deblurring that involve the normal equations show usually a slow convergence a variant of the normal equations which replaces the conjugate transpose a h of the system matrix a with a new matrix is proposed this approach which is linked with regularization preconditioning theory and reblurring processes can be applied to a wide set of iterative methods here we examine landweber steepest descent richardson lucy and image space reconstruction algorithm several computational tests show that this strategy leads to a significant improvement of the convergence speed of the methods moreover it can be naturally combined with other widely used acceleration techniques

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

an expanded theoretical treatment of iteration dependent majorize minimize algorithms the majorize minimize mm optimization technique has received considerable attention in signal and image processing applications as well as in statistics literature at each iteration of an mm algorithm one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate the next iterate is obtained by minimizing this tangent majorant function resulting in a sequence of iterates that reduces the cost function monotonically a well known special case of mm methods are expectation maximization algorithms in this paper we expand on previous analyses of mm due to fessler and hero that allowed the tangent majorants to be constructed in iteration dependent ways also this paper overcomes an error in one of those earlier analyses there are three main aspects in which our analysis builds upon previous work first our treatment relaxes many assumptions related to the structure of the cost function feasible set and tangent majorants for example the cost function can be nonconvex and the feasible set for the problem can be any convex set second we propose convergence conditions based on upper curvature bounds that can be easier to verify than more standard continuity conditions furthermore these conditions allow for considerable design freedom in the iteration dependent behavior of the algorithm finally we give an original characterization of the local region of convergence of mm algorithms based on connected e g convex tangent majorants for such algorithms cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods this expanded treatment widens the scope of the mm algorithm designs that can be considered for signal and image processing applications allows us to verify the convergent behavior of previously published algorithms and gives a fuller understanding overall of how these algorithms behave

structural optimization of guyed trusses here is a strong trend in the design of skeletal structures to wherever practical make increasing use of tensioned cables one of the best examples of the convergence of structural efficiency to a best design can probably be found when studying the evolution of the configurations of power transmission towers from the early days of cantilever truss structures these have now evolved to the much lighter cross rope type in which two separate vertical cable stayed lattice struts support the power conductors through suspension from a cross rope apart from the savings in material significant savings in transportation cost can also be obtained especially since erection often has to occur in remote areas although the cable stayed structures are easy to analyze the determination of a best or least weight design can be complex due to the variety of design variables available these are typically configuration geometry member sizing and prestraining the optimization of a configuration is complex and is best accomplished manually perhaps with the aid of layout optimization methods on the contrary geometric and prestrain optimization and sizing can be treated by direct sensitivity analysis of a chosen topology this paper describes the sizing and prestrain optimization of guyed trusses for the nonlinear analysis the total lagrangian method is used in combination with the pure newton method and adaptive arc length procedures because of the lack of stiffness of a cable structure in its undeformed state soft spring elements are used to stabilize the structure temporarily during the load stepping phase the choice of prestrain instead of prestress variables is justified by both the simplicity of inclusion of prestrain in the sensitivity analysis and by the independence of the prestrain quantity with respect to the sequence of pretensioning of the structure the prestraining of a member can also simply be seen as the lack of fit of a member in a fixed geometry and can in practice be accomplished by specifying member lengths to create a predetermined lack of fit or by specifying turns of a turnbuckle included in the member for the optimization process two optimization methods are used namely the sequential quadratic programming sqp method idesign plba authored by arora1 and the successive approximation method sam algorithm of snyman and stander2 and stander et al 3 using two methods gives optima in which greater confidence can be placed additionally some idea of the degree of difficulty of the newly formulated optimization problems is gained five examples are given to demonstrate the analysis and optimization methodology the optimization of example structures subjected to second order effects show that unexpected results can be obtained a number of papers 4 8 related to the general topic of the sensitivity analysis and optimization of structures with nonlinear geometric

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

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

computing boundary slopes of 2 bridge links we describe an algorithm for computing boundary slopes of 2bridge links as an example we work out the slopes of the links obtained by 1 k surgery on one component of the borromean rings a table of all boundary slopes of all 2 bridge links with 10 or less crossings is also included

poisson intensity estimation for tomographic data using a wavelet shrinkage approach we consider a two dimensional 2 d problem of positron emission tomography pet where the random mechanism of the generation of the tomographic data is modeled by poisson processes the goal is to estimate the intensity function which corresponds to emission density using the wavelet vaguelette decomposition wvd we propose an estimator based on thresholding of empirical vaguelette coefficients which attains the minimax rates of convergence on besov function classes furthermore we construct an adaptive estimator which attains the optimal rate of convergence up to a logarithmic term

maximum likelihood expectation maximization reconstruction of sinograms with arbitrary noise distribution using nec transformations the maximum likelihood ml expectation maximization em ml em algorithm is being widely used for image reconstruction in positron emission tomography the algorithm is strictly valid if the data are poisson distributed however it is also often applied to processed sinograms that do not meet this requirement this may sometimes lead to suboptimal results streak artifacts appear and the algorithm converges toward a lower likelihood value as a remedy the authors propose two simple pixel by pixel methods noise equivalent counts nec scaling and nec shifting in order to transform arbitrary sinogram noise into noise which is approximately poisson distributed the first and second moments of the distribution match those of the poisson distribution the convergence speed associated with both transformation methods is compared and the nec scaling method is validated with both simulations and clinical data these new methods extend the ml em algorithm to a general purpose nonnegative reconstruction algorithm

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

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

new successive approximation method for optimum structural design a new procedure using only first order explicit information and consisting of the successive solution of approximate subproblems is proposed the method differs from established sequential quadratic programming methods by not utilizing the lagrangian form and not attempting to store full and explicit hessian information instead each constraint is individually approximated by a quadratic function involving only one coefficient in the quadratic term this coefficient is determined from a two point collocation a novel feature of the method is that each quadratic subproblem is solved by a recently proposed interior feasible direction method an auxiliary problem is formulated to generate a feasible starting point for each subproblem

design sensitivity analysis and optimization of nonlinear structuralresponse using incremental procedure this paper describes procedures for design sensitivity analysis of nonlinear structural response using the incremental solution scheme the total and updated lagrangian approaches are discussed sensitivity analyses for the stress strain and displacement constraints are developed in the optimization process the buckling load constraint is also imposed numerical aspects of design sensitivity analysis are discussed three numerical examples are solved to illustrate the design sensitivity analysis and optimization process the numerical procedures are stable and work well they can be applied to more complex structures to optimize them

the space of incompressible surfaces in a 2 bridge link complement projective lamination spaces for 2 bridge link complements are computed explicitly in this paper we construct a polyhedron p s3 lp q whose rational points correspond bijectively in a natural way with the projective isotopy classes of incompressible surfaces in the exterior of a 2 bridge link lp q c s3 here projective means that we factor out by scalar multiplication taking any number of parallel copies of a surface surfaces are not assumed to be connected we expect that p s3 lp q will turn out to be the projective lamination space of s3 lp q33 as defined and studied for general compact irreducible 3 manifolds in 5 and 12 an unexpected complication not present in thurston s theory of projective lamination spaces for surfaces is the fact that p s3 lp q is frequently noncompact for example for the whitehead link l3 8 see figure 5 4 upper left hand corner however as in the general theory p s3 lp q has a natural compactification p7 53 lp q which is a finite polyhedron to construct p s3 lp q 3 we first find a fairly natural finite collection of branched surfaces bi c s3 lp q which carry all the incompressible surfaces in s3lp q to each bi is associated a convex cell ci whose rational points parametrize the projective classes of surfaces carried by bi diserent rational points of ci can determine isotopic surfaces however due to the possibility of pushing parts of surfaces across product regions in the complement of bi the analogue of digon regions in the complement of a train track on a surface this leads to a linear projection pi ci ci f ci onto another convex polyhedral cell ci such that over the interior of ci projective isotopy classes of surfaces coincide with fibers of pi however these isotopy relations may not persist over the boundary of ci namely passing to a face of ci corresponds to passing to a branched subsurface of bi and a nonproduct complementary region of this branched subsurface may be decomposed by bi into product complementary regions of bi in this case rational points of this face of ci correspond not to projective isotopy classes of incompressible surfaces but to incompressible surfaces with these limiting phantom isotopy relations the space p7 53 lp q is foed from the cells ci by identifying their faces in the most natural way distinguishing different phantom isotopies between the same sets of surfaces pz s3 lp q consists of the open cells of ps s3 lp q for which the phantom isotopies are actual isotopies received by the editors april 12 1986 1980 mathematics subject classification 1985 revtaion primary 57m25

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

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

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

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

cross reference weighted least square estimates for positron emission tomography an efficient new method termed as the cross reference weighted least square estimate wlse crwlse is proposed to integrate the incomplete local smoothness information to improve the reconstruction of positron emission tomography pet images in the presence of accidental coincidence events and attenuation the algebraic reconstruction technique art is applied to this new estimate and the convergence is proved this numerical technique is based on row operations the computational complexity is only linear in the sizes of pixels and detector tubes hence it is efficient in storage and computation for a large and sparse system moreover the easy incorporation of range limits and spatially variant penalty will not deprive the efficiency all this makes the new method practically applicable an automatically data driven selection method for this new estimate based on the generalized cross validation is also studied the monte carlo studies demonstrate the advantages of this new method

totally geodesic seifert surfaces in hyperbolic knot and link complements ii the first examples of totally geodesic seifert surfaces are constructed for hyperbolic knots and links including both free and totally knotted surfaces then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces