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

a numerical method for a class of non linear integro differential equations on the half line we design and analyse a numerical method for the solution of the following second order integro differential boundary value problem n y g y email protected 0 k x g x dx d y g y p y g 0 0 limy g y 0 which arises in the study of the kinetic theory of dusty plasmas the method we propose represents a first insight into the numerical solution of more complicated problems and consists of a discretization of the differential and integral terms and of an iteration process to solve the resulting non linear system under suitable hypotheses we prove the convergence we will show the characteristics of the method by means of some numerical simulations

analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables a study is carried out on nonlinear multimodal galloping of suspended cables a consistent model of a curved cable beam geometrically nonlinear and able to torque recently formulated by the authors is used the model accounts for quasi steady aerodynamic forces including the effect of static swing of the cable and dynamic twist of the cross section complementary solution methods are employed namely finite difference and galerkin spatial discretization followed by numerical time integration or galerkin spatial discretization in conjunction with multiple scale perturbation analysis the different techniques are applied to a cable close to the first cross over point at which a number of internal resonances exist branches of periodic solutions and their stability are evaluated as functions of wind velocity the existence of branches of quasi periodic solutions originating from narrow unstable intervals and propagating elsewhere is also proved qualitative and quantitative results furnished by the different investigation tools are compared among them and the importance of the various components of motion accounted or neglected in the reduced models is discussed

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

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

non linear galloping of sagged cables in 1 2 internal resonance the aeroelastic behaviour of a flexible elastic suspended cable driven by mean wind speed blowing perpendicularly to the cable s plane is investigated by applying the galerkin procedure to the partial differential equations of motion and using an in plane and an out of plane mode as shape functions a two d o f model is derived the discrete equations are coupled through quadratic and cubic terms arising both from geometric and aerodynamic effects the associated linear frequencies are assumed to be in an almost 1 2 ratio so that internal resonance occurs the multiple scale perturbation method is employed to obtain a set of three amplitude modulation equations whose coefficients depend on the mean wind speed which is assumed as control parameter two perturbative solutions are developed each based on a different assumption about the order of magnitude of the static displacements produced by steady state wind forces analytical results are then compared with direct numerical integrations of discrete non linear equations by performing a bifurcation analysis the existence of several equilibrium branches is proved the relative importance of geometric and aerodynamic non linearities is discussed through simplified models the influence on critical and postcritical behaviour of several parameters including geometrical cable parameters detuning and non symmetric flow effects is investigated the important role played by the steady state forces is highlighted

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

an iterative algorithm for x ray ct fluoroscopy x ray computed tomography fluoroscopy ctf enables image guidance of interventions synchronization of scanning with contrast bolus arrival and motion analysis however filtered backprojection fb the current method for ctf image reconstruction is subject to motion and metal artifacts from implants needles or other surgical instruments reduced target lesion conspicuity may result from increased image noise associated with reduced tube current in this report the authors adapt the row action expectation maximization em algorithm for ctf because time dependent variation in images is localized during ctf the row action em like algorithm allows rapid convergence more importantly this iterative ctf algorithm has fewer metal artifacts and better low contrast performance than fb

compact kahler manifolds with nonpositive bisectional curvature in this article we prove that for any compact kahler manifold mn with real analytic metric and nonpositive bisectional curvature there exists a finite cover m of m such that m is a holomorphic and metric fiber bundle over a compact kahler manifold n with nonpositive bisectional curvature and c1 n 0 and the fiber is a flat complex torus this partially confirms a conjecture of yau

on the numerical solution of some nonlinear and nonlocal boundary value problems the modeling of various physical questions often leads to nonlinear boundary value problems involving a nonlocal operator which depends on the unknown function in the entire domain rather than at a single point in order to answer an open question posed by j r cannon and d j galiffa we study the numerical solution of a special class of nonlocal nonlinear boundary value problems which involve the integral of the unknown solution over the integration domain starting from cannon and galiffa s results we provide other sufficient conditions for the unique solvability and a more general convergence theorem moreover we suggest different iterative procedures to handle the nonlocal nonlinearity of the discrete problem and show their performances by some numerical tests

choice of initial conditions in the ml reconstruction of fan beam transmission with truncated projection data the authors investigate the effects of initial conditions in the iterative maximum likelihood ml reconstruction of fan beam transmission projection data with truncation in an iterative ml reconstruction the estimate of the transmission reconstructed image in the previous iteration is multiplied by some factors to obtain the current estimate normally a flat initial condition fic or an image with equal positive pixel values is used as initial condition for an ml reconstruction usage of fic has also been perceived as a way of preventing any bias on the reconstruction which may have come from the initial condition when projection data have truncation the authors show that using are fic in an ml iterative reconstruction can introduce a bias to the reconstruction inside the densely sampled region dsr whose projection data have no truncation at any angle to reduce this bias the authors propose to use the largest right singular vector lrsv of the system matrix as the initial condition and demonstrate that the bias can be reduced with the lrsv when data truncation is reduced the lrsv approaches the fic this result does not contradict to the use of fic when projection data are not truncated the authors also demonstrate that the reconstructed transmission image using lrsv as initial condition provides a more accurate attenuation coefficient distribution than that using fic however the improvement is mostly in the area outside the dsr

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

splittings and cr structures for manifolds with nonpositive sectional curvature let m n denote the universal covering space of a compact riemannian manifold m n with sectional curvature 1 k mn 0 we show that a collection of deck transformations of m n satisfying certain metric dependent conditions determines an open dense subset of m n at every point of which there exists a local isometric splitting with nontrivial flat factor such a collection which we call an abelian structure also gives rise to an essentially canonical cr structure in the sense of buyalo i e an atlas for an injective f structure for which additional conditions hold it follows in particular that the minimal volume of m n vanishes we show that an abelian structure exists if the injectivity radius at all points of m n is less than e n 0 this yields a conjecture of buyalo as well as a strengthened version of the conclusion of gromov s gap conjecture in our special situation in addition we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric

on the numerical solution of a class of nonstandard sturm liouville boundary value problems the paper deals with the numerical solution of a nonstandard sturm liouville boundary value problem on the half line where the coefficients of the differential terms depend on the unknown function by means of a scalar integral operator by using a finite difference discretization a truncated quadrature rule and an iterative procedure we construct a numerical method whose convergence is proved the order of convergence and the truncation at infinity are also discussed finally some numerical tests are given to show the performance of the method

accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm iterative maximum likelihood ml transmission computed tomography algorithms have distinct advantages over fourier based reconstruction but unfortunately require increased computation time the convex algorithm is a relatively fast iterative ml algorithm but it is nevertheless too slow for many applications therefore an acceleration of this algorithm by using ordered subsets of projections is proposed ordered subsets convex algorithm osc osc applies the convex algorithm sequentially to subsets of projections osc was compared with the convex algorithm using simulated and physical thorax phantom data reconstructions were performed for osc using eight and 16 subsets eight and four projections subset respectively global errors image noise contrast recovery and likelihood increase were calculated results show that osc is faster than the convex algorithm the amount of acceleration being approximately proportional to the number of subsets in osc and it causes only a slight increase of noise and global errors in the reconstructions images and image profiles of the reconstructions were in good agreement in conclusion osc and the convex algorithm result in similar image quality but osc is more than an order of magnitude faster

pet regularization by envelope guided conjugate gradients the authors propose a new way to iteratively solve large scale ill posed problems and in particular the image reconstruction problem in positron emission tomography by exploiting the relation between tikhonov regularization and multiobjective optimization to obtain iteratively approximations to the tikhonov l curve and its corner monitoring the change of the approximate l curves allows the authors to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration so that the desired solution can be reconstructed with a small number of iterations

on a class of nonlocal elliptic problems via galerkin method in this paper we study existence and uniqueness of solutions to some cases of the following nonlocal elliptic problem u g x u f x u with zero dirichlet boundary conditions on a bounded and smooth domain of rn and also when rn where and are real constants

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

a theoretical study of some maximum likelihood algorithms for emission and transmission tomography this paper has the dual purpose of introducing some new algorithms for emission and transmission tomography and proving mathematically that these algorithms and related antecedent algorithms converge like the em algorithms for positron single photon and transmission tomography the algorithms provide maximum likelihood estimates of pixel concentration or linear attenuation parameters one particular innovation we discuss is a computationally practical scheme for modifying the em algorithms to include a bayesian prior the bayesian versions of the em algorithms are shown to have superior convergence properties in a vicinity of the maximum we anticipate that some of the other algorithms will also converge faster than the em algorithms

on the convergence of an em type algorithm for penalized likelihood estimation in emission tomography recently we proposed an extension of the expectation maximization em algorithm that was able to handle regularization terms in a natural way although very general convergence proofs were not valid for many possibly useful regularizations we present here a simple convergence result that is valid assuming only continuous differentiability of the penalty term and can be also extended to other methods for penalized likelihood estimation in tomography

on nonpositively curved compact riemannian manifolds with degenerate ricci tensor in this article we prove that if the ricci tensor of a compact nonpositively curved manifold m n is nowhere negative definite then it admits local flat factors to be more precise let u c m be the open subset where the ricci tensor has maximum rank r then for any connected component u a of u its universal cover u a is isometric to r n r n r a where n r a is a nonpositively curved manifold with negative ricci curvature in particular if m n is compact nonpositively curved without euclidean de rham factor and both the manifold and the metric are real analytic then its ricci tensor must be negative definite somewhere

an interior point iterative maximum likelihood reconstruction algorithm incorporating upper and lower bounds with application to spect transmission imaging the algorithm we consider here is a block iterative or ordered subset version of the inferior point algorithm for transmission reconstruction our algorithm is an interior point method because each vector of the iterative sequence x sup k k 0 1 2 satisfies the constraints a sub j x sub j sup k b sub j j 1 j because it is a block iterative algorithm that reconstructs the transmission attenuation map and places constraints above and below the pixel values of the reconstructed image we call it the bitab method computer simulations using the three dimensional mathematical cardiac and torso phantom reveal that the bitab algorithm in conjunction with reasonably selected prior upper and lower bounds has the potential to improve the accuracy of the reconstructed attenuation coefficients from truncated fan beam transmission projections by suitably selecting the bounds it is possible to restrict the over estimation of coefficients outside the fully sampled region that results from reconstructing truncated fan beam projections with iterative transmission algorithms such as the maximum likelihood gradient type algorithm

hybrid poisson polynomial objective functions for tomographic image reconstruction from transmission scans this paper describes rapidly converging algorithms for computing attenuation maps from poisson transmission measurements using penalized likelihood objective functions we demonstrate that an under relaxed cyclic coordinate ascent algorithm converges faster than the convex algorithm of lange see ibid vol 4 no 10 p 1430 1438 1995 which in turn converges faster than the expectation maximization em algorithm for transmission tomography to further reduce computation one could replace the log likelihood objective with a quadratic approximation however we show with simulations and analysis that the quadratic objective function leads to biased estimates for low count measurements therefore we introduce hybrid poisson polynomial objective functions that use the exact poisson log likelihood for detector measurements with low counts but use computationally efficient quadratic or cubic approximations for the high count detector measurements we demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias

shape based regularization of electron tomographic reconstruction we introduce a tomographic reconstruction method implemented using a shape based regularization technique spatial models of known features in the structure being reconstructed are integrated into the reconstruction process as regularizers our regularization scheme is driven locally through shape information obtained from segmentation and compared with a known spatial model we demonstrated our method on tomography data from digital phantoms simulated data and experimental electron tomography et data of virus complexes our reconstruction showed reduced blurring and an improvement in the resolution of the reconstructed volume was also measured this method also produced improved demarcation of spike boundaries in viral membranes when compared with popular techniques like weighted back projection and the algebraic reconstruction technique improved et reconstructions will provide better structure elucidation and improved feature visualization which can aid in solving key biological issues our method can also be generalized to other tomographic modalities

fixed point iterations for a class of nonstandard sturm liouville boundary value problems abstract the paper examines a particular class of nonlinear integro differential equations consisting of a sturm liouville boundary value problem on the half line where the coefficient of the differential term depends on the unknown function by means of a scalar integral operator in order to handle the nonlinearity of the problem we consider a fixed point iteration procedure which is based on considering a sequence of classical sturm liouville boundary value problems in the weak solution sense the existence of a solution and the global convergence of the fixed point iterations are stated without resorting to the banach fixed point theorem moreover the unique solvability of the problem is discussed and several examples with unique and non unique solutions are given

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

on a numerical method for a homogeneous nonlinear nonlocal elliptic boundary value problem abstract in this work we develop a numerical method for the equation 0 1 u t d t u x u x 2 n 1 0 x 0 1 u 0 a u 1 b we begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the schauder fixed point theorem from this analysis we then prove the existence and uniqueness to the problem above by defining a continuous compact mapping utilizing the a priori estimates and the brouwer fixed point theorem next we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique and that the numerical procedure converges with error h we conclude with some examples of the numerical process

accelerated penalized weighted least squares and maximum likelihood algorithms for reconstructing transmission images from pet transmission data we present penalized weighted least squares pwls and penalized maximum likelihood pml methods for reconstructing transmission images from positron emission tomography transmission data first we view the problem of minimizing the weighted least squares wls and maximum likelihood objective functions as a sequence of nonnegative least squares minimization problems this viewpoint follows from using certain quadratic functions as surrogate functions for the wls and maximum likelihood objective functions second we construct surrogate functions for a class of penalty functions that yield closed form expressions for the iterates of the pwls and pml algorithms due to the slow convergence of the pwls and pml algorithms accelerated versions of them are developed that are theoretically guaranteed to monotonically decrease their respective objective functions in experiments using real phantom data the pml images produced the most accurate attenuation correction factors on the other hand the pwls images produced images with the highest levels of contrast for low count data

globally convergent algorithms for maximum a posteriori transmission tomography this paper reviews and compares three maximum likelihood algorithms for transmission tomography one of these algorithms is the em algorithm one is based on a convexity argument devised by de pierro see ieee trans med imaging vol 12 p 328 333 1993 in the context of emission tomography and one is an ad hoc gradient algorithm the algorithms enjoy desirable local and global convergence properties and combine gracefully with bayesian smoothing priors preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the em algorithm this superiority stems from the larger number of exponentiations required by the em algorithm the convex and gradient algorithms are well adapted to parallel computing

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

on the relation between the isra and the em algorithm for positron emission tomography the image space reconstruction algorithm isra was proposed as a modification of the expectation maximization em algorithm based on physical considerations for application in volume emission computered tomography as a consequence of this modification isra searches for least squares solutions instead of maximizing poisson likelihoods as the em algorithm it is shown that both algorithms may be obtained from a common mathematical framework this fact is used to extend isra for penalized likelihood estimates

majorization minimization algorithms for wavelet based image restoration standard formulations of image signal deconvolution under wavelet based priors regularizers lead to very high dimensional optimization problems involving the following difficulties the non gaussian heavy tailed wavelet priors lead to objective functions which are nonquadratic usually nondifferentiable and sometimes even nonconvex the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet based denoising this paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems placing them in a common majorization minimization mm framework one of the classes of algorithms considered when using quadratic bounds on nondifferentiable log priors shares the infamous singularity issue si of iteratively re weighted least squares irls algorithms the possibility of having to handle infinite weights which may cause both numerical and convergence issues in this paper we prove several new results which strongly support the claim that the si does not compromise the usefulness of this class of algorithms exploiting the unified mm perspective we introduce a new algorithm resulting from using bounds for nonconvex regularizers the experiments confirm the superior performance of this method when compared to the one based on quadratic majorization finally an experimental comparison of the several algorithms reveals their relative merits for different standard types of scenarios

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

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

a gradient algorithm locally equivalent to the em algorithm in many problems of maximum likelihood estimation it is impossible to carry out either the e step or the m step of the em algorithm the present paper introduces a gradient algorithm that is closely related to the em algorithm this em gradient algorithm approximately solves the m step of the em algorithm by one iteration of newton s method since newton s method converges quickly the local properties of the em gradient algorithm are almost identical with those of the em algorithm any strict local maximum point of the observed likelihood locally attracts the em and em gradient algorithm at the same rate of convergence and near the maximum point the em gradient algorithm always produces an increase in the likelihood with proper modification the em gradient algorithm also exhibits global convergence properties that are similar to those of the em algorithm our proof of global convergence applies and improves existing theory for the em algorithm these theoretical points are reinforced by a discussion of three realistic examples illustrating how the em gradient algorithm can succeed where the em algorithm is intractable

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

on the numerical solution of a nonlocal boundary value problem we study a nonlinear boundary value problem involving a nonlocal integral operator in the coefficients of the unknown function provided sufficient conditions for the existence and uniqueness of the solution for its approximation we propose a numerical method consisting of a classical discretization of the problem and an algorithm to solve the resulting nonlocal and nonlinear algebraic system by means of some iterative procedures the second order of convergence is assured by different sufficient conditions which can be alternatively used in dependence on the given data the theoretical results are confirmed by several numerical tests

on positive solutions of nonlocal and nonvariational elliptic problems abstract we will study the nonlocal and nonvariational elliptic problem a u q u h x f u in u 0 on by considering in two different results the cases n 1 and n 2 in both cases we will use fixed point theorems

on a class of problems involving a nonlocal operator in this paper we prove a result on existence of positive solution for a nonlocal elliptic problem by using a result on fixed point index theory and we establish another existence result which is proved by sub and supersolution and a comparison principle moreover we prove the existence uniqueness and asymptotic behaviour of the solutions for the evolution case

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

a modified expectation maximization algorithm for penalized likelihood estimation in emission tomography the maximum likelihood ml expectation maximization em approach in emission tomography has been very popular in medical imaging for several years in spite of this no satisfactory convergent modifications have been proposed for the regularized approach here a modification of the em algorithm is presented the new method is a natural extension of the em for maximizing likelihood with concave priors convergence proofs are given

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

alternating minimization algorithms for transmission tomography a family of alternating minimization algorithms for finding maximum likelihood estimates of attenuation functions in transmission x ray tomography is described the model from which the algorithms are derived includes polyenergetic photon spectra background events and nonideal point spread functions the maximum likelihood image reconstruction problem is reformulated as a double minimization of the i divergence a novel application of the convex decomposition lemma results in an alternating minimization algorithm that monotonically decreases the objective function each step of the minimization is in closed form the family of algorithms includes variations that use ordered subset techniques for increasing the speed of convergence simulations demonstrate the ability to correct the cupping artifact due to beam hardening and the ability to reduce streaking artifacts that arise from beam hardening and background events

symmetry diffeomorphism group of a manifold of nonpositive curvature let m denote a complete simply connected manifold of nonpos itive sectional curvature for each point p s m let sp denote the diffeomor phism of m that fixes p and reverses all geodesies through p the symmetry diffeomorphism group g generated by all diffeomorphisms sp p m ex tends naturally to group of homeomorphisms of the boundary sphere m oo a subset x of m oo is called involutive if it is invariant under g theorem let x c m co be a proper closed involutive subset for each point p s m let n p denote the linear span in tpm of those vectors at p that are tangent to a geodesic 7 whose asymptotic equivalence class 7 00 belongs to x if n p is a proper subspace of tpm for some point p m then m splits as a riemannian product my x mi such that n is the distribution of m induced by my this result has several applications that include new results as well as great simplifications in the proofs of some known results in a sequel to this paper it is shown that if m is irreducible and m oo admits a proper closed involutive subset x then m is isometric to a symmetric space of noncompact type and rank k 2