stieltjes derivatives and b polynomial spline collocation for volterra integrodifferential equations with singularities in this paper we introduce a new tool the stieltjes derivative and new kinds of meshes which are more practical than the graded meshes based on these we discuss a nonpolynomial spline collocation method for volterra integrodifferential equations with weakly singular kernels under very weak hypotheses which allow the nonsmooth behaviour of the given functions in these equations collocation on such meshes is shown to yield optimal convergence rates an ideal super convergence result is also obtained

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

approximation of nonnegative systems by moving averages of fixed order abstract we pose the approximation problem for scalar nonnegative input output systems via impulse response convolutions of finite order i e finite order moving averages based on repeated observations of input output signal pairs the problem is converted into a nonnegative matrix factorization with special structure for which we use csiszar s i divergence as the criterion of optimality conditions are given on the input output data that guarantee the existence and uniqueness of the minimum we propose an algorithm of the alternating minimization type for i divergence minimization and present its asymptotic behaviour for the case of noisy observations we give the large sample properties of the statistical version of the minimization problem for different observation regimes numerical experiments confirm the asymptotic results and exhibit fast convergence of the proposed algorithm

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

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

iterative image reconstruction algorithms based on cross entropy minimization the related problems of minimizing the functionals f x alpha kl y px 1 alpha kl p x and g x alpha kl px y 1 alpha kl x p respectively over the set of vectors x or 0 are considered kl a b is the cross entropy or kullback leibler distance between two nonnegative vectors a and b iterative algorithms for minimizing both functionals using the method of alternating projections are derived a simultaneous version of the multiplicative algebraic reconstruction technique mart algorithm called smart is introduced and its convergence is proved

restoration of severely blurred high range images using stochastic and deterministic relaxation algorithms in compound gauss markov random fields abstract over the last few years a growing number of researchers from varied disciplines have been utilizing markov random fields mrf models for developing optimal robust algorithms for various problems such as texture analysis image synthesis classification and segmentation surface reconstruction integration of several low level vision modules sensor fusion and image restoration however no much work has been reported on the use of simulated annealing sa and iterative conditional mode icm algorithms for maximum a posteriori estimation in image restoration problems when severe blurring is present in this paper we examine the use of compound gauss markov random fields cgmrf to restore severely blurred high range images for this deblurring problem the convergence of the sa and icm algorithms has not been established we propose two new iterative restoration algorithms which can be considered as extensions of the classical sa and icm approaches and whose convergence is established finally they are tested on real and synthetic images and the results compared with the restorations obtained by other iterative schemes

generalization of median root prior reconstruction penalized iterative algorithms for image reconstruction in emission tomography contain conditions on which kind of images are accepted as solutions the penalty term has commonly been a function of pairwise pixel differences in the activity in a local neighborhood such that smooth images are favored attempts to ensure better edge and detail preservation involve difficult tailoring of parameter values or the penalty function itself the previously introduced median root prior mrp favors locally monotonic images mrp preserves sharp edges while reducing locally nonmonotonic noise at the same time quantitative properties of mrp are good because differences in the neighboring pixel values are not penalized as such the median is used as an estimate for a penalty reference against which the pixel value is compared when setting the penalty in order to generalize the class of mrp type of priors the standard median was replaced by other order statistic operations the l and finite impulse response median hybrid fmh filters they allow for smoother appearance as they apply linear weighting together with robust nonlinear operations the images reconstructed using the new mrp l and mrp fmh priors are visually more conventional good quantitative properties of mrp are not significantly altered by the new priors

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

a class of robust entropic functionals for image restoration this paper considers the concept of robust estimation in regularized image restoration robust functionals are employed for the representation of both the noise and the signal statistics such functionals allow the efficient suppression of a wide variety of noise processes and permit the reconstruction of sharper edges than their quadratic counterparts a new class of robust entropic functionals is introduced which operates only on the high frequency content of the signal and reflects sharp deviations in the signal distribution this class of functionals can also incorporate prior structural information regarding the original image in a way similar to the maximum information principle the convergence properties of robust iterative algorithms are studied for continuously and noncontinuously differentiable functionals the definition of the robust approach is completed by introducing a method for the optimal selection of the regularization parameter this method utilizes the structure of robust estimators that lack analytic specification the properties of robust algorithms are demonstrated through restoration examples in different noise environments

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

approximation of nonnegative systems by finite impulse response convolutions we pose the deterministic nonparametric approximation problem for scalar nonnegative input output systems via finite impulse response convolutions based on repeated observations of input output signal pairs the problem is converted into a nonnegative matrix factorization with special structure for which we use csiszar s i divergence as the criterion of optimality conditions are given on the input output data that guarantee the existence and uniqueness of the minimum we propose an algorithm of the alternating minimization type for i divergence minimization and study its asymptotic behavior for the case of noisy observations we give the large sample properties of the statistical version of the minimization problem numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm

a row action alternative to the em algorithm for maximizing likelihood in emission tomography the maximum likelihood ml approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography ect since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection fbp algorithm the expectation maximization em algorithm is an often used iterative approach for maximizing the poisson likelihood in ect because of its attractive theoretical and practical properties its major disadvantage is that due to its slow rate of convergence a large amount of computation is often required to achieve an acceptable image here the authors present a row action maximum likelihood algorithm ramla as an alternative to the em algorithm for maximizing the poisson likelihood in ect the authors deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of ramla increase the poisson likelihood in ect at an order of magnitude faster that the standard em algorithm specifically the authors show that from the point of view of measuring total radionuclide uptake in simulated brain phantoms iterations 1 2 3 and 4 of ramla perform at least as well as iterations 45 60 70 and 80 respectively of em moreover the authors show that iterations 1 2 3 and 4 of ramla achieve comparable likelihood values as iterations 45 60 70 and 80 respectively of em the authors also present a modified version of a recent fast ordered subsets em os em algorithm and show that ramla is a special case of this modified os em furthermore the authors show that their modification converges to a ml solution whereas the standard os em does not

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

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

on the stability of the solution of abel s integral equation it is shown that numerical differentiation can be removed from the conventional solution of abel s integral equation using the gaussian quasipotential as an example the authors show that the new inversion method is less sensitive to random errors in the input data

deblurring subject to nonnegativity constraints csiszar s i divergence is used as a discrepancy measure for deblurring subject to the constraint that all functions involved are nonnegative an iterative algorithm is proposed for minimizing this measure it is shown that every function in the sequence is nonnegative and the sequence converges monotonically to a global minimum other properties of the algorithm are shown including lower bounds on the improvement in the i divergence at each step of the algorithm and on the difference between the i difference at step k and at the limit point a method for regularizing the solution is proposed

statistical projection completion in x ray ct using consistency conditions projection data incompleteness arises in many situations relevant to x ray computed tomography ct imaging we propose a penalized maximum likelihood statistical sinogram restoration approach that incorporates the helgason ludwig hl consistency conditions to accommodate projection data incompleteness image reconstruction is performed by the filtered backprojection fbp in a second step in our problem formulation the objective function consists of the log likelihood of the x ray ct data and a penalty term the hl condition poses a linear constraint on the restored sinogram and can be implemented efficiently via fast fourier transform fft and inverse fft we derive an iterative algorithm that increases the objective function monotonically the proposed algorithm is applied to both computer simulated data and real patient data we study different factors in the problem formulation that affect the properties of the final fbp reconstructed images including the data truncation level the amount of prior knowledge on the object support as well as different approximations of the statistical distribution of the available projection data we also compare its performance with an analytical truncation artifacts reduction method the proposed method greatly improves both the accuracy and the precision of the reconstructed images within the scan field of view and to a certain extent recovers the truncated peripheral region of the object the proposed method may also be applied in areas such as limited angle tomography metal artifacts reduction and sparse sampling imaging

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

a fast algorithm for the inversion of abels transform we present a new algorithm for the computation of the inverse abel transform a problem which emerges in many areas of physics and engineering we prove that the legendre coefficients of a given function coincide with the fourier coefficients of a suitable periodic function associated with its abel transform this allows us to compute the legendre coefficients of the inverse abel transform in an easy fast and accurate way by means of a single fast fourier transform the algorithm is thus appropriate also for the inversion of abel integrals given in terms of samples representing noisy measurements rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments

asymptotic models of dilute composites with imperfectly bonded inclusions the asymptotic scheme for the analysis of dilute elastic composites which includes circular inclusions with imperfect bonding at the interface is presented interface is characterized by a discontinuous displacement field across it linearly related to the tractions the problem of a linear elastic circular inclusion with generic loading condition at infinity is solved and used to analyze effective elastic moduli of composite materials effects due to the interaction of a small circular defect and a crack are investigated it is shown that interfacial stiffness has a strong effect on the crack path and therefore may be an important design parameter for composites

roof edge preserving image smoothing based on mrfs a novel markov random field mrf model is proposed for roof edge as well as step edge preserving image smoothing image surfaces containing roof edges are represented by piecewise continuous polynomial functions governed by a few parameters piecewise smoothness constraint is imposed on these parameters rather than on the surface heights as is in traditional models for step edges in this way roof edges are preserved without the necessity to deal with unstable higher order derivatives

a new method for parameter estimation of edge preserving regularization in image restoration in image restoration the so called edge preserving regularization method is used to solve an optimization problem whose objective function has a data fidelity term and a regularization term the two terms are balanced by a parameter l in some aspect the value of l determines the quality of images in this paper we establish a new model to estimate the parameter and propose an algorithm to solve the problem in order to improve the quality of images in our algorithm an image is divided into some blocks on each block a corresponding value of l has to be determined numerical experiments are reported which show efficiency of our 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

approximate solution of abel integral equation this paper presents a new stable approximate inversion of abel integral equation by using the taylor expansion of the unknown function abel equation is approximately transformed to a system of linear equations for the unknown function together with its derivatives a desired solution can be determined by solving the resulting system according to cramer s rule this method gives a simple and closed form of approximate abel inversion which can be performed by symbolic computation the nth order approximation is exact for a polynomial of degree up to n abel integral equation is approximately expressed in terms of integrals of input data so the suggested approach is stable for experimental data with random noise an error analysis of this approach is given finally several numerical examples are given to illustrate the accuracy and stability of this method

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

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

iterative deblurring for ct metal artifact reduction iterative deblurring methods using the expectation maximization em formulation and the algebraic reconstruction technique art respectively are adapted for metal artifact reduction in medical computed tomography ct in experiments with synthetic noise free and additive noisy projection data of dental phantoms it is found that both simultaneous iterative algorithms produce superior image quality as compared to filtered backprojection after linearly fitting projection gaps furthermore the em type algorithm converges faster than the art type algorithm in terms of either the i divergence or euclidean distance between ideal and reprojected data in the authors simulation also for a given iteration number the em type deblurring method produces better image clarity but stronger noise than the art type reconstruction the computational complexity of em and art based iterative deblurring is essentially the same dominated by reprojection and backprojection relevant practical and theoretical issues are discussed

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

penalized weighted least squares image reconstruction for dual energy x ray transmission tomography presents a dual energy de transmission computed tomography ct reconstruction method it is statistically motivated and features nonnegativity constraints in the density domain a penalized weighted least squares pwls objective function has been chosen to handle the non poisson noise added by amorphous silicon asi h detectors a gauss seidel algorithm has been used to minimize the objective function the behavior of the method in terms of bias standard deviation tradeoff has been compared to that of a de method that is based on filtered back projection fbp the advantages of the de pwls method are largest for high noise and or low flux cases qualitative results suggest this as well also the reconstructed images of an object with opaque regions are presented possible applications of the method are attenuation correction for positron emission tomography pet images various quantitative computed tomography qct methods such as bone mineral densitometry bmd and the removal of metal streak artifacts

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

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

modelling of electromagnetic waves in periodic media with finite elements edge elements have been previously used to study electromagnetic waves conically propagating through a finite stack of arrays of fibres ieee trans magn 38 2 2002 this paper presents a new extension of this technique to the analysis of the propagation of such waves in a doubly periodic array of fibres in the proposed approach only the unit cell of the periodic material has to be meshed thanks to floquet bloch theory

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

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

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

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

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

roughness penalties on finite domains a class of penalty functions for use in estimation and image regularization is proposed these penalty functions are defined for vectors whose indexes are locations in a finite lattice as the discrepancy between the vector and a shifted version of itself after motivating this class of penalty functions their relationship to markov random field priors is explored one of the penalty functions proposed a divergence roughness penalty is shown to be a discretization of a penalty proposed by good and gaskins 1971 for use in density estimation one potential use in estimation problems is explored an iterative algorithm that takes advantage of induced neighborhood structures is proposed and convergence of the algorithm is proven under specified conditions examples in emission tomographic imaging and radar imaging 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

a new nonlinear filter constructed from the newton method and epr in image restoration the edge preserving regularization method epr is effective to remove gaussian noise and impulse noise in image restorations but till now we could not say that the problem has been solved very efficiently by state of the art optimization method in our paper we will construct a new nonlinear filter from the edge preserving model and the newton method the parameter estimation model in the nonlinear filter will be established and the solution existence of the model will be proved at the same time numerical results demonstrate that the nonlinear filter removes the gaussian noise from an image as effectively as the edge preserving regularization method does and the time for the nonlinear filter to restore an image is less than half time for the edge preserving regularization method

image recovery using partitioned separable paraboloidal surrogate coordinate ascent algorithms iterative coordinate ascent algorithms have been shown to be useful for image recovery but are poorly suited to parallel computing due to their sequential nature this paper presents a new fast converging parallelizable algorithm for image recovery that can be applied to a very broad class of objective functions this method is based on paraboloidal surrogate functions and a concavity technique the paraboloidal surrogates simplify the optimization problem the idea of the concavity technique is to partition pixels into subsets that can be updated in parallel to reduce the computation time for fast convergence pixels within each subset are updated sequentially using a coordinate ascent algorithm the proposed algorithm is guaranteed to monotonically increase the objective function and intrinsically accommodates nonnegativity constraints a global convergence proof is summarized simulation results show that the proposed algorithm requires less elapsed time for convergence than iterative coordinate ascent algorithms with four parallel processors the proposed algorithm yields a speedup factor of 3 77 relative to single processor coordinate ascent algorithms for a three dimensional 3 d confocal image restoration problem

developments with maximum likelihood x ray computed tomography an approach to the maximum likelihood estimation of attenuation coefficients in transmission tomography is presented as an extension of earlier theoretical work by k lange and r carson j comput assist tomography vol 8 p 306 16 1984 the reconstruction algorithm is based on the expectation maximization em algorithm several simplifying approximations are introduced which make the maximization step of the algorithm available computer simulations are presented using noise free and poisson randomized projections the images obtained with the em type method are compared to those reconstructed with the em method of lange and carson and with filtered backprojection preliminary results show that there are potential advantages in using the maximum likelihood approaches in situations where a high contrast object such as bone is embedded in low contrast soft tissue

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

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

noncommuting limits in electromagnetic scattering asymptotic analysis for an array of highly conducting inclusions we consider formulations for the helmholtz operator for periodic media containing high contrast inclusions in the limit when the wavelength outside the inclusions tends to infinity applications are to problems of electromagnetism the main focus is on the analysis of the effect of noncommuting limits an effect which indicates that linear boundary value problems of electromagnetism give formally different results for the long wavelength limits in cases where highly conducting inclusions have refractive indices of different orders of magnitude specifically the effective moduli of the homogenized material will depend on the path used to approach the origin in the coordinate space wave number normalized refractive index of the inclusions 1 this mathematical observation gives a physical subtlety which is studied in this paper the dispersion relation for the lowest frequency or acoustic mode is investigated as are the conditions for existence of an acoustic mode cases of both nondispersive and di

close form solution and parameter selection for convex minimization based edge preserving smoothing this work presents a new approach for the analysis of convex minimization based edge preserving image smoothing and the parameter selection therein the global solution that is the response of a convex smoothing model to the ideal step edge is derived in close form by analyzing the close form solution insights are drawn into how the optimal solution responds to edges in the data and how the parameter values affect resultant edges in the solution based on this a scheme is proposed for selecting parameters to achieve desirable responses at edges the theoretic results are substantiated by experiments

a fast and accurate fourier algorithm for iterative parallel beam tomography we use a series expansion approach and an operator framework to derive a new fast and accurate fourier algorithm for iterative tomographic reconstruction this algorithm is applicable for parallel ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small the conjugate gradient cg algorithm is used to minimize a regularized spectrally weighted least squares criterion and we prove that the main step in each iteration is equivalent to a 2 d discrete convolution which can be cheaply and exactly implemented via the fast fourier transform fft the proposed algorithm requires o n sup 2 logn floating point operations per iteration to reconstruct an n spl times n image from p view angles as compared to o n sup 2 p floating point operations per iteration for iterative convolution backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse and limited angle tomography under realistic sampling scenarios although the proposed algorithm cannot explicitly account for noise with nonstationary statistics additional simulations demonstrate that for low to moderate levels of nonstationary noise the quality of reconstruction is almost unaffected by assuming that the noise is stationary

line integral alternating minimization algorithm for dual energy x ray ct image reconstruction we propose a new algorithm called line integral alternating minimization liam for dual energy x ray ct image reconstruction instead of obtaining component images by minimizing the discrepancy between the data and the mean estimates liam allows for a tunable discrepancy between the basis material projections and the basis sinograms a parameter is introduced that controls the size of this discrepancy and with this parameter the new algorithm can continuously go from a two step approach to the joint estimation approach liam alternates between iteratively updating the line integrals of the component images and reconstruction of the component images using an image iterative deblurring algorithm an edge preserving penalty function can be incorporated in the iterative deblurring step to decrease the roughness in component images images from both simulated and experimentally acquired sinograms from a clinical scanner were reconstructed by liam while varying the regularization parameters to identify good choices the results from the dual energy alternating minimization algorithm applied to the same data were used for comparison using a small fraction of the computation time of dual energy alternating minimization liam achieves better accuracy of the component images in the presence of poisson noise for simulated data reconstruction and achieves the same level of accuracy for real data reconstruction

on the method of chan and lu for abel s integral equation for the method proposed recently by chan and lu 1981 for the inversion of abel s integral equation the authors show that i under a wide range of circumstances it yields numerical schemes identical with those obtained from the direct discretisation of the standard inversion formulae ii although it avoids the explicit numerical differentiation in the standard inversion formulae it does not avoid the numerical difficulties inherent in the inversion and iii it is still equivalent mathematically and numerically to a half differentiation

an approximate method for abel inversion using chebyshev polynomials many problems in physics like reconstruction of the radially distributed emissivity from the line of sight projected intensity the 3 d image reconstruction from cone beam projections in computerized tomography etc lead naturally in the case of radial symmetry to the study of abel s type integral equation in this paper a new stable algorithm based on shifted chebyshev polynomial approximation is presented and analyzed first chebyshev operational matrix of integration p is constructed and then it is used to reduce the integral equation to a system of algebraic equation which can be solved easily the method is quite accurate and stable even though the approximations are performed using polynomials of degree up to 5 some test examples from the plasma diagnostics are illustrated to demonstrate the effectiveness and stability of the proposed method

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

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

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

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

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

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

a new abel inversion by means of the integrals of an input function with noise abel s integral equations arise in many areas of natural science and engineering particularly in plasma diagnostics this paper proposes a new and effective approximation of the inversion of abel transform this algorithm can be simply implemented by symbolic computation and moreover an nth order approximation reduces to the exact solution when it is a polynomial in r 2 of degree less than or equal to n approximate abel inversion is expressed in terms of integrals of input measurement data so the suggested approach is stable for experimental data with random noise an error analysis of the approximation of abel inversion is given finally several test examples used frequently in plasma diagnostics are given to illustrate the effectiveness and stability of this method

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

statistical interior tomography this paper presents a statistical interior tomography sit approach making use of compressed sensing cs theory with the projection data modeled by the poisson distribution an objective function with a total variation tv regularization term is formulated in the maximization of a posteriori map framework to solve the interior problem an alternating minimization method is used to optimize the objective function with an initial image from the direct inversion of the truncated hilbert transform the proposed sit approach is extensively evaluated with both numerical and real datasets the results demonstrate that sit is robust with respect to data noise and down sampling and has better resolution and less bias than its deterministic counterpart in the case of low count data

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

model based iterative reconstruction for dual energy x ray ct using a joint quadratic likelihood model dual energy x ray ct dect has the potential to improve contrast and reduce artifacts as compared to traditional ct moreover by applying model based iterative reconstruction mbir to dual energy data one might also expect to reduce noise and improve resolution however the direct implementation of dual energy mbir requires the use of a nonlinear forward model which increases both complexity and computation alternatively simplified forward models have been used which treat the material decomposed channels separately but these approaches do not fully account for the statistical dependencies in the channels in this paper we present a method for joint dual energy mbir jde mbir which simplifies the forward model while still accounting for the complete statistical dependency in the material decomposed sinogram components the jde mbir approach works by using a quadratic approximation to the polychromatic log likelihood and a simple but exact nonnegativity constraint in the image domain we demonstrate that our method is particularly effective when the dect system uses fast kvp switching since in this case the model accounts for the inaccuracy of interpolated sinogram entries both phantom and clinical results show that the proposed model produces images that compare favorably in quality to previous decomposition based methods including fbp and other statistical iterative approaches

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

approximate solution of fractional integro differential equations by taylor expansion method in this paper taylor expansion approach is presented for solving approximately a class of linear fractional integro differential equations including those of fredholm and of volterra types by means of the mth order taylor expansion of the unknown function at an arbitrary point the linear fractional integro differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions this method gives a simple and closed form solution for a linear fractional integro differential equation in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method

acoustic band gaps in arrays of neutral inclusions we analyse numerically the acoustic stop band properties of an array of orthotropic coated cylinders whose elastic parameters are deduced from a geometric transform h chen c t chan acoustic cloaking in three dimensions using acoustic metamaterials appl phys lett 91 2007 183518 we find that whereas a single coated inclusion is acoustically neutral at any frequency an array of them might display some stop bands more precisely an array of freely vibrating coated voids is always neutral whereas an array of clamped coated inclusions might display a zero frequency stop band interestingly an array of radially symmetric coated inclusions behaves as local helmholtz resonators for which the eigenfield within each cloak is obtained in closed form leading to a frequency estimate associated with the lower edge of the low frequency stop band a finite phononic crystal of such coated cylinders behaves either as an invisible material or a reflector depending on the frequency of an acoustic source