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

fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

fast methods for the eikonal and related hamilton jacobi equations on unstructured meshes the fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps where m is the total number of grid points the scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstra like programming approach in this paper we discuss several extensions to this technique including higher order versions on unstructured meshes in rn and on manifolds and connections to more general static hamilton jacobi equations

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

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

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

investigation of the convergence of the mixed displacement pressure formulation for three dimensional poroelastic materials using hierarchical elements recently a mixed pressure displacement u p formulation based on biot s poroelasticity equations has been presented for porous materials this model leads to a reduction of the number of degrees of freedom required for the modeling of three dimensional porous media in comparison to classical displacement displacement u u formulations in this paper an extension of the u p formulation based on hierarchical elements is presented first a variant of the weak integral form of the u p formulation is presented and its numerical implementation using hierarchical elements is detailed together with the application of boundary and loading conditions numerical results are presented to show the accuracy and performance of the present approach in particular the importance of correctly capturing the coupling effects between the two phases is highlighted

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

comparison of different higher order finite element schemes for the simulation of lamb waves structural health monitoring shm applications call for both efficient and powerful numerical tools to predict the behavior of ultrasonic guided waves when considering waves in thin walled structures so called lamb waves conventional linear or quadratic pure displacement finite elements soon reach their limits the spatial as well as temporal discretisation required to obtain good quality results has to be very fine this results in enormous computational costs computational time and memory storage requirements when ultrasonic wave propagation problems are solved in the time domain to resolve this issue several higher order finite element methods with polynomial degrees p 2 are proposed the objective of the current article is to develop such higher order schemes and to verify their capabilities with respect to accuracy and numerical performance to the best of the authors knowledge such comparison has not been reported in literature yet specifically spectral elements based on lagarange polynomials sem p elements using the normalized integrals of the legendre polynomials p fem and isogeometric elements utilizing non uniform rational b splines nurbs n fem are discussed in this paper by solving a two dimensional benchmark problem their advantages and drawbacks with respect to lamb wave propagation are highlighted the results of the convergence studies are then used to derive guidelines for estimating the optimal element size for a given finite element type and polynomial degree template these findings serve the purpose to determine the optimal mesh configuration a priori and thus save a considerable amount of computational effort the proposed guideline is then tested on a three dimensional structure with a conical hole showing an excellent agreement with the predicted behaviour

economical symmetrical quadrature rules for complete polynomials over a square domain it is of interest in numerical analysis to develop symmetrical quadrature rules for integration of complete polynomial functions over a square domain with minimum computational effort gaussian product quadrature rules integrate such functions with maximum effort symmetrical quadrature rules are developed and presented for integration of complete polynomial functions up to 21st order with minimum computational effort

fast marching methods for stationary hamilton jacobi equations with axis aligned anisotropy the fast marching method fmm has proved to be a very efficient algorithm for solving the isotropic eikonal equation because it is a minor modification of dijkstra s algorithm for finding the shortest path through a discrete graph fmm is also easy to implement in this paper we describe a new class of hamilton jacobi hj pdes with axis aligned anisotropy which satisfy a causality condition for standard finite difference schemes on orthogonal grids and can hence be solved using the fmm the only modification required to the algorithm is in the local update equation for a node this class of hj pdes has applications in anelliptic wave propagation and robotic path planning and brief examples are included since our class of hj pdes and grids permit asymmetries we also examine some methods of improving the efficiency of the local update that do not require symmetric grids and pdes finally we include explicit update formulas for variations of the eikonal equation that use the manhattan euclidean and infinity norms on orthogonal grids of arbitrary dimension and with variable node spacing

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

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

a numerical integration scheme for finite element method based on symbolic manipulation this paper is concerned with the improvement of the efficiency and accuracy of the usual numerical integration procedures for the finite element method a modified numerical integration scheme based on both the symbolic manipulation system and the numerical integration formula is proposed its efficiency for the integration of complicated rational or irrational expressions is demonstrated through several numerical examples to verify its effectiveness in the calculation of the finite element stiffness matrix a simple numerical example is also presented

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

penalized maximum likelihood image reconstruction using space alternating generalized em algorithms most expectation maximization em type algorithms for penalized maximum likelihood image reconstruction converge slowly particularly when one incorporates additive background effects such as scatter random coincidences dark current or cosmic radiation in addition regularizing smoothness penalties or priors introduce parameter coupling rendering intractable the m steps of most em type algorithms this paper presents space alternating generalized em sage algorithms for image reconstruction which update the parameters sequentially using a sequence of small hidden data spaces rather than simultaneously using one large complete data space the sequential update decouples the m step so the maximization can typically be performed analytically we introduce new hidden data spaces that are less informative than the conventional complete data space for poisson data and that yield significant improvements in convergence rate this acceleration is due to statistical considerations not numerical overrelaxation methods so monotonic increases in the objective function are guaranteed we provide a general global convergence proof for sage methods with nonnegativity constraints

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

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

unbiased risk estimates for singular value thresholding and spectral estimators in an increasing number of applications it is of interest to recover an approximately low rank data matrix from noisy observations this paper develops an unbiased risk estimate holding in a gaussian model for any spectral estimator obeying some mild regularity assumptions in particular we give an unbiased risk estimate formula for singular value thresholding svt a popular estimation strategy that applies a soft thresholding rule to the singular values of the noisy observations among other things our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems in particular we demonstrate the utility of the unbiased risk estimation for svt based denoising of real clinical cardiac mri series data we also give new results concerning the differentiability of certain matrix valued functions

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

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

fast sweeping methods for static hamilton jacobi equations we propose a new sweeping algorithm which discretizes the legendre transform of the numerical hamiltonian using an explicit formula this formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically the minimization that is related to the legendre transform in our sweeping scheme can either be solved analytically or numerically we illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions

integration by table look up for p version finite elements on curved tetrahedra abstract we develop a method of evaluating element level matrices and vectors associated with p version finite element computations on tetrahedral meshes with curved boundaries the procedure uses optimal interpolation formula q chen i babuska approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle comput methods appl mech engrg 128 1995 405 417 q chen i babuska approximate optimal polynomial interpolation points in the tetrahedron comput methods appl mech engrg 137 1997 89 94 for non polynomial portions of integrands followed by exact integration exact integrals of product combinations of polynomial interpolants and shape functions are stored so that different integrals may be efficiently evaluated by a table look up we present error analysis for our scheme applied to curvilinear tetrahedral meshes with exact geometric representation of the boundary of the domain as defined in a geometric modelling system numerical results are presented for problems involving the helmholtz equation

radiation dose reduction in ct myocardial perfusion imaging using smart recon in this paper a newly developed statistical model based image reconstruction referred to as simultaneous multiple artifacts reduction in tomographic reconstruction smart recon is applied to low dose computer tomography ct myocardial perfusion imaging ct mpi this method uses the nuclear norm of the spatial temporal image matrix of the ct mpi images as a regularizer rather than a conventional spatial regularizer that incorporates image smoothness edge preservation or spatial sparsity into the reconstruction in addition to providing the needed noise reduction for low dose ct mpi smart recon provides images with spatial resolution and noise power spectrum nps properties which are independent of contrast and dose levels both numerical simulations and in vivo animal studies were performed to validate the proposed method in these studies it was found that 1 quantitative accuracy of perfusion maps in ct mpi was well maintained for radiation dose level as low as 10 mas per image frame compared with the reference standard of 200 mas for conventional filtered backprojection 2 flow occluded myocardium in the porcine heart was well delineated by smart recon at 10 mas per frame when compared with model based image reconstruction using spatial total variation tv as the regularizer referred to as tv sir or spatial temporal tv st tv sir the ct mpi results were confirmed with positron emission tomography imaging 3 image sharpness in smart recon images was nearly independent of image contrast level and radiation dose level in stark contrast to tv sir and st tv sir which displayed a strong dependence on both image contrast and radiation dose levels and 4 the structure of the dose normalized nps for the smart recon method did not depend on dose while the tv sir and st tv sir nps structure was dose dependent

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

high order x fem and levelsets for complex microstructures uncoupling geometry and approximation in this contribution a strategy is proposed for uncoupling geometrical description and approximation with the x fem the strategy is based on an uniform coarse mesh that defines a high order approximation of the mechanical fields and an adapted mesh that defines the geometrical features by means of levelsets the connection between the geometry and the approximation is obtained by sharing the quadtree trees of the two meshes numerical examples involving level set based parts convergence studies mechanical computations and numerical homogenization show good promise for this approach

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

efficient symmetrical cubature rules for complete polynomials of high degree over the unit cube the fully symmetrical cubature rules developed and presented are at present the most economically available for numerical integration over the unit cube of complete polynomial functions up to and including nineteenth order conventional gaussian triple product integration formulae by comparison require more than twice as many integration points and weights

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

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

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

ordered upwind methods for static hamilton jacobi equations abstract we introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static hamilton jacobi equations with dirichlet boundary conditions standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation these techniques are of complexity o m log m where m is the total number of points in the domain we consider anisotropic test problems in optimal control seismology and paths on surfaces

a theoretical study of the contrast recovery and variance of map reconstructions from pet data the authors examine the spatial resolution and variance properties of pet images reconstructed using maximum a posteriori map or penalized likelihood methods resolution is characterized by the contrast recovery coefficient crc of the local impulse response simplified approximate expressions are derived for the local impulse response crcs and variances for each voxel using these results the authors propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local crc to noise ratio in the reconstructed image

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

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

wavelet transform domain filters a spatially selective noise filtration technique wavelet transforms are multiresolution decompositions that can be used to analyze signals and images they describe a signal by the power at each scale and position edges can be located very effectively in the wavelet transform domain a spatially selective noise filtration technique based on the direct spatial correlation of the wavelet transform at several adjacent scales is introduced a high correlation is used to infer that there is a significant feature at the position that should be passed through the filter the authors have tested the technique on simulated signals phantom images and real mr images it is found that the technique can reduce noise contents in signals and images by more than 80 while maintaining at least 80 of the value of the gradient at most edges the authors did not observe any gibbs ringing or significant resolution loss on the filtered images artifacts that arose from the filtration are very small and local the noise filtration technique is quite robust there are many possible extensions of the technique the authors see its applications in spatially dependent noise filtration edge detection and enhancement image restoration and motion artifact removal they have compared the performance of the technique to that of the weiner filter and found it to be superior

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

the p version of the finite element method for three dimensional curved thin walled structures in this paper we present an implementation of a three dimensional p version for structural problems of solids with almost arbitrarily curved surfaces applying the blending function method complex structures can often be modelled by a few p elements being the basis for a higher order approximation numerical examples will demonstrate that the p version with anisotropic ansatz spaces allows to predict the structural behaviour of three dimensional plates and shells with approximately the same amount of degrees of freedom as in the two dimensional case yet significantly more accurate due to the three dimensional model furthermore it is advantageous to compute complex structures exclusively with three dimensional discretizations as no special elements are needed to model the transition from dimensionally reduced formulations like plates or shells to fully three dimensional solid elements using the p version with anisotropic ansatz spaces the whole structure can be efficiently discretized with solid elements even if the aspect ratio of the elements becomes very large copyright 2001 john wiley sons ltd

computational study of fast methods for the eikonal equation a computational study of the fast marching and the fast sweeping methods for the eikonal equation is given it is stressed that both algorithms should be considered as direct as opposed to iterative methods on realistic grids fast sweeping is faster than fast marching for problems with simple geometry for strongly nonuniform problems and or complex geometry the situation may be reversed finally fully second order generalizations of methods of this type for problems with obstacles are proposed and implemented

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

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

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

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

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

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

ray contribution masks for structure adaptive sinogram filtering the patient dose in computed tomography ct imaging is linked to measurement noise various noise reduction techniques have been developed that adapt structure preserving filters like anisotropic diffusion or bilateral filters to ct noise properties we introduce a structure adaptive sinogram sas filter that incorporates the specific properties of the ct measurement process it uses a point based forward projector to generate a local structure representation called ray contribution mask rcm the similarities between neighboring rcms are used in an enhanced variant of the bilateral filtering concept where the photometric similarity is replaced with the structural similarity we evaluate the performance in four different scenarios the robustness against reconstruction artifacts is demonstrated by a scan of a high resolution phantom without changing the modulation transfer function mtf nor introducing artifacts the sas filter reduces the noise level by 13 6 the image sharpness and noise reduction capabilities are visually assessed on in vivo patient scans and quantitatively evaluated on a simulated phantom unlike a standard bilateral filter the sas filter preserves edge information and high frequency components of organ textures well it shows a homogeneous noise reduction behavior throughout the whole frequency range the last scenario uses a simulated edge phantom to estimate the filter mtf for various contrasts the noise reduction for the simple edge phantom exceeds 80 for low contrasts at 55 hounsfield units hu the mid frequency range is slightly attenuated at higher contrasts of approximately 100 hu and above the mtf is fully preserved

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

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

an adaptive domain decomposition technique for parallelization of the fast marching method abstract the fast marching method fmm is an efficient technique to solve numerically the eikonal equation the parallelization of the fmm is not easy because of its intrinsic sequential nature in this paper we propose a novel approach to parallelize the fmm it leads to an equation dependent domain decomposition and it turns out to be particularly suitable for machines with two or four cores that are in common use today compared to other techniques in the field the proposed method is much simpler to implement and it gives a slightly better computational speed up in order to test the new method on a real world application we solve the shape from shading problem based on a hamilton jacobi equation on a standard four core machine the method confirms the good properties it shows a reasonable speedup factor of about 2 5 and it reveals its potential to good performance if the arithmetic density of the problem is high

a fast method of numerical quadrature for p version finite element matrices a new technique of numerical quadrature especially suited for quadrilateral and hexahedron p version finite element matrices is presented this new technique separates the integrand into two parts and numerically operates on each part separately the objective of this scheme is to minimize the computational cost of integrating the entire element matrix as opposed to minimizing the cost of integrating a single function the efficiency of the new technique is compared with gaussian quadrature and found to take a small fraction of the computational effort

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

combining spatial and scale space techniques for edge detection to provide a spatially adaptive wavelet based noise filtering algorithm new methods for detecting edges in an image using spatial and scale space domains are proposed a priori knowledge about geometrical characteristics of edges is used to assign a probability factor to the chance of any pixel being on an edge an improved double thresholding technique is introduced for spatial domain filtering probabilities that pixels belong to a given edge are assigned based on pixel similarity across gradient amplitudes gradient phases and edge connectivity the scale space approach uses dynamic range compression to allow wavelet correlation over a wider range of scales a probabilistic formulation is used to combine the results obtained from filtering in each domain to provide a final edge probability image which has the advantages of both spatial and scale space domain methods decomposing this edge probability image with the same wavelet as the original image permits the generation of adaptive filters that can recognize the characteristics of the edges in all wavelet detail and approximation images regardless of scale these matched filters permit significant reduction in image noise without contributing to edge distortion the spatially adaptive wavelet noise filtering algorithm is qualitatively and quantitatively compared to a frequency domain and two wavelet based noise suppression algorithms using both natural and computer generated noisy images

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

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

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

a p version finite element approach for two and three dimensional problems of the j2 flow theory with non linear isotropic hardening in this paper an implementation of a two and three dimensional p version approach to the j 2 flow theory with non linear isotropic hardening for small displacements and small strains is presented based on higher order quadrilateral and hexahedral element formulations a newton raphson iteration scheme combined with a radial return algorithm is applied to find approximate solutions for the underlying physically non linear model problem curved boundaries are taken care of with the blending function method allowing an accurate representation of geometry with only a few p elements numerical examples demonstrate that the p version supplies efficient and accurate approximations to this class of physically non linear problems

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

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

short note o n implementation of the fast marching algorithm in this note we present an implementation of the fast marching algorithm for solving eikonal equations that in practice reduces the original run time from o nlogn to linear this lower run time cost is obtained while keeping an error bound of the same order of magnitude as the original algorithm this improvement is achieved introducing the straight forward untidy priority queue obtained via a quantization of the priorities in the marching computation we present the underlying framework estimations on the error and examples showing the usefulness of the proposed approach

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

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

efficient fast marching with finsler metrics we study the discretization of the escape time problem find the length of the shortest path joining an arbitrary point z z of a domain omega to the boundary partial omega path length is measured locally via a finsler metric potentially asymmetric and strongly anisotropic this optimal control problem can be reformulated as a static hamilton jacobi partial differential equation or as a front propagation model it has numerous applications ranging from motion planning to image segmentation we introduce a new algorithm fast marching using anisotropic stencil refinement fm asr which addresses this problem on a two dimensional domain discretized on a cartesian grid the local stencils used in our discretization are produced by arithmetic means like in the fm lbr mirebeau in anisotropic fast marching on cartesian grids using lattice basis reduction preprint 2012 a method previously introduced by the author in the special case of riemannian metrics the complexity of the fm asr in an average sense over all grid orientations only depends poly logarithmically on the anisotropy ratio of the metric while most alternative approaches have a polynomial dependence numerical experiments show in several occasions that the accuracy complexity compromise is improved by an order of magnitude or more

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

tensor based dictionary learning for spectral ct reconstruction spectral computed tomography ct produces an energy discriminative attenuation map of an object extending a conventional image volume with a spectral dimension in spectral ct an image can be sparsely represented in each of multiple energy channels and are highly correlated among energy channels according to this characteristics we propose a tensor based dictionary learning method for spectral ct reconstruction in our method tensor patches are extracted from an image tensor which is reconstructed using the filtered backprojection fbp to form a training dataset with the candecomp parafac decomposition a tensor based dictionary is trained in which each atom is a rank one tensor then the trained dictionary is used to sparsely represent image tensor patches during an iterative reconstruction process and the alternating minimization scheme is adapted for optimization the effectiveness of our proposed method is validated with both numerically simulated and real preclinical mouse datasets the results demonstrate that the proposed tensor based method generally produces superior image quality and leads to more accurate material decomposition than the currently popular popular methods

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

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

markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control we consider the construction of markov chain approximations for an important class of deterministic control problems the emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties the class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost this class covers a number of interesting areas of application including problems that arise in large deviations risk sensitive and robust control robust filtering and certain problems in computer vision examples are given as well as a proof of convergence

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

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

wavelet based noise reduction in ct images using correlation analysis the projection data measured in computed tomography ct and consequently the slices reconstructed from these data are noisy we present a new wavelet based structure preserving method for noise reduction in ct images that can be used in combination with different reconstruction methods the approach is based on the assumption that data can be decomposed into information and temporally uncorrelated noise in ct two spatially identical images can be generated by reconstructions from disjoint subsets of projections using the latest generation dual source ct scanners one image can be reconstructed from the projections acquired at the first the other image from the projections acquired at the second detector for standard ct scanners the two images can be generated by splitting up the set of projections into even and odd numbered projections the resulting images show the same information but differ with respect to image noise the analysis of correlations between the wavelet representations of the input images allows separating information from noise down to a certain signal to noise level wavelet coefficients with small correlation are suppressed while those with high correlations are assumed to represent structures and are preserved the final noise suppressed image is reconstructed from the averaged and weighted wavelet coefficients of the input images the proposed method is robust of low complexity and adapts itself to the noise in the images the quantitative and qualitative evaluation based on phantom as well as real clinical data showed that high noise reduction rates of around 40 can be achieved without noticeable loss of image resolution

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

implementation of efficient algorithms for globally optimal trajectories we consider a continuous space shortest path problem in a two dimensional plane this is the problem of finding a trajectory that starts at a given point ends at the boundary of a compact set of spl rfr sup 2 and minimizes a cost function of the form spl int sub o sup t r x t dt q x t for a discretized version of this problem a dijkstra like method that requires one iteration per discretization point has been developed by tsitsiklis 1995 here we develop some new label correcting like methods based on the small label first methods of bertsekas 1993 and bertsekas et al 1996 we prove the finite termination of these methods and present computational results showing that they are competitive and often superior to the dijkstra like method and are also much faster than the traditional jacobi and gauss seidel methods

anisotropic fast marching on cartesian grids using lattice basis reduction we introduce a modification of the fast marching algorithm which solves the anisotropic eikonal equation associated to an arbitrary continuous riemannian metric cal m on a two or three dimensional domain the algorithm has a complexity cal o n ln n n ln kappa cal m where n is the discrete domain cardinality the logarithmic dependency in the maximum anisotropy ratio kappa cal m of the riemannian metric allows us to handle extreme anisotropies for a limited numerical cost we prove the convergence of the algorithm and illustrate its efficiency by numerical experiments the algorithm relies on the computation at each grid point z of a special system of coordinates a reduced basis of the lattice bbb z m with respect to the symmetric positive definite matrix cal m z encoding the desired anisotropy at this point

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed