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

a robust numerical method for a fractional differential equation abstract this paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order 0 1 first the fractional differential equation is transformed into an equivalent volterra integral equation of the second kind with a weakly singular kernel based on the a priori information about the exact solution an integral discretization scheme on an a priori chosen adapted mesh is proposed by applying the truncation error estimate techniques and a discrete analogue of gronwall s inequality it is proved that the numerical method is first order convergent in the discrete maximum norm numerical results indicate that this method is more accurate and robust than finite difference methods when is close to 0

multi focus image fusion using pcnn this paper proposes a new method for multi focus image fusion based on dual channel pulse coupled neural networks dual channel pcnn compared with previous methods our method does not decompose the input source images and need not employ more pcnns or other algorithms such as dwt this method employs the dual channel pcnn to implement multi focus image fusion two parallel source images are directly input into pcnn meanwhile focus measure is carried out for source images according to results of focus measure weighted coefficients are automatically adjusted the rule of auto adjusting depends on the specific transformation input images are combined in the dual channel pcnn four group experiments are designed to testify the performance of the proposed method several existing methods are compared with our method experimental results show our presented method outperforms existing methods in both visual effect and objective evaluation criteria finally some practical applications are given further

modulating functions based algorithm for the estimation of the coefficients and differentiation order for a space fractional advection dispersion equation in this paper a new method based on the so called modulating functions is proposed to estimate average velocity dispersion coefficient and differentiation order in a space fractional advection dispersion equation where the average velocity and the dispersion coefficient are space varying first the average velocity and the dispersion coefficient are estimated by applying the modulating functions method where the problem is transformed into a linear system of algebraic equations then the modulating functions method combined with a newton s iteration algorithm is applied to estimate the coefficients and the differentiation order simultaneously the local convergence of the proposed method is proved numerical results are presented with noisy measurements to show the effectiveness and robustness of the proposed method it is worth mentioning that this method can be extended to general fractional partial differential equations

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

smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems abstract we construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form d y t f t y t with caputo type fractional derivatives d y of order 0 using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed algorithms two numerical examples are given

convergence analysis of the multistep legendre pseudo spectral method for volterra integral equations with vanishing delays in this paper we extend the single step pseudo spectral method for second kind volterra integral equations with vanishing variable delays to the multistep pseudo spectral method we also analyze the convergence of the hp version of the multistep pseudo spectral method under the l2 norm and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner in addition it is very suitable for long time calculations and large step size situations numerical results show good agreement with the theoretical analysis

operational tau method for singular system of volterra integro differential equations the legendre spectral tau matrix formulation is proposed to approximate solution of singular system of volterra integro differential equations the existence and uniqueness solution of this system are investigated by means of the smoothing property of a volterra integral operator and some projectors the l 2 convergence of the numerical method is analyzed it is proved theoretically and demonstrated numerically that the proposed method converges exponentially finally two numerical examples illustrate the theoretical results

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

a mathematical treatment of the fluorescence capillary fill device a mathematical model in the form of two coupled diffusion equations is provided for a competitive chemical reaction between an antigen and a labeled antigen for antibody sites on a cell wall boundary conditions are such that the problem is both nonlinear and nonlocal this is then recharacterized first as a pair of coupled singular integro differential equations and then as a system of four volterra integral equations the latter permits a proof of existence and uniqueness of the solution of the original problem small and large time asymptotic solutions are derived and from the first characterization a regular perturbation solution is obtained numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin

quantitative error measures for edge detection how to measure the performance of an edge detection method is an open problem many performance measures have been presented in the literature however there seems to be no agreement on the best option and there exist no practical studies of the features of some of the most recent measures in this work we make a comprehensive overview of the different proposals followed by a practical comparison of the most representative measures on synthetic as well as natural edge images

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

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

modified asymptotic orders of the direct filon method for a class of volterra integral equations the direct filon method provides efficient approximation for solutions of volterra integral equations with highly oscillatory kernels in this paper we study the rate of convergence of this method for a class of equations with trigonometric kernels the asymptotic order is developed by examining the property of the solution numerical experiments are included to verify the given order

tensor based formulation and nuclear norm regularization for multienergy computed tomography the development of energy selective photon counting x ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation under the assumption of perfect energy resolution here we propose a tensor based iterative algorithm that simultaneously reconstructs the x ray attenuation distribution for each energy we use a multilinear image model rather than a more standard stacked vector representation in order to develop novel tensor based regularizers in particular we model the multispectral unknown as a three way tensor where the first two dimensions are space and the third dimension is energy this approach allows for the design of tensor nuclear norm regularizers which like its 2d counterpart is a convex function of the multispectral unknown the solution to the resulting convex optimization problem is obtained using an alternating direction method of multipliers approach simulation results show that the generalized tensor nuclear norm can be used as a standalone regularization technique for the energy selective spectral computed tomography problem and when combined with total variation regularization it enhances the regularization capabilities especially at low energy images where the effects of noise are most prominent

an h p version of the continuous petrov galerkin finite element method for volterra integro differential equations with smooth and nonsmooth kernels we present an h p version of the continuous petrov galerkin cpg finite element method for linear volterra integro differential equations with smooth and nonsmooth kernels we establish a priori error estimates in the l 2 h 1 and l infty norms that are completely explicit with respect to the local discretization and regularity parameters for singular solutions caused by the weakly singular kernels we prove that algebraic convergence rates can be achieved for the h p version of the cpg method with quasi uniform meshes moreover we show that exponential rates of convergence can be achieved for solutions with start up singularities by using geometric time partitions and linearly increasing polynomial degrees numerical experiments are provided to illustrate the theoretical results

structure of a large social network we study a social network consisting of over 10 4 individuals with a degree distribution exhibiting two power scaling regimes separated by a critical degree k mathrm crit and a power law relation between degree and local clustering we introduce a growing random model based on a local interaction mechanism that reproduces the observed scaling features and their exponents we suggest that the double power law originates from two very different kinds of networks that are simultaneously present in the human social network

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

a novel method for solving second order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional differential equations it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems

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

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

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

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

a complex network based approach for boundary shape analysis this paper introduces a novel methodology to shape boundary characterization where a shape is modeled into a small world complex network it uses degree and joint degree measurements in a dynamic evolution network to compose a set of shape descriptors the proposed shape characterization method has an efficient power of shape characterization it is robust noise tolerant scale invariant and rotation invariant a leaf plant classification experiment is presented on three image databases in order to evaluate the method and compare it with other descriptors in the literature fourier descriptors curvature zernike moments and multiscale fractal dimension

solving initial and boundary value problems of fractional ordinary differential equations by using collocation and fractional powers abstract this paper presents a novel approach for solving initial and boundary values problems on ordinary fractional differential equations the strategy considered is based on the use of a suitable truncated series expressed in terms of fractional powers of the independent variable and a collocation approach with this strategy accurate numerical approximations can be obtained the error criterion is based on the concepts of residue and relative error two algorithms fitted to the nature linear or nonlinear of the considered fractional differential equation are proposed finally some problems illustrating the efficiency of the proposed method are presented

solving fractional integral equations by the haar wavelet method haar wavelets for the solution of fractional integral equations are applied fractional volterra and fredholm integral equations are considered the proposed method also is used for analysing fractional harmonic vibrations the efficiency of the method is demonstrated by three numerical examples

bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations a new operational matrix for integration of bernoulli polynomials is introduced by using this new operational matrix of integration and the so called collocation method linear volterra and nonlinear volterra fredholm hammerstein integral equations are reduced to systems of algebraic equations with unknown bernoulli coefficients some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the technique a new operational matrix for integration of bernoulli polynomials is introduced a collocation method is used to reduce equations to systems of algebraic equations some error bounds and convergence results are provided for the proposed method the efficiency and applicability of the technique are checked by some examples

the max roberts operator is a hueckel type edge detector hueckel s edge detector finds the best fitting ideal step edge to a given picture neighborhood by expanding the neighborhood and step edge in terms of a set of nine basis functions a very simple case of this approach uses a 2 2 neighborhood and three basis functions this case is solved explicitly using elementary methods the magnitude of the best fitting step edge for the neighborhood ab cd turns out to be the roberts operator max a d b c

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

simulation evaluation of gibbs prior distributions for use in maximum a posteriori spect reconstructions the effects of several of gibbs prior distributions in terms of noise characteristics edge sharpness and overall quantitative accuracy of the final estimates obtained from an iterative maximum a posteriori map procedure applied to data from a realistic chest phantom are demonstrated the effects of the adjustable parameters built into the prior distribution on these properties are examined it is found that these parameter values influence the noise and edge characteristics of the final estimate and can generate reconstructions closer to the actual solution than maximum likelihood ml in addition it is found that the choice of the shape of the prior distribution affects the noise characteristics and edge sharpness in the final estimate

blow up collocation solutions of nonlinear homogeneous volterra integral equations in this paper collocation methods are used for detecting blow up solutions of nonlinear homogeneous volterra hammerstein integral equations to do this we introduce the concept of blow up collocation solution and analyze numerically some blow up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied finally we discuss the relationships between necessary conditions for blow up of collocation solutions and exact solutions

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

fully discrete spectral methods for solving time fractional nonlinear sine gordon equation with smooth and non smooth solutions abstract we consider the initial boundary value problem of the time fractional nonlinear sine gordon equation and the fractional derivative is described in caputo sense with the order 1

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

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

analysis of collocation methods for generalized auto convolution volterra integral equations we first study the existence uniqueness and regularity properties of solutions to a generalized version of the auto convolution volterra integral equation of the second kind these results are then used to establish the optimal global and local super convergence properties of piecewise polynomial collocation solutions for such integral equations the theoretical results are illustrated by extensive numerical examples

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

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical examples

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

spectral iterative method and convergence analysis for solving nonlinear fractional differential equation abstract in this study a new spectral iterative method sim based on fractional interpolation is presented for solving nonlinear fractional differential equations fdes involving caputo derivative this method is equipped with a pre algorithm to find the singularity index of solution of the problem this pre algorithm gives us a real parameter as the index of the fractional interpolation basis for which the sim achieves the highest order of convergence in comparison with some recent results about the error estimates for fractional approximations a more accurate convergence rate has been attained we have also proposed the order of convergence for fractional interpolation error under the l 2 norm finally general error analysis of sim has been considered the numerical results clearly demonstrate the capability of the proposed method

recent advances in the numerical analysis of volterra functional differential equations with variable delays the numerical analysis of volterra functional integro differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving classical delay differential equations with non vanishing delays in this paper i shall describe recent results in the analysis of optimal global and local superconvergence orders in collocation methods for such evolutionary problems following a brief survey of results for equations containing volterra integral operators with non vanishing delays the discussion will focus on pantograph type volterra integro differential equations with linear and nonlinear vanishing delays the paper concludes with a section on open problems these include the asymptotic stability of collocation solutions u h on uniform meshes for pantograph type functional equations and the analysis of collocation methods for pantograph type functional equations with advanced arguments

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

smoothing transformation and spline collocation for linear fractional boundary value problems we construct and justify a high order method for the numerical solution of multi point boundary value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed method some numerical results are given

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

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

existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous volterra integral equations we analyse collocation methods for nonlinear homogeneous volterra hammerstein integral equations with non lipschitz nonlinearity we present different kinds of existence and uniqueness of nontrivial collocation solutions and give conditions for such existence and uniqueness in some cases finally we illustrate these methods with an example of a collocation problem and give some examples of collocation problems that do not fit in the cases studied previously

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

deterministic edge preserving regularization in computed imaging many image processing problems are ill posed and must be regularized usually a roughness penalty is imposed on the solution the difficulty is to avoid the smoothing of edges which are very important attributes of the image in this paper we first give conditions for the design of such an edge preserving regularization under these conditions we show that it is possible to introduce an auxiliary variable whose role is twofold first it marks the discontinuities and ensures their preservation from smoothing second it makes the criterion half quadratic the optimization is then easier we propose a deterministic strategy based on alternate minimizations on the image and the auxiliary variable this leads to the definition of an original reconstruction algorithm called artur some theoretical properties of artur are discussed experimental results illustrate the behavior of the algorithm these results are shown in the field of 2d single photon emission tomography but this method can be applied in a large number of applications in image processing

image edge detection based on local dimension a complex networks approach abstract a novel approach based on local dimension of complex networks is proposed to address edge detecting issue in image processing a new method that the image was modeled to complex networks is proposed our method for mapping an image into complex network is based on the weighted combination of the euclidean distance and gray level similarity indices which differs from the way was done previously in several works the local dimension of node characters the local property of pixel it is observed that edge pixels obviously have lower node dimension than non edge pixels which is used in our proposed edge detection method the proposed method is applied to both synthetic and natural images the results show the efficiency of our proposed method

numerical approximations for highly oscillatory bessel transforms and applications abstract this paper presents an efficient numerical method for approximating highly oscillatory bessel transforms based on analytic continuation we transform the integrals into the problems of integrating the forms on 0 with the integrand that does not oscillate and decays exponentially fast which can be efficiently computed by using gauss laguerre quadrature rule we then derive the error of the method depending on the frequency and the node number moreover we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations preliminary numerical results show the efficiency and accuracy of numerical approximations

a multilevel finite element method for fredholm integral eigenvalue problems in this work we proposed a multigrid finite element mfe method for solving the fredholm integral eigenvalue problems the main motivation for such studies is to compute the karhunen loeve expansions of random fields which play an important role in the applications of uncertainty quantification in our mfe framework solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh then any existing efficient integration scheme can be used for the associated integration process the error estimates are provided and the computational complexity is analyzed it is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh several numerical experiments are presented to validate the efficiency of the proposed numerical method

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

integral algebraic equations theory of collocation methods i our analysis of collocation solutions for general systems of linear integral algebraic equations iaes is based on the notions of the tractability index and the nu smoothing property of a volterra integral operator these are used to decouple the given iae system into the inherent system of regular second kind volterra integral equations vies and a system of first kind vies this decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions numerical examples illustrate the theoretical results

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

regularization designs for uniform spatial resolution and noise properties in statistical image reconstruction for 3 d x ray ct statistical image reconstruction methods for x ray computed tomography ct provide improved spatial resolution and noise properties over conventional filtered back projection fbp reconstruction along with other potential advantages such as reduced patient dose and artifacts conventional regularized image reconstruction leads to spatially variant spatial resolution and noise characteristics because of interactions between the system models and the regularization previous regularization design methods aiming to solve such issues mostly rely on circulant approximations of the fisher information matrix that are very inaccurate for undersampled geometries like short scan cone beam ct this paper extends the regularization method proposed in to 3 d cone beam ct by introducing a hypothetical scanning geometry that helps address the sampling properties the proposed regularization designs were compared with the original method in with both phantom simulation and clinical reconstruction in 3 d axial x ray ct the proposed regularization methods yield improved spatial resolution or noise uniformity in statistical image reconstruction for short scan axial cone beam ct

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

a spline product quasi interpolation method for weakly singular fredholm integral equations a discrete method of accuracy o h m is constructed and justified for a class of fredholm integral equations of the second kind with kernels that may have weak diagonal and boundary singularities the method is based on i improving the boundary behavior of the kernel with the help of a change of variables and ii the product integration using quasi interpolation by smooth splines of order m

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

numerical solution of a class of fractional order integro differential algebraic equations using muntz jacobi tau method abstract this paper presents a spectral tau method based on muntz jacobi basis functions for approximating the solutions of fractional order volterra integro differential algebraic equations we examine the solvability of the considered equation and we show that the exact solutions suffer from a singularity at the origin numerical solvability and convergence analysis of the proposed approximate approach are also studied and the error bounds are obtained in a weighted l 2 norm some illustrative examples are presented to validate the theoretical results

a multistep legendre gauss spectral collocation method for nonlinear volterra integral equations we introduce a multistep legendre gauss spectral collocation method for the nonlinear volterra integral equations of the second kind this method is easy to implement and possesses high order accuracy in addition it is very suitable for long time calculations we also derive the optimal convergence of the hp version of the multistep collocation method under the l 2 norm numerical experiments confirm the theoretical expectations

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

numerical solution of linear fredholm integral equations via two dimensional modification of hat functions abstract in this work we approximate the solution of the linear fredholm integral equations by means of a new two dimensional modification of hat functions 2d mhfs and a new operational matrix of integration by this idea the basic equations will be changed into the associated systems of algebraic equations also an error analysis is provided under several mild conditions the method is computationally attractive and some numerical examples are provided to illustrate its high accuracy

piecewise polynomial collocation for linear boundary value problems of fractional differential equations we consider a class of boundary value problems for linear multi term fractional differential equations which involve caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed in particular we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by two numerical examples

numerical methods for solving two dimensional nonlinear integral equations of fractional order by using two dimensional block pulse operational matrix in this paper our purpose is to construct a two dimensional fractional integral operational matrix and its use for the numerical solution of two dimensional fractional integral equations we use these operational matrices and properties of two dimensional block pulse functions 2d bpfs to reduce two dimensional fractional integral equations 2d fies to a system of algebraic equations obtained algebraic system based on the original problem can be linear or nonlinear then we show convergence of the proposed methods and we find the error bounds to show the accuracy efficiency and speed of the proposed method linear and nonlinear examples are presented

shifted jacobi spectral galerkin method for solving fractional order initial value problems abstract in this paper the shifted jacobi spectral galerkin method is introduced to deal with fractional order initial value problems fivps in the proposed method the exact solution of the fivp is approximated by using shifted jacobi polynomials on each subinterval of the total time the main advantage of the proposed method is that the rate of convergence in l 2 norm depends on the local smoothness of solution this enables the proposed method to work for nonsmooth solutions the second merit is that the flexibility between the length of sub interval and the degree of polynomial significantly enhances the numerical accuracy we derive the spectral galerkin approximation formula for the volterra integral equation of the second kind which is equivalent to the fivp error analysis in l 2 norm for the case 0 in shifted jacobi polynomials is provided and it justifies the spectral rate of convergence with respect to both the degree and the length of the subinterval when the source function is lipschitz continuous in the second argument numerical illustrations for multi order linear and nonlinear fivps are demonstrated to confirm the convergence rate

positive almost periodic solutions of nonlinear discrete systems with finite delay the existence of positive almost periodic solutions of nonlinear discrete systems with finite delay is obtained by using a variant of hilbert s projective metric on a subset of the space of almost periodic sequences

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

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

on graded meshes for weakly singular volterra integral equations with oscillatory trigonometric kernels we compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular volterra integral equation of the second kind with oscillatory trigonometric kernels due to in general unbounded derivatives at the left endpoint of the interval of integration we should approximate the solution of the integral equation by collocation methods on graded meshes however we show that this non smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory numerical examples are given to confirm the proposed results

existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction diffusion equations with delay abstract in this paper we study the mild solutions of a class of nonlinear fractional reaction diffusion equations with delay and caputo s fractional derivatives by the measure of noncompactness the theory of resolvent operators the fixed point theorem and the banach contraction mapping principle we develop various theorems for the existence and uniqueness of the global mild solutions for the problem

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

globally convergent edge preserving regularized reconstruction an application to limited angle tomography we introduce a generalization of a deterministic relaxation algorithm for edge preserving regularization in linear inverse problems this algorithm transforms the original possibly nonconvex optimization problem into a sequence of quadratic optimization problems and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex we prove that our more general algorithm is globally convergent i e converges to a local minimum from any initialization under less restrictive conditions even when the original cost functional is nonconvex we apply this algorithm to tomographic reconstruction from limited angle data by formulating the problem as one of regularized least squares optimization the results demonstrate that the constraint of piecewise smoothness applied through the use of edge preserving regularization can provide excellent limited angle tomographic reconstructions two edge preserving regularizers one convex the other nonconvex are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited angle scenarios and to explore how factors such as the choice of error norm angular sampling rate and amount of noise affect the reconstruction quality and algorithm performance these simulation results show that for this application the nonconvex regularizer produces consistently superior results

optimal superconvergence results for delay integro differential equations of pantograph type we analyze the optimal global and local orders of superconvergence of collocation solutions u h on uniform meshes i h for delay volterra integro differential equations with proportional delay functions given by theta t qt 0 q 1 t in 0 t in particular we show that if u h is a continuous piecewise polynomial of degree m geq 2 and if collocation is at the gauss legendre points then the optimal order of local superconvergence on i h is p ast m 2 it turns out that the same order p ast holds for nonlinear strictly increasing delay functions vanishing at t 0 however on judiciously chosen geometric meshes collocation at the gauss points yields the order 2m varepsilon n where varepsilon n rightarrow 0 as the number n of mesh points tends to infinity optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis

spline collocation methods for linear multi term fractional differential equations abstract some regularity properties of the solution of linear multi term fractional differential equations are derived based on these properties the numerical solution of such equations by piecewise polynomial collocation methods is discussed the results obtained in this paper extend the results of pedas and tamme 2011 15 where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer in the present paper we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by some numerical examples

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

hierarchical characterization of complex networks while the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node valuable information about the network topological properties can be obtained by considering further neighborhoods the current work considers the concept of virtual hierarchies established around each node and the respectively defined hierarchical node degree and clustering coefficient introduced in cond mat 0408076 complemented by new hierarchical measurements in order to obtain a powerful set of topological features of complex networks the interpretation of such measurements is discussed including an analytical study of the hierarchical node degree for random networks and the potential of the suggested measurements for the characterization of complex networks is illustrated with respect to simulations of random scale free and regular network models as well as real data airports proteins and word associations the enhanced characterization of the connectivity provided by the set of hierarchical measurements also allows the use of agglomerative clustering methods in order to obtain taxonomies of relationships between nodes in a network a possibility which is also illustrated in the current article

systems based decomposition schemes for the approximate solution of multi term fractional differential equations we give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi term fractional differential equations using the caputo form of the fractional derivative the schemes we compare are based on conversion of the original problem into a system of equations we review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation