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

tomographic reconstruction using an adaptive tetrahedral mesh defined by a point cloud medical images in nuclear medicine are commonly represented in three dimensions as a stack of two dimensional images that are reconstructed from tomographic projections although natural and straightforward this may not be an optimal visual representation for performing various diagnostic tasks a method for three dimensional 3 d tomographic reconstruction is developed using a point cloud image representation a point cloud is a set of points nodes in space where each node of the point cloud is characterized by its position and intensity the density of the nodes determines the local resolution allowing for the modeling of different parts of the image with different resolution the reconstructed volume which in general could be of any resolution size shape and topology is represented by a set of nonoverlapping tetrahedra defined by the nodes the intensity at any point within the volume is defined by linearly interpolating inside a tetrahedron from the values at the four nodes that define the tetrahedron this approach creates a continuous piecewise linear intensity over the reconstruction domain the reconstruction provides a distinct multiresolution representation which is designed to accurately and efficiently represent the 3 d image the method is applicable to the acquisition of any tomographic geometry such as parallel fan and cone beam and the reconstruction procedure can also model the physics of the image detection process an efficient method for evaluating the system projection matrix is presented the system matrix is used in an iterative algorithm to reconstruct both the intensity and location of the distribution of points in the point cloud examples of the reconstruction of projection data generated by computer simulations and projection data experimentally acquired using a jaszczak cardiac torso phantom are presented this work creates a framework for voxel less multiresolution representation of images in nuclear medicine

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

parametric imaging with bayesian priors a validation study with 11c altropane pet abstract it has been suggested that bayesian estimation methods may be used to improve the signal to noise ratio of parametric images however there is little experience with the method and some of the underlying assumptions and performance properties of bayesian estimation remain to be investigated we used a sample population of 54 subjects studied previously with 11 c altropane to empirically evaluate the assumptions performance and some practical issues in forming parametric images by using normality tests we showed that the underpinning normality assumptions of data and parametric distribution apply to more than 80 of voxels the standard deviation of the binding potential can be reduced 30 50 using bayesian estimation without introducing substantial bias the sample size required to form the a priori information was found to be modest as little as ten subjects may be sufficient and the choice of specific subjects has little effect on bayesian estimation a realistic simulation study showed that detection of localized differences in parametric images e g by statistical parametric mapping spm could be made more reliable and or conducted with smaller sample size using bayesian estimation in conclusion bayesian estimation can improve the snr of parametric images and better detect localized changes in cohorts of subjects

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

operational methods and differential equations with applications to initial value problems the authors present and develop a general method of operational nature in order to explore and investigate such families of orthogonal polynomials and functions as the classical and generalized hermite and laguerre polynomials as well as their higher order and multi index cases they also consider several applications of these operational techniques to the solutions of a wide class of partial differential equations including for example the heat wave and transport equations some of which involve second order laguerre derivatives

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

extinction of superdiffusions and semilinear partial differential equations abstract a superdiffusion is a measure valued branching process associated with a nonlinear operator lu u where l is a second order elliptic differential operator and is a function from r d r to r in the case l 1 0 the so called subcritical case the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1 most results on connections between the superdiffusion and differential equations involving lu u were obtained for the subcritical case in the present paper we assume only that l 1 is a bounded function in this more general setting the probability of extinction can be smaller than 1 and we show that his happens if and only if there exists a strictly positive solution of the equation lu u 0 in r d we also establish a relation between the probability of extinction in a domain d and strictly positive solutions of equation lu u 0 in d that are equal to 0 on the boundary of d we call such solutions special for the equation c u u where is the beltrami laplace operator on a complete riemannian manifold e strictly positive solutions in e were studied in connection with a geometrical problem which two functions represent scalar curvatures of two riemannian metrics related by a conformal mapping see 1 and references there

spatiotemporal reconstruction of list mode pet data we describe a method for computing a continuous time estimate of tracer density using list mode positron emission tomography data the rate function in each voxel is modeled as an inhomogeneous poisson process whose rate function can be represented using a cubic b spline basis the rate functions are estimated by maximizing the likelihood of the arrival times of detected photon pairs over the control vertices of the spline modified by quadratic spatial and temporal smoothness penalties and a penalty term to enforce nonnegativity randoms rate functions are estimated by assuming independence between the spatial and temporal randoms distributions similarly scatter rate functions are estimated by assuming spatiotemporal independence and that the temporal distribution of the scatter is proportional to the temporal distribution of the trues a quantitative evaluation was performed using simulated data and the method is also demonstrated in a human study using sup 11 c raclopride

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

effect of overlapping projections on reconstruction image quality in multipinhole spect multipinhole single photon emission computed tomography spect imaging has several advantages over single pinhole spect imaging including an increased sensitivity and an improved sampling however the quest for a good design is challenging due to the large number of design parameters the effect of one of these the amount of overlap in the projection images on the reconstruction image quality is examined in this paper the evaluation of the quality is based on efficient approximations for the linearized local impulse response and the covariance in a voxel and on the bias of the reconstruction of the noiseless projection data two methods are proposed that remove the overlap in the projection image by blocking certain projection rays with the use of extra shielding between the pinhole plate and the detector also two measures to quantify the amount of overlap are suggested first the approximate method predicting the contrast to noise ratio cnr is validated using postsmoothed maximum likelihood expectation maximization mlem reconstructions with an imposed target resolution second designs with different amounts of overlap are evaluated to study the effect of multiplexing in addition the cnr of each pinhole design is also compared with that of the same design where overlap is removed third the results are interpreted with the overlap quantification measures fourth the two proposed overlap removal methods are compared from the results we can conclude that once the complete detector area has been used the extra sensitivity due to multiplexing is only able to compensate for the loss of information not to improve the cnr removing the overlap however improves the cnr the gain is most prominent in the central field of view though often at the cost of the cnr of some voxels at the edges since after overlap removal very little information is left for their reconstruction the reconstruction images provide insight in the multiplexing and truncation artifacts

when the telegrapher s equation furnishes a better approximation to the transport equation than the diffusion approximation it has been suggested that a solution to the transport equation which includes anisotropic scattering can be approximated by the solution to a telegrapher s equation a j ishimaru appl opt 28 2210 1989 we show that in one dimension the telegrapher s equation furnishes an exact solution to the transport equation in two dimensions we show that since the solution can become negative the telegrapher s equation will not furnish a usable approximation a comparison between simulated data in three dimensions indicates that the solution to the telegrapher s equation is a good approximation to that of the full transport equation at the times at which the diffusion equation furnishes an equally good approximation s1063 651x 97 04205 0

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

optimal rebinning of time of flight pet data time of flight tof positron emission tomography pet scanners offer the potential for significantly improved signal to noise ratio snr and lesion detectability in clinical pet however fully 3d tof pet image reconstruction is a challenging task due to the huge data size one solution to this problem is to rebin tof data into a lower dimensional format we have recently developed fourier rebinning methods for mapping tof data into non tof formats that retain substantial snr advantages relative to sinograms acquired without tof information however mappings for rebinning into non tof formats are not unique and optimization of rebinning methods has not been widely investigated in this paper we address the question of optimal rebinning in order to make full use of tof information we focus on foret 3d which approximately rebins 3d tof data into 3d non tof sinogram formats without requiring a fourier transform in the axial direction we optimize the weighting for foret 3d to minimize the variance resulting in h2 weighted foret 3d which turns out to be the best linear unbiased estimator blue under reasonable approximations and furthermore the uniformly minimum variance unbiased umvu estimator under gaussian noise assumptions this implies that any information loss due to optimal rebinning is as a result only of the approximations used in deriving the rebinning equation and developing the optimal weighting we demonstrate using simulated and real phantom tof data that the optimal rebinning method achieves variance reduction and contrast recovery improvement compared to nonoptimized rebinning weightings in our preliminary study using a simplified simulation setup the performance of the optimal rebinning method was comparable to that of fully 3d tof map

controlling familywise error rate for matched subspace detection in dynamic fdg pet detection of small lesions in fluorodeoxyglucose fdg positron emission tomography pet is limited by image resolution and low signal to noise ratio we have previously described a matched subspace detection method that uses the time activity curve to distinguish tumors from background in dynamic fdg pet applying this algorithm on a voxel by voxel basis throughout the dynamic image produces a test statistic image or ldquomaprdquo which on thresholding indicates the potential locations of secondary or metastatic tumors in this paper we describe a thresholding method that controls familywise error rate fwer for the matched subspace detection statistical map the method involves three steps first the pet image is segmented into several homogeneous regions then the statistical map is normalized to a zero mean unit variance gaussian random field finally the images are thresholded at a fixed fwer by estimating their spatial smoothness and applying a random field theory maximum statistic approach we evaluate this thresholding method using digital phantoms generated from clinical dynamic images we also present an application of the proposed approach to clinical pet data from a breast cancer patient with metastatic disease

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

noise propagation in linear and nonlinear inverse scattering the propagation of noise from the data to the reconstructed speed of sound image by inverse scattering within the framework of the lippmann schwinger integral equation of scattering is discussed the inversion algorithm that was used consisted in minimizing a tikhonov functional in the unknown speed of sound the gradient of the objective functional was computed by the method of the adjoint fields an analytical expression for the inverse scattering covariance matrix of the image noise was derived it was shown that the covariance matrix in the linear x ray computed tomography is a special case of the inverse scattering matrix derived in this paper the matrix was also analyzed in the limit of the linearized born approximation and the results were found to be in qualitative agreement with those recently reported in the literature for born inversion using filtered backpropagation algorithm finally the applicability of the analysis reported here to the obstacle problem and the physical optics approximation was discussed

evaluation of fisher information matrix based methods for fast assessment of image quality in pinhole spect the accurate determination of the local impulse response and the covariance in voxels from penalized maximum likelihood reconstructed images requires performing reconstructions from many noise realizations of the projection data as this is usually a very time consuming process efficient analytical approximations based on the fisher information matrix fim have been extensively used in pet and spect to estimate these quantities for 3d imaging however additional approximations need to be made to the fim in order to speed up the calculations the most common approach is to use the local shift invariant lsi approximation of the fim but this assumes specific conditions which are not always necessarily valid in this paper we take a single pinhole spect system and compare the accuracy of the lsi approximation against two other methods that have been more recently put forward the non uniform object space pixelation nuop and the subsampled fim these methods do not assume such restrictive conditions while still increasing the speed of the calculations considerably our results indicate that in pinhole spect the nuop and subsampled fim approaches could be more reliable than the lsi approximation especially when a high accuracy is required

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

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

statistical modeling and reconstruction of randoms precorrected pet data randoms precorrected positron emission tomography pet data is formed as the difference of two poisson random variables its exact probability mass function pmf is inconvenient for use in likelihood based iterative image reconstruction as it contains an infinite summation the shifted poisson model is a tractable approximation to this pmf but requires that negative values are truncated resulting in positively biased reconstructions in low count studies here we analyze the properties of the exact pmf and propose a simple but accurate approximation that allows negative valued data we investigate the properties of this approximation and demonstrate its application to penalized maximum likelihood image reconstruction

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

evolution of non spreading airy wavepackets in time dependent linear potentials abstract we report on the use of the algebraic methods to obtain the explicit form of the solution of the schrodinger equation with a linear potential we consider the case of the explicitly time dependent hamiltonian and formulate the general conditions that allow for the solutions to be found that are expressed in terms of airy functions yielding non spreading wave packets the relevant physical meaning of these solutions is analyzed and the examples of their applications are given the role played by the airy transform and its relevance to the problems involving linear potentials is discussed eventually we present a thorough discussion on the analogy between the airy and the gauss weierstrass transform often employed in the solutions of the heat type equations

semilinear parabolic equations diffusions and superdiffusions the semilinear equationu l0u u wherel0is a second order elliptic differential operator without zero order term and 1 1 related problems are treated with the help of diffusion processes this approach is useful even in the linear theory for instance the first boundary value problem for equationu lu fcan be investigated for a class of domains described in probabilistic terms which is substantially larger than the class considered in the literature on pdes

analytical solutions for heat diffusion beyond fourier law we obtain solutions for differential equations describing a broad range of physical problems by the operational method with recourse to inverse differential operators integral transforms and operational exponent generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions the evolutional type problems for heat transfer in various heat conduction models are studied exact analytical solutions for guyer krumhansl hyperbolic heat equation are obtained and compared with those of fourier and cattaneo equations modelling heat pulse propagation from a laser source is performed in the framework of fourier cattaneo and guyer krumhansl heat transfer models compliance of obtained solutions with the maximum principle is studied

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

single and multipinhole collimator design evaluation method for small animal spect high resolution functional imaging of small animals is often obtained by single pinhole spect with circular orbit acquisition multipinhole spect adds information due to its improved sampling and can improve the trade off between resolution and sensitivity to evaluate different pinhole collimator designs an efficient method is needed that quantifies the reconstruction image quality in this paper we propose a fast approximate method that examines the quality of individual voxels of a postsmoothed maximum likelihood expectation maximization mlem reconstruction by studying their linearized local impulse response llir and co variance for a predefined target resolution for validation the contrast to noise ratios cnrs in some voxels of a homogeneous sphere and of a realistic rat brain software phantom were calculated for many single and multipinhole designs a good agreement was observed between the cnrs obtained with the approximate method and those obtained with postsmoothed mlem reconstructions of simulated noisy projections this good agreement was quantified by a least squares fit through these results which yielded a line with slope 1 02 1 00 expected and a y intercept close to zero 0 expected 95 4 of the validation points lie within three standard deviations from that line using the approximate method the influence on the cnr of varying a parameter in realistic single and multipinhole designs was examined the investigated parameters were the aperture diameter the distance between the apertures and the axis of rotation the focal distance the acceptance angle the position of the apertures the focusing distance and the number of pinholes the results can generally be explained by the change in sensitivity the amount of postsmoothing and the amount of overlap in the projections the method was applied to multipinhole designs with apertures focusing at a single point but is also applicable to other designs

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

orthogonality properties of the hermite and related polynomials the authors present a general method of operational nature with a view to investigating the orthogonality properties of several different families of the hermite and related polynomials in particular the classical hermite polynomials and some of their higher order and multi index generalizations are considered here

operational solution for some types of second order differential equations and for relevant physical problems abstract we present an operational method to obtain solutions for differential equations describing a broad range of physical problems including ordinary non integer order and high order partial differential equations inverse differential operators are proposed to solve a variety of differential equations integral transforms and the operational exponent are used to obtain the solutions generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions examples of solutions of physical problems related to propagation of the heat and other quantities are demonstrated by the developed operational technique in particular the evolution type problems the generalizations of the black scholes of the heat conduction of the fokker planck equations are considered as well as equations involving the laguerre derivative operator

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

emission image reconstruction for randoms precorrected pet allowing negative sinogram values most positron emission tomography pet emission scans are corrected for accidental coincidence ac events by real time subtraction of delayed window coincidences leaving only the randoms precorrected data available for image reconstruction the real time randoms precorrection compensates in mean for ac events but destroys the poisson statistics the exact log likelihood for randoms precorrected data is inconvenient so practical approximations are needed for maximum likelihood or penalized likelihood image reconstruction conventional approximations involve setting negative sinogram values to zero which can induce positive systematic biases particularly for scans with low counts per ray we propose new likelihood approximations that allow negative sinogram values without requiring zero thresholding with negative sinogram values the log likelihood functions can be nonconcave complicating maximization nevertheless we develop monotonic algorithms for the new models by modifying the separable paraboloidal surrogates and the maximum likelihood expectation maximization ml em methods these algorithms ascend to local maximizers of the objective function analysis and simulation results show that the new shifted poisson sp model is nearly free of systematic bias yet keeps low variance despite its simpler implementation the new sp performs comparably to the saddle point model which has shown the best performance as to systematic bias and variance in randoms precorrected pet emission reconstruction

superdiffusions and parabolic nonlinear differential equations we establish connections between superdiffusion processes and one class of nonlinear parabolic differential equations analytic results due to brezis and friedman baras and pierre and others are used to investigate the graphs of superdiffusions a survey of the literature and general comments are presented in section 4

stochastic boundary values and boundary singularities for solutions of the equationlu uα abstract we investigate positive solutions of a nonlinear equationlu u wherelis a second order elliptic differential operator in a riemannian manifoldeand 1 2 we establish a 1 1 correspondence between the set u of positive solutions and a class z of functionals ofxwhich we call linear boundary functionals they depend only on the behavior ofxnear the martin boundarye the class z is a closed convex cone andu u is a subadditive function ofz z special roles belong to moderate solutions corresponding tozwith finite mathematical expectations and to a family of solutions determined by the range ofx a new formula is deduced connectingu zandl diffusions conditioned to hit the boundarye at a given pointy a concept of a singular boundary point foruis introduced in terms of the conditioned diffusion

a theoretical signal processing framework for linear diffusion mri implications for parameter estimation and experiment design the data measured in diffusion mri can be modeled as the fourier transform of the ensemble average propagator eap a probability distribution that summarizes the molecular diffusion behavior of the spins within each voxel this fourier relationship is potentially advantageous because of the extensive theory that has been developed to characterize the sampling requirements accuracy and stability of linear fourier reconstruction methods however existing diffusion mri data sampling and signal estimation methods have largely been developed and tuned without the benefit of such theory instead relying on approximations intuition and extensive empirical evaluation this paper aims to address this discrepancy by introducing a novel theoretical signal processing framework for diffusion mri the new framework can be used to characterize arbitrary linear diffusion estimation methods with arbitrary q space sampling and can be used to theoretically evaluate and compare the accuracy resolution and noise resilience of different data acquisition and parameter estimation techniques the framework is based on the eap and makes very limited modeling assumptions as a result the approach can even provide new insight into the behavior of model based linear diffusion estimation methods in contexts where the modeling assumptions are inaccurate the practical usefulness of the proposed framework is illustrated using both simulated and real diffusion mri data in applications such as choosing between different parameter estimation methods and choosing between different q space sampling schemes

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

a unified approach to statistical tomography using coordinate descent optimization over the past years there has been considerable interest in statistically optimal reconstruction of cross sectional images from tomographic data in particular a variety of such algorithms have been proposed for maximum a posteriori map reconstruction from emission tomographic data while map estimation requires the solution of an optimization problem most existing reconstruction algorithms take an indirect approach based on the expectation maximization em algorithm we propose a new approach to statistically optimal image reconstruction based on direct optimization of the map criterion the key to this direct optimization approach is greedy pixel wise computations known as iterative coordinate decent icd we propose a novel method for computing the icd updates which we call icd newton raphson we show that icd newton raphson requires approximately the same amount of computation per iteration as em based approaches but the new method converges much more rapidly in our experiments typically five to ten iterations other advantages of the icd newton raphson method are that it is easily applied to map estimation of transmission tomograms and typical convex constraints such as positivity are easily incorporated

convergent incremental optimization transfer algorithms application to tomography no convergent ordered subsets os type image reconstruction algorithms for transmission tomography have been proposed to date in contrast in emission tomography there are two known families of convergent os algorithms methods that use relaxation parameters and methods based on the incremental expectation maximization em approach this paper generalizes the incremental em approach by introducing a general framework incremental optimization transfer the proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters the general optimization transfer framework allows the use of a very broad family of surrogate functions enabling the development of new algorithms this paper provides the first convergent os type algorithm for nonconcave penalized likelihood pl transmission image reconstruction by using separable paraboloidal surrogates sps which yield closed form maximization steps we found it is very effective to achieve fast convergence rates by starting with an os algorithm with a large number of subsets and switching to the new transmission incremental optimization transfer triot algorithm results show that triot is faster in increasing the pl objective than nonincremental ordinary sps and even os sps yet is convergent

solutions of nonlinear differential equations on a riemannian manifold and their trace on the martin boundary let l be a second order elliptic differential operator on a riemannian manifold e with no zero order terms we say that a function h is l harmonic if lh 0 every positive l harmonic function has a unique representation

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

choquet integrals as projection operators for quantified tomographic reconstruction in this paper we propose to investigate and analyze a new method for performing quantified projection and back projection in emission tomography this method is based on using non summative kernels capacities and asymmetric choquet integral to obtain imprecise projected values i e intervals instead of usual reconstructed pixel values validation studies using numerical and physical single photon computed emission tomography spect phantoms were used to demonstrate links between the length of these reconstructed intervals and the stochastic noise level in reconstructed slices

trace on the boundary for solutions of nonlinear differential equations let l be a second order elliptic differential operator in rd with no zero order terms and let e be a bounded domain in rd with smooth boundary e we say that a function h is l harmonic if lh 0 in e every positive l harmonic function has a unique representation h x j x y i dy e where k is the poisson kernel for l and z is a finite measure on e we call z the trace of h on me our objective is to investigate positive solutions of a nonlinear equation lu u in e for 1 2 we associate with every solution u a pair if i where f is a closed subset of e and zis a radon measure on 0 e f we call f zi the trace of u on me f is empty if and only if u is dominated by an l harmonic function we call such solutions moderate a moderate solution is determined uniquely by its trace in general many solutions can have the same trace we establish necessary and sufficient conditions for a pair f z to be a trace and we give a probabilistic formula for the maximal solution with a given trace

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

generalized polynomials operational identities and their applications abstract it is shown that an appropriate combination of methods relevant to generalized operational calculus and to special functions can be a very useful tool to treat a large body of problems both in physics and mathematics we discuss operational methods associated with multivariable hermite laguerre legendre and other polynomials to derive a wealth of identities useful in quantum mechanics electromagnetism optics etc or to derive new identities between special functions as e g mehler or mixed type generating functions

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

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

iterative image reconstruction using inverse fourier rebinning for fully 3 d pet we describe a fast forward and back projector pair based on inverse fourier rebinning for use in iterative image reconstruction for fully 3 d positron emission tomography pet the projector pair is used as part of a factored system matrix that takes into account detector pair response by using shift variant sinogram blur kernels thereby combining the computational advantages of fourier rebinning with iterative reconstruction using accurate system models the forward projector consists of a 2 d projector which maps 3 d images into 2 d direct sinograms followed by exact inverse rebinning which maps the 2 d into fully 3 d sinograms the back projector is implemented as the transpose of the forward projector and differs from the true exact rebinning operator in the sense that it does not require reprojection to compute missing lines of response lors we compensate for two types of inaccuracies that arise in a cylindrical pet scanner when using inverse fourier rebinning 1 nonuniform radial sampling and 2 nonconstant oblique angles in the radial direction in a single oblique sinogram we examine the effects of these corrections on sinogram accuracy and reconstructed image quality we evaluate performance of the new projector pair for maximum a posteriori map reconstruction of simulated and in vivo data the new projector results in only a small loss in resolution towards the edge of the field of view when compared to the fully 3 d geometric projector and requires an order of magnitude less computation

theoretical analysis of penalized maximum likelihood patlak parametric image reconstruction in dynamic pet for lesion detection detecting cancerous lesions is a major clinical application of emission tomography in a previous work we studied penalized maximum likelihood pml image reconstruction for lesion detection in static pet here we extend our theoretical analysis of static pet reconstruction to dynamic pet we study both the conventional indirect reconstruction and direct reconstruction for patlak parametric image estimation in indirect reconstruction patlak parametric images are generated by first reconstructing a sequence of dynamic pet images and then performing patlak analysis on the time activity curves tacs pixel by pixel in direct reconstruction patlak parametric images are estimated directly from raw sinogram data by incorporating the patlak model into the image reconstruction procedure pml reconstruction is used in both the indirect and direct reconstruction methods we use a channelized hotelling observer cho to assess lesion detectability in patlak parametric images simplified expressions for evaluating the lesion detectability have been derived and applied to the selection of the regularization parameter value to maximize detection performance the proposed method is validated using computer based monte carlo simulations good agreements between the theoretical predictions and the monte carlo results are observed both theoretical predictions and monte carlo simulation results show the benefit of the indirect and direct methods under optimized regularization parameters in dynamic pet reconstruction for lesion detection when compared with the conventional static pet reconstruction