an improved algorithm for reprojecting rays through pixel images it is often desired to calculate line integrals through a field of reconstructed ct density pixels for the purpose of improving ct image quality two algorithms widely published and discussed in the past are known to either degrade spatial resolution or generate errors in the results due to the discontinuous square pixel modeling of the reconstructed image an algorithm is described based on linear interpolation between pixels which provides superior accuracy without unnecessary loss of resolution it was tested on simulated data for a head section and on a narrow gaussian density distribution the experimental results demonstrated improved performance the method is expected to prove useful for many types of post reconstruction processing including beam hardening missing data and noise supression algorithms

3d forward and back projection for x ray ct using separable footprints iterative methods for 3d image reconstruction have the potential to improve image quality over conventional filtered back projection fbp in x ray computed tomography ct however the computation burden of 3d cone beam forward and back projectors is one of the greatest challenges facing practical adoption of iterative methods for x ray ct moreover projector accuracy is also important for iterative methods this paper describes two new separable footprint sf projector methods that approximate the voxel footprint functions as 2d separable functions because of the separability of these footprint functions calculating their integrals over a detector cell is greatly simplified and can be implemented efficiently the sf tr projector uses trapezoid functions in the transaxial direction and rectangular functions in the axial direction whereas the sf tt projector uses trapezoid functions in both directions simulations and experiments showed that both sf projector methods are more accurate than the distance driven dd projector which is a current state of the art method in the field the sf tt projector is more accurate than the sf tr projector for rays associated with large cone angles the sf tr projector has similar computation speed with the dd projector and the sf tt projector is about two times slower

first and second order characterizations of pseudolinear functions abstract this work is concerned with first and second order characterizations of pseudolinear functions it is shown that for continuously differentiable function x 0 pseudolinearity is completely characterized by a special property of the normalized gradient x x applying some results on the quasi hessian matrices the eigenvalues and the eigenvectors of the hessian 2 x of a twice continuously differentiable pseudolinear function x is characterized

a forward regridding method with minimal oversampling for accurate and efficient iterative tomographic algorithms reconstruction of underconstrained tomographic data sets remains a major challenge standard analytical techniques frequently lead to unsatisfactory results due to insufficient information several iterative algorithms which can easily integrate a priori knowledge have been developed to tackle this problem during the last few decades most of these iterative algorithms are based on an implementation of the radon transform that acts as forward projector this operator and its adjoint the backprojector are typically called few times per iteration and represent the computational bottleneck of the reconstruction process here we present a fourier based forward projector founded on the regridding method with minimal oversampling we show that this implementation of the radon transform significantly outperforms in efficiency other state of the art operators with o n2log2n complexity despite its reduced computational cost this regridding method provides comparable accuracy to more sophisticated projectors and can therefore be exploited in iterative algorithms to substantially decrease the time required for the reconstruction of underconstrained tomographic data sets without loss in the quality of the results

accurate simulations of pure quasi p waves in complex anisotropic media abstractreverse time migration rtm in complex anisotropic media requires calculation of the propagation of a single mode wave the quasi p wave this was conventionally realized by solving a 2 2 system of second order partial differential equations the implementation of this 2 2 system required at least twice the computational resources as compared with the acoustic wave equation the s waves an introduced auxiliary function in this system were treated as artifacts in the rtm furthermore the 2 2 system suffered numerical stability problems at the places in which abrupt changes of symmetric axis of anisotropy exist which brings more challenges to real data implementation on the other hand the alkhalifah s equation which governs the pure quasi p wave propagation was hard to solve because it was a pseudodifferential equation we proposed a pure quasi p wave equation that can be easily implemented within current imaging framework our new equation was obtained by decomposing the original pseudodif

equivalent accuracy at a fraction of the cost overcoming temporal dispersion abstractnumerical dispersion in finite difference fd modeling produces coherent artifacts severely constraining the resolution of advanced imaging and inversion techniques conventionally numerical dispersion is reduced by increasing the order of accuracy of the fd operators and we resign ourselves to paying the high computational cost that is incurred assuming no spatial dispersion we have found that fd time dispersion is independent of the medium velocity and the spatial grid for propagation and only depends on the time stepping scheme and the propagation time based on this observation we have devised postpropagation filters to collapse the time dispersion effect of fd modeling our dispersion correction filters are designed by comparing the input waveform with dispersive waveforms obtained by 1d forward modeling these filters are then applied on multidimensional shot records to eliminate the time dispersion by two schemes 1 stationary filtering plus interpolation and 2 nonstationary fil

an explicit time evolution method for acoustic wave propagation cost effective waveform modeling is the key to practical reverse time migration rtm and full waveform inversion fwi implementations we evaluated an explicit time evolution ete method to efficiently simulate wave propagation in acoustic media with high temporal accuracy we started from the constant density acoustic wave equation and obtained an analytical time marching scheme in the wavenumber domain we then formulated an ete scheme in the time space domain by introducing a cosine function approximation although the ete operator appears to be similar to the second order temporal finite difference fd operator the exact nature of the ete formula ensures high accuracy in time we further introduced a set of optimum stencils and coefficients by minimizing evolution errors in a least squares sense our numerical tests indicated that ete can achieve similar waveform accuracy as fd with four times larger time steps meanwhile the compact ete operator keeps the computation efficient the efficiency and capability to handle complex velocity field make ete an attractive engine in acoustic rtm and fwi

a multilevel domain decomposition algorithm for fast o n sup 2 logn reprojection of tomographic images a novel algorithm for fast computation of tomographic image projections is presented the method comprises a decomposition of an image into subimages followed by an aggregation of projections computed for the subimages the multilevel domain decomposition algorithm is formulated as a recursive procedure the computational cost of the proposed algorithm is comparable to that of fft based techniques while it appears to be more flexible than the latter numerical results demonstrate the effectiveness of the method

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

proper efficiency in a linear fractional vector maximum problem with generalized convex constraints abstract in the present paper a vector maximum problem vmp with linear fractional objectives and generalized convex constraints is considered a necessary and sufficient condition for an efficient solution of vmp is derived in kuhn tucker s form moreover it is proved that under a certain boundedness assumption an efficient solution is properly efficient this extends the results of choo 3 for a linear fractional vector maximum problem with linear constraints to generalized convex constraints an example is given to illustrate the results

a new time space domain high order finite difference method for the acoustic wave equation a new unified methodology was proposed in finkelstein and kastner 2007 39 to derive spatial finite difference fd coefficients in the joint time space domain to reduce numerical dispersion the key idea of this method is that the dispersion relation is completely satisfied at several designated frequencies we develop this new time space domain fd method further for 1d 2d and 3d acoustic wave modeling using a plane wave theory and the taylor series expansion new spatial fd coefficients are frequency independent though they lead to a frequency dependent numerical solution we prove that the modeling accuracy is 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation however under the same discretization the new 1d method can reach 2m th order accuracy and is always stable the 2d method can reach 2m th order accuracy along eight directions and has better stability similarly the 3d method can reach 2m th order accuracy along 48 directions and also has better stability than the conventional fd method the advantages of the new method are also demonstrated by the results of dispersion analysis and numerical modeling of acoustic wave equation for homogeneous and inhomogeneous acoustic models in addition we study the influence of the fd stencil length on numerical modeling for 1d inhomogeneous media and derive an optimal fd stencil length required to balance the accuracy and efficiency of modeling a new time space domain high order staggered grid fd method for the 1d acoustic wave equation with variable densities is also developed which has similar advantages demonstrated by dispersion analysis stability analysis and modeling experiments the methodology presented in this paper can be easily extended to solve similar partial difference equations arising in other fields of science and engineering

fast approximation of algebraic reconstruction methods for tomography most reconstruction algorithms for transmission tomography can be subdivided in two classes variants of filtered backprojection fbp and iterative algebraic methods fbp is very fast and yields accurate results when a large number of projections are available with high signal to noise ratio and a full angular range algebraic methods require much more computation time yet they are more flexible in dealing with limited data problems and noise in this paper we propose an algorithm that combines the best of these two approaches for a given linear algebraic method a filter is computed that can be used within the fbp algorithm the fbp reconstructions that result from using this filter strongly resemble the algebraic reconstructions and have many of their favorable properties while the required reconstruction time is similar to standard fbp based on a series of experiments for both simulation data and experimental data we demonstrate the merits of the proposed algorithm

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

time domain explicit finite difference method based on the mixed domain function approximation for acoustic wave equation abstractexplicit finite difference fd methods with high accuracy and efficiency are preferred in full waveform inversion and reverse time migration the taylor series expansion te based fd methods can only obtain high accuracy on a small wavenumber zone we have developed a new explicit fd method with spatial arbitrary even order accuracy based on the mixed k wavenumber space domain function approximation for the acoustic wave equation and we derived the fd coefficients by minimizing the approximation error in a least squares ls sense the weighted pseudoinverse of mixed k space matrix is introduced into the ls optimization problem to improve the accuracy the new method has an exact temporal derivatives discretization in homogeneous media and also has higher temporal and spatial accuracy in heterogeneous media approximation errors and numerical dispersion analysis demonstrate that the new fd method has a higher numerical accuracy than conventional te based fd and te based time space domain disp

finite difference time dispersion transforms for wave propagation the finite difference fd wave equation is widely implemented in seismic imaging for oil exploration but the numerical dispersion due to discretization of time and space derivatives can introduce severe numerical errors we have investigated time dispersion which might distort the phase and introduce severe artifacts to the data and images especially for long time propagation we first studied precisely how the time dispersion was produced and then we developed an efficient approach the time dispersion transforms to cope with time dispersion errors in wave propagation the proposed forward time dispersion transform can predict or simulate the time dispersion errors and it can be applied to remove time dispersion errors in reverse time migration the inverse time dispersion transform can be used to eliminate time dispersion errors from synthetic data after modeling the proposed method is general and works for low and high order time fd schemes by using this method large time steps were allowed in wave propagation thus we can remove the time dispersion errors at a negligible numerical cost and the efficiency is not affected to test the effectiveness and robustness of the proposed time dispersion transforms we used isotropic and anisotropic examples as well as modeling and imaging examples

time evolution of the wave equation using rapid expansion method forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields for the case of spatially varying velocity we have worked on two approaches to evaluate the time evolution of seismic wavefields an exact solution for the constant velocity acoustic wave equation can be used to simulate the pressure response at any time for a spatially varying velocity a one step method can be developed where no intermediate time responses are required using this approach we have solved for the pressure response at intermediate times and have developed a recursive solution the solution has a very high degree of accuracy and can be reduced to various finite difference time derivative methods depending on the approximations used although the two approaches are closely related each has advantages depending on the problem being solved

technical note proper efficiency and the linear fractional vector maximum problem a vector maximum problem arises when two or more criterion functions are to be simultaneously maximized over a feasible region we prove that in linear fractional vector maximum problems every efficient solution satisfies the requirements of a restricted concept of efficiency the notion of proper efficiency

time step n tupling for wave equations abstractwe have constructed novel temporal discretizations for wave equations we first select an explicit time integrator that is of second order leading to classic time marching schemes in which the next value of the wavefield at the discrete time t t is computed from current values known at time t and the previous time t t then we determine how the time step can be doubled tripled or generally n tupled producing a new time stepping method in which the next value of the wavefield at the discrete time t n t is computed from current values known at time t and the previous time t n t in between time values of the wavefield are eliminated using the fourier method to calculate space derivatives the new time integrators allow larger stable time steps than traditional time integrators however like the lax wendroff procedure they require more computational effort per time step because the new schemes are developed from the classic second order time stepping scheme they will have the same proper

a multiresolution approach to iterative reconstruction algorithms in x ray computed tomography in computed tomography the application of iterative reconstruction methods in practical situations is impeded by their high computational demands especially in high resolution x ray computed tomography where reconstruction volumes contain a high number of volume elements several giga voxels this computational burden prevents their actual breakthrough besides the large amount of calculations iterative algorithms require the entire volume to be kept in memory during reconstruction which quickly becomes cumbersome for large data sets to overcome this obstacle we present a novel multiresolution reconstruction which greatly reduces the required amount of memory without significantly affecting the reconstructed image quality it is shown that combined with an efficient implementation on a graphical processing unit the multiresolution approach enables the application of iterative algorithms in the reconstruction of large volumes at an acceptable speed using only limited resources

maximizing the cfl number of stable time space domain explicit finite difference modeling abstract the time space domain explicit finite difference fd method numerically solves wave equation by applying fd to approximate its spatial and temporal derivatives this method always faces the stability problem i theoretically analyze stability condition of this method for the acoustic wave equation and find that the maximum courant friedrichs lewy cfl number of stable modeling depends on the maximum value of the spatial fd dispersion relation the conventional method commonly determines spatial fd coefficients by approximating the spatial fd dispersion to the exact relation within the given wavenumber area however outside this area the dispersion relation and hence the maximum cfl number is uncontrollable to make it controllable i propose to simultaneously approximate the spatial fd dispersion relation to a given function outside this area further i develop an algorithm to maximize the maximum cfl number stability analysis and modeling examples demonstrate that the proposed method has better stability than the conventional method especially when the maximum effective wavenumber frequency is small for the 2d case the conventional method performs stable simulation using a cfl number of no more than 0 707 but the new method can perform stable simulation using a large cfl number for example 1 6 4 8 and 9 0 and non long spatial fd stencils

pseudolinearity and efficiency first order and second order characterizations of pseudolinear functions are derived for a nonlinear programming problem involving pseudolinear functions only it is proved that every efficient solution is properly efficient under some mild conditions

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

averaging and metropolis iterations for positron emission tomography iterative positron emission tomography pet reconstruction computes projections between the voxel space and the lines of response lor space which are mathematically equivalent to the evaluation of multi dimensional integrals the dimension of the integration domain can be very high if scattering needs to be compensated monte carlo mc quadrature is a straightforward method to approximate high dimensional integrals as the numbers of voxels and lors can be in the order of hundred millions and the projection also depends on the measured object the quadratures cannot be precomputed but monte carlo simulation should take place on the fly during the iterative reconstruction process this paper presents modifications of the maximum likelihood expectation maximization ml em iteration scheme to reduce the reconstruction error due to the on the fly mc approximations of forward and back projections if the mc sample locations are the same in every iteration step of the ml em scheme then the approximation error will lead to a modified reconstruction result however when random estimates are statistically independent in different iteration steps then the iteration may either diverge or fluctuate around the solution our goal is to increase the accuracy and the stability of the iterative solution while keeping the number of random samples and therefore the reconstruction time low we first analyze the error behavior of ml em iteration with on the fly mc projections then propose two solutions averaging iteration and metropolis iteration averaging iteration averages forward projection estimates during the iteration sequence metropolis iteration rejects those forward projection estimates that would compromise the reconstruction and also guarantees the unbiasedness of the tracer density estimate we demonstrate that these techniques allow a significant reduction of the required number of samples and thus the reconstruction time the proposed methods are built into the teratomo system

on pseudolinear functions abstract in this paper the structure of a smooth pseudolinear function is investigated and the general form of the gradient is given explicitly

algebraic reconstruction in ct from limited views the author presents an algebraic reconstruction technique art as a viable alternative in computerized tomography ct from limited views recently algorithms of iterative reconstruction reprojection irr based on the method of convolution backprojection have been proposed for application in limited view ct reprojection was used in an iterative fashion alternating with backprojection as a means of estimating projection values within the sector of missing views in algebraic methods of reconstruction for ct only those projections corresponding to known data are required reprojection along missing views would merely serve to introduce redundant equations computer simulation studies are presented which demonstrate significantly improved reconstructed images achieved by an art algorithm as compared to irr methods

voir a volumetric image reconstruction algorithm based on fourier techniques for inversion of the 3 d radon transform a novel volumetric image reconstruction algorithm known as voir is presented for inversion of the 3 d radon transform or its radial derivative the algorithm is a direct implementation of the projection slice theorem for plane integrals it generalizes one of the most successful methods in 2 d fourier image reconstruction involving concentric square rasters to 3 d in voir the spectral data which is calculated by fast fourier techniques lie on concentric cubes and are interpolated by a bilinear method on the sides of these concentric cubes the algorithm has great computational advantages over filtered backprojection algorithms for images of side dimension n the numerical complexity of voir is o n sup 3 log n instead of o n sup 4 for backprojection techniques an evaluation of the image processing performance is reported by comparison of reconstructed images from simulated cone beam scans of a contrast and resolution test object the image processing performance is also characterized by an analysis of the edge response from the reconstructed images

one step extrapolation method for reverse time migration we have proposed a new method a one step extrapolation algorithm to solve the acoustic wave equation by introducing a square root operator the two way wave equation can be formulated as a first order partial differential equation in time which is similar to the one way wave equation to solve the new wave equation we used a stable explicit extrapolation method in the time direction and handled lateral velocity variations in the space and wavenumber domains unlike the conventional explicit finite difference schemes the new method does not suffer from numerical instability or numerical dispersion problems it can be used to design cost effective and high quality reverse time migration or modeling code

a multi level preconditioned krylov method for the efficient solution of algebraic tomographic reconstruction problems classical iterative methods for tomographic reconstruction include the class of algebraic reconstruction techniques art convergence of these stationary linear iterative methods is however notably slow in this paper we propose the use of krylov solvers for tomographic linear inversion problems these advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration causing them to be generally uncompetitive without the inclusion of a suitable preconditioner combining elements from standard multigrid mg solvers and the theory of wavelets a novel wavelet based multi level wmg preconditioner is introduced which is shown to significantly speed up krylov convergence the performance of the wmg preconditioned krylov method is analyzed through a spectral analysis and the approach is compared to existing methods like the classical simultaneous iterative reconstruction technique sirt and unpreconditioned krylov methods on a 2d tomographic benchmark problem numerical experiments are promising showing the method to be competitive with the classical algebraic reconstruction techniques in terms of convergence speed and overall performance cpu time as well as precision of the reconstruction

strip and line path integrals with a square pixel matrix a unified theory for computational ct projections calculated forward projection techniques have been used for various purposes in computerized tomography applications and several models have been proposed to simulate the tomography projection process since the area weighted strip integral is one of the best models methods to facilitate the computation of strip integrals would be very useful in particular a triangular subtraction technique with a reduced number of weighting factors is proposed for the calculation of the strip integral a unified path integral theory is also proposed to bridge various models and to clarify the integral interaction when a line or a strip passes through a square pixel matrix

errors in reprojection methods in computenzed tomography iterative tomographic reconstruction methods have been developed which can enforce various physical constraints on the reconstructed image an integral part of most of these methods is the repro jection of the reconstructed image these estimated projections are compared to the original projection data and modified according to some criteria based on a priori constraints in this paper the errors generated by such reprojection schemes are investigated bounds for these errors are derived under simple signal energy assumptions and using probabilistic assumptions on the distribution of discontinuities these bounds can be used in the enforcement of constraints in the determination of convergence of the iterative methods and in the detection of artifacts

using the rapid expansion method for accurate time stepping in modeling and reverse time migration reverse time migration is based on seismic forward modeling algorithms where spatial derivatives usually are calculated by finite differences or by the fourier method time integration in general is done by finite difference time stepping of low orders if the spatial derivatives are calculated by high order methods and time stepping is based on low order methods there is an imbalance that might require that the time step size needs to be very small to avoid numerical dispersion as a result computing times increase using the rapid expansion method rem avoids numerical dispersion if the number of expansion terms is chosen properly comparisons with analytical solutions show that the rem is preferable especially at larger propagation times for reverse time migration the rem needs to be applied in a time stepping manner this is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time this is not appropriate for reverse time migration where the sources have different time histories in reverse time migration it might be desirable to use the poynting vector information to estimate opening angles to improve the quality of the image in the solution of the wave equation this requires that one calculates not only the pressure wavefield but also its time derivative the rapid expansion method can be extended easily to provide this time derivative with negligible extra cost

prefactored optimized compact finite difference schemes for second spatial derivatives we have described systematically the processes of developing prefactored optimized compact schemes for second spatial derivatives first instead of emphasizing high resolution of a single monochromatic wave we focus on improving the representation of the compact finite difference schemes over a wide range of wavenumbers this leads to the development of the optimized compact schemes whose coefficients will be determined by fourier analysis and the least squares optimization in the wavenumber domain the resulted optimized compact schemes provide the maximum resolution in spatial directions for the simulation of wave propagations however solving for each spatial derivative using these compact schemes requires the inversion of a band matrix to resolve this issue we propose a prefactorization strategy that decomposes the original optimized compact scheme into forward and backward biased schemes which can be solved explicitly we achieve this by ensuring a property that the real numerical wavenumbers o

fast implementations of algebraic methods for three dimensional reconstruction from cone beam data the prime motivation of this work is to devise techniques that make the algebraic reconstruction technique art and related methods more efficient for routine clinical use while not compromising their accuracy since most of the computational effort of art is spent for projection backprojection operations the authors first seek to optimize the projection algorithm existing projection algorithms are surveyed and it is found that these algorithms either lack accuracy or speed or are not suitable for cone beam reconstruction the authors hence devise a new and more accurate extension to the splatting algorithm a well known voxel driven projection method they also describe a new three dimensional 3 d ray driven projector that is considerably faster than the voxel driven projector and at the same time more accurate and perfectly suited for the demands of cone beam the authors then devise caching schemes for both art and simultaneous art sart which minimize the number of redundant computations for projection and backprojection and at the same time are very memory conscious they find that with caching the cost for an art projection backprojection operation can be reduced to the equivalent cost of 1 12 projections they also find that sart due to its image based volume correction scheme is considerably harder to accelerate with caching implementations of the algorithms yield runtime ratios t sub sart t sub art between 1 5 and 1 15 depending on the amount of caching used

the evaluation of data completeness and image quality in multiplexing multi pinhole spect multi pinhole collimators are often used in pre clinical spect systems because they have a better resolution sensitivity tradeoff than parallel hole collimators when imaging small objects most multi pinhole collimators are designed to allow no or only a limited amount of overlap between the different pinhole projections because the ambiguity introduced by multiplexing pinholes can result in artifacts the origin of these artifacts is still not fully understood but previous research has already shown that data incompleteness could be part of the explanation therefore we developed a method to investigate data completeness in multiplexing multi pinhole systems and showed that a certain activity distribution can be successfully reconstructed when the nonmultiplexed data is complete or when the overlap can be sufficiently de multiplexed we validated this method using computer simulated phantom data of different multiplexing systems we also studied contrast to noise and nonprewhitening matched filter signal to noise ratio npw snr to compare the image quality in a single pinhole system with multiplexing systems we found that our method can indeed be used to evaluate data completeness in multiplexing systems and found no artifacts in the systems that had complete data sensitivity increased significantly with multiplexing but we found only small nonsignificant differences in contrast to noise ratio however the npw snr did slightly improve in the multiplexing setups we conclude that more multiplexing does not necessarily result in more artifacts and that even a high amount of multiplexing can still result in artifact free images if the nonmultiplexed data is complete or when the overlap can be sufficiently de multiplexed

recursive integral time extrapolation methods for waves a comparative review abstractwe compared several families of algorithms for recursive integral time extrapolation rite algorithms for waves in isotropic and anisotropic media these methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite difference methods these various rite algorithms all share the use of mixed space wavenumber domain operators derived from fourier integral solutions of single mode wave equations we evaluated a taxonomy for rite methods based on how they approximated the influence of medium heterogeneity one family of methods uses mixed domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps we compared several methods for deriving coefficients for such series approximations another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation such methods can b

extending the stability limit of explicit scheme with spatial filtering for solving wave equations abstract the explicit finite difference scheme is widely used for solving the wave equation however its time step is strictly constrained by the courant friedrichs lewy cfl stability limit we apply spatial filtering to reduce the high wavenumber noise caused by using a larger time step beyond the cfl stability limit we further incorporate the forward and inverse time dispersion transforms to reduce the time dispersion error this method allows us to use a time step larger than the maximum time step allowed by the cfl stability limit

semilocal pseudolinearity and efficiency abstract semilocally pseudolinear functions are defined and their first order characterization is obtained for a multiobjective programming problem involving semilocally pseudolinear functions conditions of efficiency and proper efficiency are discussed

spline driven high accuracy projectors for tomographic reconstruction from few projections tomographic iterative reconstruction methods need a very thorough modeling of data this point becomes critical when the number of available projections is limited at the core of this issue is the projector design i e the numerical model relating the representation of the object of interest to the projections on the detector voxel driven and ray driven projection models are widely used for their short execution time in spite of their coarse approximations distance driven model has an improved accuracy but makes strong approximations to project voxel basis functions cubic voxel basis functions are anisotropic accurately modeling their projection is therefore computationally expensive both smoother and more isotropic basis functions better represent the continuous functions and provide simpler projectors these considerations have led to the development of spherically symmetric volume elements called blobs set apart their isotropy blobs are often considered too computationally expensive in practice in this paper we consider using separable b splines as basis functions to represent the object and we propose to approximate the projection of these basis functions by a 2d separable model when the degree of the b splines increases their isotropy improves and projections can be computed regardless of their orientation the degree and the sampling of the b splines can be chosen according to a tradeoff between approximation quality and computational complexity we quantitatively measure the good accuracy of our model and compare it with other projectors such as the distance driven and the model proposed by long et al from the numerical experiments we demonstrate that our projector with an improved accuracy better preserves the quality of the reconstruction as the number of projections decreases our projector with cubic b splines requires about twice as many operations as a model based on voxel basis functions higher accuracy projectors can be used to improve the resolution of the existing systems or to reduce the number of projections required to reach a given resolution potentially reducing the dose absorbed by the patient

a cone beam reconstruction algorithm for circle plus arc data acquisition geometry in cone beam computerized tomography ct projections acquired with the focal spot constrained on a planar orbit cannot provide a complete set of data to reconstruct the object function exactly there are severe distortions in the reconstructed noncentral transverse planes when the cone angle is large in this work a new method is proposed which can obtain a complete set of data by acquiring cone beam projections along a circle plus arc orbit a reconstruction algorithm using this circle plus arc orbit is developed based on the radon transform and grangeat s formula this algorithm first transforms the cone beam projection data of an object to the first derivative of the three dimensional 3 d radon transform using grangeat s formula and then reconstructs the object using the inverse radon transform in order to reduce interpolation errors new rebinning equations have been derived accurately which allows one dimensional 1 d interpolation to be used in the rebinning process instead of 3 d interpolation a noise free defrise phantom and a poisson noise added shepp logan phantom were simulated and reconstructed for algorithm validation the results from the computer simulation indicate that the new cone beam data acquisition scheme can provide a complete set of projection data and the image reconstruction algorithm can achieve exact reconstruction potentially the algorithm can be applied in practice for both a standard ct gantry based volume tomographic imaging system and a c arm based cone beam tomographic imaging system with little mechanical modification required

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

fractional programming revisited abstract a new format is proposed for fractional programming problems this format gives full expression to the fact that the parametric approach to fractional programming problems is rooted in a first order necessary and sufficient optimality condition it is thus shown that although traditionally it has not been construed as such the parametric approach is in fact classical par excellence

algorithm for the reconstruction of dynamic objects in ct scanning using optical flow abstract computed tomography is a powerful imaging technique that allows non destructive visualisation of the interior of physical objects in different scientific areas in traditional reconstruction techniques the object of interest is mostly considered to be static which gives artefacts if the object is moving during the data acquisition in this paper we present a method that given only scan results of multiple successive scans can estimate the motion and correct the ct images for this motion assuming that the motion field is smooth over the complete domain using optical flow the proposed method is validated on simulated scan data the main contribution is that we show we can use the optical flow technique from imaging to correct ct scan images for motion

optimized explicit finite difference schemes for spatial derivatives using maximum norm we propose an optimized scheme using the maximum norm and the simulated annealing the maximum norm offers the largest set of possible solutions for solvers to search the explicit finite difference operator is greatly improved by our optimized scheme we use a tight error limitation to make accuracy improvement to be high and solid our optimized scheme allows greater saving of computational cost and memory demand conventional explicit finite difference methods have difficulties in handling high frequency components due to strong numerical dispersions one can reduce the numerical dispersions by optimizing the constant coefficients of the finite difference operator different from traditional optimized schemes that use the 2 norm and the least squares we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects first it provides us with much more flexibility when designing the objective functions thus we can use various possible forms and contents to make the objective functions more reasonable second it allows for tighter error limitation which is shown to be necessary to avoid rapid error accumulations for simulations on large scale models with long travel times finally it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits which means greater savings in computational efforts and memory demand

third order symplectic integration method with inverse time dispersion transform for long term simulation the symplectic integration method is popular in high accuracy numerical simulations when discretizing temporal derivatives however it still suffers from time dispersion error when the temporal interval is coarse especially for long term simulations and large scale models we employ the inverse time dispersion transform itdt to the third order symplectic integration method to reduce the time dispersion error first we adopt the pseudospectral algorithm for the spatial discretization and the third order symplectic integration method for the temporal discretization then we apply the itdt to eliminate time dispersion error from the synthetic data as a post processing method the itdt can be easily cascaded in traditional numerical simulations we implement the itdt in one typical exiting third order symplectic scheme and compare its performances with the performances of the conventional second order scheme and the rapid expansion method theoretical analyses and numerical experiments show that the itdt can significantly reduce the time dispersion error especially for long travel times the implementation of the itdt requires some additional computations on correcting the time dispersion error but it allows us to use the maximum temporal interval under stability conditions thus its final computational efficiency would be higher than that of the traditional symplectic integration method for long term simulations with the aid of the itdt we can obtain much more accurate simulation results but with a lower computational cost we use the inverse time dispersion transform to the 3rd order symplectic integration it can greatly reduce the time dispersion error especially for long travel times it is superior to the conventional 2nd and high order finite difference methods it allows the maximum temporal interval under stability conditions thus is efficient we can obtain more accurate simulation results but with a lower computational cost

pseudo analytical finite difference elastic wave extrapolation based on the k space method abstractcost effective elastic wave modeling is the key to practical elastic reverse time migration and full waveform inversion implementations we have developed an efficient elastic pseudo analytical finite difference pafd scheme for elastic wave extrapolation the elastic pafd scheme is based on a modified pseudo spectral method k space method in which a pseudo analytical operator is used to ensure the high accuracy of elastic wave extrapolation however the k space method is motivated for a pure wave mode and thus its application in coupled first order elastic wave equations may cause the elastic pseudo analytical operators to suffer from crosstalk between the p and s wavefields the approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal this is done first by performing a predictor corrector strategy in first order elastic wave equations to eliminate those errors during wave extrapolation considering the massive computational cost in

the convexity of the solution set of a pseudoconvex inequality abstract for a pseudoconvex function f on a nonempty convex set c in a real normed vector space x we present several equivalent conditions for the convexity of the set c x c c f x f c for x c these conditions turn out to be very useful in characterizing the solution set of a pseudoconvex minimization problem of f over c x and the pseudolinearity of a g teux differentiable function f we hence extend several existing results about characterizations of the solutions to a convex program and a pseudolinear program

accelerated iterative reconstruction for positron emission tomography based on the em algorithm for maximum likelihood estimation the em method that was originally developed for maximum likelihood estimation in the context of mathematical statistics may be applied to a stochastic model of positron emission tomography pet the result is an iterative algorithm for image reconstruction that is finding increasing use in pet due to its attractive theoretical and practical properties its major disadvantage is the large amount of computation that is often required due to the algorithm s slow rate of convergence this paper presents an accelerated form of the em algorithm for pet in which the changes to the image as calculated by the standard algorithm are multiplied at each iteration by an overrelaxation parameter the accelerated algorithm retains two of the important practical properties of the standard algorithm namely the selfnormalization and nonnegativity of the reconstructed images experimental results are presented using measured data obtained from a hexagonal detector system for pet the likelihood function and the norm of the data residual were monitored during the iterative process according to both of these measures the images reconstructed at iterations 7 and 11 of the accelerated algorithm are similar to those at iterations 15 and 30 of the standard algorithm for two different sets of data important theoretical properties remain to be investigated namely the convergence of the accelerated algorithm and its performance as a maximum likelihood estimator

projection space iteration reconstruction reprojection recently an iterative reconstruction reprojection irr algorithm has been suggested for application to limited view computed tomography ct in the irr the interpolation operation is performed in the object space during backprojection reprojection the errors associated with the interpolation degrade the reconstructed image and may cause divergence unless a large number of rays is used in this paper we propose a projection space iterative reconstruction reprojection psirr algorithm based on backprojection reprojection in the projection space in the process of the backprojection reprojection iteration is performed with a single equation instead of multiple equations and interpolation is eliminated computer simulation results are presented and image quality of the psirr algorithm shows a substantial improvement compared to the original irr algorithm in addition the new algorithm was applied to ultrasonic attenuation ct using a sponge phantom

method of paired transforms for reconstruction of images from projections discrete model an effective method of discrete image reconstruction from its projections is introduced the method is based on the vector and paired representations of the two dimensional 2d image with respect to the 2d discrete fourier transform such representations yield algorithms for image reconstruction by a minimal number of attenuation measurements in certain projections the proposed algorithms are described in detail for an n spl times n image when n 2 sup r r 1 the inverse formulas for image reconstruction are given the efficiency of the algorithms is expressed in the fact that they require a minimal number of multiplications or can be implemented without such at all the problem of discrete image reconstruction is also considered in three dimensional 3d space namely on the 3d torus where the reconstruction is performed by means of the nonlinear projections that are integral over 3d spirals on the torus

seismic modeling by optimizing regularized staggered grid finite difference operators using a time space domain dispersion relationship preserving method the staggered grid finite difference fd method is widely used in numerical simulation of the wave equation with stability conditions grid dispersion often exists because of the discretization of the time and the spatial derivatives in thewave equation therefore suppressing grid dispersion is a key problem for the staggered grid fd schemes to reduce the grid dispersion the traditional method uses high order staggered grid schemes in the space domain however the wave is propagated in the time and space domain simultaneously therefore some researchers proposed to derive staggered grid fd schemes based on the time space domain dispersion relationship however such methods were restricted to low frequencies and special angles of propagation we have developed a regularizing technique to tackle the ill conditioned property of the symmetric linear system and to stably provide approximate solutions of the fd coefficients for acoustic wave equations dispersion analysis and seismic numerical simulations determined that the proposed method satisfies the dispersion relationship over a much wider range of frequencies and angles of propagation and can ensure fd coefficients being solved via a well posed linear system and hence improve the forward modeling precision