improving filtered backprojection reconstruction by data dependent filtering filtered backprojection one of the most widely used reconstruction methods in tomography requires a large number of low noise projections to yield accurate reconstructions in many applications of tomography complete projection data of high quality cannot be obtained because of practical considerations algebraic methods tend to handle such problems better but are computationally more expensive in this paper we introduce a new method that improves the filtered backprojection method by using a custom data dependent filter that minimizes the projection error of the resulting reconstruction we show that the computational cost of the new method is significantly lower than that of algebraic methods experiments on both simulation and experimental data show that the method is able to produce more accurate reconstructions than filtered backprojection based on popular static filters when presented with data with a limited number of projections or statistical noise present furthermore the results show that the method produces reconstructions with similar accuracy to algebraic methods but is faster at producing them finally we show that the method can be extended to exploit certain forms of prior knowledge improving reconstruction accuracy in specific cases

computational analysis and improvement of sirt iterative x ray computed tomography ct algorithms have the potential for producing high quality images but are computationally very demanding especially when applied to high resolution problems focusing on simultaneous iterative reconstruction technique sirt we provide an eigenvalue based scheme for automatically determining a near optimal value of the relaxation parameter this accelerates the convergence rate of sirt to the point where only half the number of iterations normally required is needed we also modify the way sirt uses preconditioning to solve a weighted least squares problem the resulting algorithm which we call psirt is associated with a smaller memory footprint and calls for less data to be communicated in a distributed memory implementation experimental residual norm and timing results are provided based on cone beam micro ct mouse data including for an ordered subsets study

optimal threshold selection for segmentation of dense homogeneous objects in tomographic reconstructions in this paper we present a novel approach to segment dense homogeneous objects in a tomographic reconstruction or tomogram a popular method to extract such objects from a tomogram is global thresholding in which the threshold value is determined from the image histogram however accurate threshold selection is not straightforward since due to noise or artefacts in the reconstruction the histogram does not always contain a clear separate peak for the dense object we propose a new threshold estimation approach segmentation inconsistency minimization that exploits the available projection data to determine the optimal global threshold the proposed algorithm was tested on simulation data and on experimental ct data the results show that this method results in more accurate segmentations compared to alternative threshold selection methods

convergence studies on iterative algorithms for image reconstruction we introduce a general iterative scheme for image reconstruction based on landweber s method in our configuration a sequential block iterative seqbi version can be readily formulated from a simultaneous block iterative simbi version and vice versa this provides a mechanism to derive new algorithms from known ones it is shown that some widely used iterative algorithms such as the algebraic reconstruction technique art simultaneous art sart cimmino s and the recently designed diagonal weighting and component averaging algorithms are special examples of the general scheme we prove convergence of the general scheme under conditions more general than assumed in earlier studies for its seqbi and simbi versions in the consistent and inconsistent cases respectively our results suggest automatic relaxation strategies for the seqbi and simbi versions and characterize the dependence of the limit image on the initial guess it is found that in all cases the limit is the sum of the minimum norm solution of a weighted least squares problem and an oblique projection of the initial image onto the null space of the system matrix

quantitative stability of the entropy power inequality we establish quantitative stability results for the entropy power inequality epi specifically we show that if uniformly log concave densities nearly saturate the epi then they must be close to gaussian densities in the quadratic kantorovich wasserstein distance furthermore if one of the densities is gaussian and the other is log concave or more generally has positive spectral gap then the deficit in the epi can be controlled in terms of the l 1 kantorovich wasserstein distance or relative entropy respectively as a counterpoint an example shows that the epi can be unstable with respect to the quadratic kantorovich wasserstein distance when densities are uniformly log concave on sets of measure arbitrarily close to one our stability results can be extended to non log concave densities provided certain regularity conditions are met the proofs are based on mass transportation

convergence studies on block iterative algorithms for image reconstruction the sequential block iterative scheme based on landweber s method is a general iterative one for image reconstruction in this paper we give the finite termination convergence for this scheme which provides an approach to choose relaxation coefficients furthermore sufficient and necessary convergence conditions are also established for the sequential block iterative scheme case

rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials for the spatially homogeneous boltzmann equation with hard potentials and grad s cutoff e g hard spheres we give quantitative estimates of exponential convergence to equilibrium and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation on which we provide a lower bound our approach is based on establishing spectral gap like estimates valid near the equilibrium and then connecting the latter to the quantitative nonlinear theory this leads us to an explicit study of the linearized boltzmann collision operator in functional spaces larger than the usual linearization setting

chaos ergodicity and equilibria in a quantum kac model abstract we introduce quantum versions of the kac master equation and the kac boltzmann equation we study the steady states of each of these equations and prove a propagation of chaos theorem that relates them the quantum kac master equation qkme describes a quantum markov semigroup p n t while the kac boltzmann equation describes a non linear evolution of density matrices on the single particle state space all of the steady states of the n particle quantum system described by the qkme are separable and thus the evolution described by the qkme is entanglement breaking the results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory and a quantitative study of the rate of destruction of entanglement in a class of quantum markov semigroups describing binary interactions

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

l estimates for the space homogeneous boltzmann equation this paper studies the boundedness of solutionsf of the initial value problem for the space homogeneous boltzmann equation for inverse kth power forces whenk 5 and under angular cutoff the main result is that if the initial value isf0 0 with 1 2 0 el1 and 1 sf0e l for somes 2 then 1 s ftel fort 0 and ess tsup 1 s f t 5

a counter example to cercignani s conjecture for the d dimensional kac model kac s d dimensional model gives a linear many particle binary collision model from which under suitable conditions the celebrated boltzmann equation in its spatially homogeneous form arise as a mean field limit the ergodicity of the evolution equation leads to questions about the relaxation rate in hope that such a rate would pass on the boltzmann equation as the number of particles goes to infinity this program starting with kac and his one dimensional spectral gap conjecture at 1956 finally reached its conclusion in the maxwellian case in a series of papers by authors such as janvresse maslen carlen carvalho loss and geronimo but the hope to get a limiting relaxation rate for the boltzmann equation with this linear method was already shown to be unrealistic although the problem is still important and interesting due to its connection with the linearized boltzmann operator a less linear approach via a many particle version of cercignani s conjecture is the grounds for this paper in our paper we extend recent results by the author from the one dimensional kac model to the d dimensional one showing that the entropy entropy production ratio n still yields a very strong dependency in the number of particles of the problem when we consider the general case

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

sirt and cg type methods for the iterative solution of sparse linear least squares problems abstract sirt and cg type methods have been successfully employed for the approximate solution of least squares problems arising in tomography in this paper we study and compare their convergence and regularization properties it is pointed out that sirt methods apply an uncontrollable implicit rescaling which affects the statistical characteristics of the system whereas cg type methods do not for a large class of model problems it is shown that virtually the same solutions as obtained by sirt methods can be obtained by applying a cg type method to a properly rescaled system but with an amount of work proportional to the square root of the amount of work with sirt

on the rate of entropy production for the boltzmann equation we show that there exists a wide class of distribution functions with moments of any order as close to their equilibrium values as we like which can provide an abnormally low rate of entropy production the result is valid for the boltzmann equation with any cross section v satisfying a mild restriction the functions are constructed in an explicit form and we discuss some applications of our results

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

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

fast 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

data driven affine deformation estimation and correction in cone beam computed tomography in computed tomography ct motion and deformation during the acquisition lead to streak artefacts and blurring in the reconstructed images to remedy these artefacts we introduce an efficient algorithm to estimate and correct for global affine deformations directly on the cone beam projections the proposed technique is data driven and thus removes the need for markers and or a tracking system a relationship between affine transformations and the cone beam transform is proved and used to correct the projections the deformation parameters that describe deformation perpendicular to the projection direction are estimated for each projection by minimizing a plane based inconsistency criterion the criterion compares each projection of the main scan with all projections of a fast reference scan which is acquired prior or posterior to the main scan experiments with simulated and experimental data show that the proposed affine deformation estimation method is able to substantially reduce motion artefacts in cone beam ct images

accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm iterative maximum likelihood ml transmission computed tomography algorithms have distinct advantages over fourier based reconstruction but unfortunately require increased computation time the convex algorithm is a relatively fast iterative ml algorithm but it is nevertheless too slow for many applications therefore an acceleration of this algorithm by using ordered subsets of projections is proposed ordered subsets convex algorithm osc osc applies the convex algorithm sequentially to subsets of projections osc was compared with the convex algorithm using simulated and physical thorax phantom data reconstructions were performed for osc using eight and 16 subsets eight and four projections subset respectively global errors image noise contrast recovery and likelihood increase were calculated results show that osc is faster than the convex algorithm the amount of acceleration being approximately proportional to the number of subsets in osc and it causes only a slight increase of noise and global errors in the reconstructions images and image profiles of the reconstructions were in good agreement in conclusion osc and the convex algorithm result in similar image quality but osc is more than an order of magnitude faster

entropy production estimates for boltzmann equations with physically realistic collision kernels we establish strict entropy production bounds for the boltzmann equation with the hard sphere collision kernel using these entropy production bounds we prove results asserting that the rate at which strongl 1 convergence to equilibrium occurs is uniform in wide classes of initial data this extends our previous results in this direction which applied only to a very special collision kernel moreover the present results provide computable lower bounds compactness arguments are entirely avoided the uniformity is an important ingredient in our study of scaling limits of solutions of the non spatially homogeneous boltzmann equation and is the main focus of this paper however the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous boltzmann equation with initial data far from equilibrium to reach any given small strongl 1 neighborhood of equilibrium

asymptotic expansion of the free space green s function for the discrete 3 d poisson equation the question of the definition existence and uniqueness of the free space green s function for the discrete 3 d poisson equation on a general cartesian grid is answered by presenting three alternative definitions and demonstrating their equivalence this is a nice application of the theory of multidimensional fourier series then two methods an integral method and a taylor series method are presented for deriving the asymptotic expansion of the green s function including explicit computational formulas for several higher order terms code based on these formulas is used in the boeing transonic full potential program tranair to implement a highly accurate and computationally efficient far field absorbing boundary condition

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

sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation we derive a new lower bound for the entropy dissipation associated with the spatially homogeneous boltzmann equation this bound is expressed in terms of the relative entropy with respect to the equilibrium and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy with an explicit rate our result gives a considerable refinement of the analogous estimate by carlen and carvalho 9 10 under very little additional assumptions our proof takes advantage of the structure of boltzmann s collision operator with respect to the tensor product and its links with fokker planck and landau equations several variants are discussed

acceleration of landweber type algorithms by suppression of projection on the maximum singular vector a procedure that speeds up convergence during the initial stage the first 100 forward and backward projections of landweber type algorithms for iterative image reconstruction for positron emission tomography pet which include the landweber generalized landweber and steepest descent algorithms is discussed the procedure first identifies the singular vector associated with the maximum singular value of the pet system matrix and then suppresses projection of the data on this singular vector after a single landweber iteration it is shown that typical pet system matrices have a significant gap between their two largest singular values hence this suppression allows larger gains in subsequent iterations speeding up convergence by roughly a factor of three

some applications of the method of moments for the homogeneous boltzmann and kac equations using the method of moments we prove that any polynomial moment of the solution of the homogeneous boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially we also give partial results of convergence towards the maxwellian equilibrium in the case of soft potentials finally exponential as well as maxwellian estimates are introduced for the kac equation

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

entropy dissipation and moment production for the boltzmann equation leth f m flog f m dv be the relative entropy off and the maxwellian with the same mass momentum and energy and denote the corresponding entropy dissipation term in the boltzmann equation byd f q f f logf dv an example is presented which shows that d f h f m can be arbitrarily small this example is a sequence of isotropic functions and the estimates are very explicitly given by a simple formula ford which holds for such functions the paper also gives a simplified proof of the so called povzner inequality which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision that inequality is then used to prove that f v v s dt c t wheref is the solution of the spatially homogeneous boltzmann equation herec t is an explicitly given function dependings and the mass energy and entropy of the initial data

fast and slow convergence to equilibrium for maxwellian molecules via wild sums we consider the spatially homogeneous boltzmann equation for maxwellian molecules and general finite energy initial data positive borel measures with finite moments up to order 2 we show that the coefficients in the wild sum converge strongly to the equilibrium and quantitatively estimate the rate we show that this depends on the initial data f essentially only through on the behavior near r 0 of the function j f r v 1 r v 2 df v these estimates on the terms in the wild sum yield a quantitative estimate in the strongest physical norm on the rate at which the solution converges to equilibrium as well as a global stability estimate we show that our upper bounds are qualitatively sharp by producing examples of solutions for which the convergence is as slow as permitted by our bounds these are the first examples of solutions of the boltzmann equation that converge to equilibrium more slowly than exponentially

stability and uniqueness for the spatially homogeneous boltzmann equation with long range interactions in this paper we prove some a priori stability estimates in weighted sobolev spaces for the spatially homogeneous boltzmann equation without angular cutoff covering all physical collision kernels these estimates are conditional on some regularity estimates on the solutions and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions we then prove such regularity bounds for a class of interactions including the so called non cutoff and non mollified hard potentials and moderately soft potentials in particular we obtain the first result of global existence and uniqueness for these long range interactions

automatic parameter estimation for the discrete algebraic reconstruction technique dart computed tomography ct is a technique for noninvasive imaging of physical objects in the discrete algebraic reconstruction technique dart prior knowledge about the material s densities is exploited to obtain high quality reconstructed images from a limited number of its projections in practice this prior knowledge is typically not readily available here a fully automatic method called projection distance minimization dart pdm dart is proposed in which the optimal grey level parameters are adaptively estimated during the reconstruction process to apply pdm dart only the number of different grey levels should be known in advance simulation as well as real ct experiments show that pdm dart is capable of computing reconstructed images of which the quality is similar to reconstructions computed by conventional dart based on exact prior knowledge thereby eliminating the need for tedious and error prone user interaction