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

on a derivative free fan beam reconstruction formula we clarify that the derivative free fan beam reconstruction formula ieee trans image process 2 543 547 1993 only allows exact reconstruction of an object for a circular trajectory or at the origin of the coordinate system for a radially symmetric noncircular trajectory

fast wave computation via fourier integral operators this paper presents a numerical method for time upscaling wave equations i e performing time steps not limited by the courant friedrichs lewy cfl condition the proposed method leverages recent work on fast algorithms for pseudodierential and fourier integral operators fio this algorithmic approach is not asymptotic it is shown how to construct an exact fio propagator by 1 solving hamilton jacobi equations for the phases and 2 sampling rows and columns of low rank matrices at random for the amplitudes the setting of interest is that of scalar waves in two dimensional smooth periodic media of class c 1 over the torus where the bandlimit n of the waves goes to innity in this setting it is demonstrated that the algorithmic complexity for solving the wave equation to xed time t 1 can be as low as o n 2 logn with controlled accuracy numerical experiments show that the time complexity can be lower than that of a spectral method in certain situations of physical interest

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

optimality conditions for shape and topology optimization subject to a cone constraint this paper provides first order necessary optimality conditions for simultaneous shape and topology optimization subject to a cone constraint these conditions are expressed with the help of the shape and topological derivatives of the objective and constraint functionals several examples of applications are given

performance analysis of the optimum beamformer in the presence of correlated sources and its behavior under spatial smoothing in this paper we first study the signal cancellation and interference rejection effects of the optimum constrained least squares or minimum variance beamformer in the presence of partially and fully correlated interfering sources in particular we derive expressions for the output power and the gain in the interference direction of the beamformer in terms of the source powers correlation and the sensor noise power and show quantitatively the penalties arising from increasing correlation in several scenarios of interest next we show that spatial smoothing progressively decorrelates the sources at a rate that depends on the spacing and directions of the sources and thus relate the degree of smoothing to the improvement in signal cancellation and interference rejection behavior provided by spatial smoothing results of computer simulations are included to support our analysis

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

a fast sinc function gridding algorithm for fourier inversion in computer tomography the fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the cartesian grid required for fast numerical fourier inversion although the fourier inversion method is recognized as being computationally faster than the back projection method for parallel ray projection data the artifacts resulting from inaccurate interpolation have generally limited application of the method this paper presents a computationally efficient gridding algorithm which can be used with direct fourier transformation to achieve arbitrarily small artifact levels the method has potential for application to other measurement geometries such as fan beam projections and diffraction tomography and nmr imaging

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

numerical approximation of a free boundary problem arising in electromagnetic shaping this paper deals with the numerical approximation of the section of a column of liquid metal shaped by an external electromagnetic field the bidimensional model used here leads up to solve by a spectral method a nonlinear boundary value problem posed on the exterior of the unit disk to find a convenient conformal mapping we give accurate bounds on the errors in the approximation process some examples of shapes for both exterior and interior shaping are presented

a high speed reconstruction from projections using direct fourier method with optimized parameters an experimental analysis some issues of the direct fourier method dfm implementation are discussed a hybrid spline linear interpolation for the dfm is proposed the results of comprehensive simulation research are presented the following reconstruction problems and parameters are emphasized interpolation increasing the radial density of the polar raster filtering the 2 d inverse fourier transformation dimension and considering the cases of noiseless and noisy input data for the a priori prescribed resolution of the reconstructed image values of reconstruction parameters have been determined which are optimal with regard to reconstruction quality and computation cost the computational requirements of the dfm algorithm which correspond to distinct interpolation schemes are compared to one another for ct and mr tomography respectively the estimations obtained are compared to computational characteristics of the convolution backprojection method

iterative tomographic image reconstruction using fourier based forward and back projectors iterative image reconstruction algorithms play an increasingly important role in modern tomographic systems especially in emission tomography with the fast increase of the sizes of the tomographic data reduction of the computation demands of the reconstruction algorithms is of great importance fourier based forward and back projection methods have the potential to considerably reduce the computation time in iterative reconstruction additional substantial speed up of those approaches can be obtained utilizing powerful and cheap off the shelf fast fourier transform fft processing hardware the fourier reconstruction approaches are based on the relationship between the fourier transform of the image and fourier transformation of the parallel ray projections the critical two steps are the estimations of the samples of the projection transform on the central section through the origin of fourier space from the samples of the transform of the image and vice versa for back projection interpolation errors are a limitation of fourier based reconstruction methods we have applied min max optimized kaiser bessel interpolation within the nonuniform fft nufft framework and devised ways of incorporation of resolution models into the fourier based iterative approaches numerical and computer simulation results show that the min max nufft approach provides substantially lower approximation errors in tomographic forward and back projection than conventional interpolation methods our studies have further confirmed that fourier based projectors using the nufft approach provide accurate approximations to their space based counterparts but with about ten times faster computation and that they are viable candidates for fast iterative image reconstruction

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

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

optimality conditions for simultaneous topology and shape optimization new optimality conditions are derived for a class of shape optimization problems the conditions are established on the boundary by an application of the boundary variations technique and in the interior of an optimal domain by exploiting the topological derivative method an example is provided for which the classical second order sufficient optimality conditions are verified for an optimal simply connected domain however the value of the cost can be improved by the topology variations and therefore the optimal solution can be substantially changed by applying the topology optimization

performance analysis of the minimum variance beamformer we present an analysis of the signal to interference plus noise ratio sinr at the output of the minimum variance beamformer the analysis is based on the assumption that the signals and noise are gaussian and that the number of samples is large compared to the array size and it yields an explicit expression for the sinr in terms of the different parameters affecting the performance including signal to noise ratio snr interference to noise ratio inr signal to interference ratio sir angular separation between the desired signal and the interference array size and shape correlation between the desired signal and the interference and finite sample size

topological sensitivity analysis in heterogeneous anisotropic elasticity problem theoretical and computational aspects the topological sensitivity analysis for the heterogeneous and anisotropic elasticity problem in two dimensions is performed in this work the main result of the paper is an analytical closed form of the topological derivative for the total potential energy of the problem this derivative displays the sensitivity of the cost functional the energy in this case when a small singular perturbation is introduced in an arbitrary point of the domain in this case we consider a small disc with a completely different elastic material full mathematical justification for the derived formula and derivations of precise estimates for the remainders of the topological asymptotic expansion are provided finally the influence of the heterogeneity and anisotropy is shown through some numerical examples of structural topology optimization

on convergence in elliptic shape optimization the present paper aims at analyzing the existence and convergence of approximate solutions in shape optimization motivated by illustrative examples an abstract setting of the underlying shape optimization problem is suggested taking into account the so called two norm discrepancy a ritz galerkin type method is applied to solve the associated necessary condition existence and convergence of approximate solutions are proved provided that the infinite dimensional shape problem admits a stable second order optimizer the rate of convergence is confirmed by numerical results

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

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

fourier based reconstruction for fully 3 d pet optimization of interpolation parameters fourier based approaches for three dimensional 3 d reconstruction are based on the relationship between the 3 d fourier transform ft of the volume and the two dimensional 2 d ft of a parallel ray projection of the volume the critical step in the fourier based methods is the estimation of the samples of the 3 d transform of the image from the samples of the 2 d transforms of the projections on the planes through the origin of fourier space and vice versa for forward projection reprojection the fourier based approaches have the potential for very fast reconstruction but their straightforward implementation might lead to unsatisfactory results if careful attention is not paid to interpolation and weighting functions in our previous work we have investigated optimal interpolation parameters for the fourier based forward and back projectors for iterative image reconstruction the optimized interpolation kernels were shown to provide excellent quality comparable to the ideal sinc interpolator this work presents an optimization of interpolation parameters of the 3 d direct fourier method with fourier reprojection 3d frp for fully 3 d positron emission tomography pet data with incomplete oblique projections the reprojection step is needed for the estimation from an initial image of the missing portions of the oblique data in the 3d frp implementation we use the gridding interpolation strategy combined with proper weighting approaches in the transform and image domains we have found that while the 3 d reprojection step requires similar optimal interpolation parameters as found in our previous studies on fourier based iterative approaches the optimal interpolation parameters for the main 3d frp reconstruction stage are quite different our experimental results confirm that for the optimal interpolation parameters a very good image accuracy can be achieved even without any extra spectral oversampling which is a common practice to decrease errors caused by interpolation in fourier reconstruction

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

energy change due to the appearance of cavities in elastic solids abstract the paper presents an overview of the problem of assessing an increment of strain energy due to the appearance of small cavities in elastic solids the following approaches are discussed the compound asymptotic method by mazja et al the eshelby like method used in the classical works on the mechanics of composites the homogenization method and the topological derivative method proposed by sokolowski and ochowski the increment of energy is expressed by a quadratic form with respect to strains referring to the virgin solid all the methods lead to the same formula for the increment of energy it is expressed by a quadratic form with respect to strains referring to the virgin solid this quadratic form turns out to be unconditionally positive definite explicit formulae are derived for an elliptical hole and for a spherical cavity the results derived determine the characteristic function of the bubble method of the optimal shape design of elastic 2d and 3d structures

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

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

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

source localization using vector sensor array in a multipath environment coherent signals from distinct directions is a natural characterization of the multipath propagation effect this paper addresses the problem of coherent fully correlated source localization using vector sensor arrays the maximum likelihood ml and minimum variance distortionless response mvdr estimators for source direction of arrival doa and signal polarization parameters are derived these estimators require no search over the polarization parameters in addition a novel method for decorrelating the incident signals is presented this method is based on the polarization smoothing algorithm psa and enables the use of eigenstructure based techniques which assume uncorrelated or partially correlated signals the method is implemented as a preprocessing stage before applying eigenstructure based techniques such as music unlike other existing preprocessing techniques such as spatial smoothing and forward backward fb averaging this method is not limited to any specific array geometry the performance of the proposed psa preprocessing combined with music is evaluated and compared to the crame spl acute r rao bound crb and the ml and mvdr estimators simulation results show that the mvdr and psa music asymptotically achieve the crb for a scenario with two coherent sources with and without an uncorrelated interference source a sensitivity study of psa music to source polarization was also conducted via simulations

image reconstruction from linograms implementation and evaluation the notion of a linogram corresponds to the notion of a sinogram in the conventional representation of projection data for image reconstruction in the sinogram points which correspond to rays through a fixed point in the cross section to be reconstructed all fall on a sinusoidal curve in the linogram however these points fall on a straight line the implementation of a novel image reconstruction method using this property is discussed the implementation is of order n sup 2 log n where n is proportional to the number of pixels on a side of the reconstruction region it is demonstrated that the performance of the algorithm is superior to that of the filter backprojection method in computational speed on realistic size problems and is equivalent to filtered backprojection in accuracy of reconstruction

a fast discrete approximation algorithm for the radon transform this paper addresses fast parallel methods for the computation of the radon or hough transform the radon transform of an image is a set of projections of the image taken at different angles its computation is important in image processing and computer vision for problems such as pattern recognition and reconstruction of medical images a unique new method for combining partial results is presented from which an algorithm is constructed that computes a provably good approximation to the discrete radon transform the approximate discrete radon transform adrt algorithm computes 4 n 4 projections through an n times n image in time o n 2 mathrm lg n the majority of previous algorithms are o n 3 the method is quite simple and easy to parallelize a parallel version of the adrt requires only o mathrm lg n parallel steps on o n 2 processors ignoring communication time an additional property of the algorithm is that it can be applied directly to compute the backprojection step of the inverse rt

line detection in images through regularized hough transform the problem of determining the location and orientation of straight lines in images is of great importance in the fields of computer vision and image processing traditionally the hough transform a special case of the radon transform has been widely used to solve this problem for binary images in this paper we pose the problem of detecting straight lines in gray scale images as an inverse problem our formulation is based on use of the inverse radon operator which relates the parameters determining the location and orientation of the lines in the image to the noisy input image the advantage of this formulation is that we can then approach the problem of line detection within a regularization framework and enhance the performance of the hough based line detector through the incorporation of prior information in the form of regularization we discuss the type of regularizers that are useful for this problem and derive efficient computational schemes to solve the resulting optimization problems enabling their use in large applications finally we show how our new approach can be alternatively viewed as one of finding an optimal representation of the noisy image in terms of elements chosen from a dictionary of lines this interpretation relates the problem of hough based line finding to the body of work on adaptive signal representation

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

fast calculation of multiple line integrals a line integral is defined as the integral of two dimensional data along a one dimensional straight line of given length and orientation line integrals are used in various forms of edge and line detectors in images and in the computation of the radon transform we present a recursive algorithm which enables approximation of discretized line integrals at all lengths orientations and locations to within a prescribed error bound in at most o n log n log log n operations where n is the number of data points furthermore for most applications in particular where even small amounts of noise are present in the data all of these integrals can be computed to the desired accuracy in about 24n log n operations

a structured gradient algorithm for adaptive beamforming the constrained least mean square lms algorithm uses a noisy estimate of the required gradient to adaptively estimate the weights of an optimal antenna array the gradient is estimated by multiplying the array output with the array receiver outputs an alternative scheme for estimating the required gradient is proposed in this paper the new scheme uses a structured estimate of the array correlation matrix to estimate the gradient this structure reflects the structure of the exact array correlation matrix and is obtained by a spatial averaging of the elements of the noisy array correlation matrix used in the standard algorithm it is shown that the estimated gradient is unbiased and the array weights converge to the optimal weights in the mean sense furthermore the weights estimated by the structured gradient algorithm are less noisy than those estimated by the standard algorithm

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

risk averse shape optimization risk averse optimization has attracted much attention in finite dimensional stochastic programming in this paper we propose a risk averse approach in the infinite dimensional context of shape optimization we consider elastic materials under stochastic loading as measures of risk awareness we investigate the expected excess and the excess probability the developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets we incorporate topological derivatives to allow for topological changes in the shape optimization procedure numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios

fast tomographic reconstruction via rotation based hierarchical backprojection we introduce a family of fast algorithms for tomographic backprojection in the parallel beam geometry the algorithms reduce the computational cost of backprojecting p projections onto an n times n pixel image from the conventional o n 2 p to o n 2 log p the new algorithms aggregate the projections in a hierarchical structure with images in the hierarchy formed by the rotation and addition of other images made up of fewer projections while these algorithms are related to existing fast algorithms this work places them within the signal processing framework providing a systematic means to optimize and adjust the trade off between computational cost and accuracy rotations are performed separably in order that higher order interpolators may be used with low computational cost the same ideas are applied to create a tomographic projection algorithm which computes projections of an n times n pixel image onto p view angles at a cost of o n 2 log p

modelling of topological derivatives for contact problems the problem of topology optimization is considered for free boundary problems of thin obstacle types the formulae for the first term of asymptotics for energy functionals are derived the precision of obtained terms is verified numerically the topological differentiability of solutions to variational inequalities is established in particular the so called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis such results lead to the topological derivatives of shape functionals for contact problems the topological derivatives are used in numerical methods of simultaneous shape and topology optimization

redundancy averaging with large arrays redundancy averaging is a spatial averaging technique that can lead to improved spatial correlation matrix estimates in terms of reduced intersignal coherence and reduced variance but it has the undesirable characteristic of potentially biasing direction estimates it is shown that for linear equally spaced arrays any basis induced in the direction estimates by redundancy averaging vanishes as the number of sensors becomes large

shear based fast hierarchical backprojection for parallel beam tomography we introduce a family of fast algorithms for 2 d parallel beam tomographic backprojection they aggregate the projections in a hierarchical structure involving the shearing and addition of sparsely sampled images the algorithms achieve a computational cost of o n2 logp when backprojecting an ntimesn pixel image from p projections the algorithms provide a systematic means guided by a fourier domain interpretation to adjust and optimize the tradeoff between computational cost and accuracy in an example with n 512 and p 1458 the algorithms provide high accuracy with more than an order of magnitude reduction in operation counts

o n sup 2 log sub 2 n filtered backprojection reconstruction algorithm for tomography we present a new fast reconstruction algorithm for parallel beam tomography the new algorithm is an accelerated version of the standard filtered backprojection fbp reconstruction and uses a hierarchical decomposition of the backprojection operation to reduce the computational cost from o n sup 3 to o n sup 2 log sub 2 n we discuss the choice of the various parameters that affect the behavior of the algorithm and present numerical studies that demonstrate the cost versus distortion tradeoff comparisons with fourier reconstruction algorithms and a multilevel inversion algorithm by brandt et al both of which also have o n sup 2 log sub 2 n cost suggest that the proposed hierarchical scheme has a superior cost versus distortion performance it offers rms reconstruction errors comparable to the fbp with considerable speedup for an example with a 512 spl times 512 pixel image and 1024 views the speedup achieved with a particular implementation is over 40 fold with reconstructions visually indistinguishable from the fbp

second order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems in this paper sufficient second order optimality conditions are established for parabolic boundary control problems with nonlinear boundary condition and constraints on the control and the state the main idea is to extend the known theory for systems governed by ordinary differential equations to the case of partial differential equations this is performed by means of a semigroup approach and a two norm technique the verification of the second order conditions is discussed

beamforming in the presence of correlated arrivals using structured correlation matrix expressions are derived for the snr signal to noise ratio at the output of two antenna array processors in the presence of correlated arrivals and the effect on the output snr of the magnitude and the phase of the correlation the number of elements in the array the direction of the level of the interference source and the level of the uncorrelated noise is investigated the two processes considered are the optimal element space processor esp and the optimal postbeamformer interference canceller pic the pic processes the signal by forming two beams and adjusting the weights of one beam in contrast to the esp where the signals from all the elements are weighted and all the weights are adjusted a method is proposed for decorrelating the correlated sources and an analytical expression is derived to show the decorrelation effect of the proposed technique for a line array of equispaced elements numerical examples are included

o n sup 3 log n backprojection algorithm for the 3 d radon transform we present a novel backprojection algorithm for three dimensional 3 d radon transform data that requires o n sup 3 log sub 2 n operations for reconstruction of an n spl times n spl times n volume from o n sup 2 plane integral projections our algorithm uses a hierarchical decomposition of the 3 d radon transform to recursively decompose the backprojection operation simulations are presented demonstrating reconstruction quality comparable to the standard filtered backprojection which requires o n sup 5 computations under the same circumstances

the topological asymptotic expansion for the dirichlet problem the topological sensitivity analysis provides an asymptotic expansion of a shape function when creating a small hole inside a domain this expansion yields a descent direction which can be used for shape optimization if one wishes to keep a classical domain throughout the optimization process in this paper such an expansion is obtained for the poisson equation for a large class of cost functions and arbitrarily shaped holes in the three dimensional case this expansion depends on the shape of the hole but not on its orientation if the cost function involves only the solution u to the underlying partial differential equation whereas it may also depend on its orientation if the cost function involves the gradient nabla u in contrast the asymptotic expansion is independent of the shape in the two dimensional case a numerical example illustrates the use of the asymptotic expansion which yields a minimizing sequence of classical domains in a case where no classical solution exists

a fast and accurate multilevel inversion of the radon transform a number of imaging technologies reconstruct an image function from its radon projection using the convolution backprojection method the convolution is an o n2 log n algorithm where the image consists of n x n pixels while the backprojection is an o n3 algorithm thus constituting the major computational burden of the convolution backprojection method an o n2 log n multilevel backprojection method is presented here when implemented with a fourier domain postprocessing technique also presented here the resulting image quality is similar or superior to the image quality of the classical backprojection technique

methods for reliable topology changes for perimeter regularized geometric inverse problems this paper is devoted to the incorporation of topological derivativelike expansions into level set methods for perimeter regularized geometric inverse problems the expansions are done up to the second order with respect to the lebesgue measure of the symmetric difference they provide simpler shape functionals still including the perimeter and therefore allow the construction of steepest descent and newton type algorithms to force topology changes during the level set evolution numerous numerical examples are provided that show the strong and also the weak points of the newly developed algorithms

resolving uncorrelated and correlated sources by linear prediction the resolution and the estimation error are investigated for two linear prediction lp algorithms for bearing estimation with uniform linear array of sensors in the presence of incoherent and coherent sources the burg algorithm ba and the generalized burg algorithm gba the superresolution is easily achieved with both algorithms when the sources are incoherent for coherent sources however the ba estimate is somewhat more sensitive to the magnitude and the phase of the intersignal correlation than the gba thereby resulting in the increased error in the bearing estimates

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

pencil back projection method for sar imaging we present a high resolution method for spotlight mode sar imaging that utilizes parametric modeling of projected target reflectivity density function and tomographic reconstruction the method requires no polar to cartesian interpolation in spectral domain utilization of forward backward total least squares bandpass matrix pencil method allows super resolution to be achieved in range for a single imaging angle hence the quality of the image reconstructed by convolution back projection is also improved it is shown that the method is very resistant to noise and can generate images down to very low snr values direct formulation in terms of physical quantities such as electric field and current density is another contribution of this paper

efficient evaluation of oversampled functions abstract it has been known since long that an oversampled function can be represented by a generalized sinc series which converges much faster than the usual one we show that the speed of convergence is exponential and give numerical approximations to the kernel function involved in the generalized series the practical consequence is that a bandlimited function which is oversampled can be evaluated at any point with accuracy o l e l using o l operations

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

the gridding method for image reconstruction by fourier transformation the authors explore a computational method for reconstructing an n dimensional signal f from a sampled version of its fourier transform f spl circ the method involves a window function w spl circ and proceeds in three steps first the convolution g spl circ w spl circ f spl circ is computed numerically on a cartesian grid using the available samples of f spl circ then g wf is computed via the inverse discrete fourier transform and finally f is obtained as g w due to the smoothing effect of the convolution evaluating w spl circ f spl circ is much less error prone than merely interpolating f spl circ the method was originally devised for image reconstruction in radio astronomy but is actually applicable to a broad range of reconstructive imaging methods including magnetic resonance imaging and computed tomography in particular it provides a fast and accurate alternative to the filtered backprojection the basic method has several variants with other applications such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the radon hough transform

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

a new method for inverse electromagnetic casting problems based on the topological derivative the inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape in this paper we propose a new method for the topology design of such inductors the inverse electromagnetic casting problem is formulated as an optimization problem and topological derivatives are considered in order to locate new wires in the right position several numerical examples are presented showing that the proposed technique is effective to design suitable inductors

velocity method and lagrangian formulation for the computation of the shape hessian the object of this paper is to study the shape hessian of a shape functional by the velocity speed method it contains a review and an extension of the velocity method and its connections with methods using first or second order perturbations of the identity the key point is that all these methods yield the same shape gradient but different and unequal shape hessian since each method depends on a choice of connection however for autonomous velocity fields the velocity method yields a canonical bilinear hessian expressions obtained by other methods can be recovered by adding to that canonical term the shape gradient acting on the acceleration of the velocity field associated with the choice of perturbation of the identity the second part of the paper is an application of the lagrangian method with function space embedding to compute the shape gradient and hessian of a simple cost function associated with the nonhomogeneous dirichlet problem

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

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

an exact and fast computation of discrete fourier transform for polar and spherical grid numerous applied problems of two dimensional 2 d and 3 d imaging are formulated in continuous domain they place great emphasis on obtaining and manipulating the fourier transform in polar and spherical coordinates however the translation of continuum ideas with the discrete sampled data on a cartesian grid is problematic there exists no exact and fast solution to the problem of obtaining discrete fourier transform for polar and spherical grids in the literature in this paper we develop exact algorithms to the above problem for 2 d and 3 d which involve only 1 d equispaced fast fourier transform with no interpolation or approximation at any stage the result of the proposed approach leads to a fast solution with very high accuracy we describe the computational procedure to obtain the solution in both 2 d and 3 d which includes fast forward and inverse transforms we find the nested multilevel matrix structure of the inverse process and we propose a hybrid grid and use a preconditioned conjugate gradient method that exhibits a drastic improvement in the condition number

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

ct reconstruction from parallel and fan beam projections by a 2 d discrete radon transform the discrete radon transform drt was defined by abervuch et al as an analog of the continuous radon transform for discrete data both the drt and its inverse are computable in o n2logn operations for images of size n n in this paper we demonstrate the applicability of the inverse drt for the reconstruction of a 2 d object from its continuous projections the drt and its inverse are shown to model accurately the continuum as the number of samples increases numerical results for the reconstruction from parallel projections are presented we also show that the inverse drt can be used for reconstruction from fan beam projections with equispaced detectors

a note on redundancy averaging it is well known that the performance of direction of arrival estimation and beamforming algorithms degrades in the presence of correlated signals since techniques like spatial and weighted smoothing which were developed for decorrelating the signals suffer from reduced effective aperture several authors have recently proposed redundancy averaging as an alternative spatial averaging method considering a two source signal model and the asymptotic case the authors show analytically that this technique results in an eigenstructure which is inconsistent with that of the underlying signal model in particular the resulting signal subspace fills up the whole m dimensional space where m is the array size and the resulting covariance matrix is not guaranteed to be nonnegative definite it is shown that complete decorrelation is not possible with this method even if the array size is infinitely large

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

an intuitive discussion on the ideal ramp filter in computed tomography i in x ray computed tomography ct the ideal ramp filter is a generalized function defined by the inverse fourier transform similar to dirac s discussion on the delta function we present an intuitive discussion on the ideal ramp filter with this concise discussion one obtains a better understanding of the filter backprojection algorithm fbp and can easily construct new practical filters