a theoretical framework of regional cone beam tomography intuitively if only part of an object is to be reconstructed the information required to perform the reconstruction should be less than that required to perform global reconstruction global reconstruction is assumed by every published reconstruction formula with closed format the reduction of input information could lead to the simplification of data collection and the reduction of implementation cost the author has explored three basic issues of regional cone beam tomography 1 a completeness condition for regional reconstruction 2 an estimation formula and 3 a convergence condition the analysis of the first issue yields requirements for artifact free reconstruction and a convenient criterion for source curve design the closed format of the estimation formula developed permits efficient analysis the discussion of the third issue facilitates guidelines for selecting an appropriate interpolation technique for discrete input data

derivation and analysis of a filtered backprojection algorithm for cone beam projection data the authors address the problem of three dimensional image reconstruction from cone beam projections modifying a result due to a a kirillov soviet math dokl vol 2 p 268 9 1961 the authors derive an inversion formula for the case where the cone vertices form an unbounded curve for the special case in which the cone vertices form a circle an approximate reconstruction formula is developed and shown to be essentially equivalent to the practical cone beam algorithm of l a feldkamp et al 1984 for this approximate inverse the authors derive the resulting spatially varying point spread function examine the effect of bandlimiting due to sampling and compare the resulting image quality as a function of the radius of the circle formed by the cone vertices

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

a power series method for computing singular solutions to nonlinear analytic systems given a system of analytic equations having a singular solution we show how to develop a power series representation for the solution this series is computable and when the multiplicity of the solution is small highly accurate estimates of the solution can be generated for a moderate computational cost in this paper a theorem is proven using results from several complex variables which establishes the basis for the approach then a specific numerical method is developed and data from numerical experiments are given

solving eigenvalue problems of real nonsymmetric matrices with real homotopies the eigenvalue problem of a matrix a can be considered a system of polynomial equations az lambda z 0 in complex variables lambda in c and z in c n in this paper a homotopy continuation algorithm for solving eigenvalue problems of real nonsymmetric matrices is developed based on this point different from current homotopy continuation methods for real nonsymmetric matrices this algorithm makes use of the homotopy which consists of real polynomials hence when a complex path is followed its conjugate path is obtained as a by product moreover techniques including a double step inverse iteration are developed to eliminate complex computations completely a great amount of operations is saved with these complex to real conversions the algorithm employed the strategy of divide and conquer which makes most of the eigen paths almost straight lines and extremely easy to follow although some of the paths may contain bifurcation the situation is handled with a complexification process whic

solving real polynomial systems with real homotopies when a real homotopy is used for solving a polynomial system with real coefficients bifurcation of some of the homotopy paths at singular points is inevitable the main result of this paper shows that generically the solution set of a real homotopy contains no singular point other than a finite number of quadratic turning points at a quadratic turning point the bifurcation phenomenon is quite simple it consists of two bifurcation branches with their tangent vectors being perpendicular to each other

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

metric dependence and asymptotic minimization of the expected number of critical points of random holomorphic sections we prove the main conjecture from douglas shiffman and zelditch 2006 concerning the metric dependence and asymptotic minimization of the expected number ncrit n hof critical points of random holomorphic sections of the nth tensor power of a positive line bundle the first non topological term in the asymptotic expansion of ncrit n h is the calabi functional multiplied by the constant 2 m which depends only on the dimension of the manifold we prove that 2 m is strictly positive in all dimensions showing that the expansion is non topological for all m and that the calabi extremal metric when it exists asymptotically minimizes ncrit n h

cone beam reconstruction using the backprojection of locally filtered projections this paper describes a flexible new methodology for accurate cone beam reconstruction with source positions on a curve or set of curves the inversion formulas employed by this methodology are based on first backprojecting a simple derivative in the projection space and then applying a hilbert transform inversion in the image space the local nature of the projection space filtering distinguishes this approach from conventional filtered backprojection methods this characteristic together with a degree of flexibility in choosing the direction of the hilbert transform used for inversion offers two important features for the design of data acquisition geometries and reconstruction algorithms first the size of the detector necessary to acquire sufficient data for accurate reconstruction of a given region is often smaller than that required by previously documented approaches in other words more data truncation is allowed second redundant data can be incorporated for the purpose of noise reduction the validity of the inversion formulas along with the application of these two properties are illustrated with reconstructions from computer simulated data in particular in the helical cone beam geometry it is shown that 1 intermittent transaxial truncation has no effect on the reconstruction in a central region which means that wider patients can be accommodated on existing scanners and more importantly that radiation exposure can be reduced for region of interest imaging and 2 at maximum pitch the data outside the tam danielsson window can be used to reduce image noise and thereby improve dose utilization furthermore the degree of axial truncation tolerated by our approach for saddle trajectories is shown to be larger than that of previous methods

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

computing hopf bifurcations ii three examples from neurophysiology in guckenheimer myers and sturmfels siam j numer anal 34 1997 we present algorithms for detecting hopf bifurcations in two parameter families of vector fields based on classical algebraic constructions in addition to their utility as augmented systems for use with standard newton type continuation methods they are shown to be particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension the present study examines the performance of these methods on test problems selected from models of current research interest in neurophysiology implementation issues are examined and the numerical properties of the proposed methods are compared with several alternative algorithms for hopf pathfollowing that appear in the literature

direct least squares estimation of spatiotemporal distributions from dynamic spect projections using a spatial segmentation and temporal b splines artifacts can result when reconstructing a dynamic image sequence from inconsistent as well as insufficient and truncated cone beam single photon emission computed tomography spect projection data acquired by a slowly rotating gantry the artifacts can lead to biases in kinetic model parameters estimated from time activity curves generated by overlaying volumes of interest on the images however the biases in time activity curve estimates and subsequent kinetic parameter estimates can be reduced significantly by first modeling the spatial and temporal distribution of the radiopharmaceutical throughout the projected field of view and then estimating the time activity curves directly from the projections this approach is potentially useful for clinical spect studies involving slowly rotating gantries particularly those using a single detector system or body contouring orbits with a multidetector system the authors have implemented computationally efficient methods for fully four dimensional 4 d direct estimation of spatiotemporal distributions from dynamic spect projection data temporal b splines providing various orders of temporal continuity as well as various time samplings were used to model the time activity curves for segmented blood pool and tissue volumes in simulated cone beam and parallel beam cardiac data acquisitions least squares estimates of time activity curves were obtained quickly using a workstation given faithful spatial modeling accurate curve estimates were obtained using cubic quadratic or linear b splines and a relatively rapid time sampling during initial tracer uptake from these curves kinetic parameters were estimated accurately for noiseless data and with some bias for noisy data a preliminary study of spatial segmentation errors showed that spatial model mismatch adversely affected quantitative accuracy but also resulted in structured errors projected model versus raw data that were easily detected in the authors simulations this suggests iterative refinement of the spatial model to reduce structured errors as an area of future research

a neural network modeled by an adaptive lotka volterra system in this paper the asymptotic behavior of an n dimensional lotka volterra system with nonconstant coefficients that models the dynamics of an adaptive cellular network is analyzed each cell in the network is assumed to be connected to other cells in such a way that its output level of activity eitherinhibits or excites the growth of activity in the other cells the weight of the connection between any two cells varies over time in proportion to the average level of interaction between the two cells over the recent past if all of the connections are inhibitory it is shown that the network tends to produce a spontaneous and interesting classification of its inputs

a factorization approach for cone beam reconstruction on a circular short scan in this paper we introduce a new algorithm for 3 d image reconstruction from cone beam cb projections acquired along a partial circular scan our algorithm is based on a novel exact factorization of the initial 3 d reconstruction problem into a set of independent 2 d inversion problems each of which corresponds to finding the object density on one single plane any such 2 d inversion problem is solved numerically using a projected steepest descent iteration scheme we present a numerical evaluation of our factorization algorithm using computer simulated cb data without and with noise of the forbild head phantom and of a disk phantom first we study quantitatively the impact of the reconstruction parameters on the algorithm performance next we present reconstruction results for visual assessment of the achievable image quality and provide for comparison results obtained with two other state of the art reconstruction algorithms for the circular short scan

resolution and noise properties of map reconstruction for fully 3 d pet derives approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three dimensional 3 d positron emission tomography pet data using maximum a posteriori map estimation these expressions explicitly account for the spatially variant detector response and sensitivity of a 3 d tomograph the resulting spatially variant impulse response and covariance are computed using 3 d fourier transforms a truncated gaussian distribution is used to account for the effect on the variance of the nonnegativity constraint used in map reconstruction using monte carlo simulations and phantom data from the micropet small animal scanner the authors show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of map reconstruction for priors with quadratic energy functions they also describe how these analytical results can be used to achieve near uniform contrast recovery throughout the reconstructed volume

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

critical points and supersymmetric vacua ii asymptotics and extremal metrics motivated by the vacuum selection problem of string m theory we study a new geometric invariant of a positive hermitian line bundle l h m over a compact kahler manifold the expected distribution k crit z of critical points dlog s z h 0 of a gaussian random holomorphic section s h 0 m l with respect to h it is a measure on m whose total mass is the average number n crit h of critical points of a random holomorphic section we are interested in the metric dependence of n crit h especially metrics h which minimize n crit h we concentrate on the asymptotic minimization problem for the sequence of tensor powers l n h n m of the line bundle and their critical point densities k critn z we prove that k critn z has a complete asymptotic expansion in n whose coefficients are curvature invariants of h the first two terms in the expansion of n crit hn are topological invariants of l m the third term is a topological invariant plus a constantm 2 depending only on the dimension m of m times the calabi functional r m 2 dv olh whereis the scalar curvature of the curvature form of h we give an integral formula form 2 and show by a computer assisted calculation thatm 2 0 for m 3 hence that n crit hn is asymptotically minimized by the calabi extremal metric when one exists we conjecture thatm 2 0 in all dimensions i e that the calabi extremal metric is always the asymptotic minimizer

quasi exact filtered backprojection algorithm for long object problem in helical cone beam tomography exact reconstruction from axially truncated cone beam projections acquired with a helical vertex path is a challenging problem for which solutions are currently under investigation by some researchers this paper deals with a difficult class of this problem called the long object problem its purpose is to reconstruct a central region of interest roi of a long object when the helical path extends only a little bit above and below the roi by extending the authors recent approach based on the triangular decomposition of the grangeat formula the authors derive quasi exact reconstruction algorithms whose overall structure is of filtered backprojection fbp style unlike the previous similar approaches to the long object problem the proposed fbp algorithms do not require additional two circular scans at the ends of the helical path furthermore the algorithms require a significantly smaller detector area and achieve improved image quality even for a large pitch compared with the approximate feldkamp algorithms one drawback of the proposed algorithms is the computational time which is much longer than for the feldkamp algorithms the authors show some simulation results to demonstrate the performances of the proposed algorithms

an interval step control for continuation methods the authors present a step control for continuation methods that is deterministic in the sense that i it computationally but rigorously verifies that the corrector iteration will converge to a point on the same curve as the previous point i e the predictor corrector iteration will never jump across paths and ii each predictor step is as large as possible subject to verification that the curve is unique with the given interval extension the authors present performance data and comparisons with an approximate step control method pitcon version 6 1 a comparison of plots obtained from both step controls reveals that an approximate step control will behave erratically in situations where the interval step control leads to orderly progression along the curve this is true even if the maximum allowable stepsize for the approximate method is set to be smaller than many of the steps actually taken by the interval algorithm limitations of interval step controls are also discussed

tomographic reconstruction from experimentally obtained cone beam projections direct reconstruction in three dimensions for two dimensional projection data has been achieved by cone beam reconstruction techniques in this paper explicit formulas for a cone beam convolution and back projection reconstruction algorithm are given in a form which can be easily coded for a computer the algorithm is justified by analyzing tomographic reconstructions of a uniformly attenuating sphere from simulated noisy projection data a particular feature of this algorithm is the use of a one dimensional rather than two dimensional convolution function greatly speeding up the reconstruction the technique is applicable however large the cone angle of data capture and correctly reduces to the pure fan beam reconstruction technique in the central section of the cone the method has been applied to data captured on a cone beam ct scanner designed for bone mineral densitometry

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

computation of unmeasured third generation vct views from measured views we compute unmeasured cone beam projections from projections measured by a third generation helical volumetric computed tomography system by solving a characteristic problem for an ultrahyperbolic differential equation john 1938 by working in the fourier domain we convert the second order pde into a family of first order ordinary differential equations a simple first order integration is used to solve the odes

computing hopf bifurcations i this paper addresses the problems of detecting hopf bifurcations in systems of ordinary differential equations and following curves of hopf points in two parameter families of vector fields the established approach to this problem relies upon augmenting the equilibrium condition so that a hopf bifurcation occurs at an isolated regular point of the extended system we propose two new methods of this type based on classical algebraic results regarding the roots of polynomial equations and properties of kronecker products for matrices in addition to their utility as augmented systems for use with standard newton type continuation methods they are also particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension

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

a cone beam reconstruction algorithm using shift variant filtering and cone beam backprojection an exact inversion formula written in the form of shift variant filtered backprojection fbp is given for reconstruction from cone beam data taken from any orbit satisfying h k tuy s 1983 sufficiency conditions the method is based on a result of p grangeat 1987 involving the derivative of the three dimensional 3d radon transform but unlike grangeat s algorithm no 3d rebinning step is required data redundancy which occurs when several cone beam projections supply the same values in the radon domain is handled using an elegant weighting function and without discarding data the algorithm is expressed in a convenient cone beam detector reference frame and a specific example for the case of a dual orthogonal circular orbit is presented when the method is applied to a single circular orbit even though tuy s condition is not satisfied it is shown to be equivalent to the well known algorithm of l a feldkamp et al 1984

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

a theoretical study of the contrast recovery and variance of map reconstructions from pet data the authors examine the spatial resolution and variance properties of pet images reconstructed using maximum a posteriori map or penalized likelihood methods resolution is characterized by the contrast recovery coefficient crc of the local impulse response simplified approximate expressions are derived for the local impulse response crcs and variances for each voxel using these results the authors propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local crc to noise ratio in the reconstructed image

deflation algorithm for the multiple roots of a system of nonlinear equations abstract t ojika s watanabe and t mitsui in preparation have been developing a subroutine package naes nonlinear algebraic equations solver for the numerical solutions of the system of nonlinear equations an algorithm in the package termed the deflation algorithm for determining multiple roots for a system of nonlinear equations is presented and its effectiveness is shown by solving a numerical example

sweeping algebraic curves for singular solutions many problems give rise to polynomial systems these systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters using predictor corrector methods we track the solution paths a point along a solution path is critical when the jacobian matrix is rank deficient the simplest case of quadratic turning points is well understood but these methods no longer work for general types of singularities in order not to miss any singular solutions along a path we propose to monitor the determinant of the jacobian matrix we examine the operation range of deflation and relate the effectiveness of deflation to the winding number computational experiments on systems coming from different application fields are presented

region of interest reconstruction from truncated data in circular cone beam ct the circular scanning trajectory is one of the most widely adopted data acquisition configurations in computed tomography ct the feldkamp davis kress fdk algorithm and its various modifications have been developed for reconstructing approximately three dimensional images from circular cone beam data when data contain transverse truncations however these algorithms may reconstruct images with significant truncation artifacts it is of practical significance to develop algorithms that can reconstruct region of interest roi images from truncated circular cone beam data that are free of truncation artifacts and that have an accuracy comparable to that obtained from nontruncated cone beam data in this work we have investigated and developed a backprojection filtration bpf based algorithm for roi image reconstruction from circular cone beam data containing transverse truncations furthermore we have developed a weighted bpf algorithm to exploit redundant information in data for improving image quality in an effort to validate and evaluate the proposed bpf algorithms for circular cone beam ct we have performed numerical studies by using both computer simulation data and experimental data acquired with a radiotherapy cone beam ct system quantitative results in these studies demonstrate that the proposed bpf algorithms for circular cone beam ct can reconstruct roi images free of truncation artifacts

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

critical points and supersymmetric vacua i supersymmetric vacua universes of string m theory may be identified with certain critical points of a holomorphic section the superpotential of a hermitian holomorphic line bundle over a complex manifold an important physical problem is to determine how many vacua there are and how they are distributed as the superpotential varies over physically relevant ensembles in several papers over the last few years m r douglas and co workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor do ad dd the present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem it sets down basic results on the statistics of critical points s 0 of random holomorphic sections of hermitian holomorphic line bundles with respect to a metric connection when the sections are endowed with a gaussian measure the principal results give formulas for the expected density and number of critical points of fixed morse index of gaussian random sections relative to they are particularly concrete for riemann surfaces in our subsequent work the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points

newton s method with deflation for isolated singularities of polynomial systems we present a modification of newton s method to restore quadratic convergence for isolated singular solutions of polynomial systems our method is symbolic numeric we produce a new polynomial system which has the original multiple solution as a regular root using standard bases a tool for the symbolic computation of multiplicities we show that the number of deflation stages is bounded by the multiplicity of the isolated root our implementation performs well on a large class of applications

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

an inversion formula for cone beam reconstruction an analytic inversion formula allowing the reconstruction of a three dimensional object from x ray cone beams is given the formula is valid for the case where the source of the beams describes a bounded curve satisfying a set of weak conditions

derivation and implementation of a cone beam reconstruction algorithm for nonplanar orbits b d smith ibid vol mi 4 p 15 25 1985 opt eng vol 29 p 524 34 1990 and p grangeat these de doctorat 1987 lecture notes in mathematics 1497 p 66 97 1991 derived a cone beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used although grangeat s inversion formula is mathematically different from smith s one they have similar overall structures to each other the contribution of the present paper is two fold first based on the derivation of smith the authors point out that grangeat s inversion formula and smith s one can be conveniently described using a single formula the smith grangeat inversion formula that is in the form of space variant filtering followed by cone beam backprojection furthermore the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm second the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm as for exactness of the new algorithm the following fact can be stated the algorithm based on grangeat s intermediate function is exact for any complete orbit whereas that based on smith s intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3d space meets the orbit the validity of the new algorithm is demonstrated by simulation studies

direct reconstruction of cone beam data acquired with a vertex path containing a circle cone beam data acquired with a vertex path satisfying the data sufficiency condition of tuy can be reconstructed using exact filtered backprojection algorithms these algorithms are based on the application to each cone beam projection of a two dimensional 2 d filter that is nonstationary and therefore more complex than the one dimensional 1 d ramp filter used in the approximate algorithm of feldkamp davis and kress 1984 fdk we determine in this paper the general conditions under which the 2 d nonstationary filter reduces to a 2 d stationary filter and also give the explicit expression of the corresponding convolution kernel using this result and the redundancy of the cone beam data a composite algorithm is derived for the class of vertex paths that consist of one circle and some complementary subpath designed to guarantee data sufficiency in this algorithm the projections corresponding to vertex points along the circle are filtered using a 2 d stationary filter whereas the other projections are handled with a 2 d nonstationary filter the composite algorithm generalizes the method proposed by kudo and saito 1990 in which the circle data are processed with a 1 d ramp filter as in the fdk algorithm the advantage of the 2 d filter introduced in this paper is to guarantee that the filtered cone beam projections do not contain singularities in smooth regions of the object tests of the composite algorithm on simulated data are presented

enpit filtered back projection algorithm for helical ct using an n pi acquisition in this paper we formulate a reconstruction algorithm for an n pi acquisition where n can be any positive odd integer the algorithm is a generalization of the method presented in bontus et al 2003 it is based on the results obtained by katsevich 2004 for the algorithm different sets of filter lines have to be defined we describe the variation of these lines along the detector in some detail before we discuss how the method gives all radon plane contributions the correct weighting the different sets of filter lines are all contained within the n pi window such that a practical realization is possible reconstruction results which we present in the final section show convincing image quality

theoretical study of penalized likelihood image reconstruction for region of interest quantification region of interest roi quantification is an important task in emission tomography e g positron emission tomography and single photon emission computed tomography it is essential for exploring clinical factors such as tumor activity growth rate and the efficacy of therapeutic interventions statistical image reconstruction methods based on the penalized maximum likelihood pml or maximum a posteriori principle have been developed for emission tomography to deal with the low signal to noise ratio of the emission data similar to the filter cut off frequency in the filtered backprojection method the regularization parameter in pml reconstruction controls the resolution and noise tradeoff and hence affects roi quantification in this paper we theoretically analyze the performance of roi quantification in pml reconstructions building on previous work we derive simplified theoretical expressions for the bias variance and ensemble mean squared error emse of the estimated total activity in an roi that is surrounded by a uniform background when the mean and covariance matrix of the activity inside the roi are known the theoretical expressions are readily computable and allow for fast evaluation of image quality for roi quantification with different regularization parameters the optimum regularization parameter can then be selected to minimize the emse computer simulations are conducted for small rois with variable uniform uptake the results show that the theoretical predictions match the monte carlo results reasonably well

adaptive multiprecision path tracking this article treats numerical methods for tracking an implicitly defined path the numerical precision required to successfully track such a path is difficult to predict a priori and indeed it may change dramatically through the course of the path in current practice one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved to avoid unnecessary computational cost it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path we present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates we then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations

on artifact free projection overlaps in multi pinhole tomographic imaging in multi pinhole spect overlapping the projections from different pinholes has been used to increase sensitivity however the prevailing view is that the overall quality of the reconstructed image is not improved by the overlaps in the projections it is often stated in literatures that overlaps introduce ambiguous information which can lead to nonuniqueness of solution for the inverse problem and thus artifacts are introduced in the reconstructed image on the other hand contrary to the prevailing view a recent study on slit slat collimators shows that artifacts can be removed with the help of an extra complete nonoverlapped projection data set in this paper two types of artifact free projection overlaps are defined in general and the criteria for designing artifact free multi pinhole systems with overlaps are proposed it is shown that once the criteria are satisfied the solution of the inverse problem is unique and thus no artifact is expected in the reconstructed image via extensive simulation study various artifact free overlapping multi pinhole systems are designed and validated it is shown that overlaps in the artifact free systems can improve contrast to noise ratio cnr with a proper design the cnr for an artifact free overlapping system can be significantly higher than that for the corresponding nonoverlapping system the improved image quality is also confirmed with noisy reconstructions

exact radon rebinning algorithm for the long object problem in helical cone beam ct this paper addresses the long object problem in helical cone beam computed tomography the authors present the phi method a new algorithm for the exact reconstruction of a region of interest roi of a long object from axially truncated data extending only slightly beyond the roi the phi method is an extension of the radon method published by kudo et al in phys in med and biol vol 43 p 2885 909 1998 the key novelty of the phi method is the introduction of a virtual object f sub spl phi x for each value of the azimuthal angle spl phi in the image space with each virtual object having the property of being equal to the true object f x in some roi spl omega sub m the authors show that for each spl phi one can calculate exact radon data corresponding to the two dimensional 2 d parallel beam projection of f sub spl phi x onto the meridian plane of angle spl phi given an angular range of length spl pi of such parallel beam projections the roi spl omega sub m can be exactly reconstructed because f x is identical to f sub spl phi x in spl omega sub m simulation results are given for both the radon method and the phi method indicating that 1 for the case of short objects the radon and phi methods produce comparable image quality 2 for the case of long objects the phi method delivers the same image quality as in the short object case while the radon method fails and 3 the image quality produced by the phi method is similar for a large range of pitch values

critical points and supersymmetric vacua iii string m models a fundamental problem in contemporary string m theory is to count the number of inequivalent vacua satisfying constraints in a string theory model this article contains the first rigorous results on the number and distribution of supersymmetric vacua of type iib string theories compactified on a calabi yau 3 fold x with flux in particular complete proofs of the counting formulas in ashok douglas ad and denef douglas dd1 are given together with van der corput style remainder estimates

on the inverse of the directional derivative operator in sbb r n in this work the simplest partial differential equation in is studied and some of its properties derived it describes the infinitesimal translation of a function in a fixed direction specified by a unit vector n and driven by a source function we work out several forms of the corresponding green s function and show that the solutions may be viewed as integral transforms of known as divergent beam x ray transform in imaging science physically this connection is simply due to straight line propagation of radiation emitted by the spatial source distribution in particular we examine the special two dimensional case to point out the connection with the classical radon transform we then show how the radon transform inversion can be obtained in the context of a complex extension of this equation perspectives in higher dimensional space based on the present approach are given in the conclusion

fast predictions of variance images for fan beam transmission tomography with quadratic regularization accurate predictions of image variances can be useful for reconstruction algorithm analysis and for the design of regularization methods computing the predicted variance at every pixel using matrix based approximations is impractical even most recently adopted methods that are based on local discrete fourier approximations are impractical since they would require a forward and backprojection and two fast fourier transform fft calculations for every pixel particularly for shift variant systems like fan beam tomography this paper describes new analytical approaches to predicting the approximate variance maps of 2 d images that are reconstructed by penalized likelihood estimation with quadratic regularization in fan beam geometries the simplest of the proposed analytical approaches requires computation equivalent to one backprojection and some summations so it is computationally practical even for the data sizes in x ray computed tomography ct simulation results show that it gives accurate predictions of the variance maps the parallel beam geometry is a simple special case of the fan beam analysis the analysis is also applicable to 2 d positron emission tomography pet

asymptotics and dimensional dependence of the number of critical points of random holomorphic sections we prove two conjectures from dsz2 dsz3 concerning the expected number of critical points of random holomorphic sections of a positive line bundle we show that on average the critical points of minimal morse index are the most plentiful for holomorphic sections of mathcal o n to mathbb cp m and in an asymptotic sense for those of line bundles over general kahler manifolds we calculate the expected number of these critical points for the respective cases and use these to obtain growth rates and asymptotic bounds for the total expected number of critical points in these cases this line of research was motivated by landscape problems in string theory and spin glasses

inversion formula for the non uniformly attenuated x ray transform for emission imaging in mathbb r 3 using quaternionic analysis in this paper we present a new derivation of the inverse of the non uniformly attenuated x ray transform in three dimensions based on quaternion analysis an explicit formula is obtained using a set of three dimensional x ray projection data the result without attenuation is recovered as a special case

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

a quaternionic approach to x ray transform inversion in sbb r 3 a new derivation of the inverse of the x ray transform is presented based on quaternion analysis as pointed out by practitioners a direct inversion formula offers more efficient reconstruction algorithms than tomographic inversion it is shown that the new inverse formula is equivalent to the existing one the advantage of this approach is that it paves the way for a potential inversion of the x ray transform with a non uniform attenuation map in three dimensions which models single photon emission imaging in nuclear medicine

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 mathbb r 3 exponential x ray transform inversion in quaternion analysis in this paper we present a new derivation of the inverse of the exponential x ray transform in the three dimensions based on quaternion analysis an explicit formula is obtained using a set of three dimensional x ray projection data the result without attenuation is recovered as a special case

spatial resolution properties of penalized likelihood image reconstruction space invariant tomographs this paper examines the spatial resolution properties of penalized likelihood image reconstruction methods by analyzing the local impulse response the analysis shows that standard regularization penalties induce space variant local impulse response functions even for space invariant tomographic systems paradoxically for emission image reconstruction the local resolution is generally poorest in high count regions we show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections this analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution the modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized likelihood methods

a modified uniform cramer rao bound for multiple pinhole aperture design this paper presents a modified uniform cramer rao bound ucrb for studying estimator spatial resolution and variance tradeoffs we proposed to use a resolution constraint that is imposed on mean gradient vectors of achieved estimators and derived the minimum achievable variance for any estimator satisfies this resolution constraint this approach partially overcomes the limitations of the former ucrb approach based on a bias gradient norm constraint we applied this method in a feasibility study of using multiple pinhole apertures for small animal spect imaging applications the spect system studied was based on an existing gamma camera the achievable spatial resolution and variance tradeoffs for systems with different design parameters such as number of pinholes and pinhole size were studied

new properties of the v line radon transform and their imaging applications this paper reports on new aspects of the so called v line radon transforms rts complementing those reported in an earlier work these new properties are nicely uncovered and described with cartesian coordinates in particular we show that the v line rt belongs to the class of rts on curves in the plane which can be mapped onto the standard rt on straight lines and thereby are fully characterizable and invertible next we show that the effect of geometric inversion on the v line rt is to produce a new rt on a pair of supplementary circular arcs which provides a new access to image reconstruction in the so called norton s modality of compton scatter tomography a front runner in the race for alternatives to current emission imaging

fourier rebinning applied to multiplanar circular orbit cone beam spect the authors study the application of fourier rebinning methods to dual planar cone beam spect dual planar cone beam spect involves the use of a pair of dissimilar cone beam collimators on a dual camera spect system each collimator has its focus in a different axial plane while dual planar data is best reconstructed with fully three dimensional 3 d iterative methods these methods are slow and have prompted a search for faster reconstruction techniques fourier rebinning was developed to estimate equivalent parallel projections from 3 d pet data but it simply expresses a relationship between oblique projections taken in planes not perpendicular to the axis of rotation and direct projections taken in those that are the authors find that it is possible to put cone beam data in this context as well the rebinned data ran then be reconstructed using either filtered backprojection fbp or parallel iterative algorithms such as os em the authors compare the feldkamp algorithm and fully 3 d osem reconstruction with fourier rebinned reconstructions on realistically simulated tc 99m hmpao brain spect data they find that the fourier rebinned reconstructions exhibit much less image noise and lower variance in region of interest roi estimates than feldkamp also fourier rebinning followed by osem with nonuniform attenuation correction exhibits less bias in roi estimates than feldkamp with chang attenuation correction the fourier rebinned roi estimates exhibit bias and variance comparable to those from fully 3 d osem and require considerably less processing time however in areas off the axis of rotation the axial direction resolution of fore reconstructed images is poorer than that of images reconstructed with 3 d osem the authors conclude that fourier rebinning is a practical and potentially useful approach to reconstructing data from dual planar circular orbit cone beam systems

cone beam reconstruction with sources on a curve an inversion procedure is developed for reconstructing a function of compact support in mathbb r 3 from its divergent beam x ray transform with sources on a curve the results apply to many curves not satisfying the conditions of tuy siam j appl math 43 1983 pp 546 552 for example planar circles of sufficiently large radius in some cases estimates in sobolev norms are established for the inversion operator

a filtered backprojection algorithm for triple source helical cone beam ct multisource cone beam computed tomography ct is an attractive approach of choice for superior temporal resolution which is critically important for cardiac imaging and contrast enhanced studies in this paper we present a filtered backprojection fbp algorithm for triple source helical cone beam ct the algorithm is both exact and efficient it utilizes data from three inter helix pi arcs associated with the inter helix pi lines and the minimum detection windows defined for the triple source configuration the proof of the formula is based on the geometric relations specific to triple source helical cone beam scanning simulation results demonstrate the validity of the reconstruction algorithm this algorithm is also extended to a multisource version for 2n 1 source helical cone beam ct with parallel computing the proposed fbp algorithms can be significantly faster than our previously published multisource backprojection filtration algorithms thus the fbp algorithms are promising in applications of triple source helical cone beam ct