information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

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

adaptive estimation over anisotropic functional classes via oracle approach we address the problem of adaptive minimax estimation in white gaussian noise model under lp loss 1 p 1 on the anisotropic nikolskii classes we present the es timation procedure based on a new data driven selection scheme from the family of kernel estimators with varying bandwidths for proposed estimator we establish so called lp norm oracle inequality and use it for deriving minimax adaptive results we prove the existence of rate adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk in particular some new asymptotics of the minimax risk are discovered including necessary and sufficient conditions for existencea uniformly consistent estimator we provide also with detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation

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

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

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

learning by mirror averaging given a collection of m different estimators or classifiers we study the problem of model selection type aggregation i e we construct a new estimator or classifier called aggregate which is nearly as good as the best among them with respect to a given risk criterion we define our aggregate by a simple recursive procedure which solves an auxiliary stochastic linear programming problem related to the original non linear one and constitutes a special case of the mirror averaging algorithm we show that the aggregate satisfies sharp oracle inequalities under some general assumptions the results allow one to construct in an easy way sharp adaptive nonparametric estimators for several problems including regression classification and density estimation

component analysis with different sets of constraints on different dimensions many of the classical multivariate data analysis and multidimensional scaling techniques call for approximations by lower dimensional configurations a model is proposed in which different sets of linear constraints are imposed on different dimensions in component analysis and classical multidimensional scaling frameworks a simple efficient and monotonically convergent algorithm is presented for fitting the model to the data by least squares the basic algorithm is extended to cover across dimension constraints imposed in addition to the dimensionwise constraints and to the case of a symmetric data matrix examples are given to demonstrate the use of the method

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

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

some universal results on the behavior of increments of partial sums we establish very general one sided results on the lim sup behavior of increments of suitably normalized partial sums of i i d random variables our main results apply to arbitrary nondegenerate positive random variables which need not have any finite moments as a corollary we can show that such results also hold for not necessarily positive random variables whose negative parts have finite moment generating functions

a universal procedure for aggregating estimators in this paper we study the aggregation problem that can be formulated as follows assume that we have a family of estimators f built on the basis of available observations the goal is to construct a new estimator whose risk is as close as possible to that of the best estimator in the family we propose a general aggregation scheme that is universal in the following sense it applies for families of arbitrary estimators and a wide variety of models and global risk measures the procedure is based on comparison of empirical estimates of certain linear functionals with estimates induced by the family f we derive oracle inequalities and show that they are unimprovable in some sense numerical results demonstrate good practical behavior of the procedure

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

unmatched projector backprojector pairs in an iterative reconstruction algorithm computational burden is a major concern when an iterative algorithm is used to reconstruct a three dimensional 3 d image with attenuation detector response and scatter corrections most of the computation time is spent executing the projector and backprojector of an iterative algorithm usually the projector and the backprojector are transposed operators of each other the projector should model the imaging geometry and physics as accurately as possible some researchers have used backprojectors that are computationally less expensive than the projectors to reduce computation time this paper points out that valid backprojectors should satisfy a condition that the projector backprojector matrix must not contain negative eigenvalues this paper also investigates the effects when unmatched projector backprojector pairs are used

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

minimax adaptive spectral estimation from an ensemble of signals we develop a statistical method for estimating the spectrum from a data set that consists of several signals all of which are realizations of a common random process we first find estimates of the common spectrum using each signal then we construct m partial aggregates each partial aggregate is a linear combination of m 1 of the spectral estimates the weights are obtained from the data via a least squares criterion the final spectral estimate is the average of these m partial aggregates we show that our final estimator is minimax rate adaptive if at least two of the estimators per signal attain the optimal rate n 2alpha 2alpha 1 for spectra belonging to a generalized lipschitz ball with smoothness index alpha our simulation study strongly suggests that our procedure works well in practice and in a large variety of situations is preferable to the simple averaging of the m spectral estimates

adaptive regression by mixing adaptation over different procedures is of practical importance different procedures perform well under different conditions in many practical situations it is rather hard to assess which conditions are approximately satisfied so as to identify the best procedure for the data at hand thus automatic adaptation over various scenarios is desirable a practically feasible method named adaptive regression by mixing arm is proposed to convexly combine general candidate regression procedures under mild conditions the resulting estimator is theoretically shown to perform optimally in rates of convergence without knowing which of the original procedures work the best simulations are conducted in several settings including comparing a parametric model with nonparametric alternatives comparing a neural network with a projection pursuit in multidimensional regression and combining bandwidths in kernel regression the results clearly support the theoretical property of arm the arm algorithm assigns we

analysis of the three dimensional tongue shape using a three index factor analysis model three dimensional tongue shape during vowel production is analyzed using the three mode parafac parallel factors model three dimensional mri images of five speakers 9 vowels are analyzed sixty five virtual fleshpoints 13 segments along the rostral caudal dimension and 5 segments along the right left direction are chosen based on the interpolated tongue shape images methods used to adjust the alignment of mri images to set up the fleshpoints and to measure the position of the fleshpoints are presented parafac analysis of this 3d coordinate data results in a stable two factor solution that explains about 70 of the variance

consistency of cross validation for comparing regression procedures theoretical developments on cross validation cv have mainly focused on selecting one among a list of finite dimensional models e g subset or order selection in linear regression or selecting a smoothing parameter e g bandwidth for kernel smoothing however little is known about consistency of cross validation when applied to compare between parametric and nonparametric methods or within nonparametric methods we show that under some conditions with an appropriate choice of data splitting ratio cross validation is consistent in the sense of selecting the better procedure with probability approaching 1 our results reveal interesting behavior of cross validation when comparing two models procedures converging at the same nonparametric rate in contrast to the parametric case it turns out that the proportion of data used for evaluation in cv does not need to be dominating in size furthermore it can even be of a smaller order than the proportion for estimation while not affecting the consistency property

an efficient algorithm for tuckals3 on data with large numbers of observation units a modification of the tuckals3 algorithm is proposed that handles three way arrays of order i j k for any i when i is much larger than jk the modified algorithm needs less work space to store the data during the iterative part of the algorithm than does the original algorithm because of this and the additional feature that execution speed is higher the modified algorithm is highly suitable for use on personal computers

adaptive minimax regression estimation over sparse l q hulls given a dictionary of mn predictors in a random design regression setting with n observations we construct estimators that target the best performance among all the linear combinations of the predictors under a sparse lq norm 0 q 1 constraint on the linear coefficients besides identifying the optimal rates of convergence our universal aggregation strategies by model mixing achieve the optimal rates simultaneously over the full range of 0 q 1 for any mn and without knowledge of the lq norm of the best linear coefficients to represent the regression function to allow model misspecification our upper bound results are obtained in a framework of aggregation of estimates a striking feature is that no specific relationship among the predictors is needed to achieve the upper rates of convergence hence permitting basically arbitrary correlations between the predictors therefore whatever the true regression function assumed to be uniformly bounded our estimators automatically exploit any sparse representation of the regression function if any to the best extent possible within the lq constrained linear combinations for any 0 q 1 a sparse approximation result in the lq hulls turns out to be crucial to adaptively achieve minimax rate optimal aggregation it precisely characterizes the number of terms needed to achieve a prescribed accuracy of approximation to the best linear combination in an lq hull for 0 q 1 it offers the insight that the minimax rate of lq aggregation is basically determined by an effective model size which is a sparsity index that depends on q mn n and the lq norm bound in an easily interpretable way based on a classical model selection theory that deals with a large number of models

bandwidth selection for local linear regression smoothers summary the paper presents a general strategy for selecting the bandwidth of nonparametric regression estimators and specializes it to local linear regression smoothers the procedure requires the sample to be divided into a training sample and a testing sample using the training sample we first compute a family of regression smoothers indexed by their bandwidths next we select the bandwidth by minimizing the empirical quadratic prediction error on the testing sample the resulting bandwidth satisfies a finite sample oracle inequality which holds for all bounded regression functions this permits asymptotically optimal estimation for nearly any regression function the practical performance of the method is illustrated by a simulation study which shows good finite sample behaviour of our method compared with other bandwidth selection procedures

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

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

model free variable selection computing power has encouraged the modelling of data sets of ever increasing size data mining applications in finance marketing and bioinformatics are obvious examples a limitation of nearly all existing variable selection methods is the need to specify the correct model before selection when the number of predictors is large model formulation and validation can be difficult or even infeasible on the basis of the theory of sufficient dimension reduction we propose a new class of model free variable selection approaches the methods proposed assume no model of any form require no nonparametric smoothing and allow for general predictor effects the efficacy of the methods proposed is demonstrated via simulation and an empirical example is given

localization of vc classes beyond local rademacher complexities abstract in statistical learning the excess risk of empirical risk minimization erm is controlled by comp n f n where n is a size of a learning sample comp n f is a complexity term associated with a given class f and 1 2 1 interpolates between slow and fast learning rates in this paper we introduce an alternative localization approach for binary classification that leads to a novel complexity measure fixed points of the local empirical entropy we show that this complexity measure gives a tight control over comp n f in the upper bounds under bounded noise our results are accompanied by a minimax lower bound that involves the same quantity in particular we practically answer the question of optimality of erm under bounded noise for general vc classes

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

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

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

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

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

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

consistency for the least squares estimator in nonparametric regression we shall study the general regression model y g o x e where x and e are independent the available information about g 0 can be expressed by g 0 g for some class g as an estimator of g 0 we choose the least squares estimator we shall give necessary and sufficient conditions for consistency of this estimator in terms of basically geometric properties of g our main tool will be the theory of empirical processes

an efficient algorithm for parafac of three way data with large numbers of observation units the candecomp algorithm for the parafac analysis ofn m p three way arrays is adapted to handle arrays in whichn mp more efficiently for such arrays the adapted algorithm needs less memory space to store the data during the iterations and uses less computation time than the original candecomp algorithm the size of the arrays that can be handled by the new algorithm is in no way limited by the number of observation units n in the data

model selection in nonparametric regression model selection using a penalized data splitting device is studied in the context of nonparametric regression finite sample bounds under mild conditions are obtained the resulting estimates are adaptive for large classes of functions

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

model selection for cart regression trees the performance of the classification and regression trees cart pruning algorithm and the final discrete selection by test sample as a functional estimation procedure are considered the validation of the pruning procedure applied to gaussian and bounded regression is of primary interest on the one hand the paper shows that the complexity penalty used in the pruning algorithm is valid in both cases and on the other hand that conditionally to the construction of the maximal tree the final selection does not alter dramatically the estimation accuracy of the regression function in both cases the risk bounds that are proved obtained by using the penalized model selection validate the cart algorithm which is used in many applications such as meteorology biology medicine pollution monitoring or image coding

decompositions and biplots in three way correspondence analysis in this paper correspondence analysis for three way contingency tables is presented using three way generalisations of the singular value decomposition it is shown that in combination with lancaster s 1951 additive decomposition of interactions in three way tables a detailed analysis is possible of the deviations from independence finally biplots are shown to produce powerful graphical representations of the results from three way correspondence analyses an example from child development is used to illustrate the theoretical developments

model selection for regression on a fixed design we deal with the problem of estimating some unknown regression function involved in a regression framework with deterministic design points for this end we consider some collection of finite dimensional linear spaces models and the least squares estimator built on a data driven selected model among this collection this data driven choice is performed via the minimization of some penalized model selection criterion that generalizes on mallows c p we provide non asymptotic risk bounds for the so defined estimator from which we deduce adaptivity properties our results hold under mild moment conditions on the errors the statement and the use of a new moment inequality for empirical processes is at the heart of the techniques involved in our approach

a universally acceptable smoothing factor for kernel density estimates we define a minimum distance estimate of the smoothing factor for kernel density estimates based on a methodology first developed by yatracos it is shown that if f nh denotes the kernel density estimate on r d for an i i d sample of size n drawn from an unknown density f where h is the smoothing factor and if f n is the kernel estimate with the same kernel and with the proposed new data based smoothing factor then under a regularity condition on the kernel k formula math this is the first published smoothing factor that can be proven to have this property

information theory and mixing least squares regressions for gaussian regression we develop and analyze methods for combining estimators from various models for squared error loss an unbiased estimator of the risk of the mixture of general estimators is developed special attention is given to the case that the component estimators are least squares projections into arbitrary linear subspaces such as those spanned by subsets of explanatory variables in a given design we relate the unbiased estimate of the risk of the mixture estimator to estimates of the risks achieved by the components this results in simple and accurate bounds on the risk and its estimate in the form of sharp and exact oracle inequalities that is without advance knowledge of which model is best the resulting performance is comparable to or perhaps even superior to what is achieved by the best of the individual models furthermore in the case that the unknown parameter has a sparse representation our mixture estimator adapts to the underlying sparsity simulations show that the performance of these mixture estimators is better than that of a related model selection estimator which picks a model with the highest weight also the connection between our mixtures with bayes procedures is discussed

nonasymptotic universal smoothing factors kernel complexity and yatracos classes we introduce a method to select a smoothing factor for kernel density estimation such that for all densities in all dimensions the l 1 error of the corresponding kernel estimate is not larger than three times the error of the estimate with the optimal smoothing factor plus a constant times log n n where n is the sample size and the constant depends only on the complexity of the kernel used in the estimate the result is nonasymptotic that is the bound is valid for each n the estimate uses ideas from the minimum distance estimation work of yatracos as the inequality is uniform with respect to all densities the estimate is asymptotically minimax optimal modulo a constant over many function classes

adaptive model selection using empirical complexities given n independent replicates of a jointly distributed pair x y rd r we wish to select from a fixed sequence of model classes f1 f2 a deterministic prediction rule f rd r whose risk is small we investigate the possibility of empirically assessing the complexity of each model class that is the actual difficulty of the estimation problem within each class the estimated complexities are in turn used to define an adaptive model selection procedure which is based on complexity penalized empirical risk the available data are divided into two parts the first is used to form an empirical cover of each model class and the second is used to select a candidate rule from each cover based on empirical risk the covering radii are determined empirically to optimize a tight upper bound on the estimation error an estimate is chosen from the list of candidates in order to minimize the sum of class complexity and empirical risk a distinguishing feature of the approach is that the complexity of each model class is assessed empirically based on the size of its empirical cover finite sample performance bounds are established for the estimates and these bounds are applied to several non parametric estimation problems the estimates are shown to achieve a favorable tradeoff between approximation and estimation error and to perform as well as if the distribution dependent complexities of the model classes were known beforehand in addition it is shown that the estimate can be consistent and even possess near optimal rates of convergence when each model class has an infinite vc or pseudo dimension for regression estimation with squared loss we modify our estimate to achieve a faster rate of convergence appears in annals of statistics vol 27 1830 1864 ams 1991 subject classifications primary 62g07 62g20 secondary 62h30

sequential procedures for aggregating arbitrary estimators of a conditional mean in this correspondence a sequential procedure for aggregating linear combinations of a finite family of regression estimates is described and analyzed particular attention is given to linear combinations having coefficients in the generalized simplex the procedure is based on exponential weighting and has a computationally tractable approximation analysis of the procedure is based in part on techniques from the sequential prediction of nonrandom sequences here these techniques are applied in a stochastic setting to obtain cumulative loss bounds for the aggregation procedure from the cumulative loss bounds we derive an oracle inequality for the aggregate estimator for an unbounded response having a suitable moment generating function the inequality shows that the risk of the aggregate estimator is less than the risk of the best candidate linear combination in the generalized simplex plus a complexity term that depends on the size of the coefficient set the inequality readily yields convergence rates for aggregation over the unit simplex that are within logarithmic factors of known minimax bounds some preliminary results on model selection are also presented

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

linear model selection by cross validation abstract we consider the problem of selecting a model having the best predictive ability among a class of linear models the popular leave one out cross validation method which is asymptotically equivalent to many other model selection methods such as the akaike information criterion aic the c p and the bootstrap is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observations n we show that the inconsistency of the leave one out cross validation can be rectified by using a leave n v out cross validation with n v the number of observations reserved for validation satisfying n v n 1 as n this is a somewhat shocking discovery because nv n 1 is totally opposite to the popular leave one out recipe in cross validation motivations justifications and discussions of some practical aspects of the use of the leave n v out cross validation method are provided and results

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