equivalence of regularization and truncated iteration in the solution of iii posed image reconstruction problems abstract the equivalence of minimum norm least squares solutions of systems of linear equations and standard iterative methods of solution is well established on the other hand while it is generally understood that truncated iteration is a form of regularization comparatively few papers have formalized the relationship between direct methods of regularization and truncated iteration a brief review of such papers is presented the main result of this paper is to carry this idea one step further and prove that solutions by direct regularization are in fact identical to solutions of a certain type of truncated iterative method and conversely this equivalence is proved by construction for a very general form of regularization method in which the coefficient matrix has full rank and is rectangular

the second moment of gl 3 times gl 2 l functions at special points for a fixed sl 3 mathbb z maass form phi we consider the family of l functions l phi times u j s where u j runs over the family of hecke maass cusp forms on sl 2 mathbb z we obtain an estimate for the second moment of this family of l functions at the special points frac 1 2 it j consistent with the lindelof hypothesis we also obtain a similar upper bound on the sixth moment of the family of hecke maass cusp forms at these special points this is apparently the first occurrence of a lindelof consistent estimate for a sixth power moment of a family of gl 2 l functions

statistical analysis of maximum likelihood estimator images of human brain fdg pet studies the work presented evaluates the statistical characteristics of regional bias and expected error in reconstructions of real positron emission tomography pet data of human brain fluoro deoxiglucose fdg studies carried out by the maximum likelihood estimator mle method with a robust stopping rule and compares them with the results of filtered backprojection fbp reconstructions and with the method of sieves the task of evaluating radioisotope uptake in regions of interest rois is investigated an assessment of bias and variance in uptake measurements is carried out with simulated data then by using three different transition matrices with different degrees of accuracy and a components of variance model for statistical analysis it is shown that the characteristics obtained from real human fdg brain data are consistent with the results of the simulation studies

the sectional category of a map we study a generalization of the svarc genus of a fibre map for an arbitrary collection of spaces and a map f x y we define a numerical invariant the sectional category of f in terms of open covers of y we obtain several basic properties of sectional category including those dealing with homotopy domination and homotopy pushouts we then give three simple axioms which characterize the sectional category in the final section we obtain inequalities for the sectional category of a composition and inequalities relating the sectional category to the fadell husseini category of a map and the clapp puppe category of a map

practical tradeoffs between noise quantitation and number of iterations for maximum likelihood based reconstructions emission computerised tomography images reconstructed using a maximum likelihood expectation maximization ml based method with different reconstruction kernels and 1 200 iterations are compared to images reconstructed using filtered backprojection fbp ml based reconstructions using a single pixel sp kernel with or without a sieve filter show no quantitative advantage over fbp except in the background where a reduction of noise is possible if the number of iterations is kept small

the lusternik schnirelmann category of a lie groupoid we propose a new homotopy invariant for lie groupoids which generalizes the classical lusternik schnirelmann category for topological spaces we use a bicategorical approach to develop a notion of contraction in this context we propose a notion of homotopy between generalized maps given by the 2 arrows in a certain bicategory of fractions this notion is invariant under morita equivalence thus when the groupoid defines an orbifold we have a well defined ls category for orbifolds we prove an orbifold version of the classical lusternik schnirelmann theorem for critical points

on the validity of hypothesis testing for feasibility of image reconstructions feasible images have been defined as those images that could have generated the original data by the statistical process that governs the measurement in the case of emission tomography the statistical process of emission is poisson and it is known that feasible images resulting from the maximum likelihood estimator mle and bayesian methods with entropy priors can be of high quality tests for feasibility have been described that are based on one critical assumption the image that is being tested is independent of the data even though the reconstruction algorithm has used those data in order to obtain the image this fact could render the procedure unacceptable unless it is shown that its effects on the results of the tests are negligible experimental evidence is presented showing that images reconstructed by the mle and stopped before convergence do indeed behave as if independent of the data the results justify the use of hypothesis testing in practice although they leave the problem of analytical proof still open

corrections for accidental coincidences and attenuation in maximum likelihood image reconstruction for positron emission tomography reconstruction procedures that account for attenuation in forming maximum likelihood estimates of activity distributions in positron emission tomography are extended to include regularization constraints and accidental coincidences a mathematical model is used for these effects the corrections are incorporated into the iterations of an expectation maximization algorithm for numerically producing the maximum likelihood estimate of the distribution of radioactivity within a patient the images reconstructed with this procedure are unbiased and exhibit lower variance than those reconstructed from precorrected data

periodic solutions of symmetric autonomous newtonian systems abstract in this paper we show the existence and bifurcation of t periodic solutions of a special form for an autonomous newtonian system with symmetry if the phase space r 2 n is equipped with the structure of an orthogonal representation w w and the potential v w r is invariant then for every such a solution the set of indices of nonvanishing fourier coefficients is finite and depends on w only if the potential v depends on the squares of complex coordinates then for every such a solution t is the minimal period

appendix an effective zero free region all the notation in this appendix will be as in the preceding paper let f be a maass form which is a newform for fo n with eigenvalue a and central character x normalized so that f f 1 we have seen that the size of p l the first fourier coefficient of f is intimately related to the behavior of l s f near s 1 here l s f is the l series of f the adjoint square lift of f to gl 3 and the crucial question is whether or not l s f vanishes when s is real and close to 1 it was shown in theorem 0 1 that if l s f is nonzero in a sufficiently wide neighborhood of 1 then ip 1 12 0 where the implied constant depends on e and is ineffective about a year after the preceding paper was first circulated it developed from conversations involving the above three authors that by slightly modifying the techniques introduced in that paper the possibility of a siegel zero could be completely eliminated in many cases in particular theorem 0 1 is now true unconditionally in the generic situation when f is not a lift from gl 1 that is to say when the l series of f is not equal to a hecke l series defined over a quadratic field if we include all cusp forms we can still obtain the theorem in the a aspect but must restrict ourselves to either an ineffective constant or a weaker effective constant in the n aspect we have

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

the second moment of gl 4 gl 2 l functions at special points abstract in this paper we obtain upper bounds for the second moment of l u j 1 2 i t j where is a hecke maass form for s l 4 z and u j is taken from an orthonormal basis of hecke maass forms on s l 2 z with eigenvalue 1 4 t j 2 the bounds are consistent with the lindelof hypothesis previously these types of upper bounds are available for only g l n g l 2 where n 3

non compactness of the space of minimal hypersurfaces we show that the space of min max minimal hypersurfaces is non compact when the manifold has an analytic metric of positive ricci curvature and dimension 3 le n 1 le 7 furthermore we show that bumpy metrics with positive ricci curvature admit minimal hypersurfaces with unbounded mathrm index mathrm area when combined with the recent work fo f c marques and a neves we then deduce some new properties regarding the infinitely many minimal hypersurfaces they found

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

stopping rule for the mle algorithm based on statistical hypothesis testing it is known that when the maximum likelihood estimator mle algorithm passes a certain point it produces images that begin to deteriorate we propose a quantitative criterion with a simple probabilistic interpretation that allows the user to stop the algorithm just before this effect begins the mle algorithm searches for the image that has the maximum probability to generate the projection data the underlying assumption of the algorithm is a poisson distribution of the data therefore the best image according to the mle algorithm is the one that results in projection means which are as close to the data as possible it is shown that this goal conflicts with the assumption that the data are poisson distributed we test a statistical hypothesis whereby the projection data could have been generated by the image produced after each iteration the acceptance or rejection of the hypothesis is based on a parameter that decreases as the images improve and increases as they deteriorate we show that the best mle images which pass the test result in somewhat lower noise in regions of high activity than the filtered back projection results and much improved images in low activity regions the applicability of the proposed stopping rule to other iterative schemes is discussed

invariants of the lusternik schnirelmann type and the topology of critical sets we introduce and study in detail generalizations of the notion of lusternik schnirelmann category which give information about the topology of the critical set of a differentiable function we also improve a result of t ganea about the equality of the strong category and the category even in the classical case the category cat x of a space x in the sense of lusternik and schnirelmann 14 is the smallest number k such that there exists a covering x1 xk of x of a certain kind cf 1 2 1 for whicheach inclusion x c x is nullhomotopic the motivation for introducing this concept was that it gives a lower bound for the number of critical points of a function more precisely if m is a closed differentiable manifold and f is a differentiable real function on m then the number of critical points of f is at least cat m we propose the following generalization if v is any class of spaces we replace the condition that xj c x is nullhomotopic by requiring that it factors through some a e up to homotopy and we obtain the notion of v category v cat x if consists only of the one point space this is the classical cat x if v is the class of q connected spaces s cat is the q dimensional homotopy category introduced by fox in 7 another interesting example is the class v of q dimensional spaces if f m r is as above then v cat m does not give any new information about the number of critical points because it is less than or equal to cat m it does give however under certain conditions some new information on the topological structure of the critical set roughly one can say that either there are at least v cat m critical values of f or there is one critical value y of f such that the corresponding set of critical points k n f 1 y is not of the homotopy type of any space in v cf 2 for more details quite a number of papers have appeared on lusternik schnirelmann category and related notions cf 13 for a survey in particular there has been a revival of interest in recent years we shall present a systematic theory for our more general received by the editors november 25 1985 1980 mathematics subject classification 1985 revision primary 55m30 secondary 58e05 55p50

a fast reconstruction algorthm for stationary positron emission tomography based on a modified em algorithm an efficient iterative reconstruction method for positron emission tomography pet is presented the algorithm is basically an enhanced em expectation maximization algorithm with improved frequency response high frequency components of the ratio of measured to calculated projections are extracted and are taken into account for the iterative correction of image density in such a way that the correction is performed with a uniform efficiency over the image plane and with a flat frequency response as a result the convergence speed is not so sensitive to the image pattern or matrix size as the standard em algorithm and nonuniformity of the spatial resolution is significantly improved nonnegativity of the reconstructed image is preserved simulation studies have been made assuming two pet systems a scanning pet with ideal sampling and a stationary pet with sparse sampling in the latter a bank array of detectors is employed to improve the sampling in the object plane the new algorithm provides satisfactory images by two or three iterations starting from a flat image in either case the behavior of convergence is monitored by evaluating the root mean square of c b 1 where c b is the correction factor for pixel b in the em algorithm the value decreases rapidly and monotonically with iteration number although the theory is not accurate enough to assure the stability of convergence the algorithm is promising to achieve significant saving in computation compared to the standard em algorithm

feasible images and practical stopping rules for iterative algorithms in emission tomography the discussion of the causes of image deterioration in the maximum likelihood estimator mle method of tomographic image reconstruction initiated with the publication of a stopping rule for that iterative process e veklerov and j llacer 1987 is continued the concept of a feasible image is introduced which is a result of a reconstruction that if it were a radiation field could have generated the initial projection data by the poisson process that governs radioactive decay from the premise that the result of a reconstruction should be feasible the shape and characteristics of the region of feasibility in projection space are examined with a new rule reconstructions from real data can be tested for feasibility results of the tests and reconstructed images for the hoffman brain phantom are shown a comparative examination of the current methods of dealing with mle image deterioration is included

bounds for gl 3 gl 2 l functions and gl 3 l functions in this paper we will give the subconvexity bounds for self dual gl 3 l functions in the t aspect as well as subconvexity bounds for self dual gl 3 gl 2 l functions in the gl 2 spectral aspect

a smoothed em approach to indirect estimation problems with particular reference to stereology and emission tomography there are many practical problems where the observed data are not drawn directly from the density g of real interest but rather from another distribution derived from g by the application of an integral operator the estimation of g then entails both statistical and numerical difficulties a natural statistical approach is by maximum likelihood conveniently implemented using the em algorithm but this provides unsatisfactory reconstructions of g in this paper we modify the maximum likelihood em approach by introducing a simple smoothing step at each em iteration in our experience this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration some theoretical background is given that relates this smoothed em algorithm to a maximum penalized likelihood approach two applications are considered in detail the first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium the second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography for this problem we also suggest several technical improvements on existing methodology

relative normal modes for nonlinear hamiltonian systems an estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a hamiltonian system with continuous symmetry is given this result constitutes a generalization to the hamiltonian symmetric framework of a classical result by weinstein and moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium the estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighboring the stable relative equilibrium in question and with any prefixed spatiotemporal isotropy subgroup moreover it is easily computable in particular examples it is interesting to see how in our result the existence of non trivial relative periodic orbits requires generic conditions on the higher order terms of the taylor expansion of the hamiltonian function in contrast with the purely quadratic requirements of the weinstein moser theorem which emphasizes the highly non linear character of the relatively periodic dynamical objects

fully bayesian reconstructions from single photon emission computed tomography data abstract with suitably chosen priors bayesian models are useful in image reconstruction i consider reconstructions from single photon emission computerized tomography data using a gibbs pairwise difference prior a fully bayesian approach is presented where the prior parameters are considered drawn from hyperpriors the approach is problematical because the normalization constant in the prior is an intractable function of its parameters here the constant is estimated off line by reverse logistic regression markov chain monte carlo methods are used on simulated and real data to gain estimates of the posterior mean and pixelwise credibility bounds the resulting reconstructions are compared to those obtained by filtered back projection maximum likelihood em and the one step late solution to the fixed parameter posterior mode

vector extrapolated fast maximum likelihood estimation algorithms for emission tomography a new class of fast maximum likelihood estimation mle algorithms for emission computed tomography ect is developed in these cyclic iterative algorithms vector extrapolation techniques are integrated with the iterations in gradient based mle algorithms with the objective of accelerating the convergence of the base iterations this results in a substantial reduction in the effective number of base iterations required for obtaining an emission density estimate of specified quality the mathematical theory behind the minimal polynomial and reduced rank vector extrapolation techniques in the context of emission tomography is presented these extrapolation techniques are implemented in a positron emission tomography system the new algorithms are evaluated using computer experiments with measurements taken from simulated phantoms it is shown that with minimal additional computations the proposed approach results in substantial improvement in reconstruction

there is just one rational cone length we show that the homotopic nilpotency of the algebra of piecewise polynomial forms on a simply connected finite type cw complex coincides with the strong l s category of the rationalization of that space this is used to prove that in the rational simply connected context all reasonable notions of cone length agree both these two results are obtained as parts of a more general and functorial picture

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

fundamental groups of manifolds with s 1 category 2 a closed topological n manifold mn is of s1 category 2 if it can be covered by two open subsets w1 w2 such that the inclusions wi mn factor homotopically through maps wi s1 mn we show that the fundamental group of such an n manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation in particular if n 3 the fundamental group is cyclic

higher dimensional manifolds with s1 category 2 a closed topological n manifold mn is of s1 category 2 if it can be covered by two open subsets w1 w2 such that the inclusions wi mn factor homotopically through maps wi s1 we show that for n 3 if rm cat s 1 m n 2 then mn is homeomorphic to sn or sn 1 s1 or the non orientable sn 1 bundle over s1 we also obtain an unknotting theorem for locally flat knots of sn 2 in sn and a characterization of s1 sn 1

performance evaluation of an iterative image reconstruction algorithm for positron emission tomography an image reconstruction method motivated by positron emission tomography pet is discussed the measurements tend to be noisy and so the reconstruction method should incorporate the statistical nature of the noise the authors set up a discrete model to represent the physical situation and arrive at a nonlinear maximum a posteriori probability map formulation of the problem an iterative approach which requires the solution of simple quadratic equations is proposed the authors also present a methodology which allows them to experimentally optimize an image reconstruction method for a specific medical task and to evaluate the relative efficacy of two reconstruction methods for a particular task in a manner which meets the high standards set by the methodology of statistical hypothesis testing the new map algorithm is compared to a method which maximizes likelihood and with two variants of the filtered backprojection method

tomographic image reconstruction by eigenvector decomposition its limitations and areas of applicability this paper analyzes in detail the process of tomographic image reconstruction by pseudo inversion of the blurring matrix of a pet imaging system eigenvector and eigenvalue decomposition is used as a method to evaluate the physical reasons for the ill conditioned nature of the problem it is shown that finding an accurate pseudo inverse for even a modest pet array of 8 8 pixels is a difficult task for a computer with 48 bit mantissa the problem is caused by the strong ambiguity with which the detector system measures the activity at each pixel for a problem in which imaging with a complete detector ring is not possible and in which invariance of the point response function cannot be maintained the pseudo inverse method of reconstruction is however shown to be very useful advantage is taken of the fact that the activitiy to be measured is localized in a single plane without over or underlying activity a planar camera configuration yields very well conditioned matrices that are separable for a large number of useful cases it is even possible to define pixel sizes which are considerably smaller than the detector size and solve the problem without a substantial increase in the noise magnification factor recognizing that the above application is equivalent to a case of very well defined time of flight tof measurement the simple initial pet study is reevaluated by inclusion of tof information

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

manifolds with s1 category 2 have cyclic fundamental groups a closed topological n manifold mn is of s1 category 2 if it can be covered by two open subsets w1 w2 such that the inclusions wi mn factor homotopically through maps wi s1 we show that for n 3 the fundamental group of such an n manifold is either trivial or infinite cyclic