knot concordance whitney towers and l 2 signatures we construct many examples of nonslice knots in 3 space that cannot be distinguished from slice knots by previously known invariants using whitney towers in place of embedded disks we define a geometric filtration of the 3 dimensional topological knot concordance group the bottom part of the filtration exhibits all classical concordance invariants including the cassongordon invariants as a first step we construct an infinite sequence of new obstructions that vanish on slice knots these take values in the l theory of skew fields associated to certain universal groups finally we use the dimension theory of von neumann algebras to define an l 2 signature and use this to detect the first unknown step in our obstruction theory

approximation of nonnegative systems by moving averages of fixed order abstract we pose the approximation problem for scalar nonnegative input output systems via impulse response convolutions of finite order i e finite order moving averages based on repeated observations of input output signal pairs the problem is converted into a nonnegative matrix factorization with special structure for which we use csiszar s i divergence as the criterion of optimality conditions are given on the input output data that guarantee the existence and uniqueness of the minimum we propose an algorithm of the alternating minimization type for i divergence minimization and present its asymptotic behaviour for the case of noisy observations we give the large sample properties of the statistical version of the minimization problem for different observation regimes numerical experiments confirm the asymptotic results and exhibit fast convergence of the proposed algorithm

convergence of em image reconstruction algorithms with gibbs smoothing p j green has defined an osl one step late algorithm that retains the e step of the em algorithm for image reconstruction in emission tomography but provides an approximate solution to the m step further modifications of the osl algorithm guarantee convergence to the unique maximum of the log posterior function convergence is proved under a specific set of sufficient conditions several of these conditions concern the potential function of the gibb s prior and a number of candidate potential functions are identified generalization of the osl algorithm to transmission tomography is also considered

iterative image reconstruction algorithms based on cross entropy minimization the related problems of minimizing the functionals f x alpha kl y px 1 alpha kl p x and g x alpha kl px y 1 alpha kl x p respectively over the set of vectors x or 0 are considered kl a b is the cross entropy or kullback leibler distance between two nonnegative vectors a and b iterative algorithms for minimizing both functionals using the method of alternating projections are derived a simultaneous version of the multiplicative algebraic reconstruction technique mart algorithm called smart is introduced and its convergence is proved

globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms we present two types of globally convergent relaxed ordered subsets os algorithms for penalized likelihood image reconstruction in emission tomography modified block sequential regularized expectation maximization bsrem and relaxed os separable paraboloidal surrogates os sps the global convergence proof of the existing bsrem de pierro and yamagishi 2001 required a few a posteriori assumptions by modifying the scaling functions of bsrem we are able to prove the convergence of the modified bsrem under realistic assumptions our modification also makes stepsize selection more convenient in addition we introduce relaxation into the os sps algorithm erdogan and fessler 1999 that otherwise would converge to a limit cycle we prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed os sps is a special case main results of the proofs are from nedic and bertsekas 2001 and correa and lemarechal 1993 simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms

approximation of nonnegative systems by finite impulse response convolutions we pose the deterministic nonparametric approximation problem for scalar nonnegative input output systems via finite impulse response convolutions based on repeated observations of input output signal pairs the problem is converted into a nonnegative matrix factorization with special structure for which we use csiszar s i divergence as the criterion of optimality conditions are given on the input output data that guarantee the existence and uniqueness of the minimum we propose an algorithm of the alternating minimization type for i divergence minimization and study its asymptotic behavior for the case of noisy observations we give the large sample properties of the statistical version of the minimization problem numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm

a row action alternative to the em algorithm for maximizing likelihood in emission tomography the maximum likelihood ml approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography ect since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection fbp algorithm the expectation maximization em algorithm is an often used iterative approach for maximizing the poisson likelihood in ect because of its attractive theoretical and practical properties its major disadvantage is that due to its slow rate of convergence a large amount of computation is often required to achieve an acceptable image here the authors present a row action maximum likelihood algorithm ramla as an alternative to the em algorithm for maximizing the poisson likelihood in ect the authors deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of ramla increase the poisson likelihood in ect at an order of magnitude faster that the standard em algorithm specifically the authors show that from the point of view of measuring total radionuclide uptake in simulated brain phantoms iterations 1 2 3 and 4 of ramla perform at least as well as iterations 45 60 70 and 80 respectively of em moreover the authors show that iterations 1 2 3 and 4 of ramla achieve comparable likelihood values as iterations 45 60 70 and 80 respectively of em the authors also present a modified version of a recent fast ordered subsets em os em algorithm and show that ramla is a special case of this modified os em furthermore the authors show that their modification converges to a ml solution whereas the standard os em does not

deblurring subject to nonnegativity constraints csiszar s i divergence is used as a discrepancy measure for deblurring subject to the constraint that all functions involved are nonnegative an iterative algorithm is proposed for minimizing this measure it is shown that every function in the sequence is nonnegative and the sequence converges monotonically to a global minimum other properties of the algorithm are shown including lower bounds on the improvement in the i divergence at each step of the algorithm and on the difference between the i difference at step k and at the limit point a method for regularizing the solution is proposed

statistical projection completion in x ray ct using consistency conditions projection data incompleteness arises in many situations relevant to x ray computed tomography ct imaging we propose a penalized maximum likelihood statistical sinogram restoration approach that incorporates the helgason ludwig hl consistency conditions to accommodate projection data incompleteness image reconstruction is performed by the filtered backprojection fbp in a second step in our problem formulation the objective function consists of the log likelihood of the x ray ct data and a penalty term the hl condition poses a linear constraint on the restored sinogram and can be implemented efficiently via fast fourier transform fft and inverse fft we derive an iterative algorithm that increases the objective function monotonically the proposed algorithm is applied to both computer simulated data and real patient data we study different factors in the problem formulation that affect the properties of the final fbp reconstructed images including the data truncation level the amount of prior knowledge on the object support as well as different approximations of the statistical distribution of the available projection data we also compare its performance with an analytical truncation artifacts reduction method the proposed method greatly improves both the accuracy and the precision of the reconstructed images within the scan field of view and to a certain extent recovers the truncated peripheral region of the object the proposed method may also be applied in areas such as limited angle tomography metal artifacts reduction and sparse sampling imaging

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

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

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

penalized weighted least squares image reconstruction for dual energy x ray transmission tomography presents a dual energy de transmission computed tomography ct reconstruction method it is statistically motivated and features nonnegativity constraints in the density domain a penalized weighted least squares pwls objective function has been chosen to handle the non poisson noise added by amorphous silicon asi h detectors a gauss seidel algorithm has been used to minimize the objective function the behavior of the method in terms of bias standard deviation tradeoff has been compared to that of a de method that is based on filtered back projection fbp the advantages of the de pwls method are largest for high noise and or low flux cases qualitative results suggest this as well also the reconstructed images of an object with opaque regions are presented possible applications of the method are attenuation correction for positron emission tomography pet images various quantitative computed tomography qct methods such as bone mineral densitometry bmd and the removal of metal streak artifacts

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

l2 signatures homology localization and amenable groups aimed at geometric applications we prove the homology cobordism invariance of the l2 betti numbers and l2 signature defects associated to the class of amenable groups lying in strebel s class d r which includes some interesting infinite finite non torsion free groups this result includes the only prior known condition that is a poly torsion free abelian group or a finite p group we define a new commutator series that refines harvey s torsion free derived series of groups using the localizations of groups and rings of bousfield vogel and cohn the series called the local derived series has versions for homology with arbitrary coefficients and satisfies functoriality and an injectivity theorem we combine these two new tools to give some applications to distinct homology cobordism types within the same simple homotopy type in higher dimensions to concordance of knots in three manifolds and to spherical space forms in dimension 3 2012 wiley periodicals inc

information theoretic image formation the emergent role of information theory in image formation is surveyed unlike the subject of information theoretic communication theory information theoretic imaging is far from a mature subject the possible role of information theory in problems of image formation is to provide a rigorous framework for defining the imaging problem for defining measures of optimality used to form estimates of images for addressing issues associated with the development of algorithms based on these optimality criteria and for quantifying the quality of the approximations the definition of the imaging problem consists of an appropriate model for the data and an appropriate model for the reproduction space which is the space within which image estimates take values each problem statement has an associated optimality criterion that measures the overall quality of an estimate the optimality criteria include maximizing the likelihood function and minimizing mean squared error for stochastic problems and minimizing squared error and discrimination for deterministic problems the development of algorithms is closely tied to the definition of the imaging problem and the associated optimality criterion algorithms with a strong information theoretic motivation are obtained by the method of expectation maximization related alternating minimization algorithms are discussed in quantifying the quality of approximations global and local measures are discussed global measures include the mean squared error and discrimination between an estimate and the truth and probability of error for recognition or hypothesis testing problems local measures include fisher information

roughness penalties on finite domains a class of penalty functions for use in estimation and image regularization is proposed these penalty functions are defined for vectors whose indexes are locations in a finite lattice as the discrepancy between the vector and a shifted version of itself after motivating this class of penalty functions their relationship to markov random field priors is explored one of the penalty functions proposed a divergence roughness penalty is shown to be a discretization of a penalty proposed by good and gaskins 1971 for use in density estimation one potential use in estimation problems is explored an iterative algorithm that takes advantage of induced neighborhood structures is proposed and convergence of the algorithm is proven under specified conditions examples in emission tomographic imaging and radar imaging are given

monotonic algorithms for transmission tomography we present a framework for designing fast and monotonic algorithms for transmission tomography penalized likelihood image reconstruction the new algorithms are based on paraboloidal surrogate functions for the log likelihood due to the form of the log likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity unlike previous methods the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events such as scatter and random coincidences the gradient and the curvature of the likelihood terms are evaluated only once per iteration since the problem is simplified at each iteration the cpu time is less than that of current algorithms which directly minimize the objective yet the convergence rate is comparable the simplicity monotonicity and speed of the new algorithms are quite attractive the convergence rates of the algorithms are demonstrated using real and simulated pet transmission scans

whitney towers and gropes in 4 manifolds many open problems and important theorems in low dimensional topology have been formulated as statements about certain 2 complexes called gropes this paper describes a precise correspondence between embedded gropes in 4 manifolds and the failure of the whitney move in terms of iterated towers of whitney disks the flexibility of these whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to n solvability k cobordism and grope concordance the key observation is that the essential structure of gropes and whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and whitney moves it is shown that a whitney move in a whitney tower induces an ihx jacobi relation on the embedded trees

higher order intersections in low dimensional topology we show how to measure the failure of the whitney move in dimension 4 by constructing higher order intersection invariants of whitney towers built from iterated whitney disks on immersed surfaces in 4 manifolds for whitney towers on immersed disks in the 4 ball we identify some of these new invariants with previously known link invariants such as milnor sato levine and arf invariants we also define higher order sato levine and arf invariants and show that these invariants detect the obstructions to framing a twisted whitney tower together with milnor invariants these higher order invariants are shown to classify the existence of twisted whitney towers of increasing order in the 4 ball a conjecture regarding the nontriviality of the higher order arf invariants is formulated and related implications for filtrations of string links and 3 dimensional homology cylinders are described

a topological approach to cheeger gromov universal bounds for von neumann ρ invariants using deep analytic methods cheeger and gromov showed that for any smooth 4k 1 manifold there is a universal bound for the von neumann l2 invariants associated to arbitrary regular covers we present a proof of the existence of a universal bound for topological 4k 1 manifolds using l2 signatures of bounding 4k manifolds we give explicit linear universal bounds for 3 manifolds in terms of triangulations heegaard splittings and surgery descriptions we show that our explicit bounds are asymptotically optimal as an application we give new lower bounds of the complexity of 3 manifolds that can be arbitrarily larger than previously known lower bounds as ingredients of the proofs that seem interesting on their own we develop a geometric construction of efficient 4 dimensional bordisms of 3 manifolds over a group and develop an algebraic topological notion of uniformly controlled chain homotopies 2015 wiley periodicals inc

globally convergent algorithms for maximum a posteriori transmission tomography this paper reviews and compares three maximum likelihood algorithms for transmission tomography one of these algorithms is the em algorithm one is based on a convexity argument devised by de pierro see ieee trans med imaging vol 12 p 328 333 1993 in the context of emission tomography and one is an ad hoc gradient algorithm the algorithms enjoy desirable local and global convergence properties and combine gracefully with bayesian smoothing priors preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the em algorithm this superiority stems from the larger number of exponentiations required by the em algorithm the convex and gradient algorithms are well adapted to parallel computing

line integral alternating minimization algorithm for dual energy x ray ct image reconstruction we propose a new algorithm called line integral alternating minimization liam for dual energy x ray ct image reconstruction instead of obtaining component images by minimizing the discrepancy between the data and the mean estimates liam allows for a tunable discrepancy between the basis material projections and the basis sinograms a parameter is introduced that controls the size of this discrepancy and with this parameter the new algorithm can continuously go from a two step approach to the joint estimation approach liam alternates between iteratively updating the line integrals of the component images and reconstruction of the component images using an image iterative deblurring algorithm an edge preserving penalty function can be incorporated in the iterative deblurring step to decrease the roughness in component images images from both simulated and experimentally acquired sinograms from a clinical scanner were reconstructed by liam while varying the regularization parameters to identify good choices the results from the dual energy alternating minimization algorithm applied to the same data were used for comparison using a small fraction of the computation time of dual energy alternating minimization liam achieves better accuracy of the component images in the presence of poisson noise for simulated data reconstruction and achieves the same level of accuracy for real data reconstruction

on the relation between the isra and the em algorithm for positron emission tomography the image space reconstruction algorithm isra was proposed as a modification of the expectation maximization em algorithm based on physical considerations for application in volume emission computered tomography as a consequence of this modification isra searches for least squares solutions instead of maximizing poisson likelihoods as the em algorithm it is shown that both algorithms may be obtained from a common mathematical framework this fact is used to extend isra for penalized likelihood estimates

higher order alexander invariants and filtrations of the knot concordance group we establish certain nontriviality results for several filtrations of the smooth and topological knot concordance groups first as regards the n solvable filtration of the topological knot concordance group c defined by k orr p teichner and the first author 0 f n 5 f n f 1 5 f 1 0 f 0 5 f 0 c we refine the recent nontriviality results of cochran and teichner by including information on the alexander modules these results also extend those of c livingston and the second author we exhibit similar structure in the closely related symmetric grope filtration ofc we also show that the grope filtration of the smooth concordance group is nontrivial using examples that cannot be distinguished by the ozsvath szabo invariant nor by j rasmussen s s invariant our broader contribution is to establish in the relative case the key homological results whose analogues cochran orr teichner established in the absolute case we say two knots k 0 and k 1 are concordant modulo n solvability if k o k 1 f n our main result is that for any knot k whose classical alexander polynomial has degree greater than 2 and for any positive integer n there exist infinitely many knots k i that are concordant to k modulo n solvability but are all distinct modulo n 5 solvability moreover the k i and k share the same classical seifert matrix and alexander module as well as sharing the same higher order alexander modules and seifert presentations up to order n 1

pose estimation of known objects during transmission tomographic image reconstruction we address the problem of image formation in transmission tomography when metal objects of known composition and shape but unknown pose are present in the scan subject using an alternating minimization am algorithm derived from a model in which the detected data are viewed as poisson distributed photon counts we seek to eliminate the streaking artifacts commonly seen in filtered back projection images containing high contrast objects we show that this algorithm which minimizes the i divergence or equivalently maximizes the log likelihood between the measured data and model based estimates of the means of the data converges much faster when knowledge of the high density materials such as brachytherapy applicators or prosthetic implants is exploited the algorithm incorporates a steepest descent based method to find the position and orientation collectively called the pose of the known objects this pose is then used to constrain the image pixels to their known attenuation values or for example to form a mask on the missing projection data in the shadow of the objects results from two dimensional simulations are shown in this paper the extension of the model and methods used to three dimensions is outlined

model based iterative reconstruction for dual energy x ray ct using a joint quadratic likelihood model dual energy x ray ct dect has the potential to improve contrast and reduce artifacts as compared to traditional ct moreover by applying model based iterative reconstruction mbir to dual energy data one might also expect to reduce noise and improve resolution however the direct implementation of dual energy mbir requires the use of a nonlinear forward model which increases both complexity and computation alternatively simplified forward models have been used which treat the material decomposed channels separately but these approaches do not fully account for the statistical dependencies in the channels in this paper we present a method for joint dual energy mbir jde mbir which simplifies the forward model while still accounting for the complete statistical dependency in the material decomposed sinogram components the jde mbir approach works by using a quadratic approximation to the polychromatic log likelihood and a simple but exact nonnegativity constraint in the image domain we demonstrate that our method is particularly effective when the dect system uses fast kvp switching since in this case the model accounts for the inaccuracy of interpolated sinogram entries both phantom and clinical results show that the proposed model produces images that compare favorably in quality to previous decomposition based methods including fbp and other statistical iterative approaches

a modified expectation maximization algorithm for penalized likelihood estimation in emission tomography the maximum likelihood ml expectation maximization em approach in emission tomography has been very popular in medical imaging for several years in spite of this no satisfactory convergent modifications have been proposed for the regularized approach here a modification of the em algorithm is presented the new method is a natural extension of the em for maximizing likelihood with concave priors convergence proofs are given

statistical interior tomography this paper presents a statistical interior tomography sit approach making use of compressed sensing cs theory with the projection data modeled by the poisson distribution an objective function with a total variation tv regularization term is formulated in the maximization of a posteriori map framework to solve the interior problem an alternating minimization method is used to optimize the objective function with an initial image from the direct inversion of the truncated hilbert transform the proposed sit approach is extensively evaluated with both numerical and real datasets the results demonstrate that sit is robust with respect to data noise and down sampling and has better resolution and less bias than its deterministic counterpart in the case of low count data

alternating minimization algorithms for transmission tomography a family of alternating minimization algorithms for finding maximum likelihood estimates of attenuation functions in transmission x ray tomography is described the model from which the algorithms are derived includes polyenergetic photon spectra background events and nonideal point spread functions the maximum likelihood image reconstruction problem is reformulated as a double minimization of the i divergence a novel application of the convex decomposition lemma results in an alternating minimization algorithm that monotonically decreases the objective function each step of the minimization is in closed form the family of algorithms includes variations that use ordered subset techniques for increasing the speed of convergence simulations demonstrate the ability to correct the cupping artifact due to beam hardening and the ability to reduce streaking artifacts that arise from beam hardening and background events

knot concordance and von neumann rho invariants we present new results announced in t on the classical knot concordance group c we establish the nontriviality at all levels of the n solvable filtration fn f1 f0 c introduced in cot1 recall that this filtration is significant due to its intimate connection to tower constructions arising in work of a casson and m freedman on the topological classification problem for 4 manifolds and due to the fact that all previously known concordance invariants are reflected in the first few terms in the filtration in cot1 nontriviality at the first new level n 3 was established here we prove the nontriviality of the filtration for all n hence giving the ultimate justification to the theory a broad range of techniques is employed in our proof including cut and paste topology and analytical estimates we use the cheeger gromov estimate for von neumann invariants a deep analytic result we also introduce a number of new algebraic arguments involving noncommutative localization and blanchfield forms we have attempted to make this article accessible to readers with only passing knowledge of cot1