simplicity and superrigidity of twin building lattices kac moody groups over finite fields are finitely generated groups most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non positively curved twinned buildings those are the most important but not the only examples of twin building lattices we prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type i e they are not bruhat tits buildings many of them are finitely presented and enjoy property t our arguments explain geometrically why simplicity fails to hold only for affine kac moody groups moreover we prove that a nontrivial continuous homomorphism from a completed kac moody group is always proper we also show that kac moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non positively curved metric spaces most results apply to the general class of twin building lattices

application of the em algorithm for the multitarget multisensor tracking problem an important problem in surveillance and reconnaissance systems is the tracking of multiple moving targets in cluttered noise environments using outputs from a number of sensors possessing wide variations in individual characteristics and accuracies a number of approaches have been proposed for this multitarget multisensor tracking problem ranging from reasonably simple though ad hoc schemes to fairly complex but theoretically optimum approaches in this paper we describe an iterative procedure for time recursive multitarget multisensor tracking based on use of the expectation maximization em algorithm more specifically we pose the multitarget multisensor tracking problem as an incomplete data problem with the observable sensor outputs representing the incomplete data whereas the target associated sensor outputs constitute the complete data target updates at each time use an em based approach that calculates the maximum a posteriori map estimate of the target states under the assumption of appropriate motion models based on the outputs of disparate sensors the approach uses a markov random field mrf model of the associations between observations and targets and allows for estimation of joint association probabilities without explicit enumeration the advantage of this em based approach is that it provides a computationally efficient means for approaching the performance offered by theoretically optimum approaches that use explicit enumeration of the joint association probabilities we provide selected results illustrating the performance complexity characteristics of this em based approach compared with competing schemes

maximum likelihood parameter estimation for unsupervised stochastic model based image segmentation an unsupervised stochastic model based approach to image segmentation is described and some of its properties investigated in this approach the problem of model parameter estimation is formulated as a problem of parameter estimation from incomplete data and the expectation maximization em algorithm is used to determine a maximum likelihood ml estimate previously the use of the em algorithm in this application has encountered difficulties since an analytical expression for the conditional expectations required in the em procedure is generally unavailable except for the simplest models in this paper two solutions are proposed to solve this problem a monte carlo scheme and a scheme related to besag s 1986 iterated conditional mode icm method both schemes make use of markov random field modeling assumptions examples are provided to illustrate the implementation of the em algorithm for several general classes of image models experimental results on both synthetic and real images are provided

estimation and hypothesis testing in finite mixture models summary finite mixture models are a useful class of models for application to data when sample sizes are not large and the number of underlying densities is in question likelihood ratio tests based on joint maximum likelihood estimation of the mixing parameter x and the parameter of the underlying densities 0 are problematical our approach places a prior distribution on x and estimates 0 by maximizing the likelihood of the data given 0 with x integrated out advantages of this approach computational issues using the em algorithm and directions for further work are discussed the technique is applied to two examples

butler theory over murley groups one of the most widely studied classes of torsion free abelian groups offinite rank is the class of classical butler groups a butler group can beviewed either as a pure subgroup of a completely decomposable group offinite rank or an epimorphic image of such a group excellent sources forthe basic properties of butler groups are 9 11 12 of arnold andwxvinsonhaler vinsonhaler has also suggested an investigation of possibleextensions to the theory of butler groups to pure subgroups of

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

integration of bayes detection with target tracking existing detection systems generally are operated using a fixed threshold and optimized to the neyman pearson criterion an alternative is bayes detection in which the threshold varies according to the ratio of prior probabilities in a recursive target tracker such as the probabilistic data association filter pdaf such priors are available in the form of a predicted location and associated covariance however the information is not at present made available to the detector put another way in a standard detection tracking implementation information flows only one way from detector to tracker here we explore the idea of two way information flow in which the tracker instructs the detector where to look for a target and the detector returns what it has found more specifically we show that the bayesian detection threshold is lowered in the vicinity of the predicted measurement and we explain the appropriate modification to the pdaf the implementation is simple and the performance is remarkably good

the mean field theory in em procedures for markov random fields in many signal processing and pattern recognition applications the hidden data are modeled as markov processes and the main difficulty of using the maximisation em algorithm for these applications is the calculation of the conditional expectations of the hidden markov processes it is shown how the mean field theory from statistical mechanics can be used to calculate the conditional expectations for these problems efficiently the efficacy of the mean field theory approach is demonstrated on parameter estimation for one dimensional mixture data and two dimensional unsupervised stochastic model based image segmentation experimental results indicate that in the 1 d case the mean field theory approach provides results comparable to those obtained by baum s 1987 algorithm which is known to be optimal in the 2 d case where baum s algorithm can no longer be used the mean field theory provides good parameter estimates and image segmentation for both synthetic and real world images

codistances of 3 spherical buildings we show that a 3 spherical building in which each rank 2 residue is connected far away from a chamber and each rank 3 residue is simply 2 connected far away from a chamber admits a twinning i e is one half of a twin building as soon as it admits a codistance i e a twinning with a single chamber

curtis tits groups generalizing kac moody groups of type a n 1 abstract in 13 we define a curtis tits group as a certain generalization of a kac moody group we distinguish between orientable and non orientable curtis tits groups and identify all orientable curtis tits groups as kac moody groups associated to twin buildings in the present paper we construct all orientable as well as non orientable curtis tits groups with diagram a n 1 n 4 over a field k of size at least 4 the resulting groups are quite interesting in their own right the orientable ones are related to drinfeld s construction of vector bundles over a non commutative projective line and to the classical groups over cyclic algebras the non orientable ones are related to expander graphs 14 and have symplectic orthogonal and unitary groups as quotients

on use of the em algorithm for penalized likelihood estimation summary the em algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation this paper discusses properties of the em algorithm in such contexts concentrating on rates of conver gence and presents an alternative that is usually more practical and converges at least as quickly the em algorithm is a general approach to maximum likelihood estimation rather than a specific algorithm dempster et al 1977 discussed the method and derived basic properties demonstrating that a variety of procedures previously developed rather informally could be unified the common strand to problems where the approach is applicable is a notion of incomplete data this includes the conventional sense of missing data but is much broader than that the em algorithm demon strates its strength in situations where some hypothetical experiment yields data from which estimation is particularly convenient and economical the incomplete data actually at hand are regarded as observable functions of these complete data the resulting algorithms while usually slow to converge are often extremely simple and remain practical in large problems where no other approaches may be feasible dempster et al 1977 briefly refer to the use of the same approach to the problem of finding the posterior mode maximum a posteriori estimate in a bayesian estima

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

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

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

1 twinnings of buildings in this paper we characterize twinnings of buildings by 1 twinnings and one further condition concerning twin apartments specialized to the spherical case we obtain new characterizations of the opposition relation in such buildings we also give a new description of the standard twin building of type widetilde a n 1

a generalized em algorithm for 3 d bayesian reconstruction from poisson data using gibbs priors a generalized expectation maximization gem algorithm is developed for bayesian reconstruction based on locally correlated markov random field priors in the form of gibbs functions and on the poisson data model for the m step of the algorithm a form of coordinate gradient ascent is derived the algorithm reduces to the em maximum likelihood algorithm as the markov random field prior tends towards a uniform distribution three different gibbs function priors are examined reconstructions of 3 d images obtained from the poisson model of single photon emission computed tomography are presented