tree approximations to markov random fields methods for approximately computing the marginal probability mass functions and means of a markov random field mrf by approximating the lattice by a tree are described applied to the a posteriori mrf these methods solve bayesian spatial pattern classification and image restoration problems the methods are described several theoretical results concerning fixed point problems are proven and four numerical examples are presented including comparison with optimal estimators and the iterated conditional mode estimator and including two agricultural optical remote sensing problems

a multiscale expectation maximization semisupervised classifier suitable for badly posed image classification this paper deals with the problem of badly posed image classification although underestimated in practice bad posedness is likely to affect many real world image classification tasks where reference samples are difficult to collect e g in remote sensing rs image mapping and or spatial autocorrelation is relevant in an image classification context affected by a lack of reference samples an original inductive learning multiscale image classifier termed multiscale semisupervised expectation maximization msem is proposed the rationale behind msem is to combine useful complementary properties of two alternative data mapping procedures recently published outside of image processing literature namely the multiscale modified pappas adaptive clustering mpac algorithm and the sample based semisupervised expectation maximization sem classifier to demonstrate its potential utility msem is compared against nonstandard classifiers such as mpac sem and the single scale contextual sem csem classifier besides against well known standard classifiers in two rs image classification problems featuring few reference samples and modestly useful texture information these experiments yield weak subjective but numerous quantitative map quality indexes that are consistent with both theoretical considerations and qualitative evaluations by expert photointerpreters according to these quantitative results msem is competitive in terms of overall image mapping performance at the cost of a computational overhead three to six times superior to that of its most interesting rival sem more in general our experiments confirm that even if they rely on heavy class conditional normal distribution assumptions that may not be true in many real world problems e g in highly textured images semisupervised classifiers based on the iterative expectation maximization gaussian mixture model solution can be very powerful in practice when 1 there is a lack of reference samples with respect to the problem model complexity and 2 texture information is considered negligible i e a piecewise constant image model holds

stochastic and deterministic networks for texture segmentation several texture segmentation algorithms based on deterministic and stochastic relaxation principles and their implementation on parallel networks are described the segmentation process is posed as an optimization problem and two different optimality criteria are considered the first criterion involves maximizing the posterior distribution of the intensity field given the label field maximum a posteriori estimate the posterior distribution of the texture labels is derived by modeling the textures as gauss markov random fields gmrfs and characterizing the distribution of different texture labels by a discrete multilevel markov model a stochastic learning algorithm is proposed this iterated hill climbing algorithm combines fast convergence of deterministic relaxation with the sustained exploration of the stochastic algorithms but is guaranteed to find only a local minimum the second optimality criterion requires minimizing the expected percentage of misclassification per pixel by maximizing the posterior marginal distribution and the maximum posterior marginal algorithm is used to obtain the corresponding solution all these methods implemented on parallel networks can be easily extended for hierarchical segmentation results of the various schemes in classifying some real textured images are presented

thermodynamics and statistical analysis of gaussian random fields gaussian fields are considered as gibbsian fields thermodynamic functions are calculated for them and the variational principle is proved as an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for gaussian markov fields

generalized icm for image segmentation based on tsallis statistics abstract in this paper the iterated conditional modes optimization method of a markov random field technique for image segmentation is generalized based on tsallis statistics it is observed that for some q entropic index values the new algorithm performs better segmentation than the classical one the proposed algorithm also does not have a local minimum problem and reaches a global minimum energy point although the number of iterations remains the same as icm based on the findings of the new algorithm it can be expressed that the new technique can be used for the image segmentation processes in which the objects are gaussian or nearly gaussian distributed

numerical conditioning problems and solutions for nonparametric i i d statistical active contours in this paper we propose an active contour model based on nonparametric independent and identically distributed i i d statistics of the image that can segment an image without any a priori information about the intensity distributions of the region of interest or the background this is not however the first active contour model proposed to solve the segmentation problem under these same assumptions in contrast to prior active contour models based on nonparametric i i d statistics we do not formulate our optimization criterion according to any distance measure between estimated probability densities inside and outside the active contour instead treating the segmentation problem as a pixel wise classification problem we formulate an active contour to minimize the unbiased pixel wise average misclassification probability amp this not only simplifies the problem by avoiding the need to arbitrarily select among many sensible distance measures to measure the difference between the probability densities estimated inside and outside the active contour but it also solves a numerical conditioning problem that arises with such prior active contour models as a result the amp model exhibits faster convergence with higher accuracy and robustness when compared to active contour models previously formulated to solve the same nonparametric i i d statistical segmentation problem via probability distances to discuss this improved numerical behavior more precisely we introduce the notion of x201c conditioning ratio x201d and demonstrate that the proposed amp active contour is numerically better conditioned i e exhibits a much smaller conditioning ratio than prior probability distance based active contours

a unified approach to the optimal synthesis of fixed point state space digital filters this paper proposes two expressions of the output error variance due to coefficient quantization of fixed point state space digital filters in the time domain one is the deterministic approach which gives precisely the output error variance of state space digital filters the other is the statistical approach where the errors of coefficient quantization are assumed to be independent random variables the statistical approach gives a simple and easy way to analyze the output error variance due to coefficient quantization of state space digital filters the statistical coefficient sensitivity introduced by the statistical approach is shown to be equivalent to the round off noise power gain thus state space digital filters which are optimal with respect to both coefficient sensitivity and roundoff noise can be synthesized by the method of minimization of the roundoff noise such optimal state space digital filters which could be of any order are proved to be free of autonomous overflow limit cycles a numerical example is given to illustrate the effectiveness of the analysis of the output error variance due to coefficient quantization and the synthesis method proposed here

segmentation of color textures this paper describes an approach to perceptual segmentation of color image textures a multiscale representation of the texture image generated by a multiband smoothing algorithm based on human psychophysical measurements of color appearance is used as the input initial segmentation is achieved by applying a clustering algorithm to the image at the coarsest level of smoothing the segmented clusters are then restructured in order to isolate core clusters i e patches in which the pixels are definitely associated with the same region the image pixels representing the core clusters are used to form 3d color histograms which are then used for probabilistic assignment of all other pixels to the core clusters to form larger clusters and categorise the rest of the image the process of setting up color histograms and probabilistic reassignment of the pixels to the clusters is then propagated through finer levels of smoothing until a full segmentation is achieved at the highest level of resolution

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

a systematic approach to synthesis of limit cycle free digital filters a systematic approach to synthesis of limit cycle free digital filters under constant input conditions is proposed structures of limit cycle free digital filters under constant input conditions are easily led from structures of limit cycle free digital filters under zero input conditions there exist various realizations which can suppress limit cycles for constant input signals

multiresolution gauss markov random field models for texture segmentation this paper presents multiresolution models for gauss markov random fields gmrfs with applications to texture segmentation coarser resolution sample fields are obtained by subsampling the sample field at fine resolution although the markov property is lost under such resolution transformation coarse resolution non markov random fields can be effectively approximated by markov fields we present two techniques to estimate the gmrf parameters at coarser resolutions from the fine resolution parameters one by minimizing the kullback leibler distance and another based on local conditional distribution invariance we also allude to the fact that different gmrf parameters at the fine resolution can result in the same probability measure after subsampling and present the results for first and second order cases we apply this multiresolution model to texture segmentation different texture regions in an image are modeled by gmrfs and the associated parameters are assumed to be known parameters at lower resolutions are estimated from the fine resolution parameters the coarsest resolution data is first segmented and the segmentation results are propagated upward to the finer resolution we use the iterated conditional mode icm minimization at all resolutions our experiments with synthetic brodatz texture and real satellite images show that the multiresolution technique results in a better segmentation and requires lesser computation than the single resolution algorithm

multiple resolution segmentation of textured images a multiple resolution algorithm is presented for segmenting images into regions with differing statistical behavior in addition an algorithm is developed for determining the number of statistically distinct regions in an image and estimating the parameters of those regions both algorithms use a causal gaussian autoregressive model to describe the mean variance and spatial correlation of the image textures together the algorithms can be used to perform unsupervised texture segmentation the multiple resolution segmentation algorithm first segments images at coarse resolution and then progresses to finer resolutions until individual pixels are classified this method results in accurate segmentations and requires significantly less computation than some previously known methods the field containing the classification of each pixel in the image is modeled as a markov random field segmentation at each resolution is then performed by maximizing the a posteriori probability of this field subject to the resolution constraint at each resolution the a posteriori probability is maximized by a deterministic greedy algorithm which iteratively chooses the classification of individual pixels or pixel blocks the unsupervised parameter estimation algorithm determines both the number of textures and their parameters by minimizing a global criterion based on the aic information criterion clusters corresponding to the individual textures are formed by alternately estimating the cluster parameters and repartitioning the data into those clusters concurrently the number of distinct textures is estimated by combining clusters until a minimum of the criterion is reached

turbo segmentation of textured images we consider the problem of semi supervised segmentation of textured images existing model based approaches model the intensity field of textured images as a gauss markov random field to take into account the local spatial dependencies between the pixels classical bayesian segmentation consists of also modeling the label field as a markov random field to ensure that neighboring pixels correspond to the same texture class with high probability well known relaxation techniques are available which find the optimal label field with respect to the maximum a posteriori or the maximum posterior mode criterion but these techniques are usually computationally intensive because they require a large number of iterations to converge in this paper we propose a new bayesian framework by modeling two dimensional textured images as the concatenation of two one dimensional hidden markov autoregressive models for the lines and the columns respectively a segmentation algorithm which is similar to turbo decoding in the context of error correcting codes is obtained based on a factor graph approach the proposed method estimates the unknown parameters using the expectation maximization algorithm

on the absence of limit cycles in a class of state space digital filters which contains minimum noise realizations this paper presents a theorem for the absence of limit cycles in a class of state space digital filters such a class contains minimum noise realizations of any order as a special case that is it is proved that the transition matrix a of minimum noise realizations of any order satisfies i a t a geq 0 and thus the realizations are free of limit cycles

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

image analysis using multigrid relaxation methods image analysis problems posed mathematically as variational principles or as partial differential equations are amenable to numerical solution by relaxation algorithms that are local iterative and often parallel although they are well suited structurally for implementation on massively parallel locally interconnected computational architectures such distributed algorithms are seriously handi capped by an inherent inefficiency at propagating constraints between widely separated processing elements hence they converge extremely slowly when confronted by the large representations of early vision application of multigrid methods can overcome this drawback as we showed in previous work on 3 d surface reconstruction in this paper we develop multiresolution iterative algorithms for computing lightness shape from shading and optical flow and we examine the efficiency of these algorithms using synthetic image inputs the multigrid methodology that we describe is broadly applicable in early vision notably it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill posed image analysis problems

texture classification using logical operators in this paper a new algorithm for texture classification based on logical operators is presented operators constructed from logical building blocks are convolved with texture images an optimal set of six operators are selected based on their texture discrimination ability the responses are then converted to standard deviation matrices computed over a sliding window zonal sampling features are computed from these matrices a feature selection process is applied and the new set of features are used for texture classification classification of several natural and synthetic texture images are presented demonstrating the excellent performance of the logical operator method the computational superiority and classification accuracy of the algorithm is demonstrated by comparison with other popular methods experiments with different classifiers and feature normalization are also presented the euclidean distance classifier is found to perform best with this algorithm the algorithm involves only convolutions and simple arithmetic in the various stages which allows faster implementations the algorithm is applicable to different types of classification problems which is demonstrated by segmentation of remote sensing images compressed and reconstructed images and industrial images

on hankel singular values and reflected zeros of linear dynamical systems this note discusses a relationship between the hankel singular values and reflected zeros of linear systems our main result proves that the hankel singular values of a linear continuous time system increase decrease pointwise when one or more zeros of the transfer function are reflected with respect to the imaginary axis that is move from the left right half to the right left half of the complex plane we also derive a similar result for linear discrete time systems

passive error concealment for wavelet coded i frames with an inhomogeneous gauss markov random field model in video communication over lossy packet networks e g the internet packet loss errors can severely damage the transmitted video the damaged video can largely be repaired with passive error concealment where neighboring information is used to estimate missing information we address the problem of passive error concealment for wavelet coded data with dispersive packetization the reported techniques of this kind have many problems and usually fail in the reconstruction of high frequency content this paper presents a novel locally adaptive error concealment method for subband coded i frames based on an inhomogeneous gaussian markov random field model we estimate the parameters of this model from a local context of each lost coefficient and we interpolate the lost coefficients accordingly the results demonstrate a significant improvement over the reported related methods both in terms of objective performance measures and visually the biggest improvement of the proposed method compared to the state of the art in the field is the correct reconstruction of high frequency information such as textures and edges

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

discrete markov image modeling and inference on the quadtree noncasual markov or energy based models are widely used in early vision applications for the representation of images in high dimensional inverse problems due to their noncausal nature these models generally lead to iterative inference algorithms that are computationally demanding in this paper we consider a special class of nonlinear markov models which allow one to circumvent this drawback these models are defined as discrete markov random fields mrf attached to the nodes of a quadtree the quadtree induces causality properties which enable the design of exact noniterative inference algorithms similar to those used in the context of markov chain models we first introduce an extension of the viterbi algorithm which enables exact maximum a posteriori map estimation on the quadtree two other algorithms related to the mpm criterion and to bouman and shapiro s 1994 sequential map smap estimator are derived on the same hierarchical structure the estimation of the model hyper parameters is also addressed two expectation maximization em type algorithms allowing unsupervised inference with these models are defined the practical relevance of the different models and inference algorithms is investigated in the context of image classification problem on both synthetic and natural images

multiscale representations of markov random fields recently a framework for multiscale stochastic modeling was introduced based on coarse to fine scale recursive dynamics defined on trees this model class has some attractive characteristics which lead to extremely efficient statistically optimal signal and image processing algorithms the authors show that this model class is also quite rich in particular they describe how 1 d markov processes and 2 d markov random fields mrfs can be represented within this framework the recursive structure of 1 d markov processes makes them simple to analyze and generally leads to computationally efficient algorithms for statistical inference on the other hand 2 d mrfs are well known to be very difficult to analyze due to their noncausal structure and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification in contrast their multiscale representations are based on scale recursive models and thus lead naturally to scale recursive algorithms which can be substantially more efficient computationally than those associated with mrf models in 1 d the multiscale representation is a generalization of the midpoint deflection construction of brownian motion the representation of 2 d mrfs is based on a further generalization to a midline deflection construction the exact representations of 2 d mrfs are used to motivate a class of multiscale approximate mrf models based on one dimensional wavelet transforms they demonstrate the use of these latter models in the context of texture representation and in particular they show how they can be used as approximations for or alternatives to well known mrf texture models

simple parallel hierarchical and relaxation algorithms for segmenting noncausal markovian random fields the modeling and segmentation of images by mrf s markov random fields is treated these are two dimensional noncausal markovian stochastic processes two conceptually new algorithms are presented for segmenting textured images into regions in each of which the data are modeled as one of c mrf s the algorithms are designed to operate in real time when implemented on new parallel computer architectures that can be built with present technology a doubly stochastic representation is used in image modeling here a gaussian mrf is used to model textures in visible light and infrared images and an autobinary or autoternary etc mrf to model a priori information about the local geometry of textured image regions for image segmentation the true texture class regions are treated either as a priori completely unknown or as a realization of a binary or ternary etc mrf in the former case image segmentation is realized as true maximum likelihood estimation in the latter case it is realized as true maximum a posteriori likelihood segmentation in addition to providing a mathematically correct means for introducing geometric structure the autobinary or ternary etc mrf can be used in a generative mode to generate image geometries and artificial images and such simulations constitute a very powerful tool for studying the effects of these models and the appropriate choice of model parameters the first segmentation algorithm is hierarchical and uses a pyramid like structure in new ways that exploit the mutual dependencies among disjoint pieces of a textured region

on the existence of isotropic forms of semi simple algebraic groups over number fields with prescribed local behavior this note is a follow up on the paper a borel g harder existence of discrete cocompact subgroups of reductive groups over local fields j reine angew math 298 1978 53 64 of a borel and g harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi simple group over a number field and considering a suitable s arithmetic subgroup some years ago a lubotzky initiated a program to study the subgroup growth of arithmetic subgroups the current stage of which focuses on counting more precisely determining the asymptotics of the number of lattices of bounded covolume the finiteness of this number was established in a borel g prasad finiteness theorems for discrete subgroups of bounded covolume in semi simple groups publ math inst hautes etudes sci 69 1989 119 171 addendum publ math inst hautes etudes sci 71 1990 173 177 using the formula for the covolume developed in g prasad volumes of s arithmetic quotients of semi simple groups publ math inst hautes etudes sci 69 1989 91 117 work on this program led m belolipetsky and a lubotzky to ask questions about the existence of isotropic forms of semi simple groups over number fields with prescribed local behavior in this paper we will answer these questions a question of similar nature also arose in the work d morris real representations of semisimple lie algebras have forms in proc internat conf on algebraic groups and arithmetic december 17 22 2001 tifr mumbai 2001 pp 469 490 of d morris witte on a completely different topic we will answer that question too

multiscale image segmentation using wavelet domain hidden markov models we introduce a new image texture segmentation algorithm hmtseg based on wavelets and the hidden markov tree hmt model the hmt is a tree structured probabilistic graph that captures the statistical properties of the coefficients of the wavelet transform since the hmt is particularly well suited to images containing singularities edges and ridges it provides a good classifier for distinguishing between textures utilizing the inherent tree structure of the wavelet hmt and its fast training and likelihood computation algorithms we perform texture classification at a range of different scales we then fuse these multiscale classifications using a bayesian probabilistic graph to obtain reliable final segmentations since hmtseg works on the wavelet transform of the image it can directly segment wavelet compressed images without the need for decompression into the space domain we demonstrate the performance of hmtseg with synthetic aerial photo and document image segmentations

reply to comment on parseval relationship of nonuniform samples of one and two dimensional signals the author presents a reply to the paper by zhu and leung ieee trans signal processing vol 42 no 3 p 2183 1994 which commented on the paper by marvasti and liu ieee trans acoust speech signal processing vol 38 no 6 p 1061 3 1990 the present author discusses a typographical error in the paper by marvasti and liu it is concluded that the theorem proofs are still correct and the conclusion still valid

parallel and deterministic algorithms from mrfs surface reconstruction deterministic approximations to markov random field mrf models are derived one of the models is shown to give in a natural way the graduated nonconvexity gnc algorithm proposed by a blake and a zisserman 1987 this model can be applied to smooth a field preserving its discontinuities a class of more complex models is then proposed in order to deal with a variety of vision problems all the theoretical results are obtained in the framework of statistical mechanics and mean field techniques a parallel iterative algorithm to solve the deterministic equations of the two models is presented together with some experiments on synthetic and real images

scaling properties of image textures a detrending fluctuation analysis approach abstract the aim of this paper is to explore the application of detrended fluctuation analysis dfa to study roughness features of images unidimensional sequences at different image orientations are extracted and their average scaling exponent is estimated in this form the existence of anisotropies can be detected when considerable variations in the scaling exponent at different image orientation are observed different images from grass to solar granulation are analyzed and the underlying physics of such results is briefly commented

a model based approach for estimation of two dimensional maximum entropy power spectra a stochastic model based approach is presented for estimation of the two dimensional maximum entropy power spectrum meps from given finite uniform array data the method consists of fitting an appropriate two dimensional noncausal gaussian markov random field gmrf model to the given data using the maximum likelihood ml technique for parameter estimation the nonlinear criterion function used for ml estimation is similar in structure to the function arising in the deterministic approach of lang and mcclellan the model based approach provides new insights into the two dimensional meps estimation problem for example using the asymptotic normality of ml estimates we derive simultaneous confidence bands for the estimated meps it turns out that when the true correlations are generated by a noncausal gmrf model the two dimensional meps can be obtained by solving linear equations this approach also suggests techniques for realizing two dimensional gmrf models from the given correlation data several numerical examples are given to illustrate the usefulness of the approach

a wavelet based multiresolution statistical model for texture a multiresolution statistical model consisting of random fields in wavelet subbands is proposed for texture and has produced promising results in texture synthesis experiments the basic idea here is that a complex random field model e g one that contains long range and nonlinear spatial correlations can be constructed from several simpler ones

combining belief networks and neural networks for scene segmentation we are concerned with the problem of image segmentation in which each pixel is assigned to one of a predefined finite number of labels in bayesian image analysis this requires fusing together local predictions for the class labels with a prior model of label images following the work of bouman and shapiro 1994 we consider the use of tree structured belief networks tsbns as prior models the parameters in the tsbn are trained using a maximum likelihood objective function with the em algorithm and the resulting model is evaluated by calculating how efficiently it codes label images a number of authors have used gaussian mixture models to connect the label field to the image data we compare this approach to the scaled likelihood method of smyth 1994 and morgan and bourlard 1995 where local predictions of pixel classification from neural networks are fused with the tsbn prior our results show a higher performance is obtained with the neural networks we evaluate the classification results obtained and emphasize not only the maximum a posteriori segmentation but also the uncertainty as evidenced e g by the pixelwise posterior marginal entropies we also investigate the use of conditional maximum likelihood training for the tsbn and find that this gives rise to improved classification performance over the ml trained tsbn