distinct variants of the kdv equation with compact and noncompact structures in this paper we examine three distinct variants of the generalized kdv equation both the compact and the noncompact structures are treated for these variants we also emphasize the different physical structures of the two classes the focusing branch and the defocusing branch

patterns on liquid surfaces cnoidal waves compactons and scaling abstract localized patterns and nonlinear oscillation formations on the bounded free surface of an ideal incompressible liquid are investigated cnoidal modes solitons and compactons as traveling non axially symmetric shapes are discussed a finite difference differential generalized korteweg de vries kdv equation is shown to describe the three dimensional motion of the fluid surface and in the limit of long and shallow channels one recovers the well known kdv equation a tentative expansion formula for the representation of the general solution of a nonlinear equation for given initial conditions is introduced the model is useful in multilayer fluid dynamics cluster formation and nuclear physics since up to an overall scale these systems display a free liquid surface behavior

multiscale smoothing error models a class of multiscale stochastic models based on scale recursive dynamics on trees has recently been introduced these models are interesting because they can be used to represent a broad class of physical phenomena and because they lead to efficient algorithms for estimation and likelihood calculation in this paper we provide a complete statistical characterization of the error associated with smoothed estimates of the multiscale stochastic processes described by these models in particular we show that the smoothing error is itself a multiscale stochastic process with parameters that can be explicitly calculated

a subrate weighted minimum variance control law the paper describes the development of a weighted minimum variance control law with application to single input single output systems in which the control is applied at a slower rate than the available measurements of the plant output signal the theory is extended to incorporate dynamic compensation in the controller and to encompass generalized system models a time domain modelling approach is employed whereby the system dynamics and sampling operations are defined as a series of cyclically time varying difference equations the technique in which the cost functions penalize both the control and a set of output signals generates straightforward expressions for the closed loop characteristic polynomial and is accordant with pole placement synthesis methods the control law suffers from similar difficulties with respect to non minimum phase systems as its single rate counterpart

varying complexity in tree structured image distribution models probabilistic models of image statistics underlie many approaches in image analysis and processing an important class of such models have variables whose dependency graph is a tree if the hidden variables take values on a finite set most computations with the model can be performed exactly including the likelihood calculation training with the em algorithm etc crouse et al developed one such model the hidden markov tree hmt they took particular care to limit the complexity of their model we argue that it is beneficial to allow more complex tree structured models describe the use of information theoretic penalties to choose the model complexity and present experimental results to support these proposals for these experiments we use what we call the hierarchical image probability hip model the differences between the hip and the hmt models include the use of multivariate gaussians to model the distributions of local vectors of wavelet coefficients and the use of different numbers of hidden states at each resolution we demonstrate the broad utility of image distributions by applying the hip model to classification synthesis and compression across a variety of image types namely electrooptical synthetic aperture radar and mammograms digitized x rays in all cases we compare with the hmt

adaptive segmentation of speckled images using a hierarchical random field model a two level hierarchical random field model developed for speckled images and in particular for synthetic aperture radar sar imagery is described at the higher level is a gibbs random field governing the grouping of image pixels into regions while at the lower level are speckle processes for the different regions which are also modeled as random fields in accordance with the physical phenomena that cause speckle the speckle processes are modeled as circularly symmetric complex gaussian random fields as with real gaussian fields certain forms of autocovariance of complex gaussian field lead to markovianity properties with the assumption of a separable autocovariance for the complex gaussian random fields local joint statistics of the resulting speckle intensity fields magnitude squared of the complex field are determined the speckle model concurs with the known marginal statistics and also accounts for the autocorrelation observed in actual speckled images a map segmentation using simulated annealing and based on the hierarchical model is presented the algorithm is adaptive in that it recursively segments the image and estimates the model parameters necessary for the segmentation

multiscale autoregressive models and wavelets the multiscale autoregressive mar framework was introduced to support the development of optimal multiscale statistical signal processing its power resides in the fast and flexible algorithms to which it leads while the mar framework was originally motivated by wavelets the link between these two worlds has been previously established only in the simple case of the haar wavelet the first contribution of this paper is to provide a unification of the mar framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem the second contribution of this paper is to develop wavelet based approximate internal mar models for stochastic processes this will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients taking advantage of the statistical machinery provided by the mar framework we will illustrate the application of our models to sample path generation and estimation from noisy irregular and sparse measurements

multivariable multirate sampled data systems state space description transfer characteristics and nyquist criterion first a discrete time state space description is derived for multivariable multirate sampled data systems then the multirate impulse modulation mim and the symmetric coordinate multirate impulse modulation smim of a transfer matrix are defined and their relation to the discrete time state space description is clarified based on the above results four stability criteria are derived the contents of the paper will be useful to extend the modern results about continuous time and single rate sampled data control systems to the multirate sampled data case especially extensions of the ina method and of the robust stability conditions will be straightforward

stability analysis of solitary wave solutions for the fourth order nonlinear boussinesq water wave equation abstract in the present study the nonlinear boussinesq type equation describe the bi directional propagation of small amplitude long capillary gravity waves on the surface of shallow water by using the extended auxiliary equation method we obtained some new soliton like solutions for the two dimensional fourth order nonlinear boussinesq equation with constant coefficient these solutions include symmetrical non symmetrical kink solutions solitary pattern solutions jacobi and weierstrass elliptic function solutions and triangular function solutions the stability analysis for these solutions are discussed

multirate digital control design of an optimal regulator via singular perturbation theory abstract in this paper a multirate digital control design of an optimal regulator is investigated via singular perturbation theory it is shown that the singularity perturbed continuous time regulator leads under slow and fast sampling rates to two different discrete time versions they are decomposed into slow and fast subsystems and then these solutions are combined in a proper way within the framework of such a decomposition coordination principle a multirate control design is developed naturally furthermore the problem of the asymptotic stability of a multirate controlled system is investigated and the relationship between the original continuous time version and the multirate controlled version is discussed

the tanh and the sine cosine methods for compact and noncompact solutions of the nonlinear klein gordon equation the nonlinear klein gordon equation is used as a vehicle to employ the tanh method and the sine cosine method to formally derive a number of travelling wave solutions the study features a variety of solutions with distinct physical structures the work shows that one method complements the other and each method gives solutions of formal properties the obtained solutions include compactons solitons solitary patterns and periodic solutions

ml parameter estimation of a multiscale stochastic process using the em algorithm an algorithm for estimation of the parameters of a multiscale stochastic process based on scale recursive dynamics on trees is presented the expectation maximization algorithm is used to provide maximum likelihood estimates for the general case of a nonhomogeneous tree with no fixed structure for the process dynamics experimental results are presented using synthetic data

compactons and solitary patterns structures for variants of the kdv and the kp equations in this paper we discuss mathematical variants in higher dimensions of the kdv and the kp equations it is shown that the focusing branches of these variants exhibit compactons solitons with finite wavelength whereas the defocusing branches support solitary patterns solutions with infinite slopes or cusps the study presents a fairly complete understanding of the compact and noncompact dispersive structures

a generalized levinson algorithm for covariance extension with application to multiscale autoregressive modeling efficient computation of extensions of banded partially known covariance matrices is provided by the classical levinson algorithm one contribution of this paper is the introduction of a generalization of this algorithm that is applicable to a substantially broader class of extension problems this generalized algorithm can compute unknown covariance elements in any order that satisfies certain graph theoretic properties which we describe this flexibility which is not provided by the classical levinson algorithm is then harnessed in a second contribution of this paper the identification of a multiscale autoregressive mar model for the maximum entropy me extension of a banded partially known covariance matrix the computational complexity of mar model identification is an order of magnitude below that of explicitly computing a full covariance extension and is comparable to that required to build a standard autoregressive ar model using the classical levinson algorithm

solitary wave solutions of nonlinear evolution and wave equations using a direct method and macsyma the direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized the class of solitary wave solutions is extended to analytic rather than rational functions of the real exponential solutions of the linearized equation expanding the solutions in an infinite series in these real exponentials an exact solution of the nonlinear pde is obtained whenever the series can be summed methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed the algorithm is now suited to solving nonlinear equations by any symbolic manipulation program this direct method is illustrated by constructing exact solutions of a generalized kdv equation the kuramoto sivashinski equation and a generalized fisher equation

likelihood calculation for a class of multiscale stochastic models with application to texture discrimination a class of multiscale stochastic models based on scale recursive dynamics on trees has previously been introduced theoretical and experimental results have shown that these models provide an extremely rich framework for representing both processes which are intrinsically multiscale e g 1 f processes as well as 1d markov processes and 2d markov random fields moreover efficient optimal estimation algorithms have been developed for these models by exploiting their scale recursive structure the authors exploit this structure in order to develop a computationally efficient and parallelizable algorithm for likelihood calculation they illustrate one possible application to texture discrimination and demonstrate that likelihood based methods using the algorithm achieve performance comparable to that of gaussian markov random field based techniques which in general are prohibitively complex computationally

digital image restoration using spatial interaction models this paper is concerned with developing fast nonrecursive algorithms for the minimum mean squared error restoration of degraded images the degradation is assumed to be due to a space invariant periodic nonseparable known point spread function and additive white noise our basic approach is to represent the images by a class of spatial interaction models namely the simultaneous autoregressve models and the conditional markov models defined on toroidal lattices and develop minimum mean squared error restoration algorithms using these models the restoration algorithms are optimal if the parameters characterizing the interaction models are exactly known however in practice the parameters are estimated from the images by using spatial interaction models we develop restoration algorithms that do not require the availability of the original image or its prototype the specific structure of the underlying lattice enables the implementation of the filters using fast fourier transform fft computations several restoration examples are given

an analysis of sub rate control systems the purpose of this paper is to outline a polynomial approach to modelling and analysing multi rate control systems a certain class of system in which the data input to a digital controller are sampled at a faster rate than the data output is chosen to illustrate the technique a set of difference equations defining the propagation of variables during the repetitive time interval engenders a system description whereby the corresponding operations are defined as polynomial functions of the backward shift operator q 1 this representation circumvents difficulties associated with the z n transform approach engendering simple methods of assessing closed loop stability and assigning poles finally attention is focused upon system performance in the presence of deterministic and stochastic input signals an example is employed throughout the paper to illustrate developments in the theory

classification of textures using gaussian markov random fields the problem of texture classification arises in several disciplines such as remote sensing computer vision and image analysis in this paper we present two feature extraction methods for the classification of textures using two dimensional 2 d markov random field mrf models it is assumed that the given m m texture is generated by a gaussian mrf model in the first method the least square ls estimates of model parameters are used as features in the second method using the notion of sufficient statistics it is shown that the sample correlations over a symmetric window including the origin are optimal features for classification simple minimum distance classifiers using these two feature sets yield good classification accuracies for a seven class problem

noniterative manipulation of discrete energy based models for image analysis abstract with emphasis on the graph structure of energy based models devoted to image analysis we investigate efficient procedures for sampling and inferring we show that triangulated graphs whom trees are simple instances of always support causal models for which noniterative procedures can be devised to minimize the energy to extract probabilistic descriptions to sample from corresponding prior and posterior distributions or to infer from local marginals the relevance and efficiency of these procedures are illustrated for classification problems

classification of rotated and scaled textured images using gaussian markov random field models consideration is given to the problem of classifying a test textured image that is obtained from one of c possible parent texture classes after possibly applying unknown rotation and scale changes to the parent texture the training texture images parent classes are modeled by gaussian markov random fields gmrfs to classify a rotated and scaled test texture the rotation and scale changes are incorporated in the texture model through an appropriate transformation of the power spectral density of the gmrf for the rotated and scaled image a bona fide likelihood function that shows the explicit dependence of the likelihood function on the gmrf parameters as well as on the rotation and scale parameters is derived although in general the scaled and or rotated texture does not correspond to a finite order gmrf it is possible nonetheless to write down a likelihood function for the image data the likelihood function of the discrete fourier transform of the image data corresponds to that of a white nonstationary gaussian random field with the variance at each pixel i j being a known function of the rotation the scale the gmrf model parameters and i j the variance is an explicit function of the appropriately sampled power spectral density of the gmrf the estimation of the rotation and scale parameters is performed in the frequency domain by maximizing the likelihood function associated with the discrete fourier transform of the image data cramer rao error bounds on the scale and rotation estimates are easily computed a modified bayes decision rule is used to classify a given test image into one of c possible texture classes the classification power of the method is demonstrated through experimental results on natural textures from the brodatz album

fractal estimation using models on multiscale trees we estimate the hurst parameter h of fractional brownian motion or by extension the fractal exponent spl phi of stochastic processes having 1 f sup spl phi like spectra by applying a multiresolution framework this framework admits an efficient likelihood function evaluation allowing us to compute the maximum likelihood estimate of this fractal parameter with relative ease in addition to yielding results that compare well with other proposed methods and in contrast with other approaches our method is directly applicable with at most very simple modification in a variety of other contexts including fractal estimation given irregularly sampled data or nonstationary measurement noise and the estimation of fractal parameters for 2 d random fields

general solutions with solitary patterns for the defocusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this work the defocusing branch of the genuinely nonlinear dispersive k n n equation that exhibits solutions with solitary patterns is investigated the equation with constant coefficients to the derivatives is examined in one two and three dimensional spaces general formulas for solutions with solitary patterns are developed for all positive integers n n 1 two sets of entirely new formulas one set produces solutions for even integers n whereas the second set produces solutions in opposite directions for odd integers n n 1 are established

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

a new optimal multirate control of linear periodic and time invariant systems the optimal multirate design of linear continuous time periodic and time invariant systems is considered it is based on solving the continuous linear quadratic regulation lqr problem with the control being constrained to a certain piecewise constant feedback necessary and sufficient conditions for the asymptotic stability of the resulting closed loop system are given an explicit multirate feedback law that requires the solution of an algebraic discrete riccati equation is presented such control is simple and can be easily implemented by digital computers when applied to linear time invariant systems multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly compared to the classical single rate sampled data feedback in which the state is always sampled at the same rate the multirate system can provide a better response with a considerable reduction in the optimal cost in general the multirate scheme offers more flexibility in choosing the sampling rates

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

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

the modeling and estimation of statistically self similar processes in a multiresolution framework statistically self similar sss processes can be used to describe a variety of physical phenomena yet modeling these phenomena has proved challenging most of the proposed models for sss and approximately sss processes have power spectra that behave as 1 f sup spl gamma such as fractional brownian motion fbm fractionally differenced noise and wavelet based syntheses the most flexible framework is perhaps that based on wavelets which provides a powerful tool for the synthesis and estimation of 1 f processes but assumes a particular distribution of the measurements an alternative framework is the class of multiresolution processes proposed by chou et al 1994 which has already been shown to be useful for the identification of the parameters of fbm these multiresolution processes are defined by an autoregression in scale that makes them naturally suited to the representation of sss and approximately sss phenomena both stationary and nonstationary also this multiresolution framework is accompanied by an efficient estimator likelihood calculator and conditional simulator that make no assumptions about the distribution of the measurements we show how to use the multiscale framework to represent sss or approximately sss processes such as fbm and fractionally differenced gaussian noise the multiscale models are realized by using canonical correlations cc and by exploiting the self similarity and possible stationarity or stationary increments of the desired process a number of examples are provided to demonstrate the utility of the multiscale framework in simulating and estimating sss processes

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a canonical correlations approach to multiscale stochastic realization we develop a realization theory for a class of multiscale stochastic processes having white noise driven scale recursive dynamics that are indexed by the nodes of a tree given the correlation structure of a 1 d or 2 d random process our methods provide a systematic way to realize the given correlation as the finest scale of a multiscale process motivated by akaike s use of canonical correlation analysis to develop both exact and reduced order models for time series we too harness this tool to develop multiscale models we apply our realization scheme to build reduced order multiscale models for two applications namely linear least squares estimation and generation of random field sample paths for the numerical examples considered least squares estimates are obtained having nearly optimal mean square errors even with multiscale models of low order although both field estimates and field sample paths exhibit a visually distracting blockiness this blockiness is not an important issue in many applications for such applications our approach to multiscale stochastic realization holds promise as a valuable general tool

an overlapping tree approach to multiscale stochastic modeling and estimation recently a class of multiscale stochastic models has been introduced in which random processes and fields are described by scale recursive dynamic trees a major advantage of this framework is that it leads to an extremely efficient statistically optimal algorithm for least squares estimation in certain applications however estimates based on the types of multiscale models previously proposed may not be adequate as they have tended to exhibit a visually distracting blockiness we eliminate this blockiness by discarding the standard assumption that distinct nodes on a given level of the multiscale process correspond to disjoint portions of the image domain instead we allow a correspondence to overlapping portions of the image domain we use these so called overlapping tree models for both modeling and estimation in particular we develop an efficient multiscale algorithm for generating sample paths of a random field whose second order statistics match a prespecified covariance structure to any desired degree of fidelity furthermore we demonstrate that under easily satisfied conditions we can lift a random field estimation problem to one defined on an overlapped tree resulting in an estimation algorithm that is computationally efficient directly produces estimation error covariances and eliminates blockiness in the reconstructed imagery without any sacrifice in the resolution of fine scale detail

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

variants of the generalized kdv equation with compact and noncompact structures abstract in this paper nonlinear dispersive variants of the generalized kdv equation are investigated the compact and the noncompact physical structures are studied two general formulas for the focusing branch and the defocusing branch that are of substantial interest are formally derived

sympletic finite difference approximations of the nonlinear klein gordon equation we analyze three finite difference approximations of the nonlinear klein gordon equation and show that they are directly related to symplectic mappings two of the schemes the perring skyrme and ablowitz kruskal ladik are long established and the third is a new higher order accurate scheme we test the schemes on traveling wave and periodic breather problems over long time intervals and compare their accuracy and computational costs with those of symplectic and nonsymplectic method of lines approximations and a nonsymplectic energy conserving method

a construction of compact and noncompact solutions for nonlinear dispersive equations of even order a construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered two model equations of even orders k 4 are adopted to carry out this study the analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions

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

segmentation of textured images using a multiresolution gaussian autoregressive model we present a new algorithm for segmentation of textured images using a multiresolution bayesian approach the new algorithm uses a multiresolution gaussian autoregressive mgar model for the pyramid representation of the observed image and assumes a multiscale markov random field model for the class label pyramid the models used in this paper incorporate correlations between different levels of both the observed image pyramid and the class label pyramid the criterion used for segmentation is the minimization of the expected value of the number of misclassified nodes in the multiresolution lattice the estimate which satisfies this criterion is referred to as the multiresolution maximization of the posterior marginals mmpm estimate and is a natural extension of the single resolution maximization of the posterior marginals mpm estimate previous multiresolution segmentation techniques have been based on the maximum a posterior map estimation criterion which has been shown to be less appropriate for segmentation than the mpm criterion it is assumed that the number of distinct textures in the observed image is known the parameters of the mgar model the means prediction coefficients and prediction error variances of the different textures are unknown a modified version of the expectation maximization em algorithm is used to estimate these parameters the parameters of the gibbs distribution for the label pyramid are assumed to be known experimental results demonstrating the performance of the algorithm are presented

a study on nonlinear dispersive partial differential equations of compact and noncompact solutions in this paper we study two genuinely nonlinear dispersive partial differential equations in one and higher dimensional spaces we show that the focusing branch and the defocusing branch of these equations each leads to a different physical structure in addition we develop general solutions of compact and noncompact support which are not obtainable through classic techniques

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations

unsupervised texture segmentation using markov random field models the problem of unsupervised segmentation of textured images is considered the only explicit assumption made is that the intensity data can be modeled by a gauss markov random field gmrf the image is divided into a number of nonoverlapping regions and the gmrf parameters are computed from each of these regions a simple clustering method is used to merge these regions the parameters of the model estimated from the clustered segments are then used in two different schemes one being all approximation to the maximum a posterior estimate of the labels and the other minimizing the percentage misclassification error the proposed approach is contrasted with the algorithm of s lakshamanan and h derin 1989 which uses a simultaneous parameter estimation and segmentation scheme the results of the adaptive segmentation algorithm of lakshamanan and derin are compared with a simple nearest neighbor classification scheme to show that if enough information is available simple techniques could be used as alternatives to computationally expensive schemes