noncausal gauss markov random fields parameter structure and estimation the parameter structure of noncausal homogeneous gauss markov random fields gmrf defined on finite lattices is studied for first order nearest neighbor and a special class of second order fields a complete characterization of the parameter space and a fast implementation of the maximum likelihood estimator of the field parameters are provided for general higher order fields tight bounds for the parameter space are presented and an efficient procedure for ml estimation is described experimental results illustrate the application of the approach presented and the viability of the present method in fitting noncausal models to 2 d data

texture discrimination based upon an assumed stochastic texture model a new approach to texture discrimination is described this approach is based upon an assumed stochastic model for texture in imagery and is an approximation to the statistically optimum maximum likelihood classifier the construction and properties of the stochastic texture model are described and a digital filtering implementation of the resulting maximum likelihood texture discriminant is provided the efficacy of this approach is demonstrated through experimental results obtained with simulated texture data a comparison is provided with more conventional texture discriminants under identical conditions the implications to texture discrimination in realworld imagery are discussed

some kinds of view the fuzzy filters of bl algebras the concepts of view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras are introduced and some related properties are investigated some characterizations of these generalized fuzzy filters are derived in particular we describe the relationships among ordinary fuzzy implicative positive implicative and fantastic filters view the mathml source fuzzy implicative positive implicative and fantastic filters and view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras finally we prove that a fuzzy set f of a bl algebra l is an view the mathml source fuzzy implicative filter of l if and only if it is both an view the mathml source fuzzy positive implicative filter and an view the mathml source fuzzy fantastic filter

two dimensional spectral estimation in this paper effective methods for generating two dimensional quarter plane causal autoregressive ar and autoregressive moving average arma spectral estimation models are developed these procedures are found to provide super resolution capabilities when compared to other more classical methods such as the fourier transform the arma method involves manipulation of the model equation sum min k 0 max p 1 sum min k 0 max p 2 a km x n 1 k n 2 m sum min k 0 max q 1 sum min k 0 max q 2 b km epsilon n 1 k n 2 m and utilizes the given finite set of observations x n 1 n 2 for 1 leq n 1 leq n 1 1 leq n 2 leq n 2 in the above relationship the random excitation epsilon n 1 n 2 is taken to be white this arma model s autoregressive a km coefficients are selected to minimize a weighted least squares criterion composed of error elements while the moving average b km coefficients are obtained using an alternative approach the spectral estimation performance of the ar and arma methods will be empirically demonstrated by considering the problem of resolving two sinusoids embedded in noise

recursive estimation of images using non gaussian autoregressive models we consider recursive estimation of images modeled by non gaussian autoregressive ar models and corrupted by spatially white gaussian noise the goal is to find a recursive algorithm to compute a near minimum mean square error mmse estimate of each pixel of the scene using a fixed lookahead of d rows and d columns of the observations our method is based on a simple approximation that makes possible the development of a useful suboptimal nonlinear estimator the algorithm is first developed for a non gaussian ar time series and then generalized to two dimensions in the process we draw on the well known reduced update kalman filter kf technique of woods and radewan 1977 to circumvent computational load problems several examples demonstrate the non gaussian nature of residuals for ar image models and that our algorithm compares favorably with the kalman filtering techniques in such cases

estimation of generalized mixtures and its application in image segmentation we introduce the notion of a generalized mixture and propose some methods for estimating it along with applications to unsupervised statistical image segmentation a distribution mixture is said to be generalized when the exact nature of the components is not known but each belongs to a finite known set of families of distributions for instance we can consider a mixture of three distributions each being exponential or gaussian the problem of estimating such a mixture contains thus a new difficulty we have to label each of three components there are eight possibilities we show that the classical mixture estimation algorithms expectation maximization em stochastic em sem and iterative conditional estimation ice can be adapted to such situations once as we dispose of a method of recognition of each component separately that is when we know that a sample proceeds from one family of the set considered we have a decision rule for what family it belongs to considering the pearson system which is a set of eight families the decision rule above is defined by the use of skewness and kurtosis the different algorithms so obtained are then applied to the problem of unsupervised bayesian image segmentation we propose the adaptive versions of sem em and ice in the case of blind i e pixel by pixel segmentation global segmentation methods require modeling by hidden random markov fields and we propose adaptations of two traditional parameter estimation algorithms gibbsian em gem and ice allowing the estimation of generalized mixtures corresponding to pearson s system the efficiency of different methods is compared via numerical studies and the results of unsupervised segmentation of three real radar images by different methods are presented

new types of fuzzy filters of bl algebras this paper introduces the concepts of a b fuzzy implicative positive implicative fantastic filters and b a fuzzy implicative positive implicative fantastic filters of bl algebras where a b c q d c q d c q d a b c q d c q d c q d a c q d and b c q d and some related properties are investigated special attention is paid to c c q d fuzzy implicative positive implicative fantastic filters and c c q d fuzzy implicative positive implicative fantastic filters which are generalizations of q fuzzy implicative positive implicative fantastic filters and q fuzzy implicative positive implicative fantastic filters studied in ma et al 2008 2009 34 35 it also points out that examples 3 1 2 3 1 9 3 2 2 3 2 9 and 3 3 9 in ma et al 2009 35 are incorrect

algebraic and topological results on lifting properties in residuated lattices we define lifting properties for universal algebras which we study in this general context and then particularize to various such properties in certain classes of algebras next we focus on residuated lattices in which we investigate lifting properties for boolean and idempotent elements modulo arbitrary as well as specific kinds of filters we give topological characterizations to the lifting property for boolean elements and several properties related to it many of which we obtain by means of the reticulation

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

an efficient algorithm for two dimensional autoregressive spectrum estimation an algorithm similar to the burg algorithm is given for autoregressive estimation in one dimension coefficients of the filter used to decorrelate the signal are found recursively from lower order filters the coefficients needed to get the larger sized filters can be found at each stage of the recursion from the correlations of lower order filter outputs thus the resulting estimate does not assume knowledge of ensemble averages it offers a computational saving of 25 over the commonly used method in which an estimate of the correlation matrix is first computed and then inverted

markov modeling for bayesian restoration of two dimensional layered structures bayesian estimation of two dimensional stratified structures is described the major point addressed is the derivation of a statistical prior model that adequately describes such layered media this problem is of interest in applications in which the data are generally processed in one dimension only in order to take local interactions into account a markovian description is used the model is derived so as to fulfill a set of constraints that summarize physical and geometrical characteristics of the problem as well as practical requirements the resulting class of markov random fields presents a unilateral structure on a nonrectangular lattice and a hierarchical organization which involves a line process in addition it is shown to be an extension of one dimensional models already in use the properties of the model are investigated and its practicality is demonstrated by an application to seismic deconvolution simulation results show significant improvements with respect to the usual one dimensional methods

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

rbfn restoration of nonlinearly degraded images we investigate a technique for image restoration using nonlinear networks based on radial basis functions the technique is also based on the concept of training or learning by examples when trained properly these networks are used as spatially invariant feedforward nonlinear filters that can perform restoration of images degraded by nonlinear degradation mechanisms we examine a number of network structures including the gaussian radial basis function network rbfn and some extensions of it as well as a number of training algorithms including the stochastic gradient sg algorithm that we have proposed earlier we also propose a modified structure based on the gaussian mixture model and a learning algorithm for the modified network experimental results indicate that the radial basis function network and its extensions can be very useful in restoring images degraded by nonlinear distortion and noise

boolean products of bl algebras the representation of algebras by boolean products is a very general problem in universal algebra in this paper we shall characterize the boolean products of bl chains the weak boolean products of local bl algebras and the weak boolean products of perfect bl algebras

extreme values and high boundary crossings of locally stationary gaussian processes we consider the large values of a locally stationary gaussian process which satisfies berman s condition on the long range dependence we present some limit results on the exceedances of the process above a certain general smooth high boundary this allows deriving the limiting distribution of the maximum up to time t for example in the case of a standardized process with a constant boundary or in the case of a nonstandardized process with a smooth trend

spatial classification using fuzzy membership models in the usual statistical approach to spatial classification it is assumed that each pixel belongs to precisely one of a small number of known groups this framework is extended to include mixed pixel data then only a proportion of each pixel belongs to each group two models based on multivariate gaussian random fields are proposed to model this fuzzy membership process the problems of predicting the group membership and estimating the parameters are discussed some simulations are presented to study the properties of this approach and an example is given using landsat remote sensing data

two dimensional linear prediction models part i spectral factorization and realization in this paper we present several results for three different canonical forms of linear prediction on a plane these filters have causal semicausal and noncausal prediction geometries starting from their properties we consider the problem of realization of these filters from a given power spectral density function sdf since it is not possible in general to obtain rational spectral factors of a two dimensional sdf we propose algorithms for obtaining rational approximations which are stable and converge to their limit irrational factors as the order of approximation is increased it is also shown that the normal equations associated with the minimum variance two dimensional prediction filters give a useful algorithm for obtaining rational approximations which are stable and converge to their unique limit filters this result allows design of finite order stable filters by solving a finite number of equations while realizing the given sdf arbitrarily closely

some types of generalized fuzzy filters of bl algebras the aim of this paper is to introduce the notions of interval valued q fuzzy boolean implicative positive implicative and fantastic filters in bl algebras and to investigate some related properties some characterizations of these generalized fuzzy filters are derived finally the relations among these generalized fuzzy filters are discussed it is proved that an interval valued q fuzzy filter in bl algebras is an interval valued q fuzzy boolean implicative filter if and only if it is both an interval valued q fuzzy positive implicative filter and an interval valued q fuzzy fantastic filter

extremes of locally stationary gaussian random fields the main result of this contribution is the derivation of the exact asymptotic behavior of the supremum of a class of a t locally stationary gaussian random fields we present two applications of our result the first one deals with the extremes of aggregate multifractional brownian motions whereas the second one establishes the exact asymptotics of the supremum of the x process generated by multifractional brownian motions

robust image modeling techniques with an image restoration application a robust parameter estimation algorithm for a nonsymmetric half plane nshp autoregressive model where the driving noise is a mixture of a gaussian and an outlier process is presented the convergence of the estimation algorithm is proved an algorithm to estimate parameters and original image intensity simultaneously from the impulse noise corrupted image where the model governing the image is not available is also presented the robustness of the parameter estimates is demonstrated by simulation finally an algorithm to restore realistic images is presented the entire image generally does not obey a simple image model but a small portion e g 8 8 of the image is assumed to obey an nshp model the original image is divided into windows and the robust estimation algorithm is applied for each window the restoration algorithm is tested by comparing it to traditional methods on several different images

simultaneous confidence bands for autoregressive spectra summary a shortcoming in the use of autoregressive spectral estimation has been the lack of methods for placing confidence intervals or bands on the estimated spectral density we propose and illustrate a method for determining asymptotic simultaneous confidence bands using scheffe projections

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

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

on a parameter estimation method for gibbs markov random fields this correspondence is about a gibbs markov random field gmrf parameter estimation technique proposed by derin and elliott 1987 we will refer to this technique as the histogramming h method first the relation of the h method to the conditional maximum likelihood method is considered second a bias reduction based modification of the h method is proposed to improve its performance especially in the case of small amounts of image data

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

bayes smoothing algorithms for segmentation of binary images modeled by markov random fields a new image segmentation algorithm is presented based on recursive bayes smoothing of images modeled by markov random fields and corrupted by independent additive noise the bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel given the total noisy image in a recursive way the a posteriori distribution together with a criterion of optimality then determine a bayes estimate of the scene the algorithm presented is an extension of a 1 d bayes smoothing algorithm to 2 d and it gives the optimum bayes estimate for the scene value at each pixel computational concerns in 2 d however necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm in particular the scene noiseless image is modeled as a markov mesh random field a special class of markov random fields and the bayes smoothing algorithm is applied on overlapping strips horizontal vertical of the image consisting of several rows columns it is assumed that the signal scene values vector sequence along the strip is a vector markov chain since signal correlation in one of the dimensions is not fully used along the edges of the strip estimates are generated only along the middle sections of the strips the overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image the bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel discrete or continuous random variables

characterization and estimation of two dimensional arma models a class of finite order two dimensional autoregressive moving average arma is introduced that can represent any process with rational spectral density in this model the driving noise is correlated and need not be gaussian currently known classes of arma models or ar models are shown to be subsets of the above class the three definitions of markov property are discussed and the class of arma models are precisely stated which have the noncausal and semicausal markov property without imposing any specific boundary conditions next two approaches are considered to estimate the parameters of a model to fit a given image the first method uses only the empirical correlations and involves the solution of linear equations the second method is the likelihood approach since the exact likelihood function is difficult to compute we resort to approximations suggested by the toroidal models numerical experiments compare the quality of the two estimation schemes finally the problem of synthesizing a texture obeying an arma model is considered

maximum likelihood blind image separation using nonsymmetrical half plane markov random fields this paper presents a maximum likelihood approach for blindly separating linear instantaneous mixtures of images the spatial autocorrelation within each image is described using nonsymmetrical half plane nshp markov random fields in order to simplify the joint probability density functions of the source images a first implementation assuming stationary sources is presented it is then extended to a more realistic nonstationary image model two approaches respectively based on blocking and kernel smoothing are proposed to cope with the nonstationarity of the images the estimation of the mixing matrix is performed using an iterative equivariant version of the newton raphson algorithm moreover score functions required for the computation of the updating rule are approximated at each iteration by parametric polynomial estimators results achieved with artificial mixtures of both artificial and real world images including an astrophysical application clearly prove the high performance of our methods as compared to classical algorithms

iterative wiener filters for image restoration the iterative wiener filter which successively uses the wiener filtered signal as an improved prototype to update the covariance estimates is investigated the convergence properties of this iterative filter are analyzed it has been shown that this iterative process converges to a signal which does not correspond to the minimum mean squared error solution based on the analysis an alternate iterative filter is proposed to correct for the convergence error the theoretical performance of the filter has been shown to give minimum mean squared error in practical implementation when there is unavoidable error in the covariance computation the filter may still result in undesirable restoration its performance has been investigated and a number of experiments in a practical setting were conducted to demonstrate its effectiveness

extremes of α t locally stationary gaussian processes with non constant variances abstract with motivation from 9 in this paper we derive the exact tail asymptotics of t locally stationary gaussian processes with non constant variance functions we show that some certain variance functions lead to qualitatively new results

two dimensional spectrum estimation using noncausal autoregressive models two dimensional 2 d spectrum estimation from raw data is of interest in signal and image processing a parametric technique for spectrum estimation using 2 d noncausal autoregressive ncar models is given the ncar models characterize the statistical dependency of the observation at location s on its neighbors in all directions this modeling assumption reduces the spectrum estimation problem to two subproblems the choice of appropriate structure of the ncar model and the estimation of parameters in ncar models by assuming that the true structure of the ncar model is known we first analyze the existence and uniqueness of gaussian maximum likelihood gml estimates of ncar model parameters due to the noncausal nature of the models the computation of gml estimates is burdensome by assuming specific boundary conditions computationally tractable expressions are obtained for the likelihood function expressions for the asymptotic covariance matrix of the gml estimates as well as the simultaneous confidence bands for the estimated spectrum using gml estimates are derived finally the usefulness of the method is illustrated by computer simulation results

two dimensional robust spectrum estimation robust estimation is studied of two dimensional 2 d power spectra of signals which are adequately represented by gaussian random field models but for which there are imperfect observations two situations of particular interest occur when the contaminating noise is additive and when the contaminating noise appears in the innovations in these cases the observed data are not gaussian and conventional procedures are no longer efficient to estimate the parameters of the signal model from the contaminated data two procedures are described which were originally proposed for estimation of scale and location from independent data and adapted to one dimensional autoregression parameter estimation by previous researchers the first algorithm is a robustification of least squares and equivalent to an iterated weighted least squares problem where the weights are data dependent the second algorithm is an iterative procedure known as a filter cleaner experiments using the robust procedures with synthetic data are reported and the results compared to a conventional method of model based spectrum estimation

extremes of randomly scaled gumbel risks we investigate the product y1y2 of two independent positive risks y1 and y2 if y1 has distribution in the gumbel max domain of attraction with some auxiliary function which is regularly varying at infinity and y2 is bounded then we show that y1y2 has also distribution in the gumbel max domain of attraction if both y1 y2 have log weibullian or weibullian tail behaviour we prove that y1y2 has log weibullian or weibullian asymptotic tail behaviour respectively we present here three theoretical applications concerned with a the limit of point wise maxima of randomly scaled gaussian processes b extremes of gaussian processes over random intervals and c the tail of supremum of iterated processes

cluster expansions for the deterministic computation of bayesian estimators based on markov random fields we describe a family of approximations denoted by cluster approximations for the computation of the mean of a markov random field mrf this is a key computation in image processing when applied to the a posteriori mrf the approximation is to account exactly for only spatially local interactions application of the approximation requires the solution of a nonlinear multivariable fixed point equation for which we prove several existence uniqueness and convergence of algorithm results four numerical examples are presented including comparison with monte carlo calculations

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

compound gauss markov random fields for image estimation algorithms for obtaining approximations to statistically optimal estimates for images modeled as compound gauss markov random fields are discussed the authors consider both the maximum a posteriori probability map estimate and the minimum mean squared error mmse estimate for image estimation and image restoration compound image models consist of several submodels having different characteristics along with an underlying structure model which govern transitions between these image submodels two different compound random field models are employed the doubly stochastic gaussian dsg random field and a compound gauss markov cgm random field the authors present map estimators for dsg and cgm random fields using simulated annealing a fast converging algorithm called deterministic relaxation which however converges to only a locally optimal map estimate is also presented as an alternative for reducing computational loading on sequential machines for comparison purposes the authors include results on the fixed lag smoothing mmse estimator for the dsg field and its suboptimal m algorithm approximation

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

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

modeling 2 d ar processes with various regions of support we show that there exists a causal 2 d linear process in the nonsymmetric half plane having the same autocorrelations as a noncausal 2 d linear process in the whole plane this property is called the autocorrelation equivalence relation and can be used for practical fitting and modeling of 2 d processes some causal 2 d autoregressive ar models with various regions of support are considered such as half cross half diamond quarter plane square half square half hexagon half octagon and half circle considerations of parsimony in 2 d model fitting are then focused not only on the number of parameters in our model but also most importantly on the optimal shape of the region of support their 2 d yule walker equations are derived and a computationally efficient order recursive algorithm is proposed to solve them the autocorrelation equivalence relation and the order recursive algorithm are utilized to specify a noncausal 2 d ar process as well as its spectrum from a given realization of a random field

estimation of linear parametric models of nongaussian discrete random fields with application to texture synthesis a general possibly asymmetric noncausal and or nonminimum phase 2d autoregressive moving average random field model driven by an independent and identically distributed 2d nongaussian sequence is considered the model is restricted to be invertible i e system zeros are not allowed to lie on the unit bicircle three performance criteria are investigated for parameter estimation of the system parameters given only the output measurements image pixels the proposed criteria are functions of the higher order cumulant statistics of an inverse filter output one of these criteria is novel and the others have been considered in past only for moving average inverses and without any analysis of their consistency in the paper strong consistency of the proposed methods under the assumption that the system order is known is proved the convergence of the proposed parameter estimators under overparametrization is also analyzed experimental results involving synthesized as well as real life textures are presented to illustrate the performance of two of the considered approaches experimental results of synthesis of 128 spl times 128 textures visually resembling several real life textures in the brodatz album and other sources are presented

analysis of identified 2 d noncausal models there are two approaches to the identification of noncausal autoregressive systems in two dimensions differing in the assumed noise model for both approaches the maximum likelihood estimator formulated in the frequency domain is presented the fisher information matrix is evaluated and found to be the sum of a block toeplitz and a block hankel matrix the variance of the parameters however cannot be used for comparison of the two approaches so the variance in the frequency domain is evaluated assuming that the true system in each case can be described by a model of that type possibly high order in particular the variance of the spectrum estimate is derived if the number of parameters tends to infinity it is shown that the two approaches give the same spectrum estimate variance the question of which set of true spectra can be described by the respective approaches is discussed

texture synthesis using 2 d noncausal autoregressive models the purpose of this paper is to illustrate the usefulness of two dimensional noncausal autoregressive ncar models for the synthesis of textures these models characterize the gray level at a pixel as a linear combination of gray levels at nearby locations in all directions and an additive white noise variable we first show that the class of ncar models is capable of generating a wide variety of image patterns posessing the local replication attribute an essential ingredient of many natural textures it is also shown that the theoretical variograms of many ncar models possess an oscillatory behavior a characteristic of the variograms of many natural textures next we give experimental results of synthesis of 64 64 textures resembling several real textures in the brodatz album the synthetic textures generated by 16 parameter ncar models retain most the visual characteristics of the original textures the variograms of the original and synthetic textures are also similar

noncausal 2 d spectrum estimation for direction finding a two dimensional noncausal autoregressive ncar plus additive noise model based spectrum estimation method is presented for planar array data typical of signals encountered in array processing applications since the likelihood function for ncar plus noise data is nonlinear in the model parameters and is further complicated by the unknown variance of the additive noise computationally intensive gradient search algorithms are required for computing the estimates if a doubly periodic lattice is assumed the complexity of the approximate maximum likelihood ml equation is significantly reduced without destroying the theoretical asymptotic properties of the estimates and degrading the observed accuracy of the estimated spectra initial conditions for starting the approximate ml computation are suggested experimental results that can be used to evaluate the signal plus noise approach and compare its performance to those of signal only methods are presented for gaussian and simulated planar array data statistics of estimated spectrum parameters are given and estimated spectra for signals with close spatial frequencies are shown the approximate ml parameter estimate s asymptotic properties such as consistency and normality are established and lower bounds for the estimate s errors are derived assuming that the data are gaussian

soft bl algebras based on fuzzy sets by means of soft sets and q soft sets some characterizations of implicative positive implicative and fantastic filteristic soft bl algebras are investigated finally we prove that a soft set is an implicative filteristic soft bl algebra if and only if it is both a positive implicative filteristic soft bl algebra and a fantastic filteristic soft bl algebra

statistic based map image reconstruction from poisson data using gibbs priors a statistical method for selecting the gibbs parameter in map image restoration from poisson data using gibbs priors is presented the gibbs parameter determines the degree to which the prior influences the restoration the presented method yields a map restored image minimally influenced by the prior for which a statistic falls within an appropriate confidence interval the method assumes that a close approximation to the blurring function is known a simple iterative feedback algorithm is presented to statistically select the parameter as the map image restoration is being performed this algorithm is heuristically based on a model reference control formulation but it requires only a minimal number of iterations for the parameter to settle to its statistically specified value the performance of the statistical method for selecting the prior parameter and that of the iterative feedback algorithm are demonstrated using both 2 d and 3 d images

modeling of two dimensional random fields by parametric cepstrum a method is presented for parametric modeling of stationary random fields the class of parametric models considered allows the most general elliptic field and by linear constraints can include such special cases as isotropic quarter plane and separable fields the technique based on the cepstrum has the great advantage of requiring only the use of fast fourier transforms in the fitting process thus unlike the fitting of two dimensional autoregressions no iteration is necessary other advantages are that any wiener filters constructed from the fitted spectrum are guaranteed to be stable and that the spectrum is guaranteed to be positive statistical tests for determining various special types of field from data are developed the choice of model order is discussed as well

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

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

recursive structure of noncausal gauss markov random fields an approach is developed for noncausal gauss markov random fields gmrfs that enables the use of recursive procedures while retaining the noncausality of the field recursive representations are established that are equivalent to the original field this is achieved by first presenting a canonical representation for gmrfs that is based on the inverse of the covariance matrix which is called the potential matrix it is this matrix rather than the field covariance that reflects in a natural way the mrf structure from its properties two equivalent one sided representations are derived each of which is obtained as the successive iterates of a riccati type equation for homogeneous fields these unilateral descriptions are symmetrized versions of each other the study of only one riccati equation being required it is proven that this riccati equation converges at a geometric rate therefore the one sided representations are asymptotically invariant these unilateral representations make it possible to process the fields with well known recursive techniques such as kalman bucy filters and two point smoothers

stone like representation theorems and three valued filters in r0 algebras nilpotent minimum algebras nilpotent minimum algebras nm algebras are algebraic counterpart of a formal deductive system where conjunction is modeled by the nilpotent minimum t norm a logic also independently introduced by guo jun wang in the mid 1990s such algebras are to this logic just what boolean algebras are to the classical propositional logic in this paper by introducing respectively the stone topology and a three valued fuzzy stone topology on the set of all maximal filters in an nm algebra we first establish two analogues for an nm algebra of the well known stone representation theorem for a boolean algebra which state that the boolean skeleton of an nm algebra is isomorphic to the algebra of all clopen subsets of its stone space and the three valued skeleton is isomorphic to the algebra of all clopen fuzzy subsets of its three valued fuzzy stone space respectively then we introduce the notions of boolean filter and of three valued filter in an nm algebra and finally we prove that three valued filters and closed subsets of the stone space of an nm algebra are in one to one correspondence and boolean filters uniquely correspond to closed subsets of the subspace consisting of all ultrafilters

mosaic models for textures this paper deals with a class of image models based on random geometric processes theoretical and empirical results on properties of patterns generated using these models are summarized these properties can be used as aids in fitting the models to images

on two dimensional maximum entropy spectral estimation a novel method of two dimensional spectral estimation which the authors introduced recently 1987 using the radon transform and a one dimensional autoregressive model ar led them to investigate the maximization of entropy subject to the correlation matching constraints in the radon space instead of solving the 2d maximum entropy me spectral estimation problem the authors convert it into a problem which is easier to solve it is shown that a radial slice of the 2d me spectrum can be obtained by 1d ar modeling of the projections radon transform of a stationary random field srf the advantages and limitations of using this new duality relation to estimate the complete 2d me spectra on a polar raster are discussed

a multiscale stochastic image model for automated inspection in this paper we develop a novel multiscale stochastic image model to describe the appearance of a complex threedimensional object in a two dimensional monochrome image this formal image model is used in conjunction with bayesian estimation techniques to perform automated inspection the model is based on a stochastic tree structure in which each node is an important subassembly of the three dimensional object the data associated with each node or subassembly is modeled in a wavelet domain we use a fast multiscale search technique to compute the sequential map smap estimate of the unknown position scale factor and 2 d rotation for each subassembly the search is carried out in a manner similar to a sequential likelihood ratio test where the process advances in scale rather than time the results of this search determine whether or not the object passes inspection a similar search is used in conjunction with the em algorithm to estimate the model parameters for a given object from a set of training images the performance of the algorithm is demonstrated on two different real assemblies

on a class of residuated semilattice monoids in this paper we consider the structure of residuated join semilattice monoid and we study its first properties the class of residuated join semilattice monoids is a variety of algebras a representation of this structure as residuated mappings is given then we present the notion of normal filter together with a characterization theorem of homomorphic images based on this notion different classes of residuated join semilattice monoids are introduced

two dimensional frequency domain blind system identification using higher order statistics with application to texture synthesis in this paper shalvi and weinstein s 1993 super exponential se algorithm using higher order statistics for blind deconvolution of one dimensional 1 d linear time invariant systems is extended to a two dimensional 2 d se algorithm then a 2 d frequency domain blind system identification bsi algorithm for 2 d linear shift invariant lsi systems using the computationally efficient 2 d se algorithm and the 2 d linear prediction error filter is proposed in addition to the lsi system estimate the proposed bsi algorithm also provides a minimum mean square error mmse equalizer estimate and an mmse signal enhancement filter estimate then a texture synthesis method tsm using the proposed bsi algorithm is presented some simulation results to support the efficacy of the proposed bsi algorithm and some experimental results to support the efficacy of the proposed tsm are presented finally some conclusions are drawn