stochastic mechanics of reciprocal diffusions the dynamics and kinematics of reciprocal diffusions were examined in a previous paper j math phys 34 1846 1993 where it was shown that reciprocal diffusions admit a chain of conservation laws which close after the first two laws for two disjoint subclasses of reciprocal diffusions the markov and quantum diffusions for the case of quantum diffusions the conservation laws are equivalent to schrodinger s equation the markov diffusions were employed by schrodinger sitzungsber preuss akad wiss phys math kl 144 1931 ann inst h poincare 2 269 1932 nelson dynamical theories of brownian motion princeton university princeton nj 1967 quantum fluctuations princeton university princeton nj 1985 and other researchers to develop stochastic formulations of quantum mechanics called stochastic mechanics we propose here an alternative version of stochastic mechanics based on quantum diffusions a procedure is presented for constructing the quantum diffusion associated to a given wave function it is shown that quantum diffusions satisfy the uncertainty principle and have a locality property whereby given two dynamically uncoupled but statistically correlated particles the marginal statistics of each particle depend only on the local fields to which the particle is subjected however like wigner s joint probability distribution for the position and momentum of a particle the finite joint probability densities of quantum diffusions may take negative values

discrete time classical and quantum markovian evolutions maximum entropy problems on path space the theory of schrodinger bridges for diffusion processes is extended to classical and quantum discrete time markovian evolutions the solution of the path space maximum entropy problems is obtained from the a priori model in both cases via a suitable multiplicative functional transformation in the quantum case nonequilibrium time reversal of quantum channels is discussed and space time harmonic processes are introduced

optimal interpolation for linear stochastic systems let y satisfying dy cxdt ddw y 0 0 represent the noisy observation of the gauss markov process x generated by dx axdt bdw x 0 x 0 on the finite interval 0 t in this paper we solve the following interpolation problem determine the best least squares estimate of x t t in 0 t given the increments of y on the intervals 0 t 1 and t 2 t where 0 t 1 t 2 t and the statistics of x 0 several alternative expressions for the optimal interpolator are derived some of which are phrased in terms of two kalman bucy estimates as a by product we also solve the interpolation problem for the missing increments of y

fuzzy geometry perpendicular to fuzzy line segment abstract in this study we define the perpendicular to fuzzy line segment pfls which intersects a given fuzzy line segment fls normally at a specified fuzzy point in order to construct pfls some spatial transformations of fuzzy points such as translation rotation expansion contraction and reflection trecore are needed and have also been defined here the concept of pfls has been justified and formulated in three different ways with the help of same and inverse point theory of fuzzy points their properties and interrelations are also investigated the area of a fuzzy triangle has been calculated with the proposed methodology

on discrete algebraic riccati equations a rank characterization of solutions abstract we give a rank characterization of the solution set of discrete algebraic riccati equations dares involving real matrices our characterization involves the use of invariant subspaces of the system matrix we observe that a dare can be considered as a specific type of a non symmetric continuous time algebraic riccati equation we show how to extract the low rank solutions from the full rank solution of a homogeneous dare we also study the eigen structure of various feedback closed loop matrices associated with solutions of a dare we show as a consequence of our observations that a solution x of a dare for controllable systems and certain specific types of uncontrollable systems satisfies the inequality x m i n x x m a x we characterize conditions under which the solution set of dare is bounded unbounded we use the method of invariant subspaces to identify some invariant sets of a difference riccati equation

correspondence a note on efficient smoothing for boundary value models abstract the smoothing problem for two point boundary value models is investigated it is shown that by triangularizing the relevant hamiltonian system a new smoothing algorithm can be derived which is twice as fast as that obtained by diagonalizing the hamiltonian

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

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

variational processes and stochastic versions of mechanics the dynamical structure of any reasonable stochastic version of classical mechanics is investigated including the version created by nelson e nelson quantum fluctuations princeton u p princeton nj 1985 phys rev 150 1079 1966 for the description of quantum phenomena two different theories result from this common structure one of them is the imaginary time version of nelson s theory whose existence was unknown and yields a radically new probabilistic interpretation of the heat equation the existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of yasue k yasue j math phys 22 1010 1981

an efficient algorithm for maximum entropy extension of block circulant covariance matrices article history this paper deals with maximum entropy completion of partially specified block circulant matrices since positive definite symmet ric circulants happen to be covariance matrices of stationary peri odic processes in particular of stationary reciprocal processes this problem has applications in signal processing in particular to im age modeling in fact it is strictly related to maximum likelihood estimation of bilateral ar type representations of acausal signals subject to certain conditional independence constraints the maxi mum entropy completion problem for block circulant matrices has recently been solved by the authors although leaving open the problem of an efficient computation of the solution in this pa per we provide an efficient algorithm for computing its solution which compares very favorably with existing algorithms designed for positive definite matrix extension problems the proposed al gorithm benefits from the analysis of the relationship between our problem and the band extension problem for block toeplitz matri ces also developed in this paper

design and analysis of fuzzy morphological algorithms for image processing a general paradigm for lifting binary morphological algorithms to fuzzy algorithms is employed to construct fuzzy versions of classical binary morphological operations the lifting procedure is based upon an epistemological interpretation of both image and filter fuzzification algorithms are designed via the paradigm for various fuzzifications and their performances are analyzed to provide insight into the kind of liftings that produce suitable results algorithms are discussed for three image processing tasks shape detection edge detection and clutter removal detailed analyses are given for the effect of noise and its mitigation owing to fuzzy approaches it is demonstrated how the fuzzy hit or miss transform can be used in conjunction with a decision procedure to achieve word recognition

some comments on conditionally markov and reciprocal gaussian processes corresp mehr and mcfadden s conditionally markov processes and jamison s reciprocal processes are two independent extensions of slepian s work on a particular stationary gaussian process the relationship between them and their relationship with gauss markov processes are discussed

bayesian estimation and the kalman filter in this tutorial article we give a bayesian derivation of a basic state estimation result for discrete time markov process models with independent process and measurement noise and measurements not affecting the state we then list some properties of gaussian random vectors and show how the kalman filtering algorithm follows from the general state estimation result and a linear gaussian model definition we give some illustrative examples including a probabilistic turing machine dynamic classification and tracking a moving object

recursive filtering and smoothing for reciprocal gaussian processes with dirichlet boundary conditions the minimum mean square error mmse estimation problem for a discrete index reciprocal gaussian process impaired by additive white gaussian noise is completely solved in the general case of noisily observed dirichlet random boundary conditions finite sets of recursive equations are obtained for the computation of the filtered sequence and of the fixed point fixed interval and fixed lag smoothed sequences recursive expressions are also given for the mmse performance of the presented estimators simulation results confirm the validity of the performance analysis

malliavin calculus and euclidean quantum mechanics i functional calculus abstract we give a rigorous version of the functional calculus developed by r feynman in relation to his path integral formulation of quantum mechanics our approach is euclidean but distinct from the one founded on the feynman kac formula it uses two basic ingredients a new probabilistic interpretation of the classical heat equation introduced recently in the framework of euclidean quantum mechanics and an infinite dimensional differential calculus adapted to functionals of the diffusion processes relevant for this interpretation

model parameter estimation for reciprocal gaussian random processes the problem of estimating the model parameters of a discrete index reciprocal gaussian random process from a limited number of noisy observations is addressed the general case of a first order multivariate process is analyzed stating its basic properties and deriving a linear equation set that relates the model parameters including the unknown variance of the observation noise to the generally nonstationary autocorrelation function of the observed process it generalizes to the reciprocal processes the so called high order yule walker equations for ar processes based on these results a practical estimation algorithm is proposed

smoothing error dynamics and their use in the solution of smoothing and mapping problems martingale decomposition techniques are used to derive markovian models for the error in smoothed estimates of processes described by linear models driven by white noise these models together with some simple hilbert space decomposition ideas provide a simple unified framework for examining a variety of problems involving the efficient assimilation of spatial data which we refer to as mapping problems algorithms for several different mapping problems are derived a specific example of map updating for a two dimensional random field is included

gaussian conditionally markov sequences dynamic models and representations of reciprocal and other classes conditionally markov cm sequences are powerful mathematical tools for modeling problems one class of cm sequences is the reciprocal sequence in application we need not only cm dynamic models but also know how to design model parameters models of two important classes of nonsingular gaussian ng cm sequences called cm l and cm f models and a model of the ng reciprocal sequence called reciprocal cm l model were presented in our previous works and their applications were discussed in this paper these models are studied in more detail in particular their parameter design it is shown that every reciprocal cm l model can be induced by a markov model then parameters of each reciprocal cm l model can be obtained from those of the markov model also it is shown that an ng cm l cm f sequence can be represented by a sum of an ng markov sequence and an uncorrelated ng vector this necessary and sufficient representation provides a basis for designing parameters of a cm l cm f model from the cm viewpoint a representation is also obtained for ng reciprocal sequences this representation is simple and reveals an important property of reciprocal sequences as a result the significance of studying reciprocal sequences from the cm viewpoint is demonstrated a full spectrum of dynamic models from a cm l model to a reciprocal cm l model is also presented some examples are presented for illustration

a forgetting property of reciprocal chains reciprocal chains rc are a class of discrete index finite state stochastic processes with a generalized markov nearest neighbour property all markov chains mc are rc but not conversely it is well known that a sufficiently regular mc forgets its initial distribution in a geometric manner this article addresses the issues of forgetting for rc using the properties of inhomogeneous products of nonnegative matrices this article establishes that a rc forget its endpoint distributions in a more complicated manner in particular this article shows that the forgetting function for the initial conditions is bounded by a superposition of increasing and decreasing geometric terms implications for modeling using rc are then discussed the particular case of those rc that are mc with prescribed marginal distributions on the endpoints is considered the tightness of the forgetting bounds is illustrated with numerical examples

gaussian conditionally markov sequences singular nonsingular most existing results about modeling and characterizing gaussian markov reciprocal and conditionally markov cm processes assume nonsingularity of the processes this assumption makes the analysis easier but restricts application of these processes this paper studies models and characterizes the general singular nonsingular gaussian cm including reciprocal and markov sequence for example to our knowledge there is no dynamic model for the general singular nonsingular gaussian reciprocal sequence in the literature we obtain two such models from the cm viewpoint as a result the significance of studying reciprocal sequences from the cm viewpoint is demonstrated the results of this paper unify singular and nonsingular gaussian cm including reciprocal and markov sequences and provide tools for their application an application of cm sequences in trajectory modeling with a destination is discussed and illustrative examples are presented

minimal realization and dynamic properties of optimal smoothers smoothing algorithms of various kinds have been around for several decades however some basic issues regarding the dynamical structure and the minimal dimension of the steady state algorithm are still poorly understood in this paper we derive a realization of minimal dimension of the optimal smoother for a signal admitting a state space description of dimension n it is shown that the dimension of the smoothing algorithm can vary from n to 2n depending on the zero structure of the signal model the dynamics pole structure of the steady state smoother is also characterized explicitly and is related to the zero structure of the model we use several recent ideas from stochastic realization theory in particular a minimal markovian representation of the smoother is derived which requires solving a nonsymmetric wiener hopf factorization problem in this way the smoother is naturally expressed as the cascade of a whitening filter and a linear filter of least possible dimension whose state space is a minimal markovian subspace containing the smoothed estimate x spl i cap this among other aspects affords a very simple calculation of the error covariance matrix of the smoother a reduced order two filter implementation of the type due to mayne 1966 and fraser 1967 is obtained by solving a riccati equation of reduced dimension which is in general smaller than the dimension of the riccati equations considered in the literature

remarks on linear and nonlinear filtering this communication tries to give some insight into relationships existing between viterbi and the forward backward algorithm used in the context of hidden markov models on the one hand and kalman filtering and rauch tung striebel smoothing on the other we give a unifying view which shows how those algorithms are related and give an example of a nonlinear hybrid system that can be filtered through a mixed algorithm

probabilistic and sequential computation of optical flow using temporal coherence in the computation of dense optical flow fields spatial coherence constraints are commonly used to regularize otherwise ill posed problem formulations providing spatial integration of data we present a temporal multiframe extension of the dense optical flow estimation formulation proposed by horn and schunck 1981 in which we use a temporal coherence constraint to yield the optimal fusing of data from multiple frames of measurements conceptually standard kalman filtering algorithms are applicable to the resulting multiframe optical flow estimation problem providing a solution that is sequential and recursive in time experiments are presented to demonstrate that the resulting multiframe estimates are more robust to noise than those provided by the original single frame formulation in addition we demonstrate cases where the aperture problem of motion vision cannot be resolved satisfactorily without the temporal integration of data enabled by the proposed formulation practically the large matrix dimensions involved in the problem prohibit exact implementation of the optimal kalman filter to overcome this limitation we present a computationally efficient yet near optimal approximation of the exact filtering algorithm this approximation has a precise interpretation as the sequential estimation of a reduced order spatial model for the optical flow estimation error process at each time step and arises from an estimation theoretic treatment of the filtering problem experiments also demonstrate the efficacy of this near optimal filter

discrete time complementary models and smoothing abstract the fact that the smoothing error is a wide sense markov process is somehow surprising since smoothed estimates depend upon both past and future data in this paper we first give a simple and general proof of this fact then we use the so called complementary models introduced by weinert and desai to derive forwards and backwards markovian models for the smoothing error in state space models by exploring the structure of the complementary models we show that under certain restrictions only two simple structured models exist one that runs forwards in time and the other that runs backwards in time the forwards complementary model leads to the forward rauch tung striebel rts smoothing formula and to a backwards markovian model for the error whereas the backwards model leads to the backward rts formula and to a forwards error model the two models for the smoothing error can be derived one from the other by a forward backward transformation that preserves the sample paths finally by using

on the relative entropy of discrete time markov processes with given end point densities given a markov process x k defined over a finite interval i 0 n i spl sub z we construct a process x k with the same initial density as x but a different end point density which minimizes the relative entropy of x and x it is shown that x is a markov process in the same reciprocal class as x in the gaussian case the minimum relative entropy problem is related to a minimum energy lqg optimal control problem

numerical evaluation of coherent state path integrals in quantum dynamics the numerical evaluation of coherent state path integrals for quantum dynamical problems is discussed for one dimensional examples to propagate an initial state we use the normal and antinormal ordered coherent state path integrals combined with a split operator technique dividing the hamiltonian into harmonic and anharmonic parts for numerical purposes integrations must be approximated by quadrature formulae this leads to a matrix multiplication scheme which is systematically tested for the double well and morse potentials the method is accurate for propagation times much longer than the natural time scale of the system and it allows for short as well as long time steps without loss of stability

a new inversion free method for a rational matrix equation abstract motivated by the classical newton schulz method for finding the inverse of a nonsingular matrix we develop a new inversion free method for obtaining the minimal hermitian positive definite solution of the matrix rational equation x a x 1 a i where i is the identity matrix and a is a given nonsingular matrix we present convergence results and discuss stability properties when the method starts from the available matrix aa we also present numerical results to compare our proposal with some previously developed inversion free techniques for solving the same rational matrix equation

stationarity and reciprocity in stochastic multipoint boundary value systems investigates certain statistical properties of the state vector in a stochastic multipoint boundary value system the authors give necessary and sufficient conditions for the state vector to be stationary and prove that the state vector is locally reciprocal

state space realizations and optimal smoothing for gaussian generalized reciprocal processes this technical note derives stochastic realization and optimal smoothing algorithms for a class of gaussian generalized reciprocal processes ggrp the note exploits the interplay between reciprocal processes and markov bridges which underpin the ggrp model a forward backward algorithm for stochastic realization of ggrp is described the form on the inverse covariance matrix for the ggrp is used via cholesky factorization to derive a procedure for optimal mmse smoothing of ggrp observed in noise the note demonstrates that the associated smoothing error is also a ggrp with known covariance which may be used to assess the performance of smoothing as a function of the model parameters a numerical example is provided to illustrate the performance of the mmse smoother compared to those derived from compatible markov and reciprocal model based algorithms

the circulant rational covariance extension problem the complete solution the rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite dimensional positive definite toeplitz matrix the northwest corner of which is given the circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process which are better represented on the discrete unit circle rather than on the discrete real line the corresponding matrix completion problem then amounts to completing a finite dimensional toeplitz matrix that is circulant another important motivation for this problem is that it provides a natural approximation involving only computations based on the fast fourier transform for the ordinary rational covariance extension problem potentially leading to an efficient numerical procedure for the latter the circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general each corresponding to a bilateral arma representation of the underlying periodic process in this paper we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them a procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

reciprocal diffusions in flat space we describe the theory of reciprocal diffusions in flat space a reciprocal process is a markov random field on a one dimensional parameter space every markov process is reciprocal but not vice versa we descibe the first and second order mean differential characteristics of reciprocal diffusions this includes a new definition of stochastic acceleration we show that reciprocal diffusions satisfy stochastic differential equations of second order associated to a reciprocal diffusion is a sequence of conservation laws the first two of which are the familiar continuity and euler equations there are two cases where these laws can be closed after the first two they are the mutually exclusive subclasses of markov and quantum diffusions the latter corresponds to solutions of the schrodinger equation and may be part of a stochastic description of quantum mechanics

identification of 2 d noncausal gauss markov random fields parameter identification of multidimensional noncausal markov random fields is an important paradigm in multidimensional signal processing and modeling and the solutions to this problem are employed in many areas of image processing an original procedure for estimating the model parameters of discrete index 2 d noncausal gauss markov random fields gmrfs from noisy observations is proposed valid for both finite and infinite lattices and for any kind of boundary conditions starting from a suitable local representation of the gmrf and taking into account the symmetry property of so called field potentials a linear equation set relating the model parameters to the 2 d autocorrelation function known or estimated of the observed field is derived its solution gives the parameter estimates of the gmrf together with the estimate of the possibly unknown variance of the observation noise

analytical fuzzy plane geometry iii in this study we attempt to construct fuzzy circles in a fuzzy geometrical plane we provide a comprehensive study where we find a fuzzy number with a predetermined fuzzy distance from a given fuzzy number we formulate fuzzy circles using the conventional basic definitions of crisp circles two different formulations of fuzzy circles are proposed which depend on the information known about a fuzzy circle the interrelations between the evaluated fuzzy circles are investigated we show that the center of a fuzzy circle may not be a fuzzy point however the radius of a fuzzy circle that passes through three given fuzzy points is always a fuzzy number the concepts of same and inverse points and a fuzzy number along a line are used to define and analyze the proposed ideas our discussions and studies are supported by suitable numerical and pictorial examples

mathematical morphology a modern approach in image processing based on algebra and geometry mathematical morphology is a theory of image transformations and image functionals which is based on set theoretical geometrical and topological concepts the methodology is particularly useful for the analysis of the geometrical structure in an image the main goal of this paper is to give an impression of the underlying philosophy and the mathematical theories which are relevant to this field the following topics are discussed introduction to mathematical morphology generalization to complete lattices morphological filters and their construction by iteration geometrical aspects of morphology e g convexity distance geodesic operators granulometries metric dilations distance transform cost functions and extension of binary operators to grey scale images

on fuzzy distances and their use in image processing under imprecision abstract this paper proposesa classification of fuzzy distances with respect to the requirements needed for applications in image processing under imprecision we distinguish on the one hand distances that basically compare only the membership functions representing the concerned fuzzy objects and on the other hand distances that combine spatial distance between objects and membership functions to our point of view the second class of methods finds more general applications in image processing since these methods take into account both spatial information and information related to the imprecision attached to the image objects new distances based on mathematical morphology are proposed in this second class

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

a maximum entropy solution of the covariance extension problem for reciprocal processes stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of markov random fields restricted to one dimension nonstationary reciprocal processes have been extensively studied in the past especially by jamison et al the specialization of the nonstationary theory to the stationary case however does not seem to have been pursued in sufficient depth in the literature stationary reciprocal processes and reciprocal stochastic models are potentially useful for describing signals which naturally live in a finite region of the time or space line estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing in this paper we discuss a class of reciprocal processes which is the acausal analog of auto regressive ar processes familiar in control and signal processing we show that maximum likelihood identification of these processes leads to a covariance extension problem for block circulant covariance matrices this generalizes the famous covariance band extension problem for stationary processes on the integer line as in the usual stationary setting on the integer line the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem we show that the maximum entropy principle leads to a complete solution of the problem

representations for multivariate reciprocal gaussian processes multivariate reciprocal gaussian processes are represented as a sum of two independent processes a piecewise markov process which is also represented in terms of a wiener type process and a time dependent linear transformation of a normally distributed random vector this result is then applied to the first passage time problem

smoothing for multipoint boundary value models abstract we show how to solve the linear least squares smoothing problem for multipoint boundary value models the complementary model is derived and is used to determine the hamiltonian equations for the smoothed state estimate and its error covariance stable algorithms are obtained using an invariant imbedding multiple shooting procedure

comment on momentum in stochastic mechanics stochastic mechanics is a probabilistic description of quantum systems in terms of stochastic differential equations davidson m davidson lett math phys 5 523 1981 and de falco de martino and de siena d de falco s de martino and s de siena lett nuovo cimento 36 457 1983 have introduced momentum variables into this scheme in this paper a discussion of this attempt is presented and some difficulties concerning the physical interpretation are pointed out

optimal smoothing for finite state hidden reciprocal processes this technical note addresses modelling and estimation of a class of finite state random processes called hidden reciprocal chains hrc a hidden reciprocal chain consists of a finite state reciprocal process together with an observation process conditioned on the reciprocal process much as in the case of a hidden markov model hmm the key difference between markov models and reciprocal models is that reciprocal models are non causal the technical note presents a characterization of a hrc by a finite set of hidden markov bridges which are hmms with the final state fixed the technical note then uses this characterization to derive the optimal fixed interval smoother for a hrc performance of linear and optimal smoothers derived for both hmm and hrc are compared using simulations for a class of hrc derived from underlying markov transitions these experiments suggest that not surprisingly the performance of the optimal hmm and hrc smoothers are signifcantly better than their linear counterparts and that some performance improvement is obtained using the hrc smoothers compared to the hmm smoothers the technical note concludes by mentioning some ongoing and future work which exploits this new markov bridge characterization of a hrc

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

restriction of a markov random field on a graph and multiresolution statistical image modeling the association of statistical models and multiresolution data analysis in a consistent and tractable mathematical framework remains an intricate theoretical and practical issue several consistent approaches have been proposed previously to combine markov random field mrf models and multiresolution algorithms in image analysis renormalization group subsampling of stochastic processes mrfs defined on trees or pyramids etc for the simulation or a practical use of these models in statistical estimation an important issue is the preservation of the local markovian property of the representation at the different resolution levels it is shown that this key problem may be studied by considering the restriction of a markov random field defined on some simple finite nondirected graph to a part of its original site set several general properties of the restricted field are derived the general form of the distribution of the restriction is given locality of the field is studied by exhibiting a neighborhood structure with respect to which the restricted field is an mrf sufficient conditions for the new neighborhood structure to be minimal are derived several consequences of these general results related to various multiresolution mrf based modeling approaches in image analysis are presented

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

modelling and estimation for finite state reciprocal processes a reciprocal equation is a kind of descriptor linear discrete index stochastic system which is well known be satisfied pathwise by all gaussian reciprocal processes from a system theoretic point of view it is a kind of noncausal linear system in the sense that the solution of it cannot be determined by only an initial condition indeed requiring the terminal state as well besides all the input function between initial and terminal states also nice properties are known of a reciprocal equation such as the equivalence of it with a couple of ordinary causal dynamic systems running in opposite directions for these reasons here we assume a reciprocal equation as the target of stochastic realization for the class of finite state reciprocal processes also named reciprocal chains the central result of the present paper is showing that any canonical reciprocal chain i e valued in the canonical base of realrn n being the cardinality of the set of chain s states satisfies pathwise a reciprocal equation in a n2 dimensional canonical variable or in other word a quadratic reciprocal equation named augmented state reciprocal model asrm also for a partially observed reciprocal chain a linear optimal smoother is derived all the results here presented are based upon the idea that a reciprocal chain is a combination of markov bridges to this purpose other forms besides the asrm are presented in order to make clear the meaning of this combination as well as to prove that the linear smoother can be actually implemented as n smoothers all operating independently on each markov bridge component

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

smoothing and likelihood ratio for gaussian boundary value processes a new derivation which does not need the invertibility assumption of the covariance matrix of the boundary data is given for the smoothing of gaussian two point boundary value processes tpbvp the likelihood ratio for tpbv processes is then derived in terms of the system parameters by using the krein factorization the likelihood ratio involves the smoother of the process an alternate expression for the likelihood ratio based on the filtered estimate of the state is also given

dynamics and kinematics of reciprocal diffusions this paper examines the dynamics and kinematics of reciprocal diffusions reciprocal processes were introduced by bernstein in 1932 and were later studied in detail by jamison the reciprocal diffusions are constructed here by specifying their finite joint densities in terms of the green s function of a general heat operator and an end point density a path integral interpretation of the heat operator green s function is provided which is used to derive a stochastic form of newton s law as well as a conditional distribution for the velocity of a diffusing particle given its position these results are then employed to derive two conservation laws expressing the conservation of mass and momentum the conservation laws do not form a closed system of equations in general except for two subclasses of reciprocal diffusions the markov and quantum diffusions

on the covariance completion problem under a circulant structure covariance matrices with a circulant structure arise in the context of discrete time periodic processes and their significance stems also partly from the fact that they can be diagonalized via a fourier transformation this note deals with the problem of completion of partially specified circulant covariance matrices the particular completion that has maximal determinant i e the so called maximum entropy completion was considered in carli where it was shown that if a single band is unspecified and to be completed the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data i e it has the dempster property the purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements more specifically we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant a description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables facilitated by the circulant structure is highlighted

fuzzy relative position between objects in image processing a morphological approach in order to cope with the ambiguity of spatial relative position concepts we propose a new definition of the relative position between two objects in a fuzzy set framework this definition is based on a morphological and fuzzy pattern matching approach and consists of comparing an object to a fuzzy landscape representing the degree of satisfaction of a directional relationship to a reference object it has good formal properties it is flexible it fits the intuition and it can be used for structural pattern recognition under imprecision moreover it also applies in 3d and for fuzzy objects issued from images

edge preserving smoothing using a similarity measure in adaptive geodesic neighbourhoods this paper introduces a novel image dependent filtering approach derived from concepts known in mathematical morphology and aiming at edge preserving smoothing of natural images like other adaptive methods it assumes that the neighbourhood of a pixel contains the essential information required for the estimation of local features in the image the proposed strategy essentially consists in a weighted averaging combining both spatial and tonal information for that purpose a twofold similarity measure is calculated from local geodesic time functions this way no prior operator definition is required since a weighting neighbourhood and a weighting kernel are determined automatically from the unfiltered input data for each pixel location by designing relevant geodesic masks two adaptive filtering algorithms are derived that are particularly efficient at smoothing heterogeneous areas while preserving relevant structures in greyscale and multichannel images

telescoping recursive representations and estimation of gauss markov random fields we present telescoping recursive representations for both continuous and discrete indexed noncausal gauss markov random fields our recursions start at the boundary a hypersurface in and telescope inwards for example for images the telescoping representation reduce recursions from to i e to recursions on a single dimension under appropriate conditions the recursions for the random field are linear stochastic differential difference equations driven by white noise for which we derive recursive estimation algorithms that extend standard algorithms like the kalman bucy filter and the rauch tung striebel smoother to noncausal markov random fields

numerical path integral techniques for long time dynamics of quantum dissipative systems recent progress in numerical methods for evaluating the real time path integral in dissipative harmonic environments is reviewed quasi adiabatic propagators constructed numerically allow convergence of the path integral with large time increments integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path with an influence functional that describes nonadiabatic corrections the resulting quasi adiabatic propagator path integral is evaluated by efficient system specific quadratures in most regimes of parameter space although some cases are handled by grid monte carlo sampling exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths no uncontrolled approximations are introduced and the resulting methodology converges to the exact quantum result with modest

a new characterization for multivariate gaussian reciprocal processes established a new necessary and sufficient condition for a multivariate gaussian random process to be reciprocal this result is then used to prove that the state vector of a two point boundary value system is reciprocal the authors do not impose any nonsingularity conditions on the covariance function of the state vector

second order stochastic differential equations and non gaussian reciprocal diffusions we first prove that a markov diffusion satisfies a second order stochastic differential equation involving the invariants associated to its reciprocal class as a reciprocal process some properties of the noise term are given we also prove that this equation can be viewed as an euler lagrange equation in a problem of calculus of variations in the non markovian case a bernstein bridge is shown to satisfy the same equation but in a weak sense

maximum likelihood sequence estimation for hidden reciprocal processes this paper addresses the problem of maximum likelihood sequence estimation mlse based on a hidden reciprocal chain hrc as the underlying target model hrcs are non causal discrete time finite state stochastic processes which can be regarded as the one dimensional version of a markov random field although they are not in general markov processes this paper describes a procedure for evaluating the mlse for hrc and compares the resultant estimator with its markov model equivalent the viterbi algorithm in addition the performance of the newly proposed reciprocal mlse is compared to a hrc based optimal smoother

cyclic mathematical morphology in polar logarithmic representation we propose in this paper to perform mathematical morphology operators in a geometric transformation of an image as a result of this procedure processing images with regular structuring elements in the transformed domain is equivalent to working with deformed structuring elements in the original representation more specifically the conversion into polar logarithmic coordinates provides satisfying results in image analysis applied to round objects if they are roughly origin centered we have illustrated the interest of the derived cyclic morphology with two pattern recognition examples erythrocyte shape analysis and multiscale description of iris textures

corrections to the coherent state path integral comments upon a speculation of l s schulman intuitively the feynman path integral corresponds to a weighted sum over classical paths an interpretation that fails for the phase space path integral to address the question of whether there exists a path integral expression conforming to a sum over paths in phase space an examination of the discrete coherent state path integral cspi is undertaken via an alternative formulation of the cspi it is shown that the coherent state action for a broad class hamiltonians can be transformed from the variables q p to q q for these hamiltonians such a transformation along with the inclusion of all terms o e zi zi 1 yields an expression which for finite e can be interpreted as a sum over classical paths with gaussian weight the numerical evaluation of this expression through importance sampling monte carlo is demonstrated

2 d recursive modelling of homogeneous discrete gaussian markov fields a 2 d stochastic realization is obtained for an homogeneous discrete gaussian markov field in two dimensions in specific it is shown that such markov fields can be represented as the unique pathwise solution of a 2 d recursive system of equations with a locally correlated forcing noise the idea of stochastic dynamic system in two dimensions gets realized this way two types of domain are considered spheric and rectangular

recursive filtering and smoothing for reciprocal gaussian processes pinned boundary case the least square estimation problem for pinned to zero discrete index reciprocal gaussian processes in additive white noise is solved thus completing and extending some previous results available in the literature in particular following the innovations approach a finite set of recursive equations is obtained for the filter and for the three standard classes of smoothers fixed point fixed interval fixed lag recursive expressions for the mean square performance of the proposed estimators are also given

some new smoother implementations for discrete time gaussian reciprocal processes abstract the smoothing problem for discrete time gaussian reciprocal processes is examined for this class of process the smoother takes the form of a second order two point boundary value nearest neighbour model this smoother is non causal but admits causal implementations which extend the two filter and rauch tung striebel smoothing formulae of gauss markov processes a multi rate implementation based on a block cyclic reduction method is also described

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

geodesic balls in a fuzzy set and fuzzy geodesic mathematical morphology abstract although fuzzy operators have deserved a large attention in the euclidean case almost nothing exists concerning the geodesic case in this paper we address this question by defining fuzzy geodesic distances between points in a fuzzy set and geodesic balls in a fuzzy set based on the comparison of fuzzy numbers from which we derive fuzzy geodesic mathematical morphology operators the proposed definitions are valid in any dimension the main properties of the basic operators are demonstrated these new operations enhance the set of fuzzy morphological operators leading to transformations of a fuzzy set conditionally to another fuzzy set

transformations of diffusion and schrodinger processes a transformation by means of a new type of multiplicative functionals is given which is a generalization of doob s space time harmonic transformation in the case of arbitrary non harmonic function t x which may vanish on a subset of a b x d the transformation induces an additional singular drift term like in the case of doob s space time harmonic transformation to handle the transformation an integral equation of singular perturbations and a diffusion equation with singular potentials are discussed and the feynman kac theorem is established for a class of singular potentials the transformation is applied to schrodinger processes which are defined following an idea of e schrodinger 1931

discrete weyl heisenberg transforms the concepts of von neumann lattices and tight frames are used for defining discrete quantum mechanical transforms in the phase plane these transforms are obtained by finite shifts la and mb in the coordinate x and momentum p respectively of the weyl heisenberg group and they are called the discrete weyl heisenberg transforms la mb here ab h n with h the planck constant l and m integers and n a positive integer a construction is carried out of la mb for a general weyl heisenberg set by using the kq representation in which a useful formula is established for the frame operator the construction is illustrated on an example of the ground state of a harmonic oscillator it is shown that any physical quantity can be described by the discrete weyl heisenberg transform connections are established between la mb the bargmann representation and the husimi distribution function

reciprocal processes on a tree modeling and estimation issues motivated by multiresolution decomposition methods such as the discrete wavelet transformation the authors introduce reciprocal processes on truncated n ary trees the authors discuss the relationship between such processes and nearest neighbor models the authors show that they can derive a recursive description of the process and that all reciprocal processes on n ary trees reduce to autoregressive processes in the case of zero valued boundary values at the bottom of the tree corresponding to truncation of the tree the authors then study the smoothing equations associated with such models

extremal flows in wasserstein space we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics we develop an intrinsic geometric approach to the calculus of variations in the wasserstein space we show that the flows associated with the schrodinger bridge with general prior with optimal mass transport and with the madelung fluid can all be characterized as annihilating the first variation of a suitable action we then discuss the implications of this unified framework for stochastic mechanics it entails in particular a sort of fluid dynamic reconciliation between bohm s and nelson s stochastic mechanics

geodesic methods in quantitative image analysis abstract let x be a part of an image to be analysed given two arbitrary points x and y of x let us define the number d x x y as follows d x x y is the lower bound of the lengths of the arcs in x ending at points x and y if such arcs exist and if not the function d x is an x intrinsic distance function called geodesic distance note that if x and y belong to two disjoint connected components of x d x x y in other words d x seems to be an appropriate distance function to deal with connectivity problems in the metric space x d x all the classical morphological transformations dilation erosion skeletonization etc can be defined the geodesic distance d x also provides rigorous definitions of topological transformations which can be performed by automatic image analysers with the help of parallel iterative algorithms all these notions are illustrated by several examples definition of the length of a fibre and of an effective length factor automatic detection of cells having at least one nucleus or having one single nucleus definitions of the geodesic center and of the ends of an object without a hole etc the corresponding algorithms are described