generalized optimal designs and contingency plans in linear systems abstract to select an optimal linear design means to select the best subset from a set of possible products or opportunities for commitment and to construct the corresponding optimal contingency plans for coping with the changes of decision parameters based on lee shi and yu this paper discusses a method for generating generalized good designs from known potentially good designs we describe how to use the multi criteria and multi constraint level mc 2 simplex method to identify the generalized optimal designs and their related optimal contingency plans a procedure for effectively and systematically solving the problem of selecting linear designs is proposed

linear estimation of boundary value stochastic processes part ii 1 d smoothing problems this paper addresses the fixed interval smoothing problem for linear two point boundary value stochastic processes of the type introduced by krener 5 as these models are not markovian kalman filtering and associated smoothing algorithms are not applicable the smoothing problem for this class of noncausal processes is solved here by an application of the estimator solution which is developed in part i of this paper 3 via the method of complementary models for an n th order model this approach yields the smoother as a 2 n th order two point boundary value problem it is shown that this smoother can be realized in a stable two filter form which is remarkably similar to two filter smoothers for causal processes in addition expressions for the smoothing error and smoothing error covariance are developed these equations are employed to perform a covariance analysis of estimating the temperature and heat flow in a cooling fin

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

a representation theoretic approach to the dft with noncommutative generalizations it is known that both the one dimensional and multidimensional dfts discrete fourier transforms can be constructed as transition matrices associated with the decomposition of finite dimensional complex commutative group algebras into simple components two key attributes of these transforms orthogonality and the convolution property are inherent in such a description suggesting the possibility of enlarging the class by extending the construction to noncommutative groups in this context one speaks of a noncommutative or generalized transform the definition of which is based on the theory of semisimple rings the author reviews the ring theory and representation theory fundamental to the existence and computation of group algebra decompositions and sketches the representation theoretic construction of both the classical and noncommutative discrete fourier transforms the noncommutative transform associated with the class of dihedral groups is explicitly constructed and shown directly to exhibit both orthogonality and a noncommutative convolution property

efficient computation of fourier inversion for finite groups let g be a finite group and let s be a complex valued function on g if is a matrix representation of g of dimension d then the fourier transform of f at g if is defined to be the matrix f s g f s s let r be a complete set of inequivalent irreducible matrix representations of g the fourier transform of f with respect to r is defined as the set of matrices f r recovery of f from its fourier transform may be accomplished via the fourier inversion formula f s g d trace f s 1

complementary models and smoothing the authors apply the concept of complementary models introduced by weinert and desai 1983 to derive forwards and backwards markovian models for the smoothing error process by exploring the structure of the complementary models they show that under certain restrictions only two simple structured models exist one that runs forwards in time and another than runs backwards in time the forwards complementary model leads to a forwards rauch tung striebel rts smoothing formula and to a backwards markovian model for the error whereas the backwards model leads to a backwards rts formula and to a forwards error model the two models for the smoothing error can be derived one from the other by a forwards backwards transformation that preserves the sample paths finally by using a combination of the two complementary models another proof of the two filter smoothing formula is given

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

fast generalized fourier transforms abstract let g be a f inite group then ls g the linear complexity of a suitable wedderburn transform corresponding to g is smaller than 2 g 2 we improve this trivial upper bound by showing that ls g is smaller than min s t l t g 7 q t g 3 2 where the minimum is taken over all towers t g n g n 1 g 0 1 of subgroups of g gn and where q t resp s t is the maximum resp sum of all indices gi 1 gi corresponding to t and l t n is the length of the tower a similar upper bound is valid for the linear complexity of the inverse of such a wedderburn transform for symmetric groups our technique yields the stronger estimate ls sn 0 sn log3 sn

minimal representations of semiseparable kernels and systems with separable boundary conditions abstract the simplest representations of triangular parts of finite rank kernels are analysed criteria for uniqueness up to similarity are given the results are applied to the problem of minimal realization of systems with separable boundary conditions

fast fourier transforms for metabelian groups let g be a finite group then l s g the minimal number of arithmetic operations to evaluate a fourier transform corresponding to g is smaller than 2 cdot g 2 the fast fourier transform algorithms improve this trivial upper bound by showing that for a cyclic group g l s g leqq c cdot g cdot log g this last result is extended to metabelian groups and it is shown that these groups also have fast inverse fourier transforms in particular there are fast algorithms for the inverse fourier transforms for dihedral and generalized quaternion groups as well as for all groups of square free order

efficient computation of the fourier transform on finite groups let g be a finite group f g c a function and p an ir reducible representation of g the fourier transform is defined as f p eseeg f s p s direct computation for all irreducible representations involves order igi operations we derive fast algorithms and develop them for the symmetric group sn there n 2 is reduced to nf nf al2 where a is the constant for matrix multiplication 2 38 as of this writing variations of the algorithm allow efficient computation for small representations a practical version of the algorithm is given on sn numerical evidence is presented to show a speedup by a factor of 100 for n 9 department of mathematics harvard university cambridge massachusetts 02138 department of mathematics columbia university new york new york 10027 this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 53 00 utc all use subject to http about jstor org terms

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

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

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

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

separation of variables and the computation of fourier transforms on finite groups ii we present a general diagrammatic approach to the construction of efficient algorithms for computing the fourier transform of a function on a finite group by extending work which connects bratteli diagrams to the construction of fast fourier transform algorithms we make explicit use of the path algebra connection to the construction of gel fand tsetlin bases and work in the setting of quivers we relate this framework to the construction of a configuration space derived from a bratteli diagram in this setting the complexity of an algorithm for computing a fourier transform reduces to the calculation of the dimension of the associated configuration space our methods give improved upper bounds for computing the fourier transform for the general linear groups over finite fields the classical weyl groups and homogeneous spaces of finite groups while also recovering the best known algorithms for the symmetric group

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

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

smoothing for random fields modeled by partial differential equations the authors present an efficient numerically reliable smoothing algorithm for random fields modeled by linear constant coefficient partial differential equations the estimate is computed from discrete measurements by using the discrete fourier transform to convert the two dimensional 2d problem to a collection of uncoupled one dimensional 1d problems which are then solved using stable iterations

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

kalman filtering and riccati equations for descriptor systems a general formulation of a discrete time filtering problem for descriptor systems is considered it is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems using a dual approach to estimation the authors derive a so called 3 block form for the optimal filter and a corresponding 3 block riccati equation for a general class of time varying descriptor models which need not represent a well posed system in that the dynamics may be either over or under constrained specializing in the time invariant case they examine the asymptotic properties of the 3 block filter and in particular analyze in detail the resulting 3 block algebraic riccati equation the noncausal nature of discrete time descriptor dynamics implies that future dynamics may provide some information about the present state a modified form for the descriptor kalman filter that takes this information into account is presented

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

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

curve evolution implementation of the mumford shah functional for image segmentation denoising interpolation and magnification we first address the problem of simultaneous image segmentation and smoothing by approaching the mumford shah 1989 paradigm from a curve evolution perspective in particular we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours each gradient step involves solving an optimal estimation problem for the data within each region connecting curve evolution and the mumford shah functional with the theory of boundary value stochastic processes the resulting active contour model offers a tractable implementation of the original mumford shah model i e without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner various implementations of this algorithm are introduced to increase its speed of convergence we also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions next by generalizing the data fidelity term of the original mumford shah functional to incorporate a spatially varying penalty we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing this more general model leads us to a novel pde based approach for simultaneous image magnification segmentation and smoothing thereby extending the traditional applications of the mumford shah functional which only considers simultaneous segmentation and smoothing

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

stability stochastic stationarity and generalized lyapunov equations for two point boundary value descriptor systems the authors introduce the concept of internal stability for two point boundary value descriptor systems tpbvdss since tpbvdss are defined only over a finite interval the concept of stability is not easy to formulate the definition which is used consists of requiring that as the length of the interval of definition increases the effect of boundary conditions on states located close to the center of the interval should go to zero stochastic tpbvdss are studied and the property of stochastic stationarity is characterized in terms of a generalized lyapunov equation satisfied by the variance of the boundary vector a second generalized lyapunov equation satisfied by the state variance of a stochastically stationary tpbvds is also introduced and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability

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

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

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

generating new designs using union operations abstract in many real world problems we want to generate new designs ideas by taking unions of subsets of a given basic set of potentially good designs primitives this paper proposes a streamlined algorithm of finding distinct unions of subsets of the primitives we show how to use the algorithm to eliminate a number of redundant unions of subsets of the primitives in the process of finding all distinct unions theoretical results are derived to justify the algorithm applications of the algorithm are also discussed

a wreath product group approach to signal and image processing i multiresolution analysis we propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and in particular image processing we pay significant attention to groups constructed as wreath products of cyclic groups within this large class of groups our approach recovers the discrete fourier transform dft the haar wavelet transform various multichannel pyramid filter banks and other aspects of multiresolution analysis as special cases of a more general phenomenon in addition the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals our approach relies on a synthesis of ideas found in the early work of holmes 1987 1990 karpovsky and trachtenberg 1985 and others on noncommutative filtering as well as diaconis s 1989 spectral analysis approach to understanding data

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

multiscale recursive estimation data fusion and regularization we describe a framework for modeling stochastic phenomena at multiple scales and for their efficient estimation or reconstruction given partial and or noisy measurements which may also be at several scales in particular multiscale signal representations lead naturally to pyramidal or tree like data structures in which each level in the tree corresponds to a particular scale of representation a class of multiscale dynamic models evolving on dyadic trees is introduced the main focus of this paper is on the description analysis and application of an extremely efficient optimal estimation algorithm for this class of models this algorithm consists of a fine to coarse filtering sweep followed by a coarse to fine smoothing step corresponding to the dyadic tree generalization of kalman filtering and rauch tung striebel smoothing the kalman filtering sweep consists of the recursive application of three steps a measurement update step a fine to coarse prediction step and a fusion step we illustrate the use of our methodology for the fusion of multiresolution data and for the efficient solution of fractal regularizations of ill posed signal and image processing problems encountered

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

linear stochastic differential equations with boundary conditions we study linear stochastic differential equations with affine boundary conditions the equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution the solution is not adapted to the driving brownian motion and we use the extended stochastic calculus of nualart and pardoux 16 to analyse them we give analytical necessary and sufficient conditions for existence and uniqueness of a solution we establish sufficient conditions for the existence of probability densities using both the malliavin calculus and the co aera formula and give sufficient conditions that the solution be either a markov process or a markov field

a signal processing approach to fourier analysis of ranking data the importance of phase ranking data is a type of data obtained in some elections in customer surveys as well as from web search results such data may be considered as a type of signal defined on the group of permutations of n objects denoted sn there exists a fourier transform for sn obtained from group representation theory which is well known in the mathematics literature however previous work has not approached the transform from a signal processing perspective in particular there is no discussion of what constitutes magnitude and phase nor any analysis of what phase information might tell us beyond a well known connection to group translation this paper explores the properties of the phase spectrum of ranking data in particular a novel contribution is the formulation of the bispectrum for ranking data which may be used for studying phase linearity analysis of two well known ranking data sets shows that they are surprisingly well fit by linear phase approximations

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

internal models and recursive estimation for 2 d isotropic random fields efficient recursive smoothing algorithms are developed for isotropic random fields that can be obtained by passing white noise through rational filters the estimation problem is shown to be equivalent to a countably infinite set of 1 d separable two point boundary value smoothing problems the 1 d smoothing problems are solved using a markovianization approach followed by a standard 1 d smoothing algorithm the desired field estimate is then obtained as properly weighted sum of the 1 d smoothed estimates the 1 d two point boundary value problems are also shown to have the same asymptotic properties and yield a stable spectral factorization of the power spectrum of the isotropic random fields

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

construction and applications of discrete time smoothing error models we present a unified perspective on techniques for constructing both forward and backward markovian realizations of the error process associated with discrete time fixed interval smoothing several alternative approaches are presented each of which provides additional insight and connections with other results in linear estimation theory in addition two applications of such models are described the first is a problem in the validation of an error model for a measurement system while the second concerns the problems of updating and combining smoothed estimates we also present an example of the use of our updating solution for the mapping of random fields based on several sets of data tracks

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

trajectory planning for systems with a multiplicative stochastic uncertainty a trajectory planning problem for linear stochastic differential equations is considered in this paper the aim is to control the system such that the expected value of the output interpolates given points at given times while the variance and an integral cost of the control effort are minimized the solution is obtained by using the dynamic programming technique which allows for several possible generalizations the results of this paper can be used for control of systems with a multiplicative stochastic disturbance on the state vector and systems with a stochastic growth rate this is frequently the case in investment problems biomathematics and control theory

necessary and sufficient conditions on the parameters of 1 d gaussian markov processes the valid parameter space of infinite and finite length 1 d noncausal gaussian markov gm processes is investigated for the infinite length process the authors show that the valid parameter space admits a simple description the procedure the authors present to obtain such a description is readily implementable for any order process the authors apply this procedure to a nontrivial special class namely the second order 1 d gm processes the authors obtain a simple and complete description of the valid parameter space for this class which they observe to be considerably larger than the one implied by the previously known sufficient condition for the finite length process it is shown that the valid parameter space does not admit a simple description the set of conditions for the infinite length process however serves as a good set of sufficient conditions for the finite length case the results readily apply to 2 d gaussian markov random fields gmrfs with a separable autocorrelation

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

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

a hamiltonian formulation of risk sensitive linear quadratic gaussian control we consider the optimal control problem with time homogeneous linear and in general non markov dynamics and risk sensitive criterion appeal to the extremal principle prescribed by the risk sensitive certainty equivalence principle rscep yields a symmetric equation system indicating that the extended hamiltonian formulation generalizes naturally to the risk sensitive case the conjugate variable of a hamiltonian formulation now has an interpretation in terms of forecasted process noise and the rscep in fact provides a stochastic maximum principle for which all variables have a clear interpretation and the desired measurability properties in the infinite horizon case meaningful under generalized controllability conditions optimal control is determined explicitly in terms of a canonical factorization for the case of imperfect process observation the rscep leads to coupled equation systems that can again be solved in terms of canonical factorizations in the time invariant stationary case

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

weak approximations a malliavin calculus approach we introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating our main assumption is that the process and the initial random variable have to be smooth in the malliavin sense furthermore if the inverse of the malliavin covariance matrix associated with the process under consideration is sufficiently integrable then approximations for densities and distributions can also be achieved we apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions

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

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated

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

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 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

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

boundary value processes estimation and identification recent results obtained for boundary value processes and the associated smoothing and identification problems are presented in this paper both lumped and distributed parameter models are considered some open problems are discussed and the fundamental mathematical difficulties that arise in studying nonlinear extensions of the proposed models are mentioned

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

automatic generation of fast discrete signal transforms this paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically the class includes but is not limited to the discrete fourier and the discrete trigonometric transforms this is achieved by finding fast sparse matrix factorizations for the matrix representations of these transforms unlike previous methods the algorithm is entirely automatic and uses the defining matrix as its sole input the sparse matrix factorization algorithm consists of two steps first the symmetry of the matrix is computed in the form of a pair of group representations second the representations are stepwise decomposed giving rise to a sparse factorization of the original transform matrix we have successfully demonstrated the method by computing automatically efficient transforms in several important cases for the dft we obtain the cooley tukey 1965 fft for a class of transforms including the dct type ii the number of arithmetic operations for our fast transforms is the same as for the best known algorithms our approach provides new insights and interpretations for the structure of these signal transforms and the question of why fast algorithms exist the sparse matrix factorization algorithm is implemented within the software package arep

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

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

linear optimal designs and optimal contingency plans in optimal design problems we try to select a subset from the opportunity set for commitment for a period of time as the external parameters can vary with time optimal contingency plans must be constructed to cope with the external conditions we describe the formulation and methods for locating potentially good designs and their corresponding optimal contingency plans using a multiple criteria and multiple constraint level mc2 simplex method to solve the optimal design problems which have been shown to become amenable to the techniques of decision making under uncertainty

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

a new algorithm to compute the discrete cosine transform a new algorithm is introduced for the 2m point discrete cosine transform this algorithm reduces the number of multiplications to about half of those required by the existing efficient algorithms and it makes the system simpler

multiscale systems kalman filters and riccati equations an algorithm analogous to the rauch tung striebel algorithm spl minus consisting of a fine to coarse kalman filter like sweep followed by a coarse to fine smoothing step spl minus was developed previously by the authors ibid vol 39 no 3 p 464 78 1994 in this paper they present a detailed system theoretic analysis of this filter and of the new scale recursive riccati equation associated with it while this analysis is similar in spirit to that for standard kalman filters the structure of the dyadic tree leads to several significant differences in particular the structure of the kalman filter error dynamics leads to the formulation of an ml version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the hamiltonian for the smoothing problem in addition the notion of stability for dynamics requires some care as do the concepts of reachability and observability using these system theoretic constructs the stability and steady state behavior of the fine to coarse kalman filter and its riccati equation are analysed

wavelet packets of nonstationary random processes contributing factors for stationarity and decorrelation the paper addresses the analysis and interpretation of second order random processes by using the wavelet packet transform it is shown that statistical properties of the wavelet packet coefficients are specific to the filtering sequences characterizing wavelet packet paths these statistical properties also depend on the wavelet order and the form of the cumulants of the input random process the analysis performed points out the wavelet packet paths for which stationarization decorrelation and higher order dependency reduction are effective among the coefficients associated with these paths this analysis also highlights the presence of singular wavelet packet paths the paths such that stationarization does not occur and those for which dependency reduction is not expected through successive decompositions the focus of the paper is on understanding the role played by the parameters that govern stationarization and dependency reduction in the wavelet packet domain this is addressed with respect to semi analytical cumulant expansions for modeling different types of nonstatonarity and correlation structures the characterization obtained eases the interpretation of random signals and time series with respect to the statistical properties of their coefficients on the different wavelet packet paths

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

optimal smoothing for noncausal 2 d systems based on a descriptor model an optimal smoothing problem for noncausal two dimensional systems is formulated and solved based on the descriptor model with coefficient functions using this model a forward and backward recursion procedure is obtained it is shown that for the existence of solutions pointwise reachability and observability of the descriptor model at infinite poles are important

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

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

on complementary models and fixed interval smoothing a new algorithm is derived for the standard fixed interval linear smoothing problem in which the signal is generated by a state model the structure of this new algorithm allows the smoothed estimate to be easily updated in response to a change in the initial state covariance matrix pi 0 since unlike in existing algorifiuns the relevant riccati equation is entirely independent of pi 0 the derivation of the algorithm is based on properties of complementary models

optimal brightness functions for optical flow estimation of deformable motion estimation accuracy of horn and schunck s 1981 classical optical flow algorithm depends on many factors including the brightness pattern of the measured images since some applications can select brightness functions with which to paint the object it is desirable to know what patterns will lead to the best motion estimates the paper presents a method for determining this pattern a priori using mild assumptions about the velocity field and imaging process the method is based on formulating horn and schunck s algorithm as a linear smoother and rigorously deriving an expression for the corresponding error covariance function the authors then specify a scalar performance measure and develop an approach to select an optimal brightness function which minimizes this performance measure from within a parametrized class conditions for existence of an optimal brightness function are also given the resulting optimal performance is demonstrated using simulations and a discussion of these results and potential future research is given