fundamental limits in sensitivity minimization multiple input multiple output mimo plants this paper provides a bridge between the h sub spl infin literature and that of fundamental limits by applying operator theoretic tools developed for the standard frequency domain model matching approach to sensitivity minimization explicit closed form expressions are given for tight bounds on the h sub spl infin norm of weighted sensitivity functions for output feedback control of linear multivariable systems

optic flow based on multi scale anchor point movement and discontinuity preserving regularization we introduce a new method to determine the flow field of an image sequence using multi scale anchor points these anchor points manifest themselves in the scale space representation of an image the novelty of our method lies largely in the fact that the relation between the scale space anchor points and the flow field is formulated in terms of soft constraints in a variational method this leads to an algorithm for the computation of the flow field that differs fundamentally from previously proposed ones based on hard constraints we show a significant performance increase when our method is applied to the yosemite image sequence a standard and well established benchmark sequence in optic flow research also it is shown that this performance is not sensitive to slight changes in the two parameters used and that with the same parameter values our method yields very good results in the rubber whale image sequence as well

on degree constrained analytic interpolation with interpolation points close to the boundary in the recent article a theory for complexity constrained interpolation of contractive functions is developed in particular it is shown that any such interpolant may be obtained as the unique minimizer of a convex weighted entropy gain in this technical note we study this optimization problem in detail and describe how the minimizer depends on weight selection and on interpolation conditions we first show that if for a sequence of interpolants the values of the entropy gain of the interpolants converge to the optimum then the interpolants converge in h 2 but not in h infin this result is then used to describe the asymptotic behavior of the interpolant as an interpolation point approaches the boundary of the domain of analyticity for loop shaping to specifications in control design it might at first seem natural to place strategically additional interpolation points close to the boundary however our results indicate that such a strategy will have little effect on the shape another consequence of our results relates to model reduction based on minimum entropy principles where one should avoid placing interpolation points too close to the boundary

building blocks for computer vision with stochastic partial differential equations we discuss the basic concepts of computer vision with stochastic partial differential equations spdes in typical approaches based on partial differential equations pdes the end result in the best case is usually one value per pixel the expected value error estimates or even full probability density functions pdfs are usually not available this paper provides a framework allowing one to derive such pdfs rendering computer vision approaches into measurements fulfilling scientific standards due to full error propagation we identify the image data with random fields in order to model images and image sequences which carry uncertainty in their gray values e g due to noise in the acquisition process the noisy behaviors of gray values is modeled as stochastic processes which are approximated with the method of generalized polynomial chaos wiener askey chaos the wiener askey polynomial chaos is combined with a standard spatial approximation based upon piecewise multi linear finite elements we present the basic building blocks needed for computer vision and image processing in this stochastic setting i e we discuss the computation of stochastic moments projections gradient magnitudes edge indicators structure tensors etc finally we show applications of our framework to derive stochastic analogs of well known pdes for de noising and optical flow extraction these models are discretized with the stochastic galerkin method our selection of spde models allows us to draw connections to the classical deterministic models as well as to stochastic image processing not based on pdes several examples guide the reader through the presentation and show the usefulness of the framework

motion competition a variational approach to piecewise parametric motion segmentation we present a novel variational approach for segmenting the image plane into a set of regions of parametric motion on the basis of two consecutive frames from an image sequence our model is based on a conditional probability for the spatio temporal image gradient given a particular velocity model and on a geometric prior on the estimated motion field favoring motion boundaries of minimal length exploiting the bayesian framework we derive a cost functional which depends on parametric motion models for each of a set of regions and on the boundary separating these regions the resulting functional can be interpreted as an extension of the mumford shah functional from intensity segmentation to motion segmentation in contrast to most alternative approaches the problems of segmentation and motion estimation are jointly solved by continuous minimization of a single functional minimizing this functional with respect to its dynamic variables results in an eigenvalue problem for the motion parameters and in a gradient descent evolution for the motion discontinuity set we propose two different representations of this motion boundary an explicit spline based implementation which can be applied to the motion based tracking of a single moving object and an implicit multiphase level set implementation which allows for the segmentation of an arbitrary number of multiply connected moving objects numerical results both for simulated ground truth experiments and for real world sequences demonstrate the capacity of our approach to segment objects based exclusively on their relative motion

a complete parameterization of all positive rational extensions of a covariance sequence in this paper we formalize the observation that filtering and interpolation induce complementary or dual decompositions of the space of positive real rational functions of degree less than or equal to n from this basic result about the geometry of the space of positive real functions we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence on the one hand by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space we are able to develop estimates on the asymptotic behavior of the schur parameters 1918 of positive rational extensions on the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence indeed motivated by its application to signal processing speech processing and stochastic realization theory this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by georgiou 1983 1987 our basic result however also enables us to analyze the robustness of this parameterization with respect to variations in the problem data the methodology employed is a combination of complex analysis geometry linear systems and nonlinear dynamics

novel all pass factorization all solutions to rational matrix equation and control application for an arbitrary rational matrix rm a new descriptor realization of its all pass factorization is presented the coefficients of the factors are expressed explicitly in the solution of a linear matrix inequality which is well known in control theory using the all pass factorization necessary and sufficient conditions for existence of a solution of an equation with rms where the unknown rm has its poles in the left closed complex half plane and it can be improper are presented as well as all solutions as an application of the latter result the notion of almost disturbance decoupling i e when the infimum of the mathscr h infty norm of the transfer matrix from the disturbance input to the controlled output over the stabilizing controllers is zero is generalized the generalization is such that instead of the whole frequency axis the frequency axis with excluded intervals of small lengths is considered the controller consists of output feedback plus disturbance feedforward designed independently a necessary and sufficient solvability condition is presented which is expressed in terms of the given data the three main results are illustrated by an example

the robust regulation problem with robust stability a subspace valued function approach among the most common purposes are the tracking of reference signals and the rejection of disturbance signals in the face of uncertainties the related design problem is called the robust regulation problem here we investigate the tradeoff between the robust regulation constraint and the requirement of robust stability with respect to perturbations in the normalized coprime factors of the plant we use an approach in terms of subspace valued functions associated to plant and controller in this way necessary and sufficient conditions for robust regulation with robust stability can be related to a simple geometric property

the h control problem with zeros on the boundary of the stability domain in this paper we study the discrete and continuous time h control problems without any assumptions on the system parameters our approach yields necessary and sufficient conditions for the existence of a suitable controller

singular j j lossless factorization for strictly proper functions singular j j lossless factorization for strictly proper functions is studied the method of infinite zero compensation derived by copeland and safonov is used to extend the results of kimura to singular systems since j j lossless factorization contains both inner outer factorization and wiener hopf spectral factorization as special cases the results obtained here unify previous results on singular factorization problems

on the nevanlinna pick interpolation problem analysis of the mcmillan degree of the solutions in this paper we investigate some aspect of the nevanlinna pick and schur interpolation problem formulated for schur functions considered on the right half plane of c we consider the well established parametrization of the solution q t s s 12 22 1 s 11 21 see e g j a ball i gohberg l rodman realization and interpolation of rational matrix functions operator theory adv appl 33 1988 1 72 h dym j contractive matrix functions reproducing kernel hilbert spaces and interpolation cbms regional conference series in mathematics no 71 amer math soc providence ri 1989 where the j inner function is completely determined by the interpolation data and s is an arbitrary schur function we then compare the relations between the realizations of q and s induced by we show in particular that s generates a solution with a low mcmillan degree if and only if s satisfies some interpolation conditions formulated on the left half plane of c this analysis can be considered to be partially complementary to the results of a lindquist c byrnes et al on caratheodory functions c byrnes a lindquist on the partial stochastic realization problem ieee trans automatic control 42 1997 1049 1070 c byrnes a lindquist s v gusev a s mateev a complete parametrization of all positive rational extension of a covariance sequence ieee trans automatic control 40 1995 1841 1857 c byrnes a lindquist s v gusev a convex optimization approach to the rational covariance extension problem tria mat 1997

an algebraic approach to h output feedback control problems abstract we derive an output feedback controller which stabilizes a system and satisfies a prescribed h norm bound of the closed loop transfer function the proposed design method is available for any system i e there are no restrictions on d12 and d21 this approach utilizes only algebraic operations thus proofs are simple and clear

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

optimal boundary interpolation and one block optimal control problem with invariant zeros on the imaginary axis and infinity we formulate a matrix interpolation problem with existing interpolation points on the imaginary axis and infinity and existing equal left and right interpolation points using the concept of parametrisation of stabilising controllers then we solve the problem of obtaining all its solutions if interpolation points at infinity are absent we show that the introduced problem is equivalent to the existing one we apply this result to solve the problem of optimal interpolation with existent interpolation points on the imaginary axis and infinity we show by an example that the solution of optimal interpolation is directly applicable to the one block optimal control with existent invariant zeros on the imaginary axis and infinity it is seen from the example that not only the transfer matrix of the closed loop system is constrained on the extended imaginary axis but also its derivatives

on general ℋ2 control from frequency to time domain this article presents a general formula for discrete time 2 control it works with the regular and singular case of 2 control i e in the case of possibly non left invertible matrices g 12 and non right invertible matrices g 21 with possible unit circle invariant zeros in the generic case it can be simplified and adapted to work with the plant transfer matrix directly without invoking the matrices of the parameterisation of stabilising controllers a further result of this article is the presented necessary and sufficient conditions for state space 2 control under only stabilisability and detectability assumptions if the conditions are satisfied an observer based 2 controller is constructed the corresponding numerical algorithm consists of solving two discrete time algebraic riccati systems darss and two eigen problems

inner outer factorization for strictly proper functions with j w axis zeros abstract this paper discussed the inner outer factorization for square real rational matrices which may have zeros on the j axis including infinity a factorization method is given in terms of the descriptor form representation of the rational matrix this method uses the solution of a generalized riccati equation we also propose an efficient computation algorithm for solving the generalized riccati equation

new results and application of singular control we reconsider the singular control problem with marginal stability of the closed loop system when the transfer matrix from the input to the output can have linearly dependent columns and zeros on the extended imaginary axis we present a new theoretical existence result and a new numerical algorithm based on finding an orthogonal transformation of the matrix pencil associated to the euler lagrange differential equations into a block triangular form we present an application in linear quadratic control of descriptor systems under the constraints of physical realizability of the control and impulse free and marginally stable closed loop system

the inverse problem of analytic interpolation with degree constraint and weight selection for control synthesis the minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint and all solutions to this interpolation problem arise in this way by a suitable choice of weights selecting appropriate weights is pertinent to feedback control synthesis where interpolants represent closed loop transfer functions in this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree there are two basic issues that we address we first characterize admissible shapes of minimizers by studying the corresponding inverse problem and then we develop effective ways of shaping minimizers via suitable choices of weights this leads to a new procedure for feedback control synthesis

j j lossless factorization for descriptor systems abstract the j j lossless factorization for descriptor systems based on the theory of conjugation is studied the result on j j lossless factorization for proper functions are extended to descriptor systems and those on j lossless factorization for square matrices are extended to nonsquare cases since j j lossless factorization contains both the inner outer factorization and the wiener hopf spectral factorization as special cases and the descriptor system contains the state space system asa special case the results obtained here unify the previous results on factorization problems also j j lossless factorization can be used to solve 4 block h control problems with j axis zeros

h suboptimal control problem with boundary constraints abstract a necessary and sufficient condition is derived for the existence of a stabilizing controller such that i a specified closed loop transfer function g er s has h norm less than one and ii g er s satisfies generalized boundary constraints on the j axis

a treatment of jw axis model matching transformation zeros in the optimal h sup 2 and h sup infinity control designs a model matching transformation mmt zero is defined as a rank deficiency condition which prevents an h sup 2 or h sup infinity optimal control problem from being transformed into an equivalent model matching problem by imposing saturation constraints and accounting for additive instrument noise in the sensor and actuator signals all mmt zeros can be eliminated

all solutions of a bitangential interpolation problem that includes boundary points all solutions of a matrix interpolation problem with purely imaginary interpolation points are given by a descriptor system realization if necessary and sufficient conditions are satisfied the first step is to reduce the interpolation problem to a minimal realization no assumptions are needed in the paper

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

simplified formula for a controller in optimal control problems we derive a simplified frequency domain formula for a controller transfer matrix for lq optimal output feedback control of stochastic systems for this purpose we apply a generalization of the wiener hopf method the generalization is characterized by the following three properties 1 we generalize the usual operator cdot on rational matrices in the traditional wiener hopf approach the same algorithms as for computation of cdot are applicable to compute the new operator 2 the matrices of the youla kuccera parametrization do not appear in the optimal controller transfer matrix bf c they appear only in its derivation even if the plant is unstable 3 unlike the traditional wiener hopf method d c youla h a jabr and j j bongiorno jr ieee trans automat control ac 21 1976 pp 319 338 where the spectral factors in the two spectral factorizations are both stable and minimum phase our spectral factors need to be minimum phase only finally three state space applications of the formula are presented

from finite covariance windows to modeling filters a convex optimization approach the trigonometric moment problem is a classical moment problem with numerous applications in mathematics physics and engineering the rational covariance extension problem is a constrained version of this problem with the constraints arising from the physical realizability of the corresponding solutions although the maximum entropy method gives one well known solution in several applications a wider class of solutions is desired in a seminal paper georgiou derived an existence result for a broad class of models in this paper we review the history of this problem going back to carath eodory as well as applications to stochastic systems and signal processing in particular we present a convex optimization problem for solving the rational covariance extension problem with degree constraint given a partial covariance sequence and the desired zeros of the shaping filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s existence result and his conjecture a generalized version of which we have recently resolved using geometric methods we also survey recent related results on constrained nevanlinna pick interpolation in the context of a variational formulation of the general moment problem

algorithms for optimal control with invariant zeros on the extended imaginary axis abstract we transform an control problem with invariant zeros on the extended imaginary axis into a regular problem then using the nevanlinna pick theorem we find necessary and sufficient existence conditions for the original control problem an algorithm and examples with invariant zeros on the extended imaginary axis are given by the same approach we also solve the singular 2 control problem

piecewise smooth dense optical flow via level sets we propose a new algorithm for dense optical flow computation dense optical flow schemes are challenged by the presence of motion discontinuities in state of the art optical flow methods over smoothing of flow discontinuities accounts for most of the error a breakthrough in the performance of optical flow computation has recently been achieved by brox et al our algorithm embeds their functional within a two phase active contour segmentation framework piecewise smooth flow fields are accommodated and flow boundaries are crisp experimental results show the superiority of our algorithm with respect to alternative techniques we also study a special case of optical flow computation in which the camera is static in this case we utilize a known background image to separate the moving elements in the sequence from the static elements tests with challenging real world sequences demonstrate the performance gains made possible by incorporating the static camera assumption in our algorithm

generalized interpolation in h infty with a complexity constraint in a seminal paper sarason generalized some classical interpolation problems for h infinity functions on the unit disc to problems concerning lifting onto h 2 of an operator t that is defined on k

a convex optimization approach to the rational covariance extension problem in this paper we present a convex optimization problem for solving the rational covariance extension problem given a partial covariance sequence and the desired zeros of the modeling filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s seminal existence result and his conjecture a stronger version of which we have resolved in byrnes et al ieee trans automat control ac 40 1995 pp 1841 1857

sensitivity shaping with degree constraint by nonlinear least squares optimization this paper presents a new approach to shaping of the frequency response of the sensitivity function in this approach a desired frequency response is assumed to be specified at a finite number of frequency points a sensitivity shaping problem is formulated as an approximation problem to the desired frequency response with a function in a class of sensitivity functions with a degree bound the sensitivity shaping problem is reduced to a finite dimensional constrained nonlinear least squares optimization problem to solve the optimization problem numerically standard algorithms for an unconstrained version of nonlinear least squares problems are modified to incorporate the constraint numerical examples illustrate how these design parameters are tuned in an intuitive manner as well as how the design proceeds in actual control problems

j lossless and extended j lossless factorizations approach for domain h control the paper addresses problems of numerical conditioning of the discrete time h x control design based on j lossless factorizations of the chain scattering representation of a plant it is demonstrated that for a sufficiently small sampling period forward shift operator techniques may become ill conditioned and numerical robustness and reliability of design procedures can be significantly improved by utilizing a so called i operator form of the source problem state space models of all i domain controllers are given the solutions are obtained by considering two coupled i domain algebraic riccati equations that can be solved by applying a generalized eigenproblem formulation numerical conditioning of these solutions can effectively be evaluated by using a suitable relative condition number an extended j lossless factorization for generalized plants with zeros on the stability boundary circle is also given a so called zero compensator technique adapted to the i domain models is utilized to cancel suc

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

synthesis of positive real multivariable feedback systems this is a tutorial paper describing the application of infinity norm optimal control theory to the synthesis of sector bounded and positive real multivariable feedback systems the paper gives a concise treatment of the practical aspects of the theory describing how it has been implemented via state space oriented computer algorithms included is a concise summary of the state space approach to infinity norm control developed by doyle and by safonov and verma based on the hankel model reduction theory and incorporating the multivariable hankel results of glover

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

non standard h control problem a generalized chain scattering representation approach the non standard h x control problem is the case where the direct feedthroughs from the input to the error and from the exogenous signal to the output are not necessarily of full rank in this paper this problem is reformulated based on the generalized chain scattering representation gcsr the gcsr approach leads naturally to a generalization of the homographic transformation the state space realization for this generalized homographic transformation and a number of fundamental cascade structures of the h x control systems are further studied in a unified framework of gcsr certain sufficient conditions for the solvability of the non standard h x control problem are therefore established via a j j lossless factorization of gcsr these results present extensions to kimura s results on the chain scattering representation csr approach to the h x control in the standard case

probabilistic detection and tracking of motion boundaries we propose a bayesian framework for representing and recognizing local image motion in terms of two basic models translational motion and motion boundaries motion boundaries are represented using a non linear generative model that explicitly encodes the orientation of the boundary the velocities on either side the motion of the occluding edge over time and the appearance disappearance of pixels at the boundary we represent the posterior probability distribution over the model parameters given the image data using discrete samples this distribution is propagated over time using a particle filtering algorithm to efficiently represent such a high dimensional space we initialize samples using the responses of a low level motion discontinuity detector the formulation and computational model provide a general probabilistic framework for motion estimation with multiple non linear models

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

solution to the positive real control problem for linear time invariant systems in this paper we study the problem of synthesizing an internally stabilizing linear time invariant controller for a linear time invariant plant such that a given closed loop transfer function is extended strictly positive real necessary and sufficient conditions for the existence of a controller are obtained state space formulas for the controller design are given in terms of solutions to algebraic riccati equations or inequalities the order of the constructed controller does not exceed that of the plant

new generating functions for a class of generalized hermite polynomials the main object of this paper is to present several presumably new families of linear bilinear and mixed multilateral generating functions for a certain interesting generalization of the classical hermite and laguerre polynomials some of these generating functions are associated with the stirling numbers of the second kind numerous known or new consequences of the results derived here also considered

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

motion segmentation by multistage affine classification we present a multistage affine motion segmentation method that combines the benefits of the dominant motion and block based affine modeling approaches in particular we propose two key modifications to a recent motion segmentation algorithm developed by wang and adelson 1994 1 the adaptive k means clustering step is replaced by a merging step whereby the affine parameters of a block which has the smallest representation error rather than the respective cluster center is used to represent each layer and 2 we implement it in multiple stages where pixels belonging to a single motion model are labeled at each stage performance improvement due to the proposed modifications is demonstrated on real video frames

the singular control problem with dynamic measurement feedback this paper is concerned with the h infty problem with measurement feedback the problem is to find a dynamic feedback from the measured output to the control input such that the closed loop system has an h infty norm strictly less than some a priori given bound gamma and such that the closed loop system is internally stable necessary and sufficient conditions are given under which such a feedback exists the only assumption that must be made is that there are no invariant zeros on the imaginary axis for two subsystems contrary to recent publications no assumptions are made on the direct feedthrough matrices of the plant it turns out that this problem can be reduced to an almost disturbance decoupling problem with measurement feedback and internal stability i e the problem in which we can make the h infty norm arbitrarily small

on h optimal sensitivity theory for siso feedback systems this paper deals with the design of feedback controllers which minimize the h infty norm of the sensitivity function suitably weighted this approach to the theory of feedback design was introduced by zames 1 and developed by zames and francis 2 in this paper the theory of sarason 3 is applied to the determination of the optimal weighted sensitivity function and an upper bound on its norm the problem of achieving small sensitivity over a specified frequency band is studied and the effect of nonminimum phase is elucidated finally a method is introduced for handling plant poles and zeros on the imaginary axis

some general families of generating functions for the laguerre polynomials abstract the object of the present paper is to develop rather systematically some general families of bilinear bilateral or mixed multilateral generating functions for the classical laguerre polynomials numerous straightforward consequences of some of the results considered here frequently appear in the literature especially from the viewpoint of lie groups and lie algebras it is also pointed out how the main generating functions can be suitably applied to derive numerous further results involving laguerre polynomials and various other related polynomials

on extended j j lossless factorization abstract the extended j j lossless factorization for discrete time rational matrix functions with zeros on the unit circle is considered a necessary and sufficient condition for the existence of the extended j j lossless factorization is obtained the condition is given in terms of the original system parameters in the form of a generalized eigenvalue problem and a discrete time algebraic riccati equation state space representations for the extended j j lossless factorization are provided

the tau method and a new preconditioner in relations to the order of linear ordinary differential equations using a modified form of the chebyshev or legendre and gegenbauer polynomials some particular integral operators are introduced these are used to give a factorization of the operators arising from the application of the chebyshev or legendre tau method the new tau method presented in this article is then compared with the standard tau method and preconditioned method of cabos the new tau method shows a superior performance an analysis of error and a bound for condition number is given numerical examples applying iterative solvers show dramatic reduction in condition number and improved convergence for the tau method with the new preconditioner

a computational framework and an algorithm for the measurement of visual the robust measurement of visual motion from digitized image sequences has been an important but difficult problem in computer vision this paper describes a hierarchical computational framework for the determination of dense displacement fields from a pair of images and an algorithm consist ent with that framework our framework is based on the separation of the image intensity information as well as the process of measuring motion according to scale the large scale intensity information is first used to obtain rough estimates of image motion which are then refined by using intensity information at smaller scales the estimates are in the form of displacement or velocity vectors for pixels and are accompanied by a direction dependent confidence measure a smoothness constraint is employed to propagate the measurements with high confidence to their neigboring areas where the confidences are low at all levels the computations are pixel parallel uniform across the image and based on information from a small neighborhood of a pixel for our algorithm the local displacement vectors are determind by mini mizing the sum of squared differences ssd of intensities the confidence measures are derived from the shape of the ssd surface and the smoothness constraint is cast in the form of energy minimization results of applying our algorithm to pairs of real images are included in addition to our own

reduced order controllers for the h control problem with unstable invariant zeros this paper addresses the existence and design methods of reduced order controllers for the h control problem with unstable invariant zeros in the state space realization of the transfer function matrix from the control input to the controlled error or from the exogenous input to the observation output where the realization is induced from a stabilizable and detectable realization of the generalized plant this paper presents a new controller degree bound for the h control problem in terms of the minimal rank of the system matrix pencils of these two transfer function matrices in the unstable region when the unstable invariant zero exists this paper shows that reduced order controllers with orders strictly less than that of the generalized plant exist if the h control problem is solvable moreover this paper shows that the computational problem of finding the controllers with the new degree bound is convex by providing two linear matrix inequality based design methods algorithms for constructing the reduced order controllers the results developed in this paper are valid both for the continuous and discrete time h control problems

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

brief paper nonsmooth optimization for multiband frequency domain control design multiband frequency domain synthesis consists in the minimization of a finite family of closed loop transfer functions on prescribed frequency intervals this is an algorithmically difficult problem due to its inherent nonsmoothness and nonconvexity we extend our previous work on nonsmooth h synthesis to develop a nonsmooth optimization technique to compute local solutions to multiband synthesis problems the proposed method is shown to perform well on illustrative examples

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

interpolation with constraint on frequency region and systems control application abstract this paper elaborates on a bitangential interpolation problem with a norm constraint of the interpolant on a given frequency region and on its generalization and proves that it admits a solution a motivation from systems control for the problem is given and the solution is applied for construction of a stabilizing control for one block plants such that the closed loop transfer matrix is norm constrained on the frequency region illustrative numerical examples are given

sensitivity minimization in an h norm parametrization of all suboptimal solutions a number of problems in the control of linear feedback systems can be reduced to the following we are given three stable rational matrix functions k of sizes p 1 x q 1 p 1 x q 2 and p 2 x q 1 respectively and seek a stable rational q 2 x p 2 matrix function s so as to minimize k s we assume that p 1 q 2 p 2 q 1 and that and have maximal rank q 2 and p 2 respectively on the j axis given a tolerance level sufficiently large we obtain a linear fractional map g f 0 11 g 0 12 0 13 0 21 g 0 22 0 23 1 such that f k s with s stable and f if and only if g is a stable q 2 x p 2 matrix function with g 1 the computation of 0 0 ij 1 i 2 1 j 3 reduces to solving a pair of symmetric wiener hopf factorization problems for the special case where i q2 0 t i p2 0 and k not necessarily stable to which the general case can be reduced we provide explicit state space formulae for 0 in terms of a state space realization of k and the

h control problem with jω axis zeros abstract in this paper we discuss the so called one block h control problem in the case where the plant has j axis including infinity poles and zeros first we derive a necessary and sufficient condition for the h model matching problem to have a solution in the transfer function setting second we give a solvability condition which enables us to check the solvability by i examining matrix norm conditions related to j axis zeros and ii solving two generalized riccati equations finally we extend our result to the standard h problem

computation of component image velocity from local phase information we present a technique for the computation of 2d component velocity from image sequences initially the image sequence is represented by a family of spatiotemporal velocity tuned linear filters component velocity computed from spatiotemporal responses of identically tuned filters is expressed in terms of the local first order behavior of surfaces of constant phase justification for this definition is discussed from the perspectives of both 2d image translation and deviations from translation that are typical in perspective projections of 3d scenes the resulting technique is predominantly linear efficient and suitable for parallel processing moreover it is local in space time robust with respect to noise and permits multiple estimates within a single neighborhood promising quantiative results are reported from experiments with realistic image sequences including cases with sizeable perspective deformation

some generalizations of the euler knopp transformation the purpose of this paper is to construct a generalization of the euler knopp transformation using this one may recover previously known transformations derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well known functions

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

reduced order controllers for continuous and discrete time singular h control problems based on lmi this paper describes a method for constructing a petri net controller for a discrete event system modeled by a petri net the controller consists only of places and arcs and is computed based on the concept of petri net place invariants the size of the controller is proportional to the number of constraints that must be satisfied this method is computationally efficient and can accommodate constraints written as boolean logic formulas in the conjunctive normal form or algebraic inequalities that contain elements of the marking and or the firing vectors

mathscr h _ infty control of descriptor systems possessing invariant zeros on the imaginary axis and infinity we present an algorithm for suboptimal mathscr h infty control of descriptor systems possessing invariant zeros on the imaginary axis and infinity it is proved that the algorithm works also in an optimality case the algorithm is illustrated by examples

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

some families of generating functions associated with the stirling numbers of the second kind the object of this paper is to present a systematic introduction to and several interesting applications of a general result on generating functions associated with the stirling numbers of the second kind for a fairly wide variety of special functions and polynomials in one two and more variables the main results given below are shown to apply not only to the classical orthogonal polynomials including for example the jacobi polynomials which contain as their special cases the gegenbauer or ultraspherical polynomials the legendre or spherical polynomials and the chebyshev polynomials of the first and second kinds and the laguerre polynomials and to their various extensions and generalizations studied in recent years but indeed also to a class of generalized hypergeometric functions the lauricella polynomials in several variables and the familiar lagrange polynomials which arise in certain problems in statistics relevant connections of some of these families of generating functions with various known results are also indicated

brief paper generalized asymptotic regulation with guaranteed h2 performance an lmi solution a multi objective controller synthesis problem is considered in which a set of generalized asymptotic regulation constraints are to be satisfied while also guaranteeing a desired level of performance measured in terms of the asymptotic variance of a performance output in response to a white noise random process with zero mean and identity covariance matrix the generalized regulation constraints are expressed as bounds on the state steady peak in response to disturbances generated by an anti stable autonomous exogenous system with nonzero initial states the controllers that guarantee these constraints can be realized by replicating the dynamics of the exogenous system within a certain structure formed by gain matrices which are subject to convex constraints and an accompanying controller with which the feedback loop is to be stabilized formulating the design of the accompanying controller in a way to guarantee the additional performance objective the overall design is rendered tractable over a set of free variables in terms of which the synthesis of a suitable controller is also outlined the order of the controller is equal to the order of the plant plus the order of the exogenous system by an adaptation of the developed techniques a solution is also provided for the problem of generalized asymptotic regulation with suboptimal transient response

closed loop shaping based on nevanlinna pick interpolation with a degree bound this note presents a novel method for shaping the frequency response of a single input single output closed loop system based on the theory of nevanlinna pick interpolation with degree constraint the method imposes a degree bound on the closed loop transfer function and searches for a function with a desired frequency response numerical examples illustrate the potential of the method in designing controllers with lower degrees than the ones obtained by conventional h sup spl infin controller design methods with weighting functions

h control by state feedback for plants with zeros on the imaginary axis algebraic tests are derived for the suboptimality of some parameter in the h infty optimization problem by state feedback where the finite zero structure of the plant is not restricted as an application of these characterizations a quadratically convergent algorithm for the computation of the optimal value is presented the suboptimality tests are based on new solvability criteria for general algebraic riccati inequalities that are of independent interest

a reformulation of augmented basic interpolation problem and an application to h control abstract in this paper we present a new formulation of the augmented basic interpolation problem abip with rational matrices in terms of the stability of four rational matrices so that the abip transforms into a purely linear algebraic problem actually the existing interpolation condition given by an integral is replaced by stability of a rational matrix the new condition is applied to the h optimal control of one block plants having imaginary axis invariant zeros a new parameter in the parametrization of h optimal controllers is revealed which is given in terms of the derivative of the closed loop system at the imaginary axis invariant zeros of the plant the h optimal control algorithm is illustrated by an example and compared to the existing algorithms

a characterization of all solutions to the four block general distance problem all solutions to the four block general distance problem which arises in h infty optimal control are characterized the procedure is to embed the original problem in an all pass matrix which is constructed it is then shown that part of this all pass matrix acts as a generator of all solutions special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all pass transfer functions as an application necessary and sufficient conditions are found for the existence of an h infty optimal controller following that a descriptor representation of all solutions is derived

non euclidian metrics and the robust stabilization of systems with parameter uncertainty this paper considers from a complex function theoretic point of view certain kinds of robust synthesis problems in particular we use a certain kind of metric on the disk the hyperbolic metric which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in non euclidian geometry it is shown that several apparently different problems can be treated in a unified general framework a new result on the gain margin problem for multivariable plants is also given finally we apply our methods to systems with real zero or pole variations