poles and zeros at infinity of linear time varying systems the notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time varying systems this study is based on the notion of a newborn systems which is in a mathematical point of view a graded module extension over the noncommutative ring of differential operators it is proved to be a relevant generalization to the time varying case of the equivalence class for the so called restricted equivalence of rosenbrock s polynomial matrix descriptions the authors approach is intrinsic and unifies the definitions previously given in the literature in the time invariant case

incomplete cholesky factorizations with limited memory we propose an incomplete cholesky factorization for the solution of large scale trust region subproblems and positive definite systems of linear equations this factorization depends on a parameter p that specifies the amount of additional memory in multiples of n the dimension of the problem that is available there is no need to specify a drop tolerance our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p we also show that in contrast with drop tolerance strategies the new approach is more stable in terms of number of iterations and memory requirements

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

semi implicit finite volume scheme for image processing in 3d cylindrical geometry nowadays 3d echocardiography is a well known technique in medical diagnosis inexpensive echocardiographic acquisition devices are applied to scan 2d slices rotated along a prescribed direction then the discrete 3d image information is given on a cylindrical grid usually this original discrete image intensity function is interpolated to a uniform rectangular grid and then numerical schemes for 3d image processing operations e g nonlinear smoothing in the uniform rectangular geometry are used however due to the generally large amount of noise present in echocardiographic images the interpolation step can yield undesirable results in this paper we avoid this step and suggest a 3d finite volume method for image selective smoothing directly in the cylindrical image geometry specifically we study a semi implicit 3d cylindrical finite volume scheme for solving a perona malik type nonlinear diffusion equation and apply the scheme to 3d cylindrical echocardiographic images the l stability and convergence of the scheme to the weak solution of the regularized perona malik equation is proved

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

wu s method and its application to perspective viewing abstract an algebraic method for proving a class of geometry theorems recently proposed by wu is informally discussed key concepts relevant to the method are explained an application of this method to perspective viewing in image understanding is discussed finally it is outlined how the method can be helpful to prove geometry theorems involving inequalities to capture the relation of a point being in between two given points this is also illustrated using an example from image understanding

a new image flux conduction model and its application to selective image smoothing a discrete image flux conduction equation which is completely new in this field is proposed the new approach starts with formulating a discrete image flux conduction equation based on the concept of heat conduction theory based on this discrete equation the status change at a time point can be directly computed from its spatial neighborhood to more accurately estimate an image flux we have used an orthogonal wavelet basis to approximate the gradient of the intensity at each point since the proposed approach is discrete by nature it is not necessary to formulate a continuous pde to fit the discrete image data set furthermore introduction of different numerical methods to solve the pde can also be avoided since the proposed approach does not require that a pde be solved it is therefore more efficient and accurate than the conventional methods experimental results obtained using both synthetic signals and real images have demonstrated that the proposed model could effectively handle the selective image smoothing problem

dead beat control of linear periodic discrete time systems in this paper the dead beat control of linear periodic discrete time systems subject to periodic disturbances is studied necessary and sufficient conditions are given for the existence of a linear periodic controller such that both output dead beat control under periodic disturbances and state dead beat control under no disturbance are performed moreover synthesis procedures of the controller as well as of state dead beat observers are given which can be applied also to a class of bilinear systems

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

semi cancellable fractions in system theory in this paper we propose to define the transfer function of a system in a new way namely as a semi cancellable fraction scf i e a fraction in which unstable pole zero cancellations are not allowed within this new framework it is possible to be rigorous and coherent in dealing with the stabilizability of a system from an input output viewpoint not focusing on the stabilizability of infinite dimensional input output systems

discrete time linear periodic systems gramian and modal criteria for reachability and controllability this paper deals with the reachability and controllability of periodic discrete time systems first we supply two necessary and sufficient complete reachability conditions which apply to reversible and non reversible systems respectively then a necessary and sufficient complete controllability condition is provided this condition as well as the complete reachability criteria is given in terms of the reachability gramian matrix equivalent modal criteria for reachability and controllability are established in the second part of the paper

brief minimal representations of mimo time varying systems and realization of cyclostationary models this paper describes a family of minimally parameterized state space models for completely n step observable multi input multi output time varying systems and their algebraical links with equivalent input output models these results are then used to develop an algorithm for the realization of generic input output sequences generated by a periodic system a complete numerical example is also developed to illustrate all steps of the algorithm described in the paper

semi implicit covolume method in 3d image segmentation we introduce a three dimensional 3d semi implicit complementary volume numerical scheme for solving the level set formulation of riemannian mean curvature flow problem we apply the scheme to segmentation of objects with interrupted edges in 3d images the method is unconditionally stable and efficient regarding computational times the study of its experimental order of convergence on 3d examples shows its second order accuracy for smooth solutions and first order accuracy for highly singular solutions with vanishing gradients as arising in image segmentation

system equivalence for periodic models and systems in this paper the problem is considered of obtaining a periodic description in state space form of a linear process which can be modelled by linear difference equations with periodic coefficients on the basis of a polynomial time invariant description of a linear periodic process system equivalence between two such processes is introduced and studied for a given periodic causal process under an additional assumption a periodic state space description is found which is system equivalent to it it is shown that the order the characteristic multipliers and the stacked transfer matrix at any initial time of the periodic system thus obtained coincide with those of the original periodic process and that the asymptotic stability the reachability the observability the controllability the reconstructibility the stabilizability the detectability and even the jordan form of the monodromy matrix of such a system are determined by the original periodic model as well as the existence of a solution of the robust tracking and regulation problem

semi implicit complementary volume scheme for solving level set like equations in image processing and curve evolution we introduce linear semi implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems we study discretization of image selective smoothing equation of mean curvature flow type given by alvarez lions and morel 3 solution of the level set equation of osher and sethian 26 30 is also included in the study we prove l infty and w 1 1 estimates for the proposed scheme and give existence of its generalized solution in every discrete time scale step efficiency of the scheme is given by its linearity and stability preconditioned iterative solvers are used for computing arising linear systems we present computational results related to image processing and plane curve evolution

nonlinear multiscale analysis of three dimensional echocardiographic sequences introduces a new model for multiscale analysis of space time echocardiographic sequences the proposed nonlinear partial differential equation representing the multiscale analysis filters the sequence while keeping the space time coherent structures it combines the ideas of regularized perona malik anisotropic diffusion and the galilean invariant movie multiscale analysis of alvarez et al arch rat mech anal vol 123 p 200 57 1993 a numerical method for solving the proposed partial differential equation is suggested and its stability is shown computational results on synthesized and real sequences are provided a qualitative and quantitative evaluation of the accuracy of the method is presented

iterative parameter choice and multigrid methods for anisotropic diffusion denoising anisotropic diffusion methods are well known for giving good qualitative results for image denoising this paper gives a review of the anisotropic diffusion methodology and its application to image denoising we propose a fixed point iteration using a multigrid solver to solve a regularized anisotropic diffusion equation which is not only well posed but also has a nontrivial steady state solution a new regularization parameter choice method brent ncp combining brent s method and the normalized cumulative periodogram information of the misfit is also introduced we test our algorithm on several common test images with different noise levels the experimental results demonstrate the effectiveness of the anisotropic diffusion with a multigrid approach and the broad applicability of the brent ncp parameter choice algorithm

a discrete model for an ill posed nonlinear parabolic pde we study a finite difference discretization of an ill posed nonlinear parabolic partial differential equation the pde is the one dimensional version of a simplified two dimensional model for the formation of shear bands via anti plane shear of a granular medium for the discretized initial value problem we derive analytically and observed numerically a two stage evolution leading to a steady state i an initial growth of grid scale instabilities and ii coarsening dynamics elaborating the second phase at any fixed time the solution has a piecewise linear profile with a finite number of shear bands in this coarsening phase one shear band after another collapses until a steady state with just one jump discontinuity is achieved the amplitude of this steady state shear band is derived analytically but due to the ill posedness of the underlying problem its position exhibits sensitive dependence analyzing data from the simulations we observe that the number of shear bands at time t decays like t 1 3 from this scaling law we show that the time scale of the coarsening phase in the evolution of this model for granular media critically depends on the discreteness of the model our analysis also has implications to related ill posed nonlinear pdes for the one dimensional perona malik equation in image processing and to models for clustering instabilities in granular materials 2001 elsevier science b v all rights reserved

alternating krylov subspace image restoration methods alternating methods for image deblurring and denoising have recently received considerable attention the simplest of these methods are two way methods that restore contaminated images by alternating between deblurring and denoising this paper describes krylov subspace based two way alternating iterative methods that allow the application of regularization operators different from the identity in both the deblurring and the denoising steps numerical examples show that this can improve the quality of the computed restorations the methods are particularly attractive when matrix vector products with a discrete blurring operator and its transpose can be evaluated rapidly but the structure of these operators does not allow inexpensive diagonalization

the perona malik paradox the perona malik equation is a formally ill posed parabolic equation for which simple discretizations are nevertheless numerically found to be stable after discussing the background of this parad

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

evolution of convex plane curves describing anistropic motions of phase interfaces the numerical approximation schemes for solving the nonlinear initial value problem partial t b v a v x x bv x dv g with periodic boundary conditions are presented we assume that the function b is increasing and asymptotically b s 0 and b s infty so that the model describes in a sense both slow and fast diffusions the solution also may blow up in a finite time this problem arises from the evolving curves theory which was used in the construction of models of motion of phase interface in multiphase thermomechanics by angenent and gurtin arch rational mech anal 108 1989 pp 323 391 the so called curve shortening equation is included in the model our approximating solutions converge strongly in l 2 i v space to the weak solution we also derive an error estimate for semidiscretization which implies uniqueness the numerical experiments in various situations of the anisotropic curve shortening are discussed

brief polynomial approach to the control of siso periodic systems subject to input constraint the problem of stabilization of single input single output discrete time periodic systems subject to input constraint is discussed in this paper precisely the initial state of the system is assumed to belong to a given polyhedron and the scalar input is constrained to be bounded it is shown how this problem can be posed in terms of the polynomial description of the system the solution can be found by using a linear programming optimization algorithm

periodic polynomial interpretation for structural properties of linear periodic discrete time systems the structural properties of linear periodic discrete time systems are analyzed in the periodic polynomial representation it is shown that the classical polynomial approach for linear time invariant systems can be extended to periodic systems new definitions and properties are given in terms of skew polynomial rings and periodic difference algebra necessary and sufficient conditions for the characterization of reachability controllability observability and reconstructibility are given in this framework

algorithms for computing motion by mean curvature we propose a finite element algorithm for computing the motion of a surface moving by mean curvature the algorithm uses the level set formulation so that changes in topology of the surface can be accommodated stability is deduced by showing that the discrete solutions satisfy both l infty and w 1 1 bounds existence of discrete solutions and connections with brakke flows are established some numerical examples and application to related problems such as the phase field equations are also presented

an analysis of the perona malik scheme the perona malik scheme is a numerical procedure for de noising digital pictures in this work we investigate how it evolves piecewise smooth initial data in one space dimension by scaling a natural parameter that appears in the scheme in an appropriate way with respect to the grid size we obtain a meaningful continuum limit the resulting evolution is unusual it involves solving a system of heat equations coupled to each other through nonlinear boundary conditions that become singular at special times leading to gradient blow up for the solutions however the scheme suggests a natural continuation beyond each of the singular times that involves a change in the p d e system we give a convergence proof that applies under some technical restrictions on any bounded interval of time which might include singular times furthermore the proposed continuum limit has some of the essential features observed in applications of the numerical scheme

a backward forward regularization of the perona malik equation abstract it is shown that the perona malik equation pme admits a natural regularization by forward backward diffusions possessing better analytical properties than pme itself well posedness of the regularizing problem along with a complete understanding of its long time behavior can be obtained by resorting to weak young measure valued solutions in the spirit of kinderlehrer and pedregal 1992 1 and demoulini 1996 2 solutions are unique to an extent to be specified but can exhibit micro oscillations in the sense of minimizing sequences and in the spirit of material science between preferred gradient states in the limit of vanishing regularization the preferred gradients have size 0 or thus explaining the well known phenomenon of staircasing the theoretical results do completely confirm and or predict numerical observations concerning the generic behavior of solutions

output dead beat controllers and function dead beat observers for linear periodic discrete time systems in this paper two problems are considered that of finding a possibly dynamic linear periodic controller such that both the regulated output and the control input of a linear periodic discrete time system become and remain zero and that of designing a dead beat observer of a linear function of the state of the system for both these problems necessary and sufficient conditions for the existence of a solution are given and synthesis procedures of both the controller and the observer are provided by the sufficiency proofs such conditions are very simple and concise when expressed in terms of controllable and or unreconstructible states

finite poles and zeros of linear systems an intrinsic approach several kinds of finite poles and zeros are defined and their relationships are studied in an intrinsic manner some of these definitions are more precise than the usual ones all kinds of multiplicities of an input output decoupling zero are defined here in addition we believe that our approach clarifies those notions and it makes it possible to very concisely establish fundamental relationships between various poles and zeros some new relationships are derived the notions of input output stability and transfer stability are also introduced several examples illustrate the theory

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

application of grobner bases to problems of movement of a particle abstract the classical method of grobner bases for multivariate polynomials in computer algebra and the related buchberger s algorithm and its modifications for the computation of such bases are applied to some elementary problems of kinematics as well as to the classical kepler newton problem in celestial mechanics where beyond the variables in the polynomials the differential operator d appears as well the popular computer algebra system maple v and the related standard package were used for this purpose and several possibilities of using grobner bases for the proof and or the derivation of formulae in mechanics are illustrated the present results generalize well known results for the proof derivation of geometric theorems by using classical grobner bases and related techniques and they illustrate the power of commercial computer algebra systems in the aforementioned tasks in kinematics modifications and generalizations of the present approach are also possible

convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing in this article we design the semiimplicit finite volume scheme for coherence enhancing diffusion in image processing and prove its convergence to the weak solution of the problem the finite volume methods are natural tools for image processing applications since they use piecewise constant representation of approximate solutions similarly to the structure of digital images they have been successfully applied in image processing e g for solving the perona malik equation or curvature driven level set equations where the nonlinearities are represented by a scalar function dependent on a solution gradient design of suitable finite volume schemes for tensor diffusion is a nontrivial task here we present the first such scheme with a convergence proof for the practical nonlinear model used in coherence enhancing image smoothing we provide basic information about this type of nonlinear diffusion including a construction of its diffusion tensor and we derive a semiimplicit finite volume scheme for this nonlinear model with the help of covolume mesh this method is well known as the diamond cell method owing to the choice of covolume as a diamond shaped polygon further we prove a convergence of a discrete solution given by our scheme to the weak solution of the problem the proof is based on kolmogorov s compactness theorem and a bounding of a gradient in the tangential direction by using a gradient in the normal direction finally computational results illustrated in figures are discussed

stability properties of the perona malik scheme the perona malik scheme is a numerical technique for denoising digital images without blurring object boundaries edges in general solutions generated by this scheme do not satisfy a comparison principle we identify conditions under which two solutions initially ordered remain ordered and we state restricted comparison principles these allow us to study stability properties of the scheme we also explore what these results say in the limit as the discretization size goes to 0

reversible linear and nonlinear discrete time dynamics it is shown that any causal discrete time system can be realized with a reversible or invertible state variable representation this bridges the gap with continuous time where flows always satisfy this property and settles an old debate on the relevance of invertibility in nonlinear discrete dynamics as the result applies to the linear case it is shown how the classic approach should be slightly modified and several examples of the benefits that can be obtained are given difference algebra is employed for nonlinear systems it leads to a most elegant characterization of causality which enables reversibility to be demonstrated module theory suffices in the linear case which is supplemented by elementary matrix manipulations

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

nonlinear regularized reaction diffusion filters for denoising of images with textures denoising is always a challenging problem in natural imaging and geophysical data processing in this paper we consider the denoising of texture images using a nonlinear reaction diffusion equation and directional wavelet frames in our model a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures we derive a digital reaction diffusion filter that lives on graphs and show convergence of the corresponding iteration process experimental results and comparisons show very good performance of the proposed model for texture preserving denoising

young measure solutions and instability of the one dimensional perona malik equation we apply a variational approach to the one dimensional version of the widely used perona malik equation in image processing we rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2 x 2 matrices m 2x2 we then use the solutions of some heat equations as the centre of the mass for the young measure valued solutions to construct the approximate solutions by using simple laminates the approximate solutions can be viewed as solutions of a perturbation problem by w 1 w p or w 1 w infinity functions the sequences of the approximate solutions generates young measure valued solutions our results also show that the solutions of the one dimensional perona malik equation are unstable under small w 1 w infinity perturbations

gradient estimates for the perona malik equation we consider the cauchy problem for the perona malik equation u t mathrm div left frac nabla u 1 nabla u 2 right in a bounded open set omega subseteq mathbb r n with neumann boundary conditions

semi implicit finite volume scheme for solving nonlinear diffusion equations in image processing we propose and prove a convergence of the semi implicit finite volume approximation scheme for the numerical solution of the modified in the sense of catte lions morel and coll perona malik nonlinear image selective smoothing equation called anisotropic diffusion in the image processing the proof is based on l 2 a priori estimates and kolmogorov s compactness theorem the implementation aspects and computational results are discussed

cylindrical algebraic decomposition ii an adjacency algorithm for the plane in part i of the present paper we defined cylindrical algebraic decompositions cad s and described an algorithm for cad construction in part ii we give an algorithm that provides information about the topological structure of a cad of the plane informally two disjoint cells in e r r 1 are adjacent if they touch each other formally they are adjacent if their union is connected in a picture of a cad eg fig 2 of part i it is obvious to the eye which pairs of cells are adjacent however the cad algorithm of part i does not actually produce this information nor did the original version of that algorithm in collins 1975

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

a new nonlocal nonlinear diffusion of image processing abstract a novel nonlocal nonlinear diffusion is analyzed which has proven useful as a denoising tool in image processing the equation can be viewed as a new paradigm for the regularization of the well known perona malik equation the regularization is implemented via nonlinearity intensity reduction through fractional derivatives well posedness in the weak setting is established global existence and convergence to the average holds in the purely diffusive limit whereas an interesting dynamic behavior is engendered by the presence of nontrivial equilibria as the intensity of the nonlinearity is increased and comes close to the one of perona malik

a canonical decomposition of linear periodic discrete time systems in this paper a linear periodic discrete time system is studied on the basis of simple properties of the subspaces of controllable states and of unreconstructible ones a canonical structure theorem is derived this generalizes 1o such a system the classical kalman decomposition while preserving the constant dimensionality of the four subsystems which arise when a periodic continuous time system is decomposed the dynamic matrices of the non controllable and or non reconstructible subsystems are shown to be non singular at each time instant as those for a time invariant discrete time system are

a multi level geometric reasoning system for vision abstract geometry is known to play an important role in image understanding and machine vision geometric reasoning is employed in different forms in model formation and model matching problems in model based vision research geometric representations and reasoning methods are often used implicitly in the design of vision systems and algorithms used there an approach towards model based vision in which geometric and algebraic reasoning is explicit is discussed a multi level reasoning system for machine vision based on this approach is being designed and developed at general electric corporate research and development three key components of this system a hierarchical organization of geometric knowledge for its systematic and efficient use as well as to control the search space labeling algorithms and algebraic reasoning algorithms are presented the use of such a system for deriving a three dimensional model from two dimensional images as well as for using a two dimensional image for matching against another two dimensional image called view consistency problems is discussed

pole assignment problem for discrete time linear periodic systems abstract this paper considers the pole assignment problem for discrete time linear periodic systems through the use of linear periodic state variable feedback control it is shown that if the n periodic system with m inputs and n states is completely reachable then the problem can be reduced to the pole assignment problem for a discrete time linear invariant system with nm inputs and n states

model matching for discrete time periodic systems with time varying relative degree and order abstract this paper extends the well known solution for the linear time invariant model matching problem to discrete time periodic systems with time varying relative degree and order it is shown that a key step to the design of a periodic output feedback controller is to compute the stable inverse of the periodic system using input output equations this problem is solved and model matching is achieved with system internal stability