a study on the spectrum of the sampled data transfer operator with application to robust exponential stability problems this paper begins by studying some spectral properties of the transfer operators of sampled data systems described by applying the lifting technique through a nonasymptotic characterization of the transfer operator its spectrum is determined in terms of finite dimensional eigenvalue problems then it is shown that a close connection with such eigenvalue problems and the exponential stability condition can be exploited to study the robust internal exponential stability problem of sampled data systems since the transfer operator is relevant to input output characteristics the relationship between input output stability and internal stability is also discussed in the context of sampled data systems

on a fractional reaction diffusion system applied to image decomposition and restoration abstract in this paper we propose a new model using a fractional reaction diffusion system for image restoration and image decomposition which is motivated by work of s osher a sole and l vese s our model decomposes the degraded image into cartoon component belonging to a fractional sobolev space and textured component belonging to a negative hilbert space we are interested in the existence and uniqueness of weak solutions of the system obtained from the model to overcome the difficulties of the singularity in the system we use a regularization method we also give a numerical algorithm for solving the system and present numerical experimental results to show the efficiency of our model

on the essential spectrum of the transfer operator for expanding markov maps the essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail to this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inc k is a disk whose radius is related to the free energy of the liapunov exponent

eventual disconjugacy of self adjoint difference equations abstract given an eventually disconjugate second order self adjoint difference equation c n e n a n e n 1 0 where c n 0 and is not assumed to be positive this paper studies the eventual disconjugacy of the perturbed difference equation c n y n a n f n y n 1 0 analogous to a theorem of willett for corresponding differential equations a necessary and sufficient condition for eventual disconjugacy in terms of a convergent sequence of averaged sums of f n is obtained

riccati techniques and discrete oscillations abstract oscillation and nonoscillation criteria are obtained for the second order difference equation r n x n p n x n 1 0 where x n x n 1 x n is the forward difference operator and r n p n are sequences of real numbers the main techniques involved are riccati equations of the first and second levels

iterated function systems ruelle operators and invariant projective measures we introduce a fourier based harmonic analysis for a class of discrete dynamical systems which arise from iterated function systems our starting point is the following pair of special features of these systems 1 we assume that a measurable space x comes with a finite to one endomorphism r x x which is onto but not one to one 2 in the case of affine iterated function systems ifss in r d this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets b l in r d of the same cardinality which generate complex hadamard matrices our harmonic analysis for these iterated function systems ifs x is based on a markov process on certain paths the probabilities are determined by a weight function w on x from w we define a transition operator r w acting on functions on x and a corresponding class h of continuous r w harmonic functions the properties of the functions in h are analyzed and they determine the spectral theory of l 2 for affine ifss we establish orthogonal bases in l 2 these bases are generated by paths with infinite repetition of finite words we use this in the last section to analyze tiles in r d

robust stabilization of sampled data systems to structured lti perturbations sampled data systems with stable structured lti linear time invariant perturbations in the half plane algebra applied to the continuous time plant are considered necessary and sufficient conditions are derived for robust l sub 2 stability of such systems an example is provided to illustrate the results which shows that the small gain theorem can be an extremely conservative robustness test in this sampled data context

dimension estimation of discrete time fractional brownian motion with applications to image texture classification fractional brownian motion fbm is a suitable description model for a large number of natural shapes and phenomena in applications it is imperative to estimate the fractal dimension from sampled data namely discrete time fbm dfbm to this aim the increment of dfbm referred to as discrete time fractional gaussian noise dfgn is invoked as an auxiliary tool the regular part of dfgn is first filtered out via levinson s algorithm the power spectral density of the regular process is found to satisfy a power law that its exponent can be well fitted by a quadratic function of fractal dimension a new method is then proposed to estimate the fractal dimension of dfbm from the given data set the computational complexity and statistical properties are investigated moreover the proposed algorithm is robust with respect to amplitude scaling and shifting as well as time shifting on the data finally the effectiveness of this estimator is demonstrated via a classification problem of natural texture images

a problem of hartman and wintner approximation for discrete perturbations abstract consider the second order self adjoint difference equation cn xn an fn xn 1 0 as a perturbation of the eventually disconjugate difference equation cn zn an zn 1 0 where cn 0 asymptotic approximation for the fundamental system of solutions of the perturbed equation are expressed explicitly in terms of the coefficients and the principal or recessive solution of the unperturbed equation in particular the coefficient cn is allowed to be oscillatory and we do not assume absolute summability conditions on fn results obtained are applied to a second order poincare difference equation whose unperturbed equation has a double characteristic root nonlinear and nonhomogeneous perturbations are also considered an example is given for illustration

estimation of 1 f noise several models have emerged for describing 1 f sup spl gamma noise processes based on these various techniques for estimating the properties of such processes have been developed this paper provides theoretical analysis of a new wavelet based approach which has the advantages of having low computational complexity and being able to handle the case where the 1 f sup spl gamma noise might be embedded in a further white noise process however the analysis conducted here shows that these advantages are balanced by the fact that the wavelet based scheme is only consistent for spectral exponents spl gamma in the range spl gamma spl isin 0 1 this is in contradiction to the results suggested in previous empirical studies when spl gamma spl isin 0 1 this paper also establishes that wavelet based maximum likelihood methods are asymptotically gaussian and efficient finally the asymptotic rate of mean square convergence of the parameter estimates is established and is shown to slow as spl gamma approaches one combined with a survey of non wavelet based methods these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1 f sup spl gamma noise processes

fast cartoon texture image filters can images be decomposed into the sum of a geometric part and a textural part in a theoretical breakthrough y meyer oscillating patterns in image processing and nonlinear evolution equations providence ri american mathematical society 2001 proposed variational models that force the geometric part into the space of functions with bounded variation and the textural part into a space of oscillatory distributions meyer s models are simple minimization problems extending the famous total variation model however their numerical solution has proved challenging it is the object of a literature rich in variants and numerical attempts this paper starts with the linear model which reduces to a low pass high pass filter pair a simple conversion of the linear filter pair into a nonlinear filter pair involving the total variation is introduced this new proposed nonlinear filter pair retains both the essential features of meyer s models and the simplicity and rapidity of the linear model it depends upon only one transparent parameter the texture scale measured in pixel mesh comparative experiments show a better and faster separation of cartoon from texture one application is illustrated edge detection

a fractional laplace equation regularity of solutions and finite element approximations this paper deals with the integral version of the dirichlet homogeneous fractional laplace equation for this problem weighted and fractional sobolev a priori estimates are provided in terms of the holder regularity of the data by relying on these results optimal order of convergence for the standard linear finite element method is proved for quasi uniform as well as graded meshes some numerical examples are given showing results in agreement with the theoretical predictions

robust stability and performance analysis of sampled data systems the authors consider the robust stability and performance analysis of a general linear interconnection of a continuous time plant and a discrete time controller via sample and hold devices they obtain necessary and sufficient conditions for the robust stability of the feedback system consisting of an uncertain plant represented as a known linear time invariant nominal model with time varying norm bounded uncertainty and a shift invariant discrete time controller these results show that the condition arising from the small gain theorem is necessary and sufficient when the uncertainties belong to a class of linear systems also similar results are derived for a case of structured uncertainty both l sub 2 and l sub infinity signal norms are considered robust performance analysis problem is solved by converting it to a robust stability analysis problem

the algebra of harmonic functions for a matrix valued transfer operator abstract we analyze matrix valued transfer operators we prove that the fixed points of transfer operators form a finite dimensional c algebra for matrix weights satisfying a low pass condition we identify the minimal projections in this algebra as correlations of scaling functions i e limits of cascade algorithms

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

an iterative method with guaranteed convergence for robust performance analysis of sampled data systems against parameter uncertainties this paper is concerned with robust performance analysis of sampled data systems in particular we consider the uncertainty of parameters in the continuous time plant and study the problem of determining the allowable range of the uncertainty around the origin or equivalently the allowable range of the parameter around the nominal value over which the sampled data system remains internally stable and a prescribed l2 induced norm or equivalently h performance level is retained we provide an iterative procedure with guaranteed convergence that gives an exact allowable parameter range together with rigorous arguments for the convergence this method is free from the assumption that an associated sampled data system has a compact transfer operator and thus is readily applicable also to such cases when the uncertain parameter appears in the system dynamics e g in polynomial forms or linear fractional forms

positive realness analysis of sampled data systems and its applications in this paper we first study strong positive realness of sampled data systems and introduce a measure called positive realness gap index we show that this index can be computed efficiently with a bisection method and provide state space formulas for its computation the importance of this index lies in that it is useful for robust stability analysis of sampled data systems an iterative procedure for computing an exact robust stability margin is given and illustrated through a numerical example

oscillation results for second order scalar and matrix difference equations we consider the self adjoint second order scalar difference equation rn xn pnxn 1 0 1 and the matrix system rn xn pnxn 1 0 2 where rn 0 pn 0 rn 0 pn 0 are sequences of real numbers d d hermitian matrices with rn 0 rn 0 we obtain some new oscillation criteria through change of variable techniques combined with consideration of 1 and 2 as perturbations of the cases pn 0 pn 0

adaptive smoothing respecting feature directions accurate extraction of the image feature directions is essential to image smoothing and other image processing tasks we show that the gradient based feature direction extraction method can be very erroneous the gradient is too local and it cannot detect oscillations we have developed two new methods the hessian method an approach using higher order differentiations and the gabor 1946 method an approach using space frequency analysis

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

noise removal with gauss curvature driven diffusion in this paper we propose the use of the gauss curvature in a gauss curvature driven diffusion equation for noise removal the proposed scheme uses the gauss curvature as the conductance term and controls the amount of diffusion the main advantage of the scheme is that it preserves important structures such as straight edges curvy edges ramps corners small scaled features etc

adaptive smoothing via contextual and local discontinuities a novel adaptive smoothing approach is proposed for noise removal and feature preservation where two distinct measures are simultaneously adopted to detect discontinuities in an image inhomogeneity underlying an image is employed as a multiscale measure to detect contextual discontinuities for feature preservation and control of the smoothing speed while local spatial gradient is used for detection of variable local discontinuities during smoothing unlike previous adaptive smoothing approaches two discontinuity measures are combined in our algorithm for synergy in preserving nontrivial features which leads to a constrained anisotropic diffusion process that inhomogeneity offers intrinsic constraints for selective smoothing thanks to the use of intrinsic constraints our smoothing scheme is insensitive to termination times and the resultant images in a wide range of iterations are applicable to achieve nearly identical results for various early vision tasks our algorithm is formally analyzed and related to anisotropic diffusion comparative results indicate that our algorithm yields favorable smoothing results and its application in extraction of hydrographic objects demonstrates its usefulness as a tool for early vision

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

numerical approximations for a nonlocal evolution equation in this paper we study numerical approximations of continuous solutions to the nonlocal p laplacian type diffusion equation u t t x int omega j x y u t y u t x p 2 u t y u t x dy first we find that a semidiscretization in space of this problem gives rise to an ode system whose solutions converge uniformly to the continuous one as the mesh size goes to zero moreover the semidiscrete approximation shares some properties of the continuous problem it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity next we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties in addition we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile finally we present some numerical experiments that illustrate our results

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

tight frames of compactly supported affine wavelets this paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by daubechies commun pure appl math 41 909 1988 for each integer n 1 a family of wavelet functions having support 0 2n 1 is constructed such that jk x 2j 2 2jx k kj k z is a tight frame of l2 r i e for every f l2 r f c jk jk f jk for some c 0 this family is parametrized by an algebraic subset vn of r4n furthermore for n 2 a proper algebraic subset wn of vn is specified such that all points in vn outside of wn yield orthonormal bases the relationship between these tight frames and the theory of group representations and coherent states is discussed

frames generated by actions of countable discrete groups we consider dual frames generated by actions of countable discrete groups on a hilbert space module frames in a class of modules over a group algebra are shown to coincide with a class of ordinary frames in a representation of the group this has applications to shift invariant spaces and wavelet theory one of the main findings in this paper is that whenever a shift invariant subspace in l 2 n has compactly supported dual frame generators then it also has compactly supported bi orthogonal generators the crucial part in the proof is a theorem by swan that states that every finitely generated projective module over the laurent polynomials in n variables is free

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

a new iterated two band diffusion equation theory and its application we propose an iterated two band filtering method to solve the selective image smoothing problem we prove that a discrete computation step in an iterated nonlinear diffusion based filtering algorithm is equivalent to a sequence of operations including decomposition regularization and then reconstruction in the proposed two band filtering scheme to correctly separate the high frequency components from the low frequency ones in the decomposition process we adopt a dyadic wavelet based approximation scheme in the regularization process we use a diffusivity function as a guide to retain useful data and suppress noises finally the signal of the next stage which is a smoother version of the signal at the previous stage can be computed by reconstructing the decomposed low frequency component and the regularized high frequency component based on the proposed scheme the smoothing operation can be applied to the correct targets experimental results show that our new approach is really efficient in noise removing

disconjugacy disfocality and oscillation of second order difference equations abstract we obtain necessary and sufficient conditions for the disconjugacy disfocality or oscillation of the second order linear difference equation the conditions are given explicitly in terms of the coefficients in the equation

riccati techniques and approximation for a second order poincare difference equation approximation results are obtained for second order self adjoint difference equations and applied to a second order poincare difference equation whose unperturbed equation has a double characteristic root

input output stability of sampled data systems the authors study the l sub p input output stability of a continuous time controller using the usual arrangement of periodic sampling and zero order hold it is noted that even if the hybrid system is exponentially stable this arrangement does not yield l sub p 1

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples