interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

differentiable interpolants for high order runge kutta methods for a particular family of pairs of explicit runge kutta methods of orders p 1 and p sets of efficient continuously differentiable interpolants of several orders up to p are characters zed algorithmically in terms of several arbitrary parameters the approach can be applied in an obvious way to yield interpolants for other types of explicit runge kutta methods as well as families of other types of explicit and implicit methods derivative evaluations required for each pair of methods are reused and additional derivative evaluations are selected in an attempt to minimize the total number of stages required the analysis provides a lower bound on the number of stages required and indicates for example why twelve stages are required to provide interpolants for eight stage pairs of orders 5 and 6 in contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of methods of orders 6 and 7 only sixteen stages are required by the proposed derivation details for interpola

weight assignment for adaptive image restoration by neural networks this paper presents a scheme for adaptively training the weights in terms of varying the regularization parameter in a neural network for the restoration of digital images the flexibility of neural network based image restoration algorithms easily allow the variation of restoration parameters such as blur statistics and regularization value spatially and temporally within the image this paper focuses on spatial variation of the regularization parameter we first show that the previously proposed neural network method based on gradient descent can only find suboptimal solutions and then introduce a regional processing approach based on local statistics a method is presented to vary the regularization parameter spatially this method is applied to a number of images degraded by various levels of noise and the results are examined the method is also applied to an image degraded by spatially variant blur in all cases the proposed method provides visually satisfactory results in an efficient way

solution of inverse problems in image processing by wavelet expansion we describe a wavelet based approach to linear inverse problems in image processing in this approach both the images and the linear operator to be inverted are represented by wavelet expansions leading to a multiresolution sparse matrix representation of the inverse problem the constraints for a regularized solution are enforced through wavelet expansion coefficients a unique feature of the wavelet approach is a general and consistent scheme for representing an operator in different resolutions an important problem in multigrid multiresolution processing this and the sparseness of the representation induce a multigrid algorithm the proposed approach was tested on image restoration problems and produced good results

oscillation results for linear hamiltonian systems for linear hamiltonian systems this paper established some new oscillation criteria which improve previous results even for self adjoint second order differential systems the main tool used is a generalized riccati transformation

nonlinear functionals and a theorem of sun we use a recently developed theory of nonlinear functionals in the study of oscillations of second order symmetric vector differential systems to extend a number of theorems of sun new kamenev type theorems for second order linear matrix differential systems appl math lett 2004 in press under a common theme the criteria presented here are of the form the integral of the coefficient matrix is bounded at infinity in a sense to be made explicit in the paper and bounded away from a positive absolute constant implies oscillation at infinity

interpolating high order runge kutta formulas abstract a runge kutta formula becomes inefficient when the step size must be reduced often to produce answers at specified points recently a lot of effort has been devoted to providing runge kutta formulas with an interpolation capability so that answers can be produced inexpensively throughout the span of a step though quite successful at low and moderate orders interpolation of high order formulas is still unsatisfactory a new approach to the task is presented in the context of interpolating an important pair of orders 7 and 8 due to prince and dormand j comput appl math 7 67 75 1981

new kamenev type oscillation criteria for linear matrix hamiltonian systems some new kamenev type criteria have been obtained for the oscillation of the linear matrix hamiltonian system x a t x b t y y c t x a t y under the hypothesis a t b t b t 0 and c t c t are nxn real continuous matrix functions on the interval t 0 t 0 our results are different from most known ones in the sense that they are given in the form of lim t supg const rather than in the form of lim t sup l 1 where g is a positive linear functional on the linear space of nxn matrices with real entries consequently our results improve some previous results to some extent which can be seen by the examples given at the end of this paper

new oscillation criteria for linear matrix hamiltonian systems abstract some new oscillation criteria are established for the matrix linear hamiltonian system x a t x b t y y c t x a t y under the hypothesis a t b t b t 0 and c t c t are n n real continuous matrix functions on the interval t 0 t 0 these results are sharper than some previous results even for self adjoint second order matrix differential systems

regularized color clustering in medical image database a regularized color clustering algorithm is proposed to solve the color clustering problem in medical image database by incorporating both measures of cluster separability and cluster compactness regularized color clustering allows the automatic extraction of significant color groups with varying populations experimental results in different color spaces show that the regularized color clustering gives superior results in extracting significant distinct abnormal color clusters without significant increases in cluster compactness furthermore results of color clustering in different color spaces show that the luv color space is more suitable for color clustering methods for selecting the regularization constants have also been suggested

a unified approach to optimal image interpolation problems based on linear partial differential equation models the unified approach to optimal image interpolation problems presented provides a constructive procedure for finding explicit and closed form optimal solutions to image interpolation problems when the type of interpolation can be either spatial or temporal spatial the unknown image is reconstructed from a finite set of sampled data in such a way that a mean square error is minimized by first expressing the solution in terms of the reproducing kernel of a related hilbert space and then constructing this kernel using the fundamental solution of an induced linear partial differential equation or the green s function of the corresponding self adjoint operator it is proved that in most cases closed form fundamental solutions or green s functions for the corresponding linear partial differential operators can be found in the general image reconstruction problem described by a first or second order linear partial differential operator an efficient method for obtaining the corresponding closed form fundamental solutions or green s functions of the operators is presented a computer simulation demonstrates the reconstruction procedure

comments on two dimensional interpolation by generalized spline filters based on partial differential equation image models the generalized spline formula corresponding to the separable semicausal partial differential equation pde image model is given in its correct form in addition the correct formula for the second order b spline is derived finally it is shown that there is no reason to consider separable pde image models only at least when describing the concept of generalized splines

new oscillation criteria for linear matrix hamiltonian systems some new criteria have been obtained for the oscillation of the linear matrix hamiltonian system x a t x b t y y c t x a t y under the hypothesis a t b t b t 0 and c t c t are nxn real continuous matrix functions on the interval t 0 t 0 our results are independent of the main results obtained in our recent paper appl math comput 131 2002 357 in the sense that they are given in terms of the positive linear functional on the linear space of nxn matrices with real entries at the same time our results improve some previous results to a great extent which can be seen by the examples given at the end of this paper

a global energy approach to facet model and its minimization using weighted least squares algorithm abstract a global energy approach and a weighted least squares facet algorithm have been developed using the facet model the global energy method gives a near optimal estimation of the ideal image as predicted by the facet model using a regularization framework the weighted least squares facet algorithm improves the estimation of the facet parameters by taking into account the variances of estimations from different resolution cells experimental results show the validity of the global energy approach and the superiority of the wls facet algorithms in both synthetic images and real range images

synthesis and estimation of random fields using long correlation models random field models have arisen in important applications in many diverse fields including image synthesis and analysis the automatic description and processing of images is limited by the types of underlying models that are hypothesized for the image hence new random field models and associated techniques are developed the new class of models includes the traditional simultaneous autoregressive and moving average models but extends these to comprise a substantially larger new class thus new image textures are representable with convenient small order parameterizations in particular representations with significant correlations between distant locations are achieved the algorithms for synthesis and estimation however require only about the same amount of time as the classical models the development is facilitated by a new mathematical formulation suitable for problems involving high dimensional transformations

integral inequalities and second order linear oscillation on the interval 0 a u 0 below by a solution we mean a non trivial solution a solution is oscillatory if it has an infinite number of zeros numerous oscillation criteria are known and diversified approaches have proved fruitful among these is the use of the associated riccati equation although the theory of differential and integral inequalities has been used by some investigators see e g hartman 4 71 the authors believe that it has not been fully exploited the non oscillation of 1 1 is equivalent to the existence of a solution of the associated riccati equation on a half line thus one technique of proving oscillation is to show that any solution of the riccati equation diverges to i co at a finite point the theory of differential and integral inequalities comes in handy in the estimation of the growth of such a solution below we describe some known results that are relevant to results established in this paper although these are stated here for the case i co in the rest of the paper we consider both cases a 0 and a 00 it is well known that if q is large in the mean then 1 1 is oscillatory a typical example of such a criterion is due to fite leighton and wintner if

sequential and parallel image restoration neural network implementations sequential and parallel image restoration algorithms and their implementations on neural networks are proposed for images degraded by linear blur and contaminated by additive white gaussian noise maximum a posteriori map estimation and regularization theory lead to the same high dimension convex optimization problem the commonly adopted strategy in using neural networks for image restoration is to map the objective function of the optimization problem into the energy of a predefined network taking advantage of its energy minimization properties departing from this approach we propose neural implementations of iterative minimization algorithms which are first proved to converge the developed schemes are based on modified hopfield 1985 networks of graded elements with both sequential and parallel updating schedules an algorithm supported on a fully standard hopfield network binary elements and zero autoconnections is also considered robustness with respect to finite numerical precision is studied and examples with real images are presented

weighted means and oscillation conditions for second order matrix differential equations on considere le systeme d equations differentielles d ordre 2 p t y t q t y t 0 pour t 0 ou p et q sont des matrices symetriques n n reelles sur 0 et y un vecteur colonne on suppose que p t est definie positive pour tout t tel que p t 1 on donne des conditions d oscillations

interval oscillation criteria for matrix differential systems with damping abstract this paper concerns the oscillation of solutions to leighton type for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t and r t r t are n n matrices of real valued continuous functions on the interval t 0

regularization theory in image restoration the stabilizing functional approach several aspects of the application of regularization theory in image restoration are presented this is accomplished by extending the applicability of the stabilizing functional approach to 2 d ill posed inverse problems inverse restoration is formulated as the constrained minimization of a stabilizing functional the choice of a particular quadratic functional to be minimized is related to the a priori knowledge regarding the original object through a formulation of image restoration as a maximum a posteriori estimation problem this formulation is based on image representation by certain stochastic partial differential equation image models the analytical study and computational treatment of the resulting optimization problem are subsequently presented as a result a variety of regularizing filters and iterative regularizing algorithms are proposed a relationship between the regularized solutions proposed and optimal wiener estimation is identified the filters and algorithms proposed are evaluated through several experimental results

oscillation results for matrix differential systems with damping abstract in this paper we establish several new oscillation theorems of leighton type for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t and r t r t are n n matrices of real valued continuous functions on the interval t 0 our results are not contained in known results

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

image restoration using total variation with overlapping group sparsity image restoration is one of the most fundamental issues in imaging science total variation regularization is widely used in image restoration problems for its capability to preserve edges in the literature however it is also well known for producing staircase artifacts in this work we extend the total variation with overlapping group sparsity which we previously developed for one dimension signal processing to image restoration a convex cost function is given and an efficient algorithm is proposed for solving the corresponding minimization problem in the experiments we compare our method with several state of the art methods the results illustrate the efficiency and effectiveness of the proposed method in terms of psnr and computing time

oscillation results for linear second order matrix differential systems with damping in this paper we present some new oscillation criteria in terms of the coefficient functions for the linear second order matrix differential system with damping p t x r t x q t x 0

regularization of inverse visual problems involving discontinuities inverse problems such as the reconstruction problems that arise in early vision tend to be mathematically ill posed through regularization they may be reformulated as well posed variational principles whose solutions are computable standard regularization theory employs quadratic stabilizing functionals that impose global smoothness constraints on possible solutions discontinuities present serious difficulties to standard regularization however since their reconstruction requires a precise spatial control over the smoothing properties of stabilizers this paper proposes a general class of controlled continuity stabilizers which provide the necessary control over smoothness these nonquadratic stabilizing functionals comprise multiple generalized spline kernels combined with noncontinuous continuity control functions in the context of computational vision they may be thought of as controlled continuity constraints these generic constraints are applicable to visual reconstruction problems that involve both continuous regions and discontinuities for which global smoothness constraints fail

oscillation results for selfadjoint differential systems abstract we consider the second order differential system 1 r t y q t y 0 where r q y are n n matrices with r t q t symmetric and r t positive definite for t a r t 0 t a we establish sufficient conditions in order that all prepared solutions y t of 1 are oscillatory that is det y t vanishes infinitely often on a the conditions involve the smallest and largest eigen values n r 1 t and 1 a t q s ds respectively the results obtained can be regarded as generalizing well known results of leighton and others in the scalar case

simple algorithms and architectures for b spline interpolation it is proved that the toeplitz binary value matrix inversion associated with mth order b spline interpolation can be implemented using only 2 m 1 additions pipelined architectures are developed for real time b spline interpolation based on simple running average filters it is shown that an ideal interpolating function which is approximated by a truncated sinc function with m half cycles can be implemented using b splines with m 2 multiplies with insignificant loss of performance the coefficients at the knots of the truncated sinc function can be approximated using coefficients which are powers of two the resulting implementation requires only m 4m 6 additions it is believed that the truncated sinc function approximated by zero order b spline functions actually achieves the best visual performance

adaptive image resizing based on continuous domain stochastic modeling we introduce an adaptive continuous domain modeling approach to texture and natural images the continuous domain image is assumed to be a smooth function and we embed it in a parameterized sobolev space we point out a link between sobolev spaces and stochastic auto regressive models and exploit it for optimally choosing sobolev parameters from available pixel values to this aim we use exact continuous to discrete mapping of the auto regressive model that is based on symmetric exponential splines the mapping is computationally efficient and we exploit it for maximizing an approximated gaussian likelihood function we account for non gaussian levy type processes by deriving a more robust estimator that is based on the sample auto correlation sequence both estimators use multiple initialization values for overcoming the local minima structure of the fitting criteria experimental image resizing results indicate that the auto correlation criterion can cope better with non gaussian processes and model mismatch our work demonstrates the importance of the auto correlation function in adaptive image interpolation and image modeling tasks and we believe it is instrumental in other image processing tasks as well

nonideal sampling and interpolation from noisy observations in shift invariant spaces digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous time signal from a sequence of corrupted discrete time samples in this paper a general formulation of this problem is developed that treats the interpolation problem from ideal noisy samples and the deconvolution problem in which the signal is filtered prior to sampling in a unified way the signal reconstruction is performed in a shift invariant subspace spanned by the integer shifts of a generating function where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter several alternative approaches to designing the correction filter are suggested which differ in their assumptions on the signal and noise the classical deconvolution solutions least squares tikhonov and wiener are adapted to our particular situation and new methods that are optimal in a minimax sense are also proposed the solutions often have a similar structure and can be computed simply and efficiently by digital filtering some concrete examples of reconstruction filters are presented as well as simple guidelines for selecting the free parameters e g regularization of the various algorithms

nonideal sampling and regularization theory shannon s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some shift invariant space which may or may not be bandlimited in this paper we present some further justification for this type of representation while addressing the issue of the specification of the best reconstruction space we consider a realistic setting where a multidimensional signal is prefiltered prior to sampling and the samples are corrupted by additive noise we adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a tikhonov like continuous domain l2 regularization to obtain the continuous space solution we present theoretical justification for the minimization of this cost functional and show that the globally minimal continuous space solution belongs to a shift invariant space generated by a function generalized b spline that is generally not bandlimited when the sampling is ideal we recover some of the classical smoothing spline estimators the optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a b spline like basis to make the scheme practical we specify the generating functions corresponding to the most popular families of regularization operators derivatives iterated laplacian as well as a new generalized one that leads to a new brand of matern splines we conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error mmse wiener solutions of the generalized sampling problem

generalized smoothing splines and the optimal discretization of the wiener filter we introduce an extended class of cardinal l sup l splines where l is a pseudo differential operator satisfying some admissibility conditions we show that the l sup l spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional spl par ls spl par sub l2 sup 2 subject to the interpolation constraint next we consider the corresponding regularized least squares estimation problem which is more appropriate for dealing with noisy data the criterion to be minimized is the sum of a quadratic data term which forces the solution to be close to the input samples and a smoothness term that privileges solutions with small spline energies here too we find that the optimal solution among all possible functions is a cardinal l sup l spline we show that this smoothing spline estimator has a stable representation in a b spline like basis and that its coefficients can be computed by digital filtering of the input signal we describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of l is rational which corresponds to the case of exponential splines we justify these algorithms statistically by establishing an equivalence between l sup l smoothing splines and the minimum mean square error mmse estimation of a stationary signal corrupted by white gaussian noise in this model based formulation the optimum operator l is the whitening filter of the process and the regularization parameter is proportional to the noise variance thus the proposed formalism yields the optimal discretization of the classical wiener filter together with a fast recursive algorithm it extends the standard wiener solution by providing the optimal interpolation space we also present a bayesian interpretation of the algorithm

some practical runge kutta formulas a new selection is made of the most practical of the many explicit runge kutta formulas of order 4 which have been proposed a new formula is considered formulas are modified to improve their quality and efficiency in agreement with improved understanding of the issues and formulas are derived which permit interpolation it is possible to do a lot better than the pair of fehlberg currently regarded as best

an adaptive image interpolation algorithm for image video processing image interpolation is one of the key technologies in image video processing in this study a new adaptive image interpolation algorithm is proposed the objective of the proposed approach is to recover up sampled image frames from the corresponding decimated low resolution image frames in the proposed approach within each iteration two proposed nonlinear filters are utilized to iteratively generate high frequency components lost within the decimation procedure finally a post processing procedure is adopted to reduce the blocking artifacts within the interpolated images based on the experimental results obtained in this study in terms of the average psnrp peak signal to noise ratio in db and subjective measure of the quality of the interpolated images the interpolation results by the proposed approach are better than that by three existing interpolation approaches for comparison

runge kutta triples abstract the addition of a third runge kutta rk formula to an embedded pair forming an rk triple can yield an algorithm which gives solutions of appropriate asymptotic accuracy at points within the normal step intervals in the numerical solution of the initial value problem two modes of implementation are possible and it is shown that these can be regarded as equivalent when certain conditions are satisfied an optimized rk5 4 triple based on a particularly efficient embedded pair is presented and tested the results indicate the efficiency of the dense output technique

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

2 d discrete signal interpolation and its image resampling application using fuzzy rule based inference this paper describes an interpolation algorithm for two dimensional 2 d discrete signals using fuzzy rule based inference the original signal is estimated by the main surface function in the interpolation region and four sub plane functions surrounding the interpolation region the main surface is a bilinearly interpolated function passing through four signal samples in the interpolation region and the four sub planes reect the tendencies of pixels from the left right up and down of the interpolation region drawing fuzzy inferences about signals from these ve functions we can estimate original signals very well even when the signals are buried in noise we veried the method by computer simulations of some assumed 2 d signals and by resampling of the actual image data c 2000 elsevier science b v all rights reserved

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

oscillation of two dimensional linear second order differential systems abstract this article is concerned with the oscillatory behavior at infinity of the solution y a r 2 of a system of two second order differential equations y t q t y t 0 t a q is a continuous matrix valued function on a whose values are real symmetric matrices of order 2 it is shown that the solution is oscillatory at infinity if the largest eigenvalue of the matrix at q s ds tends to infinity as t this proves a conjecture of d hinton and r t lewis rocky mountain j math 10 1980 751 766 for the two dimensional case furthermore it is shown that considerably weaker forms of the condition still suffice for oscillatory behavior at infinity

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

two dimensional interpolation by generalized spline filters based on partial differential equation image models optimal image interpolation schemes based on partial differential equation pde models for the image signal are presented these schemes use the generalized spline interpolation approach minimum mean square error mmse is achieved given an image pde model and a finite set of the image data samples two new types of two dimensional 2 d interpolation techniques are presented for the implementation of these schemes these are the generalized cardinal spline digital filter and the generalized b spline interpolator computer simulations are described in addition to possessing a meaningful theoretical basis the results show a practical improvement of the approach presented over the existing schemes

oscillation criteria for second order linear matrix differential systems with damping using a generalization of sturm s comparison theorem some new oscillation criteria are established for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t y t are nxn matrices of real valued continuous functions on the interval t 0 and r t r t c 1 t 0 r n 2 our results are sharper than some previous results

convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods we have recently developed a generic approach to solving retarded and neutral delay differential equations ddes the approach is based on the use of an explicit continuous runge kutta formula and employs defect control the approach can be applied to problems with state dependent delays and vanishing delays in this paper we determine convergence properties for the numerical solution associated with methods that implement this approach we first analyze such properties for retarded ddes and then the analysis is extended to the case of neutral ddes ndes such an extension is particularly important since ndes have received little attention in the literature the main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect our analysis can also be applied directly to other numerical methods which use defect error control based on runge kutta formulas for ddes

oscillation of linear hamiltonian systems abstract in this paper sufficient conditions have been obtained for the oscillation of a class of linear hamiltonian systems

a general formulation for iterative restoration methods information theory based techniques for signal or image restoration and resolution enhancement are now considered a viable alternative to other popular techniques in this correspondence we present a general iterative mapping derived using an information theoretic criterion of optimality a mathematical structure is defined followed by an analysis of the mapping based on this structure the convergence of the mapping is considered both in the general formulation and for a particular example some popular techniques are shown to be special cases of this general mapping

riccati techniques and oscillation for self adjoint matrix hamiltonian systems the purpose of this paper is to develop a generalized matrix riccati technique for the self adjoint matrix hamiltonian system u a x u b x v v c x u a x v together with the integral averaging technique and monotone functionals new oscillation and interval oscillation criteria are established for the system these criteria extend improve complement a number of existing results and handle some cases not covered by known criteria in particular two interesting examples are included to illustrate the versatility of our results

polyharmonic smoothing splines and the multidimensional wiener filtering of fractal like signals motivated by the fractal like behavior of natural images we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the laplacian this type of functional was initially introduced by duchon for the approximation of nonuniformily sampled multidimensional data he proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions rbfs unfortunately this is tedious to implement for images because of the poor conditioning of rbfs and their lack of decay here we present a much more efficient method for the special case of a uniform grid the key idea is to express duchon s solution in a fractional polyharmonic b spline basis that spans the same space as the rbfs this allows us to derive an algorithm where the smoothing is performed by filtering in the fourier domain next we prove that the above smoothing spline can be optimally tuned to provide the mmse estimation of a fractional brownian field corrupted by white noise this is a strong result that not only yields the best linear filter wiener solution but also the optimal interpolation space which is not bandlimited it also suggests a way of using the noisy data to identify the optimal parameters order of the spline and smoothing strength which yields a fully automatic smoothing procedure we evaluate the performance of our algorithm by comparing it against an oracle wiener filter which requires the knowledge of the true noiseless power spectrum of the signal we find that our approach performs almost as well as the oracle solution over a wide range of conditions