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

higher order method for singular boundary value problems by using spline function abstract in problems of physics and engineering we often come across singular boundary value problems which cannot be solved by the usual numerical methods special methods for solving such problems have been developed these methods lead to banded systems linear and non linear depending upon the nature of boundary value problem in this paper we discuss the construction of a spline function for a class of singular two point boundary value problems x x u f x u 0 x 1 u 0 a u 1 b five point finite difference method using the above splines is obtained it is shown to be order h4 convergent for all 0 1 the method is illustrated computationally

approach to an irregular time series on the basis of the fractal theory we present a technique to measure the fractal dimension of the set of points t f t forming the graph of a function f defined on the unit interval first we apply it to a fractional brownian function 1 which has a property of self similarity for all scales and we can get the stable and precise fractal dimension this technique is also applied to the observational data of natural phenomena it does not show self similarity all over the scale but has a different self similarity across the characteristic time scale the present method gives us a stable characteristic time scale as well as the fractal dimension

deconvolution filter design for fractal signal transmission systems a multiscale kalman filter bank approach a deconvolution filtering design is proposed for the 1 f fractal signal transmission systems with its design philosophy being based on multiscale kalman deconvolution filter bank equipped in the analysis synthesis wavelet filter bank the role of wavelet transformation for 1 f fractal signal process is exploited as a multiscale whitening filter for removing the properties of self similarity and long range dependence from the fractal signals

optimized estimation of moments for nonstationary signals the behavior of nonstationary moment estimators that are based on linear time invariant filtering is analyzed the performance of such estimators is evaluated in terms of their time averaged variance and time averaged squared bias optimal estimators that minimize a convex combination of bias and variance are derived the superiority of such optimally weighted filtering over the conventional exponentially windowed moment estimation technique is demonstrated by means of a simple example the same example also serves to illustrate the difficulties encountered when the construction of optimal estimators relies on uncertain prior information as well as to demonstrate the feasibility of overcoming such difficulties by using appropriately designed robust optimal estimators we also include a brief discussion of the relevance of the proposed moment estimation technique to identification of linear time variant systems and to estimation of time variant spectra

optimized wavelet denoising for self similar alpha stable processes we investigate the performance of wavelet shrinkage methods for the denoising of symmetric alpha stable text s alpha text s self similar stochastic processes corrupted by additive white gaussian noise awgn where alpha is tied to the sparsity of the process the wavelet transform is assumed to be orthonormal and the shrinkage function minimizes the mean square approximation error mmse estimator we derive the corresponding formula for the expected value of the averaged estimation error we show that the predicted mmse is a monotone function of a simple criterion that depends on the wavelet and the statistical parameters of the process using the calculus of variations we then optimize this criterion to find the best performing wavelet within the extended family of meyer wavelets which are bandlimited these are compared with the daubechies wavelets which are compactly supported in time we find that the wavelets that are shorter in time in particular the haar basis are better suited to denoise the sparser processes say alpha while the bandlimited ones including the held and shannon wavelets offer the best performance for alpha 1 6 the limit corresponding to the gaussian case fbm with alpha 2

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

a wavelet based kl like expansion for wide sense stationary random processes we describe a wavelet based series expansion for wide sense stationary processes the expansion coefficients are uncorrelated random variables a property similar to that of a karhunen loeve kl expansion unlike the kl expansion however the wavelet based expansion does not require the solution of the eigen equation and does not require that the process be time limited this expansion also has advantages over fourier series which is often used as an approximation to the kl expansion in that it completely eliminates correlation and that the computation for its coefficients are more stable over large time intervals the basis functions of this expansion can be obtained easily from wavelets of the lemaire meyer 1990 type and the power spectral density of the process finally the expansion can be extended to some nonstationary processes such as those with wide sense stationary increments

wavelet decomposition of harmonizable random processes the discrete wavelet decomposition of second order harmonizable random processes is considered the deterministic wavelet decomposition of a complex exponential function is examined where its pointwise and bounded convergence to the function is proved this result is then used for establishing the stochastic wavelet decomposition of harmonizable processes the similarities and differences between the wavelet decompositions of general harmonizable processes and a subclass of processes having no spectral mass at zero frequency e g those that are wide sense stationary and have continuous power spectral densities are also investigated the relationships between the harmonization of a process and that of its wavelet decomposition are examined finally certain linear operations such as addition differentiation and linear filtering on stochastic wavelet decompositions are considered it is shown that certain linear operations can be performed term by term with the decomposition

boundary value problems for second order nonlinear differential equations on infinite intervals in this paper we consider boundary value problems for nonlinear differential equations on the semi axis 0 and also on the whole axis under the assumption that the left hand side being a second order linear differential expression belongs to the weyl limit circle case the boundary value problems are considered in the hilbert spaces l2 0 and l2 and include boundary conditions at infinity the existence and uniqueness results for solutions of the considered boundary value problems are established

solvability of singular second order m point boundary value problems abstract let f 0 1 r 2 r be a function satisfying caratheodory s conditions and 1 t e t l 1 0 1 let i 0 1 a i r i 1 m 2 0 1 2 m 2 1 be given this paper is concerned with the problem of existence of a c 1 0 1 solution for the m point boundary value problem x f t x t x t e t 0 t 1 0 1 x 0 0 x 1 i 1 m 2 a i x i the proof of our main result is based upon the leray schauder continuation theorem

multiscale wiener filter for the restoration of fractal signals wavelet filter bank approach a filter bank design based on orthonormal wavelets and equipped with a multiscale wiener filter is proposed in this paper for signal restoration of 1 f family of fractal signals which are distorted by the transmission channel and corrupted by external noise first the fractal signal transmission process is transformed via the analysis filter bank into multiscale convolution subsystems in time scale domain based on orthonormal wavelets some nonstationary properties e g self similarity long term dependency of fractal signals are attenuated in each subband by wavelet multiresolution decomposition so that the wiener filter bank can be applied to estimate the multiscale input signals then the estimated multiscale input signals are synthesized to obtain the estimated input signal some simulation examples are given for testing the performance of the proposed algorithm with this multiscale analysis synthesis design via the technique of the wavelet filter bank the multiscale wiener filter can be applied to treat the signal restoration problem for nonstationary 1 f fractal signals

multiresolution analysis of a class of nonstationary processes processing nonstationary signals is an important and challenging problem we focus on the class of nonstationary processes with stationary increments of an arbitrary order and place them in a multiscale framework unlike other related studies we concentrate on the discrete time analysis and derive a number of new results in addition to placing the related existing ones in the same framework we extend the study to various parametric models for which we derive the resulting multiresolution description we show that wide sense stationarity may be achieved by adequately selecting the analysis wavelet after generalizing the study to wavelet packet analysis we show that the latter possesses additional properties which are useful in the presence of other types of nonstationarities

on the optimal choice of a wavelet for signal representation two techniques for finding the discrete orthogonal wavelet of support less than or equal to some given integer that leads to the best approximation to a given finite support signal up to a desired scale are presented the techniques are based on optimizing certain cost functions the first technique consists of minimizing an upper bound that is derived on the l sub 2 norm of error in approximating the signal up to the desired scale it is shown that a solution to the problem of minimizing that bound does exist and it is explained how the constrained minimization over the parameters that define discrete finite support orthogonal wavelets can be turned into an unconstrained one the second technique is based on maximizing an approximation to the norm of the projection of the signal on the space spanned by translates and dilates of the analyzing discrete orthogonal wavelet up to the desired scale both techniques can be implemented much faster than the optimization of the l sub 2 norm of either the approximation to the given signal up to the desired scale or that of the error in that approximation

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

a uniform mesh finite difference method for a class of singular two point boundary value problems a new finite difference method based on uniform mesh is given for the weakly singular two point boundary value problem x alpha y f x y y 0 a y 1 b 0 alpha 1 under quite general conditions on f and f we show that our present method based on uniform mesh provides o h 2 convergent approximations for all alpha in 0 1 our method is based on one evaluation of f and for alpha 0 it reduces to the classical second order method for y f x y

a novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems this paper deals with two new recursive numerical schemes for the solution of a class of doubly singular two point boundary value problems with different types of boundary conditions we first transform the original problem into an equivalent integral equation to overcome the singular behaviour at the origin and then establish recursive schemes by employing homotopy perturbation method for the solution of the resultant equation we establish the convergence results of the new numerical methods four numerical examples are considered to demonstrate the efficiency and accuracy of the proposed methods the present methods are shown their advantage over some existing methods chawla and katti 1982 pandey and singh 2007 pandey and singh 2004 caglar et al 2009 kumar and aziz 2004 khuri and sayfy 2010 developed for the solution of singular two point boundary value problems moreover the new methods are used to obtain the approximate solutions of the singular boundary value problems arising in various physical problems

wavelet generalized least squares a new blu estimator of linear regression models with 1 f errors long memory noise is common to many areas of signal processing and can seriously confound estimation of linear regression model parameters and their standard errors classical autoregressive moving average arma methods can adequately address the problem of linear time invariant short memory errors but may be inefficient and or insufficient to secure type 1 error control in the context of fractal or scale invariant noise with a more slowly decaying autocorrelation function here we introduce a novel method called wavelet generalized least squares wls which is to a good approximation the best linear unbiased blu estimator of regression model parameters in the context of long memory errors the method also provides maximum likelihood ml estimates of the hurst exponent which can be readily translated to the fractal dimension or spectral exponent characterizing the correlational structure of the errors and the error variance the algorithm exploits the whitening or kar

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

local stationarity and simulation of self affine intrinsic random functions gaussian intrinsic random functions with power law generalized covariance functions which in one dimension essentially correspond to fractional and integrated fractional brownian motions form a class of self affine models for random fields with a wide range of smoothness properties these random fields are nonstationary but appropriately filtered versions of them are stationary this work proves that most such random functions are locally stationary in a certain well defined sense this result yields an efficient and exact method of simulating all fractional and integrated fractional brownian motions

estimation of fractional brownian motion embedded in a noisy environment using nonorthogonal wavelets we show that nonorthogonal wavelets can characterize the fractional brownian motion fbm that is in white noise we demonstrate the point that discriminating the parameter of fbm from that of noise is equivalent to discriminating the composite singularity formed by superimposing a peak singularity on a dirac singularity we characterize the composite singularity by formalizing this problem as a nonlinear optimization problem this yields our parameter estimation algorithm for fractal signal estimation wiener filtering is explicitly formulated as a function of the signal and noise parameters and the wavelets we show that the estimated signal is a 1 f process comparative studies through numerical simulations of our methods with those of wornell and oppenheim 1992 are presented

low rank variance approximation in gmrf models single and multiscale approaches we present a versatile framework for tractable computation of approximate variances in large scale gaussian markov random field estimation problems in addition to its efficiency and simplicity it also provides accuracy guarantees our approach relies on the construction of a certain low rank aliasing matrix with respect to the markov graph of the model we first construct this matrix for single scale models with short range correlations and then introduce spliced wavelets and propose a construction for the long range correlation case and also for multiscale models we describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse irregular and noisy measurements and to a gravity inversion problem

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

an improved method for 2 d self similar image synthesis we propose a new method called incremental fourier synthesis to generate 2 d self similar images based on a 2d fractional brownian motion fbm model with this method the stationary increments of fbm are created by a fourier synthesis method and the increments are added up to generate the nonstationary 2d fbm process since the new method takes advantage of the fft its computational complexity is only o n sup 2 log sub 2 n and its memory requirement is only o n sup 2 for a self similar image of size n spl times n

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

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

a fourth order spline method for singular two point boundary value problems this paper describes a new fourth order spline approximation method for the class of singular two point boundary value problems y frac alpha x y f x y 0 0 x 1 y 0 0 y 1 a for all alpha geqslant 1

nonstationary spectral analysis based on time frequency operator symbols and underspread approximations we present a unified framework for time varying or time frequency tf spectra of nonstationary random processes in terms of tf operator symbols we provide axiomatic definitions and tf operator symbol formulations for two broad classes of tf spectra one of which is new these classes contain all major existing tf spectra such as the wigner ville evolutionary instantaneous power and physical spectrum our subsequent analysis focuses on the practically important case of nonstationary processes with negligible high lag tf correlations so called underspread processes we demonstrate that for underspread processes all tf spectra yield effectively identical results and satisfy several desirable properties at least approximately we also show that gabor frames provide approximate karhunen loeve kl functions of underspread processes and tf spectra provide a corresponding approximate kl spectrum finally we formulate simple approximate input output relations for the tf spectra of underspread processes that are passed through underspread linear time varying systems all approximations are substantiated mathematically by upper bounds on the associated approximation errors our results establish a tf calculus for the second order analysis and time varying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes

existence of positive solutions for sturm liouville boundary value problems on the half line in this paper we establish sufficient conditions to guarantee the existence of at least one positive solution a unique positive solution and multiple positive solutions for the sturm liouville boundary value problem on the half line by using an effective operator the fixed point theorems in cone especially krasnoselskii fixed point theorem can be applied to such systems and then existence criteria are established the interesting point of the results is that the nonlinear term f can be sign changing

a fourth order finite difference method for a class of singular two point boundary value problems a method is described based on a non uniform mesh for the class of singular two point boundary value problems x ay f x y 0

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

on the convergence of finite difference methods for weakly regular singular boundary value problems the second order finite difference methods m 1 based on a non uniform mesh and m 2 based on an uniform mesh developed by chawla and katti finite difference methods and their convergence for a class of singular two point boundary value problems numer math 39 1982 341 350 for weakly regular singular boundary value problems p x y f x y 0

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

a matrix valued wavelet kl like expansion for wide sense stationary random processes matrix valued wavelet series expansions for wide sense stationary processes are studied in this paper the expansion coefficients a are uncorrelated matrix random process which is a property similar to that of a matrix karhunen loe spl grave ve mkl expansion unlike the mkl expansion however the matrix wavelet expansion does not require the solution of the eigen equation this expansion also has advantages over the fourier series which is often used as an approximation to the mkl expansion in that it completely eliminates correlation the basis functions of this expansion can be obtained easily from wavelets of the matrix valued lemarie spl acute meyer type and the power spectral density of the process

multiscale autoregressive processes ii lattice structures for whitening and modeling for pt i see ibid vol 40 no 8 p 1915 34 1992 lattice structures for the whitening and modeling of isotropic processes on trees are developed and a result relating the stability properties of these models to the reflection coefficient sequence introduced in pt i is presented this framework allows one to obtain a detailed analysis of the wold decomposition of processes on trees one interesting aspect of this is that there is a significantly larger class of singular processes on dyadic trees than on the integers

third order boundary value problems with sign changing solutions we consider the third order nonlinear differential equation u t f t u t u t u t a e t 0 1 satisfying u 0 u 0 u 1 0 and u 0 u 1 u 1 0 where f 0 1 r3 r is lp caratheodory 1 p 1 and p 1

self similarity part i splines and operators the central theme of this pair of papers parts i and ii in this issue is self similarity which is used as a bridge for connecting splines and fractals the first part of the investigation is deterministic and the context is that of l splines these are defined in the following terms s t is a cardinal l spline iff l s t sigma kisinza k delta t k where l is a suitable pseudodifferential operator our starting point for the construction of self similar splines is the identification of the class of differential operators l that are both translation and scale invariant this results into a two parameter family of generalized fractional derivatives parttau gamma where gamma is the order of the derivative and tau is an additional phase factor we specify the corresponding l splines which yield an extended class of fractional splines the operator parttau gamma is used to define a scale invariant energy measure the squared l2 norm of the gammath derivative of the signal which provides a regularization functional for interpolating or fitting the noisy samples of a signal we prove that the corresponding variational or smoothing spline estimator is a cardinal fractional spline of order 2gamma which admits a stable representation in a b spline basis we characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical butterworth filter of order 2gamma we also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter omega0aplambda 2gamma finally we present an efficient computational solution to the fractional smoothing spline problem it uses the fast fourier transform and takes advantage of the multiresolution properties of the underlying basis functions

analysis and synthesis of 1 f processes via shannon wavelets 1 f processes can be very useful in modeling processes with long term correlation we propose analysis and synthesis procedures to express these processes in terms of the shannon wavelet unlike previous techniques our analysis procedure generates uncorrelated decomposition coefficients for the 1 f process this is done by taking onto account and then removing the residual correlation between the wavelet components the analysis procedure is the major contribution of this work the proposed synthesis algorithm which is a byproduct of the proposed analysis algorithm is competitive with other techniques

wavelet packets of fractional brownian motion asymptotic analysis and spectrum estimation this paper provides asymptotic properties of the autocorrelation functions of the wavelet packet coefficients of a fractional brownian motion it also discusses the convergence speed to the limit autocorrelation function when the input random process is either a fractional brownian motion or a wide sense stationary second order random process the analysis concerns some families of wavelet paraunitary filters that converge almost everywhere to the shannon paraunitary filters from this analysis we derive wavelet packet based spectrum estimation for fractional brownian motions and wide sense stationary random processes experimental tests show good results for estimating the spectrum of 1 f processes

mra of processes synthesized by fractional integration in this correspondence we show that the multiresolution analysis mra of the reproducing kernel hilbert space associated with processes synthesized by fractional integration of white processes leads to an orthogonal multiresolution decomposition of the process we show that the decomposition converges almost surely and uniformly on finite intervals we also show that the random coefficients in the decomposition can be evaluated almost surely

wavelet based representations for a class of self similar signals with application to fractal modulation a potentially important family of self similar signals based upon a deterministic scale invariance characterization is introduced these signals which are referred to as dy homogeneous signals because they generalize the well known homogeneous functions have highly convenient representations in terms of orthonormal wavelet bases in particular wavelet representations can be exploited to construct orthonormal self similar bases for these signals the spectral and fractal characteristics of dy homogeneous signals make them appealing candidates for use in a number of applications as one potential example their use in a communications based context is considered specifically a strategy for embedding information into a dy homogeneous waveform on multiple time scales is developed this multirate modulation strategy called fractal modulation is potentially well suited for use with noisy channels of simultaneously unknown duration and bandwidth

a multifractal wavelet model with application to network traffic we develop a new multiscale modeling framework for characterizing positive valued data with long range dependent correlations 1 f noise using the haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results the model provides a rapid o n cascade algorithm for synthesizing n point data sets we study both the second order and multifractal properties of the model the latter after a tutorial overview of multifractal analysis we derive a scheme for matching the model to real data observations and to demonstrate its effectiveness apply the model to network traffic synthesis the flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics variance time plots and moment scaling and queuing behavior although for illustrative purposes we focus on applications in network traffic modeling the multifractal wavelet model could be useful in a number of other areas involving positive data including image processing finance and geophysics

a fourth order finite difference method based on uniform mesh for singular two point boundary value problems we present a new fourth order finite difference method based on uniform mesh for the weakly singular two point boundary value problem x y f x y 0 x 1 y 0 a y 1 b 0 1 our method provides o h4 convergent approximations for all 0 1 for 0 it reduces to the well known fourth order method of numerov for y x y

fractal estimation from noisy data via discrete fractional gaussian noise dfgn and the haar basis the authors show that the application of the discrete wavelet transform dwt using the haar basis to the increments of fractional brownian motion fbm also known as discrete fractional gaussian noise dfgn yields coefficients which are weakly correlated and have a variance that is exponentially related to scale similar results were derived by flandrin 1989 tewfik and kim for a continuous time fbm going through a continuous wavelet transform cwt the new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by wornell and oppenheim the performance of the new algorithm is compared with that of wornell and oppenheim s see ieee trans signal processing vol 40 p 611 623 mar 1992 algorithm in numerical simulation

some distributional properties of the continuous wavelet transform of random processes without finite moment conditions some properties of random processes such as stationarity and self similarity are characterized via corresponding properties of their wavelet transform anyone of these distributional properties of the wavelet transform characterizes the corresponding property of the increments of the random process of order equal to the order of regularity of the analyzing wavelet extensions of these results to random fields are then indicated

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

fractional brownian motion a maximum likelihood estimator and its application to image texture fractals have been shown to be useful in characterizing texture in a variety of contexts use of this methodology normally involves measurement of a parameter h which is directly related to fractal dimension in this work the basic theory of fractional brownian motion is extended to the discrete case it is shown that the power spectral density of such a discrete process is only approximately proportional to f a instead of in direct proportion as in the continuous case an asymptotic cramer rao bound is derived for the variance of an estimate of h subsequently a maximum likelihood estimator mle is developed to estimate h it is shown that the variance of this estimator nearly achieves the minimum bound a generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the mle when the discrete fractional brownian process is contaminated with additive gaussian noise the results show that even at signal to noise ratios of 30 db significant errors in estimation of h can result when noise is present the mle is then applied to x ray images of the human calcaneus to demonstrate how the line to line formulation can be applied to the two dimensional case these results indicate that it has strong potential for quantifying texture

new fast smoothers for multiscale systems we consider the smoothing problem for multiscale stochastic systems based on the wavelet transform these models involve processes indexed by the nodes of a dyadic tree each level of the dyadic tree represents one scale or resolution of the process therefore moving upward on the tree divides the resolution by 2 whereas moving downward multiplies it by 2 the processes are built according to a recursion in scale from coarse to fine to which random details are added to operate the change in scale one must perform an interpolation this is achieved using the qmf pair of operators attached to a wavelet transform these models have proved to be of great value to capture textures or fractal like processes as well as to perform multiresolution sensor fusion an example of which is given here up to now however only subclasses of multiscale systems were amenable to fast algorithms and through different formalisms those relying on haar s wavelet and those involving only one of the two wavelet interpolators we provide here a unifying framework that handles any system based on orthogonal wavelets a smoothing theory is presented to define the field of fast algorithms for markov random fields and give intuition on how to design them this theory reveals the difficulties arising with general multiscale systems we then prove that orthogonality properties of wavelets are the gate to fastness

a note on the discrete wavelet transform of second order processes some properties of second order random processes such as stationarity and scaling are characterized via corresponding multiscale properties of their discrete wavelet transform the single scale characterizations valid in the continuous setting do not extend to the discrete setting some extensions of these results are also indicated

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

matching wavelet packets to gaussian random processes we consider the problem of approximating a set of arbitrary discrete time gaussian random processes by a single representative wavelet based gaussian process we measure the similarity between the original processes and the wavelet based process with the bhattacharyya 1943 coefficient by manipulating the bhattacharyya coefficient we reduce the task of defining the representative process to finding an optimal unitary matrix of wavelet based eigenvectors an associated diagonal matrix of eigenvalues and a mean vector the matching algorithm we derive maximizes the nonadditive bhattacharyya coefficient in three steps a migration algorithm that determines the best basis by searching through a wavelet packet tree for the optimal unitary matrix of wavelet based eigenvectors and two separate fixed point algorithms that derive an appropriate set of eigenvalues and a mean vector we illustrate the method with two different classes of processes first order markov and bandlimited the technique is also applied to the problem of robust terrain classification in polarimetric sar images

wavelet transform inversion with prior scale information the concept of scale is of central importance in all problems related to the interpretation of earth data inversion of geophysical data does not at present however allow the incorporation of scale dependent information directly this article presents an application of the wavelet transform to traveltime inversion where we are specifically concerned with the selection and application of a priori scale information in the wavelet domain to reconstruct tomographic models because of the nature of the problem the wavelet basis chosen for this example is the haar basis the results are compared with the classic damped least squares dls method which is a special case of the wavelet transform approach when all the scales are treated equally we show that with scale dependent prior information most of the artifacts created by the dls technique are diminished considerably

finite difference methods and their convergence for a class of singular two point boundary value problems we discuss the construction of three point finite difference approximations and their convergence for the class of singular two point boundary value problems x y f x y y 0 a y 1 b 0 1 we first establish a certain identity based on general non uniform mesh from which various methods can be derived to obtain a method having order two for all 0 1 we investigate three possibilities by employing an appropriate non uniform mesh over 0 1 we obtain a methodm1 based on just one evaluation off for uniform mesh we obtain two methodsm2 andm3 each based on three evaluations off for 0 m1 andm2 both reduce to the classical second order method based on one evaluation off these three methods are investigated theiro h2 convergence established and illustrated by numerical examples

a multiscale guide to brownian motion we revise the levy s construction of brownian motion as a simple though rigorous approach to operate with various gaussian processes a brownian path is explicitly constructed as a linear combination of wavelet based geometrical features at multiple length scales with random weights such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of brownian paths this formula elucidates many classical results about brownian motion e g non differentiability of its path providing intuitive feeling for non mathematicians the illustrative character of the wavelet representation along with the simple structure of the underlying probability space is different from the usual presentation of most classical textbooks similar concepts are discussed for brownian bridge fractional brownian motion ornstein uhlenbeck process gaussian free field and fractional gaussian fields wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate gaussian processes and fields including brownian motion and other diffusive processes in confining domains

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

fractal function estimation via wavelet shrinkage in scientific studies objects are often very rough mathematically these rough objects are modelled by fractal functions and the fractal dimension is usually used to measure their roughness this paper investigates fractal function estimation by wavelet shrinkage it is shown that wavelet shrinkage can estimate fractal functions with their fractal dimensions virtually preserved

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

1 f noise in diffuse optical imaging and wavelet based response estimation in diffuse optical imaging doi data analysis the functional response is contaminated with physiological noise as in functional magnetic resonance imaging fmri in this work we extend a previously proposed method for fmri to estimate the parameters of a linear model of doi time series the regression is performed in the wavelet domain to infer drift coefficients at different scales and to estimate the strength of the hemodynamic response function hrf this multiresolution approach benefits from the whitening property of the discrete wavelet transform dwt which approximately decorrelates long memory noise processes we also show that a more accurate estimation is obtained by removing some regressors correlating with the protocol moreover we observe that this improvement is related to a quantitative measure of 1 f noise the performances of the method are first evaluated against a standard spline cosine drift approach with simulated hrf and real background physiology lastly the technique is applied to experimental event related data acquired by near infrared spectroscopy nirs

wavelet based model for stochastic analysis of beam structures a wavelet based model for stochastic analysis of beam structures is presented in this model the random processes representing the stochastic material and geometric properties are treated as stationary gaussian processes with specified mean and correlation functions using the karhunen loeve expansion the process is represented as a linear sum of orthonormal eigenfunctions with uncorrelated random coefficients the correlation and the eigenfunctions are approximated as truncated linear sums of compactly supported orthogonal wavelets and the integral eigenvalue problem is converted to a finite dimensional eigenvalue problem the energy principle based finite element approach is used to obtain the equilibrium and boundary conditions neumann expansion of the stiffness matrix is used to write the nodal displacement vector in terms of random coefficients the expectation operator is applied to the nodal displacements and their squares to obtain the mean and standard deviation of the displacements studies show that the results obtained using this method compare well with monte carlo and semianalytical techniques

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling

wavelet estimation of fractional brownian motion embedded in a noisy environment this correspondence proposes a wavelet based fractional brownian motion fbm signal estimation scheme despite the fact that wavelet transform approximately whitens the fbm processes it is observed that statistical dependencies still exist across adjacent wavelet scales and between neighboring wavelet coefficients these dependencies can be exploited to improve the estimation of fbm signals embedded into noise the idea is to reorganize the wavelet coefficients into a scale time mixture model and then carry out the minimum mean square error estimation mmse using the model experiments show that the proposed scheme obtains better estimates than wornell and oppenheim s algorithm in which the wavelet dependencies are not utilized

construction of multivariate surrogate sets from nonlinear data using the wavelet transform abstract the use of surrogate data has become a crucial first step in the study of nonlinearity in time series data a widely used technique to construct surrogate data is to randomize the phases of the data in the fourier domain in this paper an alternative technique based on the resampling of wavelet coefficients is discussed this approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data it proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties removal of nonlinear structure and computational demands however the wavelet technique may have specific and distinct advantages in the application to complex data sets such as numerical analysis of turbulence and experimental brain imaging data where wavelets give a more parsimonious representation of spatio temporal patterns than fourier modes it is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints

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

second order boundary value problems on an unbounded domain abstract we consider the second order nonlinear differential equation p t u t f t u t u t a e in 0 satisfying two sets of boundary conditions u 0 0 lim t u t 0 and u 0 0 lim t u t 0 where f 0 r 2 r is caratheodory with respect to l 1 0 p c 0 c 1 0 and p t 0 for all t 0 we obtain the existence of at least one solution using the leray schauder continuation principle

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

wavelet based estimation of a semiparametric generalized linear model of fmri time series addresses the problem of detecting significant changes in fmri time series that are correlated to a stimulus time course this paper provides a new approach to estimate the parameters of a semiparametric generalized linear model of the fmri time series the fmri signal is described as the sum of two effects a smooth trend and the response to the stimulus the trend belongs to a subspace spanned by large scale wavelets the wavelet transform provides an approximation to the karhunen loeve transform for the long memory noise and we have developed a scale space regression that permits one to carry out the regression in the wavelet domain while omitting the scales that are contaminated by the trend in order to demonstrate that our approach outperforms the state of the art detrending technique we evaluated our method against a smoothing spline approach experiments with simulated data and experimental fmri data demonstrate that our approach can infer and remove drifts that cannot be adequately represented with splines

a hybrid detrending method for fractional gaussian noise abstract determining trend and implementing detrending operations are important steps in data analysis yet there is neither precise definition of trend nor any logical algorithm for extracting it in this paper we propose a hybrid detrending method hdm which is based on the empirical mode decomposition emd and the detrended fluctuation analysis dfa our method can remove the polynomial and sinusoidal trends from the fractional gaussian noise fgn background we illustrate the method by selected examples from artificial time series and climate data in comparison with existing frequency domain based detrending methods our method is a posteriori the trend defined by our method is only derived from the data further our method also preserves the scaling behavior of the original signals

a dirichelet neumann m point bvp with a p laplacian like operator abstract let be odd increasing homeomorphisms from r onto r satisfying 0 0 0 and let f a b r r r be a function satisfying carath e odory s conditions let i r i a b i 1 m 2 a 1 2 m 2 b be given we are interested in the problem of existence of solutions for the m point boundary value problem u f t u u t a b u a 0 u b i 1 m 2 i u i in the resonance and non resonance cases we say that this problem is at resonance if the associated problem u 0 t a b u a 0 u b i 1 m 2 i u i has a non trivial solutions this is the case if and only if i 1 m 2 i 1 our results use topological degree methods interestingly enough in the non resonance case i e when i 1 m 2 i 1 the sign of degree for the relevant operator depends on whether i 1 m 2 i 1 or i 1 m 2 i 1

on positive solutions of boundary value problems for second order functional differential equations on infinite intervals we study the existence of positive solutions for the following boundary value problem on infinite interval for second order functional differential equations x t px t qx t f t xt 0 t 0 x t x t t t 0 limt x t 0 and x t px t qx t f t xt x t 0 t 0 x t x t t t 0 limt x t 0 where p 0 2 2 0 and q 0 the fixed point theorem on cone is used

on a class of regular singular two point boundary value problems abstract existence of a unique solution for a class of regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established wherep x satisfies i p x 0 on 0 b ii p x c1 0 r and for somer b iii xp x p x is analytic in z z xp x p x b 0 b 1 x b 0 1 and with quite general conditions onf x y these conditions onf x y are sharp which is seen through one example regions for multiple solutions have also been determined

function estimation via wavelet shrinkage for long memory data in this article we study function estimation via wavelet shrinkage for data with long range dependence we propose a fractional gaussian noise model to approximate nonparametric regression with long range dependence and establish asymptotics for minimax risks because of long range dependence the minimax risk and the minimax linear risk converge to 0 at rates that differ from those for data with independence or short range dependence wavelet estimates with best selection of resolution leveldependent threshold achieve minimax rates over a wide range of spaces cross validation for dependent data is proposed to select the optimal threshold the wavelet estimates significantly outperform linear estimates the key to proving the asymptotic results is a wavelet vaguelette decomposition which decorrelates fractional gaussian noise such wavelet vaguelette decomposition is also very useful in fractal signal processing

a second order spline finite difference method for singular two point boundary value problems a second order spline finite difference method based on a non uniform mesh for singular two point boundary value problems of the form x a x au f x u 0

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

estimation of 2 d noisy fractional brownian motion and its applications using wavelets the two dimensional 2 d fractional brownian motion fbm model is useful in describing natural scenes and textures most fractal estimation algorithms for 2 d isotropic fbm images are simple extensions of the one dimensional 1 d fbm estimation method this method does not perform well when the image size is small say 32 spl times 32 we propose a new algorithm that estimates the fractal parameter from the decay of the variance of the wavelet coefficients across scales our method places no restriction on the wavelets also it provides a robust parameter estimation for small noisy fractal images for image denoising a wiener filter is constructed by our algorithm using the estimated parameters and is then applied to the noisy wavelet coefficients at each scale we show that the averaged power spectrum of the denoised image is isotropic and is a nearly 1 f process the performance of our algorithm is shown by numerical simulation for both the fractal parameter and the image estimation applications to coastline detection and texture segmentation in a noisy environment are also demonstrated

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed