zero crossings of a wavelet transform the completeness stability and application to pattern recognition of a multiscale representation based on zero crossings is discussed an alternative projection algorithm is described that reconstructs a signal from a zero crossing representation which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero crossings the reconstruction algorithm has a fast convergence and each iteration requires o n log sup 2 n computation for a signal of n samples the zero crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems as an example the implementation and results of a coarse to fine stereo matching algorithm are described

a filter bank for the directional decomposition of images theory and design the authors introduce a directionally oriented 2 d filter bank with the property that the individual channels may be critically sampled without loss of information the passband regions of the component filters are wedge shaped and thus provide directional information it is shown that these filter bank outputs may be maximally decimated to achieve a minimum sample representation in a way that permits the original signal to be exactly reconstructed the authors discuss the theory for directional decomposition and reconstruction in addition implementation issues are addressed where realizations based on both recursive and nonrecursive filters are considered

on a complex beam beam interaction model with random forcing abstract in recent years many studies have been devoted to complex differential equations cde which appear in very important applications in physics and engineering this paper aims to investigate one such cde containing a random forcing term z o 2 z e 2 g z e 2 f z z p o t n t where z t x t iy t a complex function i 1 n t is a broad band process with zero mean and e2 and small real parameters in particular we use eq to model the interaction between two colliding beams in particle accelerators setting g z z f z z z z 2 and p o t cos o t and extend the work we had started in an earlier publication mahmoud physica a 216 1995 445 we apply the stochastic averaging method to derive a fokker planck kolmogorov equation for this equation and obtain analytically the exact stationary probability density function and the first and second moments in the amplitude of the solutions numerical simulations are carried out to compare with the theoretical ones and excellent agreement is found

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

analysis of self organized criticality in ehrenfest s dog flea model the self organized criticality in ehrenfest s historical dog flea model is analyzed by simulating the underlying stochastic process the fluctuations around the thermal equilibrium in the model are treated as avalanches we show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q gaussian the distribution which is obtained naturally in the context of nonextensive statistical mechanics if one avoids the finite size effects by increasing the system size we provide clear numerical evidence that the relation between the exponent ensuremath tau of avalanche size distribution obtained by maximum likelihood estimation and the q value of appropriate q gaussian obeys the analytical result recently introduced by caruso et al phys rev e 75 055101 r 2007 this allows us to determine the value of q parameter a priori from one of the well known exponents of such dynamical systems

design of directional filter banks with arbitrary number of subbands this correspondence proposes a novel method for designing directional filter banks dfbs with arbitrary number of subbands its key feature is the ability to decompose images into arbitrary directionally oriented subbands the proposed approach is based on the pseudo polar fourier transform ppft and one dimensional 1 d filter banks fbs we take some modifications on the ppft and then employ 1 d fbs to the modified ppft with these operations the two dimensional 2 d dfbs are obtained and the design of them is converted to that of 1 d fbs plus a modified ppft since the number of channels of 1 d fbs can be arbitrary the 2 d dfbs with arbitrary number of subbands can be achieved which is highly expected for directional representations of images two examples on directional feature extractions illustrate that the proposed non 2n channel dfbs can capture directional information more flexibly than the existing methods

a new family of nonredundant transforms using hybrid wavelets and directional filter banks we propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks we convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks where we form a nonredundant transform family which exhibits both directional and nondirectional basis functions we demonstrate the potential of the proposed transforms using nonlinear approximation in addition we employ the proposed family in two key image processing applications image coding and denoising and show its efficiency for these applications

motion estimation using a complex valued wavelet transform this paper describes a new motion estimation algorithm that is potentially useful for both computer vision and video compression applications it is hierarchical in structure using a separable two dimensional 2 d discrete wavelet transform dwt on each frame to efficiently construct a multiresolution pyramid of subimages the dwt is based on a complex valued pair of four tap fir filters with gabor like characteristics the resulting complex dwt cdwt effectively implements an analysis by an ensemble of gabor like filters with a variety of orientations and scales the phase difference between the subband coefficients of each frame at a given subpel bears a predictable relation to a local translation in the region of the reference frame subtended by that subpel that relation is used to estimate the displacement field at the coarsest scale of the multiresolution pyramid each estimate is accompanied by a directional confidence measure in the form of the parameters of a quadratic matching surface the initial estimate field is progressively refined by a coarse to fine strategy in which finer scale information is appropriately incorporated at each stage the accuracy efficiency and robustness of the new algorithm are demonstrated in comparison testing against hierarchical implementations of intensity gradient based and fractional precision block matching motion estimators

on the generalized averaging method of a class of strongly nonlinear forced oscillators we present a modified version of the generalized averaging method for studying the periodic solutions of a class of strongly nonlinear forced oscillators of the form x 2x ef x p t 0 where f x is a nonlinear function of x p t is a periodic function of t e need not be small and is a constant parameter this equation can be used to describe e g a pendulum with a vibrating length or the displacements of colliding particle beams in high energy accelerators the new version is based on defining a new parameter e and a linear transformation of the time this version is applied for the cases f x x3 and f x x4 with p t cos t and excellent agreement is found with the results of numerical experiments for large values of e

motion among random obstacles on a hyperbolic space we consider the motion of a particle along the geodesic lines of the poincare half plane the particle is specularly reflected when it hits randomly distributed obstacles that are assumed to be motionless this is the hyperbolic version of the well known lorentz process studied in the euclidean context we analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes under a suitable scaling we prove that our process converges to a markovian process namely a random flight on the hyperbolic manifold

stability regions for coupled hill s equations abstract in this paper we extend well known results for one hill s equation and present the stability analysis of two coupled hill s equations for which the general theory is not readily available approximate expressions are derived in the context of peturbation theory for the boundaries between bounded and unbounded periodic solutions with frequencies n m n and m are positive integers of both linear and nonlinear coupled mathieu equations as examples excellent agreement is found between theoretical predictions and numerical computations over large ranges of parameter values and initial conditions these periodic solutions are important because they correspond to some of the lowest order resonances of the system and when they are stable they turn out to have large regions of regular motion around them in phase space coupled mathieu equations appear in numerous important physical applications in problems of accelerator dynamics electrohydrodynamics and mechanics

wedgelets nearly minimax estimation of edges we study a simple horizon model for the problem of recovering an image from noisy data in this model the image has an edge with fi holder regularity adopting the viewpoint of computational harmonic analysis we develop an overcomplete collection of atoms called wedgelets dyadically organized indicator functions with a variety of locations scales and orientations the wedgelet representation provides nearly optimal representations of objects in the horizon model as measured by minimax description length we show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet decorated recursive partition which minimizes a complexity penalized sum of squares this estimate using su cient sub pixel resolution achieves nearly the minimax mean squared error in the horizon model in fact the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon 1 fi 2 wedgelet analysis and de noising may be used successfully outside the horizon model we study images modelled as indicators of star shaped sets with smooth bound aries and show that complexity penalized wedgelet partitioning achieves nearly the minimax risk in that setting also

critically sampled wavelets with composite dilations wavelets with composite dilations provide a general framework for the construction of waveforms defined not only at various scales and locations as traditional wavelets but also at various orientations and with different scaling factors in each coordinate as a result they are useful to analyze the geometric information that often dominate multidimensional data much more efficiently than traditional wavelets the shearlet system for example is a particular well known realization of this framework which provides optimally sparse representations of images with edges in this paper we further investigate the constructions derived from this approach to develop critically sampled wavelets with composite dilations for the purpose of image coding not only do we show that many nonredundant directional constructions recently introduced in the literature can be derived within this setting but we also introduce new critically sampled discrete transforms that achieve much better nonlinear approximation rates than traditional discrete wavelet transforms and outperform the other critically sampled multiscale transforms recently proposed

beam beam interaction models under narrow band random excitation abstract the aim of this paper is to continue our previous investigations on beam beam interaction models in particle accelerators by considering these models under narrow band random excitation in this investigation the multiple scale and the moment methods are used the equations of modulation of the amplitude and the phase are derived and perturbation techniques are used to seek approximate steady state solutions two special cases are considered to illustrate this study and excellent agreement is found between analytical and numerical results other cases can be similarly studied the effects of noise and detuning parameters are examined numerically and theoretically and we find that the system demonstrates a diffused limit cycle as the intensity of noise increases the numerical results show that the multiple scale method is effective for these stochastic systems

multiple multidimensional morse wavelets this paper defines a set of operators that localize a radial image in space and radial frequency simultaneously the eigenfunctions of the operator are determined and a nonseparable orthogonal set of radial wavelet functions are found the eigenfunctions are optimally concentrated over a given region of radial space and scale space defined via a triplet of parameters analytic forms for the energy concentration of the functions over the region are given the radial function localization operator can be generalised to an operator localizing any l2 ropf2 function it is demonstrated that the latter operator given an appropriate choice of localization region approximately has the same radial eigenfunctions as the radial operator based on a given radial wavelet function a quaternionic wavelet is defined that can extract the local orientation of discontinuous signals as well as amplitude orientation and phase structure of locally oscillatory signals the full set of quaternionic wavelet functions are component by component orthogonal their statistical properties are tractable and forms for the variability of the estimators of the local phase and orientation are given as well as the local energy of the image by averaging estimators across wavelets a substantial reduction in the variance is achieved

unraveling the structure of treelike networks from first passage times of lazy random walkers we study the problem of random search in finite networks with a tree topology where it is expected that the distribution of the first passage time f t decays exponentially we show that the slope of the exponential tail is independent of the initial conditions of entering the tree in general and scales exponentially or as a power law with the extent of the tree l depending on the tendency p to jump toward the target node it is unfeasible to uniquely determine l and p from measuring the tail slope or the mean first passage time mfpt of an ordinary diffusion along the tree to unravel the structure we consider lazy random walkers that take steps with probability m when jumping on the nodes and return with probability q from the leaves by deriving an exact analytical expression for the mfpt of the intermittent random walk we verify that the structural information of the tree can be uniquely extracted by measuring the mfpt for two randomly chosen types of tracer particles with distinct experimental parameters m and q we also address the applicability of our approach in the presence of disorder in the structure of the tree or statistical uncertainty in the experimental parameters

cascades of particles moving at finite velocity in hyperbolic spaces a branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space poincare half plane and poincare disk is examined each particle can split into two particles only once at poisson spaced times and deviates orthogonally when splitted at time t after n t poisson events there are n t 1 particles moving along different geodesic lines we are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles we derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as t increases and for different values of the parameters c hyperbolic velocity of motion and rate of reproduction the mean hyperbolic distance of each moving particle is also examined and a useful representation as the distance of a randomly stopped particle moving over the main geodesic line is presented

multidimensional mapping based complex wavelet transforms although the discrete wavelet transform dwt is a powerful tool for signal and image processing it has three serious disadvantages shift sensitivity poor directionality and lack of phase information to overcome these disadvantages we introduce multidimensional mapping based complex wavelet transforms that consist of a mapping onto a complex function space followed by a dwt of the complex mapping unlike other popular transforms that also mitigate dwt shortcomings the decoupled implementation of our transforms has two important advantages first the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy this allows us to create a directional nonredundant complex wavelet transform with potential benefits for image coding systems to the best of our knowledge no other complex wavelet transform is simultaneously directional and nonredundant the second advantage of our approach is the flexibility to use any dwt in the transform implementation as an example we exploit this flexibility to create the complex double density dwt a shift insensitive directional complex wavelet transform with a low redundancy of 3 sup m 1 2 sup m 1 in m dimensions no other transform achieves all these properties at a lower redundancy to the best of our knowledge by exploiting the advantages of our multidimensional mapping based complex wavelet transforms in seismic signal processing applications we have demonstrated state of the art results

a parametric texture model based on joint statistics of complex wavelet coefficients we present a universal statistical model for texture images in the context of an overcomplete complex wavelet transform the model is parameterized by a set of statistics computed on pairs of coefficients corresponding to basis functions at adjacent spatial locations orientations and scales we develop an efficient algorithm for synthesizing random images subject to these constraints by iteratively projecting onto the set of images satisfying each constraint and we use this to test the perceptual validity of the model in particular we demonstrate the necessity of subgroups of the parameter set by showing examples of texture synthesis that fail when those parameters are removed from the set we also demonstrate the power of our model by successfully synthesizing examples drawn from a diverse collection of artificial and natural textures

brownian bridge on hyperbolic spaces and on homogeneous trees let b be the brownian motion on a noncompact non euclidean rank one symmetric space h a typical examples is an hyperbolic space hn n 2 for 0 the brownian bridge b of length on h is the process bt 0 t conditioned by b0 b o where o is an origin in h it is proved that the process converges weakly to the brownian excursion when the brownian excursion is the radial part of the brownian bridge on 3 the same result holds for the simple random walk on an homogeneous tree

anderson transition on the cayley tree as a traveling wave critical point for various probability distributions for anderson localization on the cayley tree we study the statistics of various observables as a function of the disorder strength w and the number n of generations we first consider the landauer transmission tn in the localized phase its logarithm follows the traveling wave form where i the disorder averaged value moves linearly and the localization length diverges as with loc 1 and ii the variable t is a fixed random variable with a power law tail p t 1 t 1 w for large t with 0 w 1 2 so that all integer moments of tn are governed by rare events in the delocalized phase the transmission tn remains a finite random variable as n and we measure near criticality the essential singularity with t 0 25 we then consider the statistical properties of normalized eigenstates x x 2 1 in particular the entropy s x x 2ln x 2 and the inverse participation ratios ipr iq x x 2q in the localized phase the typical entropy diverges as with s 1 5 whereas it grows linearly as styp n n in the delocalized phase finally for the ipr we explain how closely related variables propagate as traveling waves in the delocalized phase in conclusion both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions and the anderson localization delocalization transition then corresponds to a traveling non traveling critical point moreover our results point toward the existence of several length scales that diverge with different exponents at criticality

beam beam interaction models with a small stochastic perturbation in this work we study a class of differential equations which may be used to model the beam beam interaction in particle accelerators in the presence of a small stochastic perturbation z t x 02x 2 g dotx 2 f x p 0t 2z t the method of stochastic averaging is used to derive a fokker planck kolmogorov equation describing the probability density for the amplitude of the solutions in the case g dotx dotx an odd polynomial f x k3x3 k5x5 and p 0t cos 0t we obtain the exact stationary probability density function and the first and second moments for the amplitude of the solutions numerical simulation shows very good agreement with the analytical results of this study

the monogenic wavelet transform this paper extends the 1 d analytic wavelet transform to the 2 d monogenic wavelet transform the transformation requires care in its specification to ensure suitable transform coefficients are calculated and it is constructed so that the wavelet transform may be considered as both local and monogenic this is consistent with defining the transform as a real wavelet transform of a monogenic signal in analogy with the analytic wavelet transform classes of monogenic wavelets are proposed with suitable local properties it is shown that the monogenic wavelet annihilates anti monogenic signals that the monogenic wavelet transform is phase shift covariant and that the transform magnitude is phase shift invariant a simple form for the magnitude and orientation of the isotropic transform coefficients of a unidirectional signal when observed in a rotated frame of reference is derived the monogenic wavelet ridges of local plane waves are given

random walks on the bethe lattice we obtain random walk statistics for a nearest neighbor polya walk on a bethe lattice infinite cayley tree of coordination numberz and show how a random walk problem for a particular inhomogeneous bethe lattice may be solved exactly we question the common assertion that the bethe lattice is an infinite dimensional system

scaling functions robust to translations the discrete wavelet transform dwt is popular in a wide variety of applications its sparse sampling eliminates redundancy in the representation of signals and leads to efficient processing however the dwt lacks translation invariance this makes it ill suited for many problems where the received signal is the superposition of arbitrarily shifted replicas of a transmitted signal as when multipath occurs for example the paper develops algorithms for the design of orthogonal and biorthogonal compact support scaling functions that are robust to translations our approach is to maintain the critical sampling of the dwt while designing multiresolution representations for which the coefficient energy redistributes itself mostly within each subband and not across the entire time scale plane we obtain expedite algorithms by decoupling the optimization from the constraints on the scaling function examples illustrate that the designed scaling function significantly improves the robustness of the representation

the double density dual tree dwt this paper introduces the double density dual tree discrete wavelet transform dwt which is a dwt that combines the double density dwt and the dual tree dwt each of which has its own characteristics and advantages the transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets one pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair simultaneously one pair of wavelets are designed to be approximate hilbert transforms of the other pair of wavelets so that two complex approximately analytic wavelets can be formed therefore they can be used to implement complex and directional wavelet transforms the paper develops a design procedure to obtain finite impulse response fir filters that satisfy the numerous constraints imposed this design procedure employs a fractional delay allpass filter spectral factorization and filterbank completion the solutions have vanishing moments compact support a high degree of smoothness and are nearly shift invariant

travelling randomly on the poincare half plane with a pythagorean compass a random motion on the poincare half plane is studied a particle runs on the geodesic lines changing direction at poisson paced times the hyperbolic distance is analyzed also in the case where returns to the starting point are admitted the main results concern the mean hyperbolic distance and also the conditional mean distance in all versions of the motion envisaged also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated

shiftable multiscale transforms one of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance the content of wavelet subbands is unstable under translations of the input signal wavelet transforms are also unstable with respect to dilations of the input signal and in two dimensions rotations of the input signal the authors formalize these problems by defining a type of translation invariance called shiftability in the spatial domain shiftability corresponds to a lack of aliasing thus the conditions under which the property holds are specified by the sampling theorem shiftability may also be applied in the context of other domains particularly orientation and scale jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored two examples of jointly shiftable transforms are designed and implemented a 1 d transform that is jointly shiftable in position and scale and a 2 d transform that is jointly shiftable in position and orientation the usefulness of these image representations for scale space analysis stereo disparity measurement and image enhancement is demonstrated

a translation invariant wavelet representation algorithm with applications we address the time varying problem of wavelet transforms and a new translation invariant wavelet representation algorithm is proposed using the algorithm introduced by beylkin see siam j numer anal vol 29 p 1716 1740 1992 we compute the wavelet transform for all the circular time shifts of a length n signal in o n log n operations the wavelet coefficients of the time shift with minimal cost are selected as the best representation of the signal using a binary tree search algorithm with an appropriate cost function we apply the translation invariant representation algorithm to a geoacoustic data compression application the results show that the new algorithm can reduce the distortion the squared error in our case substantially if the input signals are transients that are sensitive to time shifts

random walk kernels and learning curves for gaussian process regression on random graphs we consider learning on graphs guided by kernels that encode similarity between vertices our focus is on random walk kernels the analogues of squared exponential kernels in euclidean spaces we show that on large locally treelike graphs these have some counter intuitive properties specifically in the limit of large kernel lengthscales we consider using these kernels as covariance functions of gaussian processes in this situation one typically scales the prior globally to normalise the average of the prior variance across vertices we demonstrate that in contrast to the euclidean case this generically leads to significant variation in the prior variance across vertices which is undesirable from a probabilistic modelling point of view we suggest the random walk kernel should be normalised locally so that each vertex has the same prior variance and analyse the consequences of this by studying learning curves for gaussian process regression numerical calculations as well as novel theoretical predictions for the learning curves using belief propagation show that one obtains distinctly different probabilistic models depending on the choice of normalisation our method for predicting the learning curves using belief propagation is significantly more accurate than previous approximations and should become exact in the limit of large random graphs

diffusion in a one dimensional random medium and hyperbolic brownian motion classical diffusion in a random medium involves an exponential functional of brownian motion this functional also appears in the study of brownian diffusion on a riemann surface of constant negative curvature we analyse in detail this relationship and study various distributions using stochastic calculus and functional integration

exact closed form of the return probability on the bethe lattice an exact closed form solution for the return probability of a random walk on the bethe lattice is given the long time asymptotic form confirms a previously known expression it is however shown that this exact result reduces to the proper expression when the bethe lattice degenerates on a line unlike the asymptotic result which is singular this is shown to be an artefact of the asymptotic expansion the density of states is also calculated

quantum oscillations in mesoscopic rings and anomalous diffusion we consider the weak localization correction to the conductance of a ring connected to a network we analyse the harmonics content of the al tshuler aronov spivak aas oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by aas the physical origin of this behaviour is the anomalous diffusion of brownian trajectories around the ring due to the diffusion in the wires we show that this problem is related to the anomalous diffusion along the skeleton of a comb we study in detail the winding properties of brownian curves around a ring connected to an arbitrary network our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales a general expression of this length is derived for arbitrary networks more specifically we consider the case of a ring connected to wires to a square network and to a bethe lattice

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

random walk on the bethe lattice and hyperbolic brownian motion we give the exact solution to the problem of a random walk on the bethe lattice through a mapping on an asymmetric random walk on the half line we also study the continuous limit of this model and discuss in detail the relation between the random walk on the bethe lattice and brownian motion on a space of constant negative curvature

diffusion on a heptagonal lattice we study the diffusion phenomena on the negatively curved surface made up of congruent heptagons unlike the usual two dimensional plane this structure makes the boundary increase exponentially with the distance from the center and hence the displacement of a classical random walker increases linearly in time the diffusion of a quantum particle put on the heptagonal lattice is also studied in the framework of the tight binding model hamiltonian and we again find the linear diffusion like the classical random walk a comparison with diffusion on complex networks is also made

on the analytic wavelet transform an exact and general expression for the analytic wavelet transform of a real valued signal is constructed resolving the time dependent effects of nonnegligible amplitude and frequency modulation the analytic signal is first locally represented as a modulated oscillation demodulated by its own instantaneous frequency and then taylor expanded at each point in time the terms in this expansion called the instantaneous modulation functions are time varying functions which quantify at increasingly higher orders the local departures of the signal from a uniform sinusoidal oscillation closed form expressions for these functions are found in terms of bell polynomials and derivatives of the signal s instantaneous frequency and bandwidth the analytic wavelet transform is shown to depend upon the interaction between the signal s instantaneous modulation functions and frequency domain derivatives of the wavelet inducing a hierarchy of departures of the transform away from a perfect representation of the signal the form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal in which case our expressions simplify considerably one may then quantify the time varying bias associated with signal estimation via wavelet ridge analysis and choose wavelets to minimize this bias

a new framework for complex wavelet transforms although the discrete wavelet transform dwt is a powerful tool for signal and image processing it has three serious disadvantages shift sensitivity poor directionality and lack of phase information to overcome these disadvantages we introduce two stage mapping based complex wavelet transforms that consist of a mapping onto a complex function space followed by a dwt of the complex mapping unlike other popular transforms that also mitigate dwt shortcomings the decoupled implementation of our transforms has two important advantages first the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy this allows us to create a directional non redundant complex wavelet transform with potential benefits for image coding systems to the best of our knowledge no other complex wavelet transform is simultaneously directional and non redundant the second advantage of our approach is the flexibility to use any dwt in the transform implementation as an example we can exploit this flexibility to create the complex double density dwt cddwt a shift insensitive directional complex wavelet transform with a low redundancy of 3 sup m 1 2 sup m 1 in m dimensions to the best of our knowledge no other transform achieves all these properties at a lower redundancy

on stochastic complex beam beam interaction models with gaussian colored noise abstract this paper is to continue our study on complex beam beam interaction models in particle accelerators with random excitations y xu w xu g m mahmoud on a complex beam beam interaction model with random forcing physica a 336 2004 347 360 the random noise is taken as the form of exponentially correlated gaussian colored noise and the transition probability density function is obtained in terms of a perturbation expansion of the parameter then the method of stochastic averaging based on perturbation technique is used to derive a fokker planck equation for the transition probability density function the solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order e also the exact stationary probability density and the first and second moments of the amplitude are obtained and one can find when the correlation time equals to zero the result is identical to that derived from the stratonovich khasminskii theorem for the same model under a broad band excitation in our previous work

a class of multiresolution directional filter banks in this paper we introduced a class of directional filter banks dfbs having the previously proposed uniform dfb udfb as a special case except for the udfb each dfb in this class can be used to decompose an image yielding up to 12 directions while maintaining perfect reconstruction and maximal decimation a multiresolution representation can be obtained by repeating the same decomposition at the lowpass band the permissible property of the filter banks in cases of being implemented by a tree structure and by direct implementation is discussed the result shows that only one dfb in the class called the uniform quincunx dfb uqdfb satisfies the permissible property when being implemented directly without using the tree structure the nonuniform quincunx dfb nuqdfb is then constructed from the uqdfb by merging its two lowpass subbands an alternative structure for constructing the nuqdfb is presented the new structure while yielding the same frequency partitioning allows the dfb to be realized with complexity comparable to that of the separable wavelet filter bank the connection between the discrete filter bank and the continuous directional wavelet is also established numerical experiments on directional feature extractions image denoising and nonlinear approximation are presented at the end of the paper to demonstrate the potential of the nuqdfb

the design of approximate hilbert transform pairs of wavelet bases several authors have demonstrated that significant improvements can be obtained in wavelet based signal processing by utilizing a pair of wavelet transforms where the wavelets form a hilbert transform pair this paper describes design procedures based on spectral factorization for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate hilbert transform pair both orthogonal and biorthogonal fir solutions are presented as well as iir solutions in each case the solution depends on an allpass filter having a flat delay response the design procedure allows for an arbitrary number of vanishing wavelet moments to be specified a matlab program for the procedure is given and examples are also given to illustrate the results

transient response of non linear systems to random excitation abstract an approximate method for computing the transient response of non linear oscillators to suddenly applied wide band random excitation is developed the technique is derived initially for systems with non linear damping by using the method of stochastic averaging and then generalized to deal with oscillators with arbitrary non linear restoring forces predictions from the theory are compared with corresponding digital simulation estimates in a number of specific cases