parametrization of compactly supported orthonormal wavelets the parametrization of two channel reconstruction quadrature mirror filter banks is extended to obtain a single characterization of discrete orthonormal wavelets in particular it is shown that discrete orthonormal wavelets of support less than 2n 1 where n is an even integer are parametrized by n 1 real parameters each taking values in 0 2 pi a parametrization of discrete orthonormal wavelets with a number of vanishing moments greater than one is also given

harmonic wavelet analysis a new harmonic wavelet is suggested unlike wavelets generated by discrete dilation equations whose shape cannot be expressed in functional form harmonic wavelets have the simple structure w x exp i4 pi x exp i2 pi x i2 pi x this function w x is concentrated locally around x 0 and is orthogonal to its own unit translations and octave dilations its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain rather than in the x domain an efficient implementation of a discrete transform using this wavelet is based on the fast fourier transform fft fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the fft the same process works backwards for the inverse transform

on the stability of the classical collocation method for cauchy type singular integral equations abstract the classical collocation method for cauchy type singular integral equations of the second kind on an interval using gaussian quadrature rules with respect to jacobi nodes is investigated in general the nodes and the weights of the quadrature rules can only be evaluated approximately in the present paper the question is answered how precise our knowledge about them must be in order to ensure that the corresponding algebraic system of equations is solvable yet and its solution is close to the solution of the exact system

rational sampling filter banks based on iir filters the problem of splitting the spectrum of a digital signal by using nonuniform infinite impulse response iir filter banks is addressed near perfect reconstruction npr is considered the method uses the modulation of different iir prototypes the cancellation of the main aliasing components constrains the prototypes to be dependent on each other by using this approach linear phase prototypes are needed and noncausal filtering is required numerical examples of filter bank design are given and the computational complexity is compared with the finite impulse response fir case

biorthogonal spline type wavelets let f be an orthonormal scaling function with approximation degree p 1 and let b nbe the b spline of order n then spline type scaling functions defined by n b n n 1 2 possess higher approximation order p n 1 and compact support the corresponding biorthogonal wavelet functions are also constructed this technique is extended to the case of biorthogonal scaling function system as an application of the method supplied in this paper one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function in particular if both the method and the lifting scheme of sweldens see 1 are applied then all pairs of biorthogonal spline type scaling functions shown in references 2 and 3 can be constructed from the haar scaling function

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

optimal design of cosine modulated nonuniform linear phase fir filter bank via both stretching and shifting frequency response of single prototype filter this paper designs an optimal cosine modulated nonuniform linear phase finite impulse response fir filter bank the frequency responses of all the analysis filters and the synthesis filters of the filter bank are derived based on both stretching and shifting the frequency response of a single prototype filter the total aliasing error of the filter bank is minimized subject to specifications on the maximum magnitude distortion of the filter bank and the maximum ripple magnitudes of the prototype filter over both the passband and the stopband this paper proposes a joint constraint transcription and modified filled function method for solving the optimization problem in particular the functional inequality constraints are converted to discrete constraints via the constraint transcription method the globally optimal solution of the nonconvex optimization problem can be found efficiently via the modified filled function method computer numerical simulation results show that our design outperforms existing designs

numerical solution of systems of cauchy singular integral equations with constant coefficients abstract this paper deals with the numerical solution of a class of systems of cauchy singular integral equations with constant coefficients the proposed procedure consists of two basic steps the first one is to consider a modified problem equivalent to the original one under suitable conditions the second one is to approximate its solution by means of a vector of polynomial functions such array is constructed by applying a quadrature type method based on gaussian rules that leads to solve a determined and well conditioned linear system the convergence and stability of the method are proved in weighted l 2 spaces some numerical tests are also shown

wavelet multigrid analysis of squeeze film characteristics of poroelastic bearings the combined study of effects of poroelasticity and couple stresses on the performance of lubrication aspects of porelastic bearings in general and that of synovial joints in particular are analyzed the modified form of reynolds equation which incorporates the elastic nature of cartilage and stokes couple stress fluid as lubricant is derived and solved using a recently developed wavelet multigrid method this method has greatest advantage of minimizing the errors using wavelet transforms in obtaining accurate solution as grid size tends to zero it is found that 6 7 cycles are required to obtain a reasonably accurate solution in the multigrid scheme whereas only one cycle is required to obtain the solution in the wavelet multigrid method also matrix of discrete wavelet transform acts as a natural preconditioner producing rapid convergence it is observed that the poroelastic bearings with couple stress fluid as lubricant provide enhancement in pressure and ensure the increased load carrying capacity compared with viscous fluids this may be one of the reasons in the efficient lubrication and proper functioning of synovial joints

near perfect reconstruction pseudo qmf banks a novel approach to the design of m channel pseudo quadrature mirror filter qmf banks is presented in this approach the prototype filter is constrained to be a linear phase spectral factor of a 2mth band filter as a result the overall transfer function of the analysis synthesis system is a delay moreover the aliasing cancellation ac constraint is derived such that all the significant aliasing terms are canceled consequently the aliasing level at the output is comparable to the stopband attenuation of the prototype filter in other words the only error at the output of the analysis synthesis system is the aliasing error which is at the level of stopband attenuation using this approach it is possible to design a pseudo qmf bank where the stopband attenuation of the analysis and thus synthesis filters is on the order of spl minus 100 db moreover the resulting reconstruction error is also on the order of spl minus 100 db several examples are included

the shift invariant discrete wavelet transform and application to speech waveform analysis the discrete wavelet transform may be used as a signal processing tool for visualization and analysis of nonstationary time sampled waveforms the highly desirable property of shift invariance can be obtained at the cost of a moderate increase in computational complexity and accepting a least squares inverse pseudoinverse in place of a true inverse a new algorithm for the pseudoinverse of the shift invariant transform that is easier to implement in array oriented scripting languages than existing algorithms is presented together with self contained proofs representing only one of the many and varied potential applications a recorded speech waveform illustrates the benefits of shift invariance with pseudoinvertibility visualization shows the glottal modulation of vowel formants and frication noise revealing secondary glottal pulses and other waveform irregularities additionally performing sound waveform editing operations i e cutting and pasting sections on the shift invariant wavelet representation automatically produces quiet click free section boundaries in the resulting sound the capabilities of this wavelet domain editing technique are demonstrated by changing the rate of a recorded spoken word individual pitch periods are repeated to obtain a half speed result and alternate individual pitch periods are removed to obtain a double speed result the original pitch and formant frequencies are preserved in informal listening tests the results are clear and understandable

design of pseudo qmf banks with rational sampling factors using several prototype filters a method is presented for designing filter banks that allow the spectrum of a signal to be split into nonuniform width subbands the filters of the banks are obtained by the cosine modulation of more than one prototype the method uses the cancellation of the main aliasing component of the reconstructed signal this imposes constraints on the prototypes that become dependent on each other the procedure can be applied to banks with both integer and rational decimation factors numerical examples are presented to show the effectiveness of the design procedure

wavelet transforms associated with finite cyclic groups multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers an important class of multiresolution decompositions are the laplacian pyramid schemes in which the resolution is successively halved by recursively low pass filtering the signal under analysis and decimating it by a factor of two in general the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete time fourier analysis of sequences of complex numbers an analogous framework is developed for the multiresolution analysis of finite length sequences of elements from arbitrary fields attention is restricted to sequences of length 2 sup n for n a positive integer so that the resolution may be recursively halved to completion as in finite length fourier analysis a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields

adaptive learning for event modeling and characterization adaptive learning of specific patterns or events of interest has been an area of significant research for various applications in the last two decades in developing diagnostic evaluation and safety monitoring applications of a propulsion system it is critical to detect characterize and model events of interest it is a challenging task since the detection system should allow adaptive characterization of potential events of interest and correlate them to learn new models for future detection for online health monitoring and diagnostic evaluation in this paper a novel framework is established using a hierarchical adaptive clustering approach with fuzzy membership functions to characterize specific events of interest from the measured and processed features raw engine measurement data is first analyzed using the wavelet transform to provide features for localization of frequency information for use in the classification system a method combining hierarchical and fuzzy k means clustering is then applied to a set of selected measurements and computed features to determine the events of interest during engine operations experimental results have shown that the proposed approach is effective and computationally efficient to detect characterize and model new events of interest from data collected through continuous operations

on the solution of a nonlinear integral equation on the basis of a fixed point technique and cubic b spline scaling functions in this paper we apply the fixed point method to solve some nonlinear functional volterra integral equations which appear in many physical chemical and biological problems in each iteration of this method cubic semi orthogonal compactly supported b spline wavelets are used as basis functions to approximate the solution also the convergence of this numerical method is investigated and some examples are presented to show the accuracy and convergence of the method

the convergence of a collocation method for a class of cauchy singular integral equations in several recent papers we have studied the numerical solution of cauchy singular integral equations of the first kind by a polynomial collocation method 4 61 this technique has proved to be quite efftcient for many problems even when the kernel but not the right hand side is discontinuous 4 61 the l convergence and stability was established in s and 6 in 121 ioakimidis extended this method to solve cauchy singular integral equations of the second kind with constant coefficients although some discussion of convergence was given it appears at least to this author that the proof is incomplete and the conditions on the kernel and the right hand side are somewhat more restrictive than necessary in this paper we will show that the convergence analysis given in 5 6 can be generalized to establish the l convergence of collocation under rather mild restrictions on the data although a variety of numerical techniques such as galerkin s method and direct quadrature have been used extensively to solve cauchy singular equations they seem to be generally suited for problems with smooth kernels for problems with discontinuous kernels collocation methods appear to be a reasonable compromise between time consuming galerkin methods 2 and direct quadrature methods which require the evaluation of the kernel at pairs of points where the kernel may become unbounded 3 131 it is well known that for fredholm equations collocation is related to product integration quadrature methods 11 since such methods seem not to have been developed for the solution of cauchy singular equations collocation appears to be a good way of dealing with fairly general integral equations of this type 4 71

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

design of linear phase nonuniform filter banks with partial cosine modulation the majority of the existing work on designing nonuniform filter banks nufbs cannot achieve linear phase lp property because of the high complexity associated with phase distribution this correspondence proposes an idea of partial cosine modulation to obtain the lp property of nufbs with rational sampling factors it makes the efficient modulation technique possible to be used to design lp nufbs except the separately designed lowpass and highpass analysis synthesis filters we obtain the bandpass by cosine modulation of several prototypes worth of the name partial cosine modulation by analyzing the phase issue of significant aliasing terms we derive the matching conditions to make the required lp nufb achievable with these criteria being satisfied the design problem becomes that of several prototypes as well as the lowpass and highpass filters leading to a less design effort by using the proposed method near perfect reconstruction lp nufbs can be obtained in a simple and efficient way as demonstrated by examples

the discrete wavelet transform wedding the a trous and mallat algorithms two separately motivated implementations of the wavelet transform are brought together it is observed that these algorithms are both special cases of a single filter bank structure the discrete wavelet transform the behavior of which is governed by the choice of filters in fact the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact moreover it is shown that the commonly used lagrange a trous filters are in one to one correspondence with the convolutional squares of the daubechies filters for orthonormal wavelets of compact support a systematic framework for the discrete wavelet transform is provided and conditions are derived under which it computes the continuous wavelet transform exactly suitable filter constraints for finite energy and boundedness of the discrete transform are also derived relevant signal processing parameters are examined and it is observed that orthonormality is balanced by restrictions on resolution

complete to overcomplete discrete wavelet transforms theory and applications a new transform is proposed that derives the overcomplete discrete wavelet transform odwt subbands from the critically sampled dwt subbands complete representation this complete to overcomplete dwt codwt has certain advantages in comparison to the conventional approach that performs the inverse dwt to reconstruct the input signal followed by the a spl grave trous or the lowband shift algorithm specifically the computation of the input signal is not required as a result the minimum number of downsampling operations is performed and the use of upsampling is avoided the proposed codwt computes the odwt subbands by using a set of prediction filter matrices and filtering and downsampling operators applied to the dwt this formulation demonstrates a clear separation between the single rate and multirate components of the transform this can be especially significant when the codwt is used in resource constrained environments such as resolution scalable image and video codecs to illustrate the applicability of the proposed transform in these emerging applications a new scheme for the transform calculation is proposed and existing coding techniques that benefit from its usage are surveyed the analysis of the proposed codwt in terms of arithmetic complexity and delay reveals significant gains as compared with the conventional approach

random affine iterated function systems curve generation and wavelets it is shown how refinement methods for smooth curve generation can be carried out efficiently through iterated function systems ifs affine transformations are constructed so that when composed randomly they generate the desired smooth curve underlying this random algorithm is the tree traversal property of ifs under a refinement method the points on the curve correspond to leaves on some n tree ifs theory enables one to generate all of these leaves through a single orbit of an appropriate markov chain applications include bezier curves splines wavelets and various interpolants

the near shift invariance of the dual tree complex wavelet transform revisited abstract the dual tree complex wavelet transform dt c wt is an enhancement of the conventional discrete wavelet transform dwt due to a higher degree of shift invariance and a greater directional selectivity finding its applications in signal and image processing this paper presents a quantitative proof of the superiority of the dt c wt over the dwt in case of modulated wavelets

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

the design of nonuniform modulated filterbanks we present an approach to nonuniform filterbank design based on modulated filters and the principle of adjacent channel aliasing cancellation the approach is a generalization of pseudo qmf designs to nonuniform channel arrangements and the essential idea is to form nonuniform filterbanks from uniform sections joined by transition filters all channel filters are formed by modulating lowpass prototypes to meet the aliasing cancellation conditions the transition filters are modulated from complex lowpass prototypes although the resulting channel filters are still real the approach provides a computationally simple method for designing filterbanks with large numbers of channels because of the fact that channel filters are modulated from a few lowpass prototypes and because the lowpass prototype designs can be scaled that is designs for narrow channel spacing can be produced from designs for wider channel spacing by interpolating the lowpass prototype impulse response the development is limited to the design of nonuniform filterbanks with integer decimation factors although we demonstrate that the technique can be used to produce excellent rational decimation factor designs by combining channel outputs using an inverse polyphase transform

phase shifting for nonseparable 2 d haar wavelets in this paper we present a novel and efficient solution to phase shifting 2 d nonseparable haar wavelet coefficients while other methods either modify existing wavelets or introduce new ones to handle the lack of shift invariance we derive the explicit relationships between the coefficients of the shifted signal and those of the unshifted one we then establish their computational complexity and compare and demonstrate the superior performance of the proposed approach against classical interpolation tools in terms of accumulation of errors under successive shifting

perfect reconstruction filter banks with rational sampling factors an open problem namely how to construct perfect reconstruction filter banks with rational sampling factors is solved such filter banks have n branches each one having a sampling factor of p i qi and their sum equals one in this way the well known theory of filter banks with uniform band splitting is extended to allow for nonuniform divisions of the spectrum this can be very useful in the analysis of speech and music the theory relies on two transforms the first transform leads to uniform filter banks having polyphase components as individual filters the other results in a uniform filter bank containing shifted versions of same filters this in turn introduces dependencies in design and is left for future work as an illustration several design examples for the 2 3 1 3 case are given filter banks are then classified according to the possible ways in which they can be built it is shown that some cases cannot be solved even with ideal filters with real coefficients

wavelet packet division multiplexing and wavelet packet design under timing error effects wavelet packet division multiplexing wpdm is a multiple signal transmission technique in which the message signals are waveform coded onto wavelet packet basis functions for transmission the overlapping nature of such waveforms in time and frequency provides a capacity improvement over the commonly used frequency division multiplexing fdm and time division multiplexing tdm schemes while their orthogonality properties ensure that the overlapping message signals can be separated by a simple correlator receiver the interference caused by timing offset in transmission is examined a design procedure that exploits the inherent degrees of freedom in the wpdm structure to mitigate the effects of timing error is introduced and a waveform that minimizes the energy of the timing error interference is designed an expression for the probability of error due to the presence of gaussian noise and timing error for the transmission of binary data is derived the performance advantages of the designed waveform over standard wavelet packet basis functions are demonstrated by both analytical and simulation methods the capacity improvement of wpdm its simple implementation and the possibility of having optimum waveform designs indicate that wpdm holds considerable promise as a multiple signal transmission technique

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

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

on the choice of smoothing parameter threshold and truncation in nonparametric regression by non linear wavelet methods summary concise asymptotic theory is developed for non linear wavelet estimators of regression means in the context of general error distributions general designs general normalizations in the case of stochastic design and non structural assumptions about the mean the influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters mainly the tail weight is described in an extremely simple way by a moment condition previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal different approaches to correction for stochastic design are suggested these include conventional kernel estimation of the design density in which case the interaction between the smoothing parameters of the non linear wavelet estimator and the linear kernel method is described

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

condition numbers for singular integral equations in weighted l 2 spaces the convergence and stability of a discrete collocation method for cauchy singular integral equations in some weighted besov spaces are studied this numerical method results in solving a linear system in order to determine the unknown coefficients of the approximate solution the author proves that this linear system is well conditioned

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

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

cosine modulated fir filter banks satisfying perfect reconstruction the authors obtain a necessary and sufficient condition on the 2m m number of channels polyphase components of a linear phase prototype filter of length n 2 mm where m an arbitrary positive integer such that the polyphase component matrix of the modulated filter is lossless the losslessness of the polyphase component matrix in turn is sufficient to ensure that the analysis synthesis system satisfies perfect reconstruction pr using this result a novel design procedure is presented based on the two channel lossless lattice this enables the design of a large class of fir finite impulse response pr filter banks and includes the n 2m case it is shown that this approach requires fewer parameters to be optimized than in the pseudo qmf quadrature mirror filter designs and in the lossless lattice based pr qmf designs for equal length filters in the three designs this advantage becomes significant when designing long filters for large m the design procedure and its other advantages are described in detail design examples and comparisons are included

algorithms for designing wavelets to match a specified signal algorithms for designing a mother wavelet spl psi x such that it matches a signal of interest and such that the family of wavelets 2 sup j 2 spl psi 2 sup j x k forms an orthonormal riesz basis of l sup 2 spl rscr are developed the algorithms are based on a closed form solution for finding the scaling function spectrum from the wavelet spectrum many applications require wavelets that are matched to a signal of interest most current design techniques however do not design the wavelet directly they either build a composite wavelet from a library of previously designed wavelets modify the bases in an existing multiresolution analysis or design a scaling function that generates a multiresolution analysis with some desired properties in this paper two sets of equations are developed that allow us to design the wavelet directly from the signal of interest both sets impose bandlimitedness resulting in closed form solutions the first set derives expressions for continuous matched wavelet spectrum amplitudes the second set of equations provides a direct discrete algorithm for calculating close approximations to the optimal complex wavelet spectrum the discrete solution for the matched wavelet spectrum amplitude is identical to that of the continuous solution at the sampled frequencies an interesting byproduct of this work is the result that meyer s spectrum amplitude construction for an orthonormal bandlimited wavelet is not only sufficient but necessary specific examples are given which demonstrate the performance of the wavelet matching algorithms for both known orthonormal wavelets and arbitrary signals

a shift invariant discrete wavelet transform this article presents a unifying approach to the derivation and implementation of a shift invariant wavelet transform of one and two dimensional 1 d and 2 d discrete signals starting with mallat s 1989 multiresolution wavelet representation mrwar it presents an analytical process through which a shift invariant orthogonal discrete wavelet transform called the multiscale wavelet representation mswar is obtained the coefficients in the mswar are shown to be inclusive of those in the mrwar with the implication that the derived representation is invertible the computational complexity of the mswar is quantified in terms of the required convolutions and its implementation is shown to be equivalent to the filter upsampling technique

digital filter bank design quadratic constrained formulation formulate the filter bank design problem as an quadratic constrained least squares minimization problem the solution of the minimization problem converges very quickly since the cost function as well as the constraints are quadratic functions with respect to the unknown parameters the formulations of the perfect reconstruction cosine modulated filter bank of the near perfect reconstruction pseudo qmf bank and of the two channel biorthogonal linear phase filter bank are derived using the proposed approach compared with other design methods the proposed technique yields pr filter banks with much higher stopband attenuation the proposed technique can also be extended to design multidimensional filter banks

wavelet transform based adaptive filters analysis and new results in this paper the wavelet transform is used in an adaptive filtering structure the coefficients of the adaptive filter are updated by the help of the least mean square lms algorithm first the wavelet transform based adaptive filter wtaf is described and it is analyzed for its wiener optimal solution then the performance of the wtaf is studied by the help of learning curves for three different convergence factors 1 constant convergence factor 2 time varying convergence factor and 3 exponentially weighted convergence factor the exponentially weighted convergence factor is proposed to introduce scale based variation to the weight update equation it is shown for two different sets of data that the rate of convergence increases significantly for all three wtaf structures as compared to that of time domain lms the high convergence rates of the wtaf give us reason to expect that it will perform well in tracking rapid changes in a signal

feature extraction across individual time series observations with spikes using wavelet principal component analysis clinical studies frequently include repeated measurements of individuals often for long periods we present a methodology for extracting common temporal features across a set of individual time series observations in particular the methodology explores extreme observations within the time series such as spikes as a possible common temporal phenomenon wavelet basis functions are attractive in this sense as they are localized in both time and frequency domains simultaneously allowing for localized feature extraction from a time varying signal we apply wavelet basis function decomposition of individual time series with corresponding wavelet shrinkage to remove noise we then extract common temporal features using linear principal component analysis on the wavelet coefficients before inverse transformation back to the time domain for clinical interpretation we demonstrate the methodology on a subset of a large fetal activity study aiming to identify temporal patterns in fetal movement fm count data in order to explore formal fm counting as a screening tool for identifying fetal compromise and thus preventing adverse birth outcomes copyright 2013 john wiley sons ltd

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

fast positive definite linear system solvers the authors show that the m band wavelet transforms of a wide class of covariance matrices consist of subblocks that are essentially banded furthermore they prove that the cholesky factors of the transformed covariance matrices also consist of subblocks that are essentially banded they combine these two observations to construct a fast o n sup 2 algorithm for solving the n spl times n linear positive definite systems of equations that arise in statistical signal processing finally they provide an error analysis of the proposed linear positive definite system solver

the root unroot algorithm for density estimation as implemented via wavelet block thresholding we propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem this regression problem is created by binning the original observations into many small size bins and by then applying a suitable form of root transformation to the binned data counts in principle many common nonparametric regression estimators could then be applied to the transformed data we propose use of a wavelet block thresholding estimator in this paper finally the estimated regression function is un rooted by squaring and normalizing the density estimation procedure achieves simultaneously three objectives computational efficiency adaptivity and spatial adaptivity a numerical example and a practical data example are discussed to illustrate and explain the use of this procedure theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the besov classes the estimator also automatically adapts to the local smoothness of the underlying function and attains the local adaptive minimax rate for estimating functions at a point there are three key steps in the technical argument poissonization quantile coupling and oracle risk bound for block thresholding in the non gaussian setting some of the technical results may be of independent interest

the advanced oca for 2 d discrete periodized wavelet transformation we develop a novel computation method for the 2 d discrete periodized wavelet transformation dpwt the new algorithm is based on the operator correlation algorithm oca compared with the classical 2 d pyramid algorithm the new advanced operator correlation algorithm aoca has two major advantages it requires half the number of multiplications and can yield the same output snr with half the precision of the dpwt filter coefficients furthermore the modular structure of the adca makes it particularly suitable for a vlsi implementation

an introduction to wavelets with applications to andrews plots a brief introduction to wavelets targeted to the statistician is given several types of wavelets are described and compared with the fourier functions it is shown how wavelets might extend the multivariate clustering technique of andrews plots

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

deblurring of bilevel waveforms an algorithm for processing closely spaced edges and accurately restoring their locations is presented the convolution distortion model is based on interacting edges the restoration algorithm takes three edges as input and forces the effects of two of them to cancel each other thus the third edge appears to be isolated and is located using a traditional edge detector experiments on bar code waveforms reveal that this approach outperforms the current commercial bar code readers

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