an optimum solution for scale invariant object recognition based on the multiresolution approximation this paper presents a multiresolution approximation approach to obtaining boundary representations for object recognition our technique combines a multiresolution approximation and the curvature scale space representation for obtaining representations our research consists of two main parts in the first part of our research we introduce the continuous multiresolution approximation cma in terms of the continuous wavelet transform cwt then we implement a fast algorithm to compute the cma we apply the cma to a boundary to obtain approximations of the boundary at various resolutions the cma provides a consistent interpretation of objects with scale variations moreover we can quickly compute our representations by using the fast algorithm for the cma in the second part we propose three representations for object recognition which cover most boundary based object recognition problems all three representations use the approximations obtained by the cma each representation has different features and covers different types of matching problems but all representations are constructed by using curvature zero crossings of the approximations our representations provide a general but reliable solution to most boundary based object matching problems finally we investigate the properties of our representations such as validity efficiency and reliability we verified our results experimentally to demonstrate the feasibility of using our representations for object recognition

wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

fast implementation of the continuous wavelet transform with integer scales we describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m in this approach the input signal and the analyzing wavelet are both represented by polynomial splines the algorithm uses a combination of moving sum and zero padded filters and its complexity per scale is o n where n is the signal length the computation is exact and the implementation is noniterative across scales we also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection first and second derivative operators or gabor like time frequency signal analysis

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

scalability of 2 d wavelet transform algorithms analytical and experimental results on mpps this paper studies the scalability of two dimensional 2 d discrete wavelet transform dwt algorithms on massively parallel processors mpps the principal operation in the 2 d dwt is the filtering operation used to implement the filter banks of the 2 d subband decomposition this filtering operation can be implemented as a convolution in the time domain or as a multiplication in the frequency domain we demonstrate that there exist combinations of machine size image size and wavelet kernel size for which the time domain algorithms outperform the frequency domain algorithms and vice versa we therefore demonstrate that a hybrid approach that combines time and frequency domain approaches can yield linear scalability for a broad range of problem and machine sizes furthermore we show the effect of processor speed versus communication overhead and the use of separable versus nonseparable wavelets on the crossover points between the algorithm approaches

joint space frequency segmentation using balanced wavelet packet trees for least cost image representation we examine the question of how to choose a space varying filterbank tree representation that minimizes some additive cost function for an image the idea is that for a particular cost function e g energy compaction or quantization distortion some tree structures perform better than others while the wavelet tree represents a good choice for many signals it is generally outperformed by the best tree from the library of wavelet packet frequency selective trees the double tree library of bases performs better still by allowing different wavelet packet trees over all binary spatial segments of the image we build on this foundation and present efficient new pruning algorithms for both one and two dimensional 1 d and 2 d trees that will find the best basis from a library that is many times larger than the library of the single tree or double tree algorithms the augmentation of the library of bases overcomes the constrained nature of the spatial variation in the double tree bases and is a significant enhancement in practice use of these algorithms to select the least cost expansion for images with a rate distortion cost function gives a very effective signal adaptive compression scheme this scheme is universal in the sense that without assuming a model for the signal or making use of training data it performs very well over a large class of signal types in experiments it achieves compression rates that are competitive with the best training based schemes

relative complexity changes in time series using information measures abstract estimation of complexity is of great interest in nonlinear signal and system analysis several complexity measures have been proposed lyapunov exponents lempel and ziv approximate entropy in the present study complexity measures derived from shannon entropy harvda charvat darovczy tsallis q entropies and their corresponding relative information measures are presented and evaluated in the context of nonlinear systems presenting abrupt complexity changes the performance of the proposed measures in the presence of controlled complexity is evaluated through numerical experiments using nonlinear models an example with heart rate variability signals is presented the results obtained show that the entropic and the relative complexity measures approach allow to discern complexity changes in a similar qualitative way compared against classical techniques but with much less computational cost and less amount of data in the presence of noise the relative complexity measures behave as robust tools for relative complexity changes detection time scale complexity analyses are presented using the continuous multiresolution entropies the assessment of time scale complexity changes is also discussed

multiwavelet moments and projection prefilters an efficient projection procedure is derived for use of orthogonal multiwavelets in the analysis of discrete data sequences a family of simple prefilters corresponding to numerical quadrature evaluation of the projection integrals provides exact results for locally polynomial data the full approximation order of the multiwavelet basis can thus always be enabled for nonpolynomial signals the prefilters provide approximations to the coefficients of the multiwavelet series whose convergence accelerates quickly with increase in sampling rate comparison is also made with previous time invariant multiwavelet prefilters

fast integral wavelet transform on a dense set of the time scale domain the objective of this paper is to introduce a fast algorithm for computing the integral wavelet transform iwt on a dense set of points in the time scale domain by applying the duality principle and using a compactly supported spline wavelet as the analyzing wavelet this fast integral wavelet transform fiwt is realized by applying only fir moving average operations and can be implemented in parallel since this computational procedure is based on a local optimal order spline interpolation scheme and the fir filters are exact the iwt values so obtained are guaranteed to have zero moments up to the order of the cardinal spline functions the semi orthogonal s o spline wavelets used here cannot be replaced by any other biorthogonal wavelet spline or otherwise which is not s o since the duality principle must be applied to some subspace of the multiresolution analysis under consideration in contrast with the existing procedures based on direct numerical integration or an fft based multi voice per octave scheme the computational complexity of our fiwt algorithm does not increase with the increasing number of values of the scale parameter

a new prefilter design for discrete multiwavelet transforms in conventional wavelet transforms prefiltering is not necessary due to the lowpass property of a scaling function this is no longer true for multiwavelet transforms a few research papers on the design of prefilters have appeared but the existing prefilters are usually not orthogonal which often causes problems in coding moreover the condition on the prefilters was imposed based on the first step discrete multiwavelet decomposition we propose a new prefilter design that combines the ideas of the conventional wavelet transforms and multiwavelet transforms the prefilters are orthogonal but nonmaximally decimated they are derived from a very natural calculation of multiwavelet transform coefficients in this new prefilter design multiple step discrete multiwavelet decomposition is taken into account our numerical examples by taking care of the redundant prefiltering indicate that the energy compaction ratio with the geronimo hardin massopust 1994 2 wavelet transform and our new prefiltering is better than the one with daubechies d sub 4 wavelet transform

frequency domain volume rendering by the wavelet x ray transform we describe a wavelet based x ray rendering method in the frequency domain with a smaller time complexity than wavelet splatting standard fourier volume rendering is summarized and interpolation and accuracy issues are briefly discussed we review the implementation of the fast wavelet transform in the frequency domain the wavelet x ray transform is derived and the corresponding fourier wavelet volume rendering algorithm fwvr is introduced fwvr uses haar or b spline wavelets and linear or cubic spline interpolation various combinations are tested and compared with wavelet splatting ws we use medical mr and ct scan data as well as a 3 d analytical phantom to assess the accuracy time complexity and memory cost of both fwvr and ws the differences between both methods are enumerated

characterization of analytic wavelet transforms and a new phaseless reconstruction algorithm we obtain a characterization of all wavelets leading to analytic wavelet transforms wt the characterization is obtained as a byproduct of the theoretical foundations of a new method for wavelet phase reconstruction from magnitude only coefficients the cornerstone of our analysis is an expression of the partial derivatives of the continuous wt which results in phase magnitude relationships similar to the short time fourier transform setting and valid for the generalized family of cauchy wavelets we show that the existence of such relations is equivalent to analyticity of the wt up to a multiplicative weight and a scaling of the mother wavelet the implementation of the new phaseless reconstruction method is considered in detail and compared to previous methods it is shown that the proposed method provides significant performance gains and a great flexibility regarding accuracy versus complexity in addition we discuss the relation between scalogram reassignment operators and the wavelet transform phase gradient and present an observation on the phase around zeros of the wt

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

subintegrality invertible modules and polynomial extensions motivated by a result of traverso and swan on seminormality we prove that for a ring extension a b a is subintegrally closed in b if and only if the group of invertible a submodules of b is canonically isomorphic to the group of invertible a x submodules of b x we also examine a relationship between these two groups in the general case i e when a may not be subintegrally closed in b

matched sampling systems relation to wavelets and implementation using prcc filter banks this paper deals with two types of sampling systems namely the interpolation and approximation sampling systems closed form expressions are derived for the frequency responses of the filters used in these systems that are matched to the input process in the mean squared sense closed form expressions are also derived for the mean squared error between the input and the reconstructed processes for these matched sampling systems using these expressions it is shown that the meyer scaling function and wavelet or functions derived from these arise naturally in the context of subsampled bandlimited processes to implement these systems the perfect reconstruction circular convolution prcc filter bank is proposed as a framework for the frequency sampled implementation of these systems examples of matched interpolation and approximation sampling systems are provided and their performance is compared with some standard interpolators to demonstrate their efficacy

the recursive pyramid algorithm for the discrete wavelet transform the recursive pyramid algorithm rpa is a reformulation of the classical pyramid algorithm pa for computing the discrete wavelet transform dwt the rpa computes the n point dwt in real time running dwt using just l log n spl minus 1 words of storage as compared with o n words required by the pa l is the length of the wavelet filter the rpa is combined with the short length fir filter algorithms to reduce the number of multiplications and additions

segmentation of bright targets using wavelets and adaptive thresholding a general systematic method for the detection and segmentation of bright targets is developed we use the term bright target to mean a connected cohesive object which has an average intensity distribution above that of the rest of the image we develop an analytic model for the segmentation of targets which uses a novel multiresolution analysis in concert with a bayes classifier to identify the possible target areas a method is developed which adaptively chooses thresholds to segment targets from background by using a multiscale analysis of the image probability density function pdf a performance analysis based on a gaussian distribution model is used to show that the obtained adaptive threshold is often close to the bayes threshold the method has proven robust even when the image distribution is unknown examples are presented to demonstrate the efficiency of the technique on a variety of targets

a new class of biorthogonal wavelet systems for image transform coding we construct general biorthogonal coifman wavelet systems a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair a time domain design method is employed and closed form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived the resulting filter coefficients are all dyadic fractions which is an attractive feature in the realization of multiplication free discrete wavelet transform even ordered systems in this family are symmetric which correspond to linear phase dual filters in particular three filterbanks fbs in this family are systematically verified to have competitive compression potential to the 9 7 tap biorthogonal wavelet fb by cohen et al 1992 which is currently the most widely used one in the field of wavelet transform coding in addition the proposed fb s have much smaller computational complexity in terms of floating point operations required in transformation and therefore indicate a better tradeoff between compression performance and computational complexity

filter design and comparison for two fast cwt algorithms the continuous wavelet transform cwt is a powerful technique for signal analysis direct cwt computation by fft requires o n log sub 2 n operations per scale where n is the data length the a trous algorithm and the shensa 1992 algorithm are two fast methods to compute cwt recursively that require only o n operations per scale both of them can be described by the multiresolution analysis mra structure but with different mra filters this paper proposes methods to design the mra filters of the two algorithms to improve their accuracy on cwt computation we begin with the formulation of the cwt computation error using the mra structure the mra filters of the two algorithms are then designed to minimize the error in either algorithm both the lowpass and bandpass mra filters can be optimized the a trous algorithm has closed form solutions for the two filters the shensa algorithm on the other hand has an analytic solution for the bandpass filter only finding the optimum lowpass filter requires a multidimensional numerical search simulation studies show that by using the proposed optimum filters the shensa algorithm in general outperforms the a trous algorithm

extraction of shift invariant wavelet features for classification of images with different sizes an effective shift invariant wavelet feature extraction method for classification of images with different sizes is proposed the feature extraction process involves a normalization followed by an adaptive shift invariant wavelet packet transform an energy signature is computed for each subband of these invariant wavelet coefficients a reduced subset of energy signatures is selected as the feature vector for classification of images with different sizes experimental results show that the proposed method can achieve high classification accuracy of 98 5 percent and outperforms the other two image classification methods

m band compactly supported orthogonal symmetric interpolating scaling functions in many applications wavelets are usually expected to have the following properties compact support orthogonality linear phase regularity and interpolation to construct such wavelets it is crucial designing scaling functions with the above properties in two and three band cases except for the haar functions there exists no scaling function with the above five properties in m band case m spl ges 4 more free degrees available in design enable us to construct such scaling functions a novel approach to designing such scaling functions is proposed first we extend the two band dubuc 1986 filters to the m band case next the m band fir regular symmetric interpolating scaling filters are parameterized and then m band fir regular orthogonal symmetric interpolating scaling filters osisfs are designed via optimal selection of parameters finally two family of four band and five band osisfs and scaling functions are developed and their smoothness are estimated

a radar target discrimination scheme using the discrete wavelet transform for reduced data storage a correlative radar target discrimination scheme using the transient scattered field response is proposed this scheme uses a one dimensional discrete wavelet transform on the temporal response to reduce the amount of data that must be stored for each anticipated aspect angle experimental results show that a reduction in stored data of sixteen to one still allows accurate discrimination in adverse noise situations with signal to noise ratios as low as 5 db

an orthogonal family of quincunx wavelets with continuously adjustable order we present a new family of two dimensional and three dimensional orthogonal wavelets which uses quincunx sampling the orthogonal refinement filters have a simple analytical expression in the fourier domain as a function of the order spl lambda which may be noninteger we can also prove that they yield wavelet bases of l sub 2 r sup 2 for any spl lambda 0 the wavelets are fractional in the sense that the approximation error at a given scale a decays like o a sup spl lambda they also essentially behave like fractional derivative operators to make our construction practical we propose a fast fourier transform based implementation that turns out to be surprisingly fast in fact our method is almost as efficient as the standard mallat algorithm for separable wavelets

noise covariance properties in dual tree wavelet decompositions dual tree wavelet decompositions have recently gained much popularity mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy when the decomposition of a random process is performed which occurs in particular when an additive noise is corrupting the signal to be analyzed it is useful to characterize the statistical properties of the dual tree wavelet coefficients of this process as dual tree decompositions constitute over complete frame expansions correlation structures are introduced among the coefficients even when a white noise is analyzed in this paper we show that it is possible to provide an accurate description of the covariance properties of the dual tree coefficients of a wide sense stationary process the expressions of the cross covariance sequences of the coefficients are derived in the one and two dimensional cases asymptotic results are also provided allowing to predict the behavior of the second order moments for large lag values or at coarse resolution in addition the cross correlations between the primal and dual wavelets which play a primary role in our theoretical analysis are calculated for a number of classical wavelet families simulation results are finally provided to validate these results

fast algorithms for discrete and continuous wavelet transforms several algorithms are reviewed for computing various types of wavelet transforms the mallat algorithm 1989 the a trous algorithm and their generalizations by shensa the goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform although this general method is already efficient it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques such as the fft fast fourier transform in a suitable manner the modified algorithms are termed fast because of their ability to reduce the computational complexity per computed coefficient from l to log l within a small constant factor for large filter lengths l for short filters smaller gains are obtained fast running fir finite impulse response filtering techniques allow one to achieve typically 30 savings in computations

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

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

quadrature prefilters for the discrete wavelet transform discrepancies between the discrete wavelet transform and the coefficients of the wavelet series are known to be reducible by initialization of input data prefilters based on lagrange interpolants are derived here for biorthogonal compact support wavelet systems providing exact subspace projection in cases of local polynomial smoothness the resulting divergence acceleration in a nonpolynomial test case is examined irregular sampling rates are also accommodated

a maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform in this paper a maximum likelihood ml method is presented for joint estimation of amplitude phase time delay and data symbols in a single user direct sequence spread spectrum communication system since maximization of the likelihood function is analytically intractable a novel coordinate ascent algorithm is used to obtain sequential updates of the data symbols and all unknown nuisance parameters the novelty of the algorithm is due to the use of a multiresolution expansion of the received signal and the use of polynomial rooting in the complex plane in place of a line search over the signal delay parameter the multiresolution structure of the algorithm is exploited to reduce sensitivity to impulsive noise via wavelet thresholding computer simulations of the single user system show that the algorithm has fast convergence and comparison with theoretical lower bounds establishes that the algorithm achieves nearly optimal error performance

the coding gain of multiplexed wavelet transforms evangelista 1994 1993 introduced the multiplexed wavelet transform mwt and pointed out its potential applications to the analysis synthesis processing and coding of pseudo periodic signals such as voiced speech and music coders based on the mwt have been shown to outperform the conventional subband coders when a reliable pitch parameter can be extracted from data in this paper we investigate the effects of uniform quantization of the mwt coefficients we compare the performance of the new coders with that of wt block dct and klt coders in terms of the coding gain achieved when optimal bit allocation schemes are adopted

fast adaptive wavelet packet image compression wavelets are ill suited to represent oscillatory patterns rapid variations of intensity can only be described by the small scale wavelet coefficients which are often quantized to zero even at high bit rates our goal is to provide a fast numerical implementation of the best wavelet packet algorithm in order to demonstrate that an advantage can be gained by constructing a basis adapted to a target image emphasis is placed on developing algorithms that are computationally efficient we developed a new fast two dimensional 2 d convolution decimation algorithm with factorized nonseparable 2 d filters the algorithm is four times faster than a standard convolution decimation an extensive evaluation of the algorithm was performed on a large class of textured images because of its ability to reproduce textures so well the wavelet packet coder significantly out performs one of the best wavelet coder on images such as barbara and fingerprints both visually and in term of psnr

fast fir algorithms for the continuous wavelet transform from constrained least squares new algorithms for the continuous wavelet transform are developed that are easy to apply each consisting of a single pass finite impulse response fir filter and several times faster than the fastest existing algorithms the single pass filter named wt fir 1 is made possible by applying constraint equations to least squares estimation of filter coefficients which removes the need for separate low pass and high pass filters non dyadic two scale relations are developed and it is shown that filters based on them can work more efficiently than dyadic ones example applications to the mexican hat wavelet are presented

vector valued wavelets and vector filter banks in this paper we introduce vector valued multiresolution analysis and vector valued wavelets for vector valued signal spaces we construct vector valued wavelets by using paraunitary vector filter bank theory in particular we construct vector valued meyer wavelets that are band limited we classify and construct vector valued wavelets with sampling property as an application of vector valued wavelets multiwavelets can be constructed from vector valued wavelets we show that certain linear combinations of known scalar valued wavelets may yield multiwavelets we then present discrete vector wavelet transforms for discrete time vector valued or blocked signals which can be thought of as a family of unitary vector transforms

a novel approach to the 2 d blind deconvolution problem in medical ultrasound the finite frequency bandwidth of ultrasound transducers and the nonnegligible width of transmitted acoustic beams are the most significant factors that limit the resolution of medical ultrasound imaging consequently in order to recover diagnostically important image details obscured due to the resolution limitations an image restoration procedure should be applied the present study addresses the problem of ultrasound image restoration by means of the blind deconvolution techniques given an acquired ultrasound image algorithms of this kind perform either concurrent or successive estimation of the point spread function psf of the imaging system and the original image a blind deconvolution algorithm is proposed in which the psf is recovered as a preliminary stage of the restoration problem as the accuracy of this estimation affects all the following stages of the image restoration it is considered as the most fundamental and important problem the contribution of the present study is twofold first it introduces a novel approach to the problem of estimating the psf which is based on a generalization of several fundamental concepts of the homomorphic deconvolution it is shown that a useful estimate of the spectrum of the psf can be obtained by applying a proper smoothing operator to both log magnitude and phase of the spectra of acquired radio frequency rf images it is demonstrated that the proposed approach performs considerably better than the existing homomorphic cepstrum based deconvolution methods second the study shows that given a reliable estimate of the psf it is possible to deconvolve it out of the rf image and obtain an estimate of the true tissue reflectivity function which is relatively independent of the properties of the imaging system the deconvolution was performed using the maximum a posteriori map estimation framework for a number of statistical priors assumed for the reflectivity function it is shown in a series of in vivo experiments that reconstructions based on the priors which tend to emphasize the sparseness of the tissue structure result in solutions of higher resolution and contrast

an efficient computational scheme for the two dimensional overcomplete wavelet transform we have studied the computational complexity associated with the overcomplete wavelet transform for the commonly used spline wavelet family by deriving general expressions for the computational complexity using the conventional filtering implementation we show that the inverse transform is significantly more costly in computation than the forward transform to reduce this computational complexity we propose a new spatial implementation based on the exploitation of the correlation between the lowpass and the bandpass outputs that is inherent in the overcomplete representation both theoretical studies and experimental findings show that the proposed spatial implementation can greatly simplify the computations associated with the inverse transform in particular the complexity of the inverse transform using the proposed implementation can be reduced to slightly less than that of the forward transform using the conventional filtering implementation we also demonstrate that the proposed scheme allows the use of an arbitrary boundary extension method while maintaining the ease of the inverse transform

an iterative algorithm for two scale wavelet decomposition this correspondence proposes an iterative method to decompose an arbitrary mother wavelet into a bandpass filter and a lowpass filter where the filter pair will reproduce the mother wavelet through the two scale equations this problem is not straightforward because the two scale relationship between the filter pair and the mother wavelet is nonlinear the method finds the filter pair by minimizing an objective function with respect to one filter in an iteration and the other in the next iteration until convergence the algorithm requires least squares minimization only and is computationally efficient the performance of the proposed algorithm is supported by simulations

rapid computation of the continuous wavelet transform by oblique projections we introduce a fast simple method for computing the real continuous wavelet transform cwt the approach has the following attractive features it achieves o n complexity per scale the filter coefficients can be analytically obtained by a simple integration and the algorithm is faster than a least squares approach with negligible loss in accuracy our method is to use p wavelets per octave and to approximate them with their oblique projection onto a space defined by a compactly supported scaling function the wavelet templates are expanded to larger sizes octaves using the two scale relation and zero padded filtering error bounds are presented to justify the use of an oblique projection over an orthogonal one all the filters are fir with the exception of one filter which is implemented using a fast recursive algorithm

numerical simulation of scattering from rough surfaces a wavelet based approach in this paper a preliminary study is carried out to demonstrate the application of wavelets for improving the computation time and reducing computational memory required for evaluating the statistics of the scattered field from rough surfaces using the method of moments mom in conjunction with a monte carlo simulation in specific haar and the first order b spline wavelet basis functions are applied to the mom formulation of one dimensional rough surfaces in order to compare the computation time and sparsity for wavelets in the same family but of higher order since the scattering coefficient the second moment of the backscatter field per unit area is a gentle function of the surface parameters and the radar attributes it is demonstrated that a relatively high thresholding level can be applied to the impedance matrix which leads to a sparser impedance matrix and faster computation time it is also shown that applying a high threshold level the coefficients of the high order wavelets would increase out of proportion however the effect of these current components averages out when computing the scattering coefficients the resulting sparse impedance matrices are solved efficiently using fast search routines such as the conjugate gradient method a systematic study is carried out to investigate the effect of different threshold levels on the accuracy versus computing speed criterion the computed scattering coefficients are compared to previous results computed using a conventional pulse basis function as well as the existing theoretical solutions for rough surfaces it is shown that wavelet basis functions provide substantial reductions in both memory requirements and computation time

design of prefilters for discrete multiwavelet transforms the pyramid algorithm for computing single wavelet transform coefficients is well known the pyramid algorithm can be implemented by using tree structured multirate filter banks the authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets the proposed algorithm can be thought of as a discrete vector valued wavelet transform for certain discrete time vector valued signals the proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete time signals the authors then present some numerical experiments to illustrate the performance of the algorithm which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms

running fir and iir filtering using multirate filter banks the computational complexity of running fir finite impulse response and iir infinite impulse response filtering using multirate filter banks is considered no restrictions are put on signal filter or block lengths it is shown how to map long running convolutions into smaller ones by using filter banks based on aperiodic convolution algorithms and short time fourier transforms with the proposed approach good tradeoffs among computational complexity system architecture and input output delay are achieved

impedance matrix compression using adaptively constructed basis functions wavelet expansions have been employed recently in numerical solutions of commonly used frequency domain integral equations in this paper we propose a novel method for integrating wavelet based transforms into existing numerical solvers the newly proposed method differs from the presently used ones in two ways first the transformation is affected by means of a digital filtering approach this approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand second the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the yet unknown current are retained and subsequently derived numerical results for a few tm scattering problems are included to demonstrate the advantages of the proposed method over the presently used ones

a discrete time multiresolution theory multiresolution analysis and synthesis for discrete time signals is described concepts of scale and resolution are first reviewed in discrete time the resulting framework allows one to treat the discrete wavelet transform octave band perfect reconstruction filter banks and pyramid transforms from a unified standpoint this approach is very close to previous work on multiresolution decomposition of functions of a continuous variable and the connection between these two approaches is made it is shown that they share many mathematical properties such as biorthogonality orthonormality and regularity however the discrete time formalism is well suited to practical tasks in digital signal processing and does not require the use of functional spaces as an intermediate step

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

wavelet coefficient computation with optimal prefiltering discrete wavelet transform dwt is often used to approximate wavelet series transform wst and continuous wavelet transform cwt since it can be computed numerically in this research we first study the accuracy of the computed dwt coefficients obtained from the shensa see ibid vol 40 no 10 p 2464 2482 1992 algorithm as an approximate of the wst coefficients based on the accuracy analysis we then propose a procedure to design optimal fir prefilters used in the shensa algorithm to reduce the approximation error finally numerical examples are presented to demonstrate the performance of the optimal fir prefilters

wavelet transforms for detecting microcalcifications in mammograms clusters of fine granular microcalcifications in mammograms may be an early sign of disease individual grains are difficult to detect and segment due to size and shape variability and because the background mammogram texture is typically inhomogeneous the authors develop a 2 stage method based on wavelet transforms for detecting and segmenting calcifications the first stage is based on an undecimated wavelet transform which is simply the conventional filter bank implementation without downsampling so that the low low ll low high lh high low hl and high high hh sub bands remain at full size detection takes place in hh and the combination lh hl four octaves are computed with 2 inter octave voices for finer scale resolution by appropriate selection of the wavelet basis the detection of microcalcifications in the relevant size range can be nearly optimized in fact the filters which transform the input image into hh and lh hl are closely related to prewhitening matched filters for detecting gaussian objects idealized microcalcifications in 2 common forms of markov background noise the second stage is designed to overcome the limitations of the simplistic gaussian assumption and provides an accurate segmentation of calcification boundaries detected pixel sites in hh and lh hl are dilated then weighted before computing the inverse wavelet transform individual microcalcifications are greatly enhanced in the output image to the point where straightforward thresholding can be applied to segment them frog curves are computed from tests using a freely distributed database of digitized mammograms

estimating the joint statistics of images using nonparametric windows with application to registration using mutual information recently the non parametric np windows has been proposed to estimate the statistics of real 1d and 2d signals np windows is accurate because it is equivalent to sampling images at a high infinite resolution for an assumed interpolation model this paper extends the proposed approach to consider joint distributions of image pairs secondly green s theorem is used to simplify the previous np windows algorithm finally a resolution aware np windows algorithm is proposed to improve robustness to relative scaling between an image pair comparative testing of 2d image registration was performed using translation only and affine transformations although more expensive than other methods np windows frequently demonstrated superior performance for bias distance between ground truth and global maximum and frequency of convergence unlike other methods the number of samples and histogram bin size has little effect on np windows and the prior selection of a kernel is not required

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

directional dyadic wavelet transforms design and algorithms we propose a simple and efficient technique for designing translation invariant dyadic wavelet transforms dwts in two dimensions our technique relies on an extension of the work of duval destin et al 1993 where dyadic decompositions are constructed starting from the continuous wavelet transform the main advantage of this framework is that it allows for a lot of freedom in designing two dimensional 2 d dyadic wavelets we use this property to construct directional wavelets whose orientation filtering capabilities are very important in image processing we address the efficient implementation of these decompositions by constructing approximate qmfs through an l sup 2 optimization we also propose and study an efficient implementation in the fourier domain for dealing with large filters

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

fast cwt computation at integer scales by the generalized mra structure this article proposes a fast algorithm for continuous wavelet transform cwt at linear scale without decimation by using the generalized multiresolution analysis mra structure the constraints required on the lowpass and bandpass filters in the generalized mra structure are derived a possible solution for the lowpass filters and a least squares design of the bandpass filters are given the computational complexity of the algorithm is o n per scale where n is the data length the fast algorithm is verified by computer simulations

the distortion rate function of sampled wiener processes we consider the recovery of a continuous time wiener process from a quantized or a lossy compressed version of its uniform samples under limited bitrate and sampling rate we derive a closed form expression for the optimal tradeoff among sampling rate bitrate and quadratic distortion in this setting this expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite rank operators associated with the optimal estimator of the wiener process from its samples we show that the ratio between this expression and the standard distortion rate function of the wiener process describing the optimal tradeoff between bitrate and distortion without a sampling constraint is only a function of the number of bits per sample we also consider a sub optimal lossy compression scheme in which the continuous time process is estimated from the output of an encoder that is optimal with respect to the discrete time samples we show that the latter is strictly greater than the distortion under optimal encoding but only by at most 3 we therefore conclude that near optimal performance is attained even if the encoder is unaware of the continuous time origin of the samples

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