pattern clustering based on noise modeling in wavelet space we describe an effective approach to object or feature detection in point patterns via noise modeling this is based on use of a redundant or non pyramidal wavelet transform noise modeling is based on a poisson process we illustrate this new method with a range of examples we use the close relationship between image pixelated and point representations to achieve the result of a clustering method with constant time computational cost

analysis and vlsi architecture for 1 d and 2 d discrete wavelet transform in this paper a detailed analysis of very large scale integration vlsi architectures for the one dimensional 1 d and two dimensional 2 d discrete wavelet transform dwt is presented in many aspects and three related architectures are proposed as well the 1 d dwt and inverse dwt idwt architectures are classified into three categories convolution based lifting based and b spline based they are discussed in terms of hardware complexity critical path and registers as for the 2 d dwt the large amount of the frame memory access and the die area occupied by the embedded internal buffer become the most critical issues the 2 d dwt architectures are categorized and analyzed by different external memory scan methods the implementation issues of the internal buffer are also discussed and some real life experiments are given to show that the area and power for the internal buffer are highly related to memory technology and working frequency instead of the required memory size only besides the analysis the b spline based idwt architecture and the overlapped stripe based scan method are also proposed last we propose a flexible and efficient architecture for a one level 2 d dwt that exploits many advantages of the presented analysis

a review of geometric transformations for nonrigid body registration this paper provides a comprehensive and quantitative review of spatial transformations models for nonrigid image registration it explains the theoretical foundation of the models and classifies them according to this basis this results in two categories physically based models described by partial differential equations of continuum mechanics e g linear elasticity and fluid flow and basis function expansions derived from interpolation and approximation theory e g radial basis functions b splines and wavelets recent work on constraining the transformation so that it preserves the topology or is diffeomorphic is also described the final section reviews some recent evaluation studies the paper concludes by explaining under what conditions a particular transformation model is appropriate

the solution of ill conditioned symmetric toeplitz systems via two grid and wavelet methods abstract in this paper we present a generalized multigrid method combined with wavelet filters for solving ill conditioned symmetric toeplitz systems t n x b where t n r n n is generated by nonnegative functions with zeros first we propose the construction of general cohen daubechies and feauveau cdf 9 7 biorthogonal wavelet systems so that a new class of compactly supported biorthogonal wavelet systems gcdf are achieved with specified vanishing moments for scaling functions in order to solve ill conditioned toeplitz systems by using the two grid method tgm we use the constructed gcdf wavelets to get prolongation and restriction operators as a result the proposed tgm is proved by numerical experiment to be more efficient than the classic tgm especially when t n is seriously ill conditioned for the vanishing moments being n experimental tests illustrate that the tgm with damped jacobi smoother converges when the generating function has zeros of order less than or equal to 2n n 8 besides we prove in theory that the proposed method converges for toeplitz systems that are generated by functions with zeros of order less than or equal to four

a wavelet or lifting scheme based imputation method summary the paper proposes a new approach to imputation using the expected sparse representation of a surface in a wavelet or lifting scheme basis our method incorporates a bayesian mixture prior for these wavelet coefficients into a gibbs sampler to generate a complete posterior distribution for the variable of interest intuitively the estimator operates by borrowing strength from those observed neighbouring values to impute at the unobserved sites we demonstrate the strong performance of our estimator in both one and two dimensional imputation problems where we also compare its application with the standard imputation techniques of kriging and thin plate splines

polynomial wavelet regression for images with irregular boundaries in this paper we focus on denoising images for which observations are equally spaced except around the boundaries which are irregular such images are very common in many fields for example in geophysics the advantages of adding a low order polynomial term when implementing a wavelet regression for such images are presented besides removing the classical restriction of having a dyadic of number of observations this strategy reduces the bias at the edges without significantly increasing the risk in addition this method is simple to implement fast and efficient its utility is illustrated with simulation studies and a real example

a multiresolution study of two dimensional scattering by metallic cylinders the application of wavelet transforms in method of moments mom solutions for scattering problems is extended to cases involving metallic cylinders whose periphery contains a variety of length scale features ranging from smoothly varying large scale features characterized by a radius of curvature of several wavelengths to rapidly varying small scale ones characterized by a radius of curvature that is small compared with the wavelength the basic idea is to first consider a periodic extension of the equivalent current in the arc length variable with a period identical to the scatterer circumference and then to expand this representation using a set of periodic wavelets derived from a conventional basis of wavelets by a periodic extension using a galerkin method and subsequently applying a threshold operation a substantial reduction in the number of elements of the moment method matrix is attained without virtually affecting the solution accuracy the proposed extension is illustrated by a numerical analysis of tm transverse magnetic and te transverse electric scattering from a cylinder of elliptic cross section a thorough study is carried out showing how the solution accuracy improves with increasing resolution level and how this accuracy is affected by a thresholding process which renders the moment matrix sparsely populated

analytical footprints compact representation of elementary singularities in wavelet bases we introduce a family of elementary singularities that are point holder regular these singularities are self similar and are the green functions of fractional derivative operators i e by suitable fractional differentiation one retrieves a dirac function at the exact location of the singularity we propose to use fractional operator like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain we show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed form expression termed the analytical footprint practically this means that the dictionary of wavelet footprints is embodied in a single analytical form we show that the wavelet coefficients of the nonredundant decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity we propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations

a functional wavelet kernel approach for time series prediction summary we consider the prediction problem of a time series on a whole time interval in terms of its past the approach that we adopt is based on functional kernel nonparametric regression estimation techniques where observations are discrete recordings of segments of an underlying stochastic process considered as curves these curves are assumed to lie within the space of continuous functions and the discretized time series data set consists of a relatively small compared with the number of segments number of measurements made at regular times we estimate conditional expectations by using appropriate wavelet decompositions of the segmented sample paths a notion of similarity based on wavelet decompositions is used to calibrate the prediction asymptotic properties when the number of segments grows to 1 are investigated under mild conditions and a nonparametric resampling procedure is used to generate in a flexible way valid asymptotic pointwise prediction intervals for the trajectories predicted we illustrate the usefulness of the proposed functional wavelet kernel methodology in finite sample situations by means of a simulated example and two real life data sets and we compare the resulting predictions with those obtained by three other methods in the literature in particular with a smoothing spline method with an exponential smoothing procedure and with a seasonal autoregressive integrated moving average model

two dimensional filter bank design for optimal reconstruction using limited subband information in this correspondence we propose design techniques for analysis and synthesis filters of 2 d perfect reconstruction filter banks prfb s that perform optimal reconstruction when a reduced number of subband signals is used based on the minimization of the squared error between the original signal and some low resolution representation of it the 2 d filters are optimally adjusted to the statistics of the input images so that most of the signal s energy is concentrated in the first few subband components this property makes the optimal prfb s efficient for image compression and pattern representations at lower resolutions for classification purposes by extending recently introduced ideas from frequency domain principal component analysis to two dimensions we present results for general 2 d discrete nonstationary and stationary second order processes showing that the optimal filters are nonseparable particular attention is paid to separable random fields proving that only the first and last filters of the optimal prfb are separable in this case simulation results that illustrate the theoretical achievements are presented

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

wavelet based ilu preconditioners for the numerical solution by pufem of high frequency elastic wave scattering daubechies family of wavelets combined to the incomplete lower upper ilu factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the partition of unity finite element method pufem after applying the discrete wavelet transform dwt to each dense block in the final matrix and the known right hand side due to the local enrichment by pressure p and shear s plane waves the resulting linear system is solved by the restarted generalized minimum residual method gmres with ilu preconditioners allowing fill in elements in the l and u matrix factors a reordering algorithm of the vertices based on gibbs method is also introduced it leads to a significant reduction of the bandwidth of the wave based finite element fe matrix and enables the ilu preconditioners to be more effective to study the performance of the proposed preconditioners a problem of a horizontal s plane wave scattered by a rigid circular body in an infinite elastic medium is considered the numerical tests show the good performance of the dwt based ilu preconditioners in improving the rate of convergence of gmres for high numbers of approximating p and s plane waves on coarse mesh grids containing multi wavelength sized elements moreover the haar dwt enhances the scalability with respect to the problem size when the number of the nodal points increases of the ilu preconditioner which uses the threshold strategy in the control of the fill in elements despite the high level of the conditioning a good accuracy may be achieved for a discretization level of around 1 9 degrees of freedom per s wavelength which is far below the rule of thumb of 10 nodal points per wavelength adopted for the conventional fe

new solution to the pressure transient equation in a two layer reservoir with crossflow abstract this paper presents a new mathematical model and an analytical solution to the pressure transient equation of a uniform flux which fully penetrates the vertical well in a two layer petroleum reservoir with crossflow this new model and solution provide an accurate and fast tool to 1 evaluate a vertical well performance in a two layer reservoir 2 estimate the effects of formation properties on pressure behavior at the locations both far away from and around the well the solution is obtained from the model by using laplace transform double fourier transform and green s functions method when all the corresponding parameters are identical in each layer the proposed solution for a two layer reservoir is naturally reduced to the solution for a single layer reservoir based on the analytical solution the pressure of the vertical well which produces with a constant rate in the two layer reservoir can be examined in detail more importantly a dimensionless crossflow coefficient is proposed to describe the crossflow in a two layer reservoir quantitatively it is verified that the crossflow coefficient is not constant but a function of the time and distance away from the wellbore in the numerical experiments the pressure response is observed in three stages 1 the first one is similar to the commingled system because the crossflow is very weak at the beginning of the production 2 the second one is the transition stage when the crossflow becomes stronger with the production 3 the third stage is close to the performance of the homogeneous system when the crossflow coefficient approaches zero

a novel integration method for weak singularity arising in two dimensional scattering problems in this paper we introduce a novel mechanical quadrature method for an efficient solution of weakly singular integral equations arising in two dimensional electromagnetic scattering problems this approach is based on and adapted from the recently proposed mechanical quadrature methods in extrapolation algorithms for solving mixed boundary integral equations of the helmholtz equation by mechanical quadrature methods siam j sci comput vol 31 4115 4129 2009 we report experiments for solving tm polarized induced currents and scattered fields to show its superiority to the classical method of moments when accuracy is a concern moreover additional numerical experiments made with an extrapolation algorithm suggest that the accuracy of the present method can be further improved dramatically by means of the extrapolation algorithm to some extent

information theoretic wavelet noise removal for inverse elastic wave scattering theory a discussion of noise removal in ultrasound elastic wave scattering for nondestructive evaluation is given the methods used in this paper include a useful suboptimal wiener filter information theory and orthonormal wavelets the multiresolution analysis mra due to mallat is the key wavelet feature used here whereas fourier transforms have a translational symmetry wavelets have a dilation or affine symmetry which consists of the semi direct product of a translation with a change of scale of the variable the mra describes the scale change features of orthonormal wavelet families first an empirical method of noise removal from scattered elastic waves using wavelets is shown to markedly improve the l 1 and l 2 error norms this suggests that the wavelet scale can act as dial to tune out noise maximization of the kullback liebler information is also shown to provide a scale dependent noise removal technique that supports but does not prove the intuition that certain small energy coefficients that are retained contain large information content the wavelet mra thereby locates islands of information in the phase space of the signal it is conjectured that this method holds more generally

analysis of low bit rate image transform coding calculations based on high resolution quantizations prove that the distortion rate d r of an image transform coding is proportional to 2 sup 2r when r is large enough in wavelet and block cosine bases we show that if r 1 bit pixel then d r varies like r sup 1 2 spl gamma where spl gamma remains of the order of 1 for most natural images the improved performance of embedded codings in wavelet bases is analyzed at low bit rates we show that the compression performance of an orthonormal basis depends mostly on its ability to approximate images with a few nonzero vectors

discriminative wavelet packet filter bank selection for pattern recognition this paper addresses the problem of discriminative wavelet packet wp filter bank selection for pattern recognition the problem is formulated as a complexity regularized optimization criterion where the tree indexed structure of the wp bases is explored to find conditions for reducing this criterion to a type of minimum cost tree pruning a method well understood in regression and classification trees cart for estimating the conditional mutual information adopted to compute the fidelity criterion of the minimum cost tree pruning problem a nonparametric approach based on product adaptive partitions is proposed extending the darbellay vajda data dependent partition algorithm finally experimental evaluation within an automatic speech recognition asr task shows that proposed solutions for the wp decomposition problem are consistent with well understood empirically determined acoustic features and the derived feature representations yield competitive performances with respect to standard feature extraction techniques

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

ranging through gabor logons a consistent hierarchical approach in this work the correspondence problem in stereo vision is handled by matching two sets of dense feature vectors inspired by biological evidence these feature vectors are generated by a correlation between a bank of gabor sensors and the intensity image the sensors consist of two dimensional gabor filters at various scales spatial frequencies and orientations which bear close resemblance to the receptive field profiles of simple v1 cells in visual cortex a hierarchical stochastic relaxation method is then used to obtain the dense stereo disparities unlike traditional hierarchical methods for stereo feature based hierarchical processing yields consistent disparities to avoid false matchings due to static occlusion a dual matching based on the imaging geometry is used

image enhancement based on a nonlinear multiscale method an image enhancement method that reduces speckle noise and preserves edges is introduced the method is based on a new nonlinear multiscale reconstruction scheme that is obtained by successively combining each coarser scale image with the corresponding modified interscale image simulation results are included to demonstrate the performance of the proposed method

modeling and an immersed finite element method for an interface wave equation abstract the electromagnetic field which is governed by maxwell s equation plays a key role in plasma simulation in this article we first derive the interface conditions when we rewrite the interface maxwell s equation whose problem domain involves complex media such as objects of different materials into a parabolic hyperbolic type of interface model which is a modified wave equation by adding a first order time derivative term due to the lossy medium based on the interface conditions and the existing bilinear immersed finite element space for the interface poisson equation we propose an immersed finite element method for the spatial discretization of this parabolic hyperbolic interface equation on a cartesian mesh independent of the interface then we use a second order finite difference method for the temporal discretization in order to develop the full discretization scheme compared with the unstructured body fitting mesh which is needed by the traditional finite element method for interface problems the cartesian mesh independent of the interface will significantly benefit the plasma simulation in the electromagnetic field especially the particle models such as the particle in cell method this method provides a new efficient way to study varying electromagnetic field with objects of different materials on a cartesian mesh independent of the interface hence builds a solid foundation for the further study on the motion of plasma in this electromagnetic field numerical examples are provided to demonstrate the features of the proposed method

wavelet processes and adaptive estimation of the evolutionary wavelet spectrum this article defines and studies a new class of non stationary random processes constructed from discrete non decimated wavelets which generalizes the cramer fourier representation of stationary time series we define an evolutionary wavelet spectrum ews which quantifies how process power varies locally over time and scale we show how the ews may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time localized autocovariance we illustrate our theory with a pedagogical example based on discrete non decimated haar wavelets and also a real medical time series example

image reconstruction from projections under wavelet constraints first the authors discuss how the wavelet transform can be used to perform spatially varying filtering of an image suppressing noise locally in smooth regions of the image and discuss detection of such regions in a noise corrupted image second they show how to compute the minimum mean square estimate of an image given 1 noisy projections of the image 2 statistics of additive noise in the projections and 3 constraints on wavelet coefficients of the image examples illustrate the resulting procedure

high accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems we investigate the mechanical quadrature methods mqms for solving boundary integral equations bies of the three dimensional axisymmetric elasticity equations with dirichlet s problems the convergence of the mqms is proved based on anselone s collective compact theory moreover an asymptotic error expansion of the mqms shows that the approximation order is of o h 3 numerical examples are tested and the results verify the theoretical analysis

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

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

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

wavelet based parameter estimation for polynomial contaminated fractionally differenced processes we consider the problem of estimating the parameters for a stochastic process using a time series containing a trend component trend i e large scale variations in the series that are best modeled outside of a stochastic framework is often confounded with low frequency stochastic fluctuations this problem is particularly evident in models such as fractionally differenced fd processes which exhibit slowly decaying autocorrelations and can be extended to encompass nonstationary processes with substantial low frequency components we use the discrete wavelet transform dwt to estimate parameters for stationary and nonstationary fd processes in a model of polynomial trend plus fd noise using daubechies wavelet filters allows for automatic elimination of polynomial trends due to embedded differencing operations parameter estimation is based on an approximate maximum likelihood approach made possible by the fact that the dwt decorrelates fd processes approximately we consider this decorrelation in detail examining the between and within scale wavelet correlations separately better between scale decorrelation can be achieved by increasing the length of the wavelet filter whereas the within scale correlations can be handled via explicit modeling by a low order autoregressive process we demonstrate our methodology by applying it to a popular climate dataset

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

a bound on the proportion of pure strategy equilibria in generic games in a generic finite normal form game with 2 1 nash equilibria at least of the equilibria are nondegenerate mixed strategy equilibria that is they involve randomization by some players

unsupervised texture segmentation using gabor filters abstract this paper presents a texture segmentation algorithm inspired by the multi channel filtering theory for visual information processing in the early stages of human visual system the channels are characterized by a bank of gabor filters that nearly uniformly covers the spatial frequency domain and a systematic filter selection scheme is proposed which is based on reconstruction of the input image from the filtered images texture features are obtained by subjecting each selected filtered image to a nonlinear transformation and computing a measure of energy in a window around each pixel a square error clustering algorithm is then used to integrate the feature images and produce a segmentation a simple procedure to incorporate spatial information in the clustering process is proposed a relative index is used to estimate the true number of texture categories

anisotropic motion estimation on edge preserving riesz wavelets for robust video mosaicing image mosaicing is a technique widely used for extending the field of view of industrial medical outdoor or indoor scenes however image registration can be very challenging e g due to large texture variability illumination changes image blur and camera perspective changes in this paper a total variational optical flow approach is investigated to estimate dense point correspondences between image pairs an edge preserving riesz wavelet scale space combined with a novel tv regularizer is proposed for preserving motion discontinuities along the edges of weak textures and for handling strong in plane rotations present in image sequences an anisotropic weighted median filtering is implemented for minimizing outliers quantitative evaluation of the method on the middlebury image database and simulated sequences with known ground truth demonstrates high accuracy of the proposed method in comparison with other state of the art methods including a robust graph cut method and a patch matching approach qualitative results on video sequences of difficult real scenes demonstrate the robustness of the proposed method highlightsmosaicing of images with strong texture and illumination variability fast robust and accurate optical flow computation tv l1 method on a second order riesz wavelet basis for preserving texture discontinuities novel anisotropic tv regularizer accurate results for both the middlebury data set and various complicated scenes

structure theorems for game trees kohlberg and mertens kohlberg e mertens j 1986 econometrica 54 1003 1039 proved that the graph of the nash equilibrium correspondence is homeomorphic to its domain when the domain is the space of payoffs in normal form games a counterexample disproves the analog for the equilibrium outcome correspondence over the space of payoffs in extensive form games but we prove an analog when the space of behavior strategies is perturbed so that every path in the game tree has nonzero probability without such perturbations the graph is the closure of the union of a finite collection of its subsets each diffeomorphic to a corresponding path connected open subset of the space of payoffs as an application we construct an algorithm for computing equilibria of an extensive form game with a perturbed strategy space and thus approximate equilibria of the unperturbed game

optimum discrete wavelet scaling and its application to delay and doppler estimation this paper studies the scaling of an arbitrary waveform from its samples the scaling problem is formulated as a mean square minimization and the resulting estimator consists of two parts noise filtering and sinc function scaling sinc function scaling is a time dependent process and requires o n sup 2 operations where n is the data length a fast algorithm based on the fft is proposed to reduce the complexity to o nlog sub 2 n this new algorithm is applied to wideband time delay and doppler estimation where the optimum wavelet is one of the received signal samples that has no analytic form the scaling method is found to be very effective in that the estimation accuracy achieves the cramer rao lower bound crlb

on denoising and best signal representation we propose a best basis algorithm for signal enhancement in white gaussian noise the best basis search is performed in families of orthonormal bases constructed with wavelet packets or local cosine bases we base our search for the best basis on a criterion of minimal reconstruction error of the underlying signal this approach is intuitively appealing because the enhanced or estimated signal has an associated measure of performance namely the resulting mean square error previous approaches in this framework have focused on obtaining the most compact signal representations which consequently contribute to effective denoising these approaches however do not possess the inherent measure of performance which our algorithm provides we first propose an estimator of the mean square error based on a heuristic argument and subsequently compare the reconstruction performance based upon it to that based on the stein 1981 unbiased risk estimator we compare the two proposed estimators by providing both qualitative and quantitative analyses of the bias term having two estimators of the mean square error we incorporate these cost functions into the search for the best basis and subsequently provide a substantiating example to demonstrate their performance

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

improved transient signal detection using a wavepacket based detector with an extended translation invariant wavelet transform this paper presents the theory of m band extended translation invariant eti wavelet transforms the eti generalizes the translation invariant wavelet transform of weiss 1933 it is shown that iteration of the eti in a tree structure provides a signal decomposition into an orthonormal wavepacket basis other properties such as translation invariance and invertibility of the transform are proven the theory is then applied to transient signal detection through development of a family of translation invariant wavepacket based detectors this family of detectors provides improved the performance over previously defined wavepacket based detectors a performance analysis is conducted roc curves generated by monte carlo simulation are presented indicating the detector performance the detector performance is demonstrated to be independent of the signal translation

minimax estimation with thresholding and its application to wavelet analysis many statistical practices involve choosing between a full model and reduced models where some coefficients are reduced to zero data were used to select a model with estimated coefficients is it possible to do so and still come up with an estimator always better than the traditional estimator based on the full model the james stein estimator is such an estimator having a property called minimaxity however the estimator considers only one reduced model namely the origin hence it reduces no coefficient estimator to zero or every coefficient estimator to zero in many applications including wavelet analysis what should be more desirable is to reduce to zero only the estimators smaller than a threshold called thresholding in this paper is it possible to construct this kind of estimators which are minimax in this paper we construct such minimax estimators which perform thresholding we apply our recommended estimator to the wavelet analysis and show that it performs the best among the well known estimators aiming simultaneously at estimation and model selection some of our estimators are also shown to be asymptotically optimal

automatic generation of rbf networks using wavelets abstract learning can be viewed as mapping from an input space to an output space examples of these mappings are used to construct a continuous function that approximates the given data and generalizes for intermediate instances radial basis function rbf networks are used to formulate this approximating function a novel method is introduced that automatically constructs a generalized radial basis function grbf network for a given mapping and error bound this network is shown to be the smallest network within the error bound for the given mapping the integral wavelet transform is used to determine the parameters of the network simple one dimensional examples are used to demonstrate how the network constructed using the transform is superior to that constructed using standard ad hoc optimization techniques the paper concludes with the automatic generation of grbf networks for a multi dimensional problem namely real time 3d object recognition and pose estimation the results of this application are favorable

image registration using the walsh transform this paper presents a new algorithm which can be used to register images of the same or different modalities e g images with multiple channels such as x rays temperature or elevation or simply images of different spectral bands in particular a correlation based scheme is used but instead of gray values it correlates numbers formulated by different combinations of the extracted local walsh coefficients of the images each image patch is expanded in terms of walsh basis functions each walsh basis function can be thought of as measuring a different aspect of local structure e g horizontal edge corner etc the coefficients of the expansion therefore can be thought of as dense local features estimating at each point the degree of presence of for example a horizontal edge a corner with contrast of a certain type etc these coefficients are normalized and used as digits in a chosen number system which allows one to create a unique number for each type of local structure the choice of the basis of the number system allows one to give different emphasis to different types of local feature e g corners versus edges and thus the method we present forms a unified framework in terms of which several feature matching methods may be interpreted the algorithm is compared with wavelet and contour based approaches using simulated and real images the two images are assumed to differ from each other by a rotation and a translation only

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

wavelet and matching pursuit estimates of the transient evoked otoacoustic emission latency different time frequency techniques may be used to investigate the relation between latency and frequency of transient evoked otoacoustic emissions in this work the optimization of these techniques and the interpretation of the experimental result are discussed time frequency analysis of click evoked otoacoustic emissions of 42 normal hearing young subjects has been performed using both wavelet and matching pursuit algorithms wavelet techniques are very effective to provide fast and reliable evaluation of the average latency of large samples of subjects a major advantage of the matching pursuit technique as observed by jedrzejczak et al j acoust soc am 115 2148 2158 2004 is to provide detailed information about the time evolution of the response of single ears at selected frequencies a hybrid matching pursuit algorithm that includes fourier spectral information was developed capable of speeding up computation times and of identifying spurious atoms whose latency frequency relation i

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

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

significance linked connected component analysis for wavelet image coding the success in wavelet image coding is mainly attributed to a recognition of the importance of data organization and representation there have been several very competitive wavelet coders developed namely shapiro s 1993 embedded zerotree wavelets ezw servetto et al s 1995 morphological representation of wavelet data mrwd and said and pearlman s see ieee trans circuits syst video technol vol 6 p 245 50 1996 set partitioning in hierarchical trees spiht we develop a novel wavelet image coder called significance linked connected component analysis slcca of wavelet coefficients that extends mrwd by exploiting both within subband clustering of significant coefficients and cross subband dependency in significant fields extensive computer experiments on both natural and texture images show convincingly that the proposed slcca outperforms ezw mrwd and spiht for example for the barbara image at 0 25 b pixel slcca outperforms ezw mrwd and spiht by 1 41 db 0 32 db and 0 60 db in psnr respectively it is also observed that slcca works extremely well for images with a large portion of texture for eight typical 256 spl times 256 grayscale texture images compressed at 0 40 b pixel slcca outperforms spiht by 0 16 db 0 63 db in psnr this performance is achieved without using any optimal bit allocation procedure thus both the encoding and decoding procedures are fast

dynamic multiscale spatiotemporal models for gaussian areal data summary we introduce a new class of dynamic multiscale models for spatiotemporal processes arising from gaussian areal data specifically we use nested geographical structures to decompose the original process into multiscale coefficients which evolve through time following state space equations our approach naturally accommodates data that are observed on irregular grids as well as heteroscedasticity moreover we propose a multiscale spatiotemporal clustering algorithm that facilitates estimation of the nested geographical multiscale structure in addition we present a singular forward filter backward sampler for efficient bayesian estimation our multiscale spatiotemporal methodology decomposes large data analysis problems into many smaller components and thus leads to scalable and highly efficient computational procedures finally we illustrate the utility and flexibility of our dynamic multiscale framework through two spatiotemporal applications the first example considers mortality ratios in the state of missouri whereas the second example examines agricultural production in espirito santo state brazil

quantum field theory in ninety minutes on presente divers aspects de la theorie quantique des champs la theorie p 2 a 2 dimensions la theorie 3 4 a 3 dimensions la theorie de yang mills a 4 dimensions

on orthonormal wavelets and paraunitary filter banks the known result that a binary tree structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis is generalized to binary trees having different paraunitary matrices on each level a converse result that every orthonormal wavelet basis can be generated by a tree structured filter bank having paraunitary polyphase matrices is then proved the concept of orthonormal bases is extended to generalized nonbinary tree structures and it is seen that a close relationship exists between orthonormality and paraunitariness it is proved that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis since not all phases can be generated by tree structured filter banks it is proved that if an orthonormal basis can be generated using a tree structure it can be generated specifically by a paraunitary tree

combining spatial and scale space techniques for edge detection to provide a spatially adaptive wavelet based noise filtering algorithm new methods for detecting edges in an image using spatial and scale space domains are proposed a priori knowledge about geometrical characteristics of edges is used to assign a probability factor to the chance of any pixel being on an edge an improved double thresholding technique is introduced for spatial domain filtering probabilities that pixels belong to a given edge are assigned based on pixel similarity across gradient amplitudes gradient phases and edge connectivity the scale space approach uses dynamic range compression to allow wavelet correlation over a wider range of scales a probabilistic formulation is used to combine the results obtained from filtering in each domain to provide a final edge probability image which has the advantages of both spatial and scale space domain methods decomposing this edge probability image with the same wavelet as the original image permits the generation of adaptive filters that can recognize the characteristics of the edges in all wavelet detail and approximation images regardless of scale these matched filters permit significant reduction in image noise without contributing to edge distortion the spatially adaptive wavelet noise filtering algorithm is qualitatively and quantitatively compared to a frequency domain and two wavelet based noise suppression algorithms using both natural and computer generated noisy images

recursive quadrature mirror filters criteria specification and design method criteria are found for a pair of recursive infinite impluse response iir filters which have mirror image amplitude responses and whose outputs are in phase quadrature at all frequencies a method is suggested for designing filters with these characteristics and their performance and processing requirements are compared with a typical nonrecursive finite impluse response fir realization a particular filter is described in more detail and it is shown how it could be incorporated in a multiband processing scheme it is concluded that recursive quadrature mirror filters could offer considerable savings in terms of signal processing compared with nonrecursive filters with similar performances additionally it is suggested that because they can be designed to have relatively short absolute delays the performance of the iir filters may prove superior for certain applications

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

narrowband interference suppression using undecimated wavelet packets in direct sequence spread spectrum receivers a new algorithm for narrowband interference suppression in direct sequence spread spectrum ds ss communications is presented the algorithm combines the undecimated wavelet packet transform uwpt with frequency shifts to confine the interference energy in a subband that is subsequently eliminated computer simulation shows a robust performance that appears to be independent of the interference frequency

extrapolation algorithms for solving mixed boundary integral equations of the helmholtz equation by mechanical quadrature methods this paper presents mechanical quadrature methods with high accuracy for solving mixed boundary integral equations of the helmholtz equation by estimating the range of eigenvalues for the discretization matrix of the integral equations and applying the collectively compact convergent theory we prove the stability and convergence of numerical solutions which is a challenging task for this method moreover the asymptotic error expansions show the method is of order h 3 hence extrapolation algorithms can be introduced to achieve higher approximation accuracy degree mathcal o h 5 meanwhile an a posteriori asymptotic error estimate is derived which can be used to construct self adaptive algorithms the numerical examples support our theoretical analysis

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

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

testing for homogeneity of variance in time series long memory wavelets and the nile river 1 we consider the problem of testing for homogeneity of variance in a time series with long memorystructure wedemonstratethat atest whose nullhypothesis isdesigned tobewhite noisecan in fact be applied on a scale by scale basis to the discrete wavelet transform of long memory processes in particular we show that evaluating a normalized cumulative sum of squares test statistic using critical levels for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractionally differenced process the point at which the test statistic using a nondecimated version of the discrete wavelet transform achieves its maximum value can be used to estimate the time of the unknown variance change we apply our proposed test statistic on five time series derived from the historical record of nile river yearly minimum water levels covering 622 1922 a d each series exhibiting various degrees of serial correlation including long memory in the longest subseries spanning 622 1284 a d the test confirms an inhomogeneity of variance at short time scales and identifies the change point around 720 a d which coincides closely with the construction of a new device around 715 a d for measuring the nile river the test also detects a change in variance for a record of only 36 years index terms 1829 hydrology groundwater hydrology 1869 hydrology stochastic processes 3299 mathematical geophysics general or miscellaneous

improving computational efficiency of model predictive control algorithm using wavelet transformation we propose an mpc algorithm with desirable stability and constraint handling properties yet which retains computational simplicity the closed loop stability and constraint handling are achieved b

infinite hidden markov models for unusual event detection in video we address the problem of unusual event detection in a video sequence invariant subspace analysis isa is used to extract features from the video and the time evolving properties of these features are modeled via an infinite hidden markov model ihmm which is trained using ldquonormalrdquo ldquotypicalrdquo video the ihmm retains a full posterior density function on all model parameters including the number of underlying hmm states anomalies unusual events are detected subsequently if a low likelihood is observed when associated sequential features are submitted to the trained ihmm a hierarchical dirichlet process framework is employed in the formulation of the ihmm the evaluation of posterior distributions for the ihmm is achieved in two ways via markov chain monte carlo and using a variational bayes formulation comparisons are made to modeling based on conventional maximum likelihood based hmms as well as to dirichlet process based gaussian mixture models

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

biorthogonal partners and applications two digital filters h z and f z are said to be biorthogonal partners of each other if their cascade h z f z satisfies the nyquist or zero crossing property biorthogonal partners arise in many different contexts such as filterbank theory exact and least squares digital interpolation and multiresolution theory they also play a central role in the theory of equalization especially fractionally spaced equalizers in digital communications we first develop several theoretical properties of biorthogonal partners we also develop conditions for the existence of biorthogonal partners and fir biorthogonal pairs and establish the connections to the riesz basis property we then explain how these results play a role in many of the above mentioned applications

m ary wavelet transform and formulation for perfect reconstruction in m band filter bank the binary wavelet transform is generalized and extended to the m ary biorthonormal case the computational equivalence between the discrete wavelet analysis and the m band multirate signal filtering is indicated the equivalence allows the perfect reconstruction requirement in a filter bank to be investigated from the vector space decomposition reconstruction in wavelet analysis from the construction of the biorthonormal wavelet bases the necessary and sufficient condition for the filters in a perfect reconstruction filter bank is formulated under this formulation an additional optimization procedure is then used to model the frequency domain requirement in filter bank design

texture features for classification of ultrasonic liver images the classification of ultrasonic liver images is studied making use of the spatial gray level dependence matrices the fourier power spectrum the gray level difference statistics and the laws texture energy measures features of these types are used to classify three sets of ultrasonic liver images normal liver hepatoma and cirrhosis 30 samples each the bayes classifier and the hotelling trace criterion are employed to evaluate the performance of these features from the viewpoint of speed and accuracy of classification it is found that these features do not perform well enough hence a new texture feature set multiresolution fractal features based on multiple resolution imagery and the fractional brownian motion model is proposed to detect diffuse liver diseases quickly and accurately fractal dimensions estimated at various resolutions of the image are gathered to form the feature vector texture information contained in the proposed feature vector is discussed a real time implementation of the algorithm produces about 90 correct classification for the three sets of ultrasonic liver images

hierarchical active shape models using the wavelet transform active shape models asms are often limited by the inability of relatively few eigenvectors to capture the full range of biological shape variability this paper presents a method that overcomes this limitation by using a hierarchical formulation of active shape models using the wavelet transform the statistical properties of the wavelet transform of a deformable contour are analyzed via principal component analysis and used as priors in the contour s deformation some of these priors reflect relatively global shape characteristics of the object boundaries whereas some of them capture local and high frequency shape characteristics and thus serve as local smoothness constraints this formulation achieves two objectives first it is robust when only a limited number of training samples is available second by using local statistics as smoothness constraints it eliminates the need for adopting ad hoc physical models such as elasticity or other smoothness models which do not necessarily reflect true biological variability examples on magnetic resonance images of the corpus callosum and hand contours demonstrate that good and fully automated segmentations can be achieved even with as few as five training samples

multiresolution analysis of printed antennas and circuits a dual isoscalar approach a new approach is presented for the integral equation analysis of multilayer printed circuits antennas and arrays it is based on the introduction of new multiresolution vector functions with properties similar to those of the scalar wavelets and flexible enough to accommodate for general shapes the resulting method of moments mom impedance matrix shows a good conditioning that allows its safe sparsification and the efficient use of iterative methods for the solution of the linear system the results of the analysis of various real life structures show the advantage of the method

tomographic reconstruction and estimation based on multiscale natural pixel bases we use a natural pixel type representation of an object originally developed for incomplete data tomography problems to construct nearly orthonormal multiscale basis functions the nearly orthonormal behavior of the multiscale basis functions results in a system matrix relating the input the object coefficients and the output the projection data which is extremely sparse in addition the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system we exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well conditioned and sparse matrix this enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions in case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability map multiscale reconstruction estimates based on a certain self similar prior on the multiscale object coefficients the framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete irregular and noisy and in object feature recognition directly from projection data

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 general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

symmetry based scalable lossless compression of 3d medical image data we propose a novel symmetry based technique for scalable lossless compression of 3d medical image data the proposed method employs the 2d integer wavelet transform to decorrelate the data and an intraband prediction method to reduce the energy of the sub bands by exploiting the anatomical symmetries typically present in structural medical images a modified version of the embedded block coder with optimized truncation ebcot tailored according to the characteristics of the data encodes the residual data generated after prediction to provide resolution and quality scalability performance evaluations on a wide range of real 3d medical images show an average improvement of 15 in lossless compression ratios when compared to other state of the art lossless compression methods that also provide resolution and quality scalability including 3d jpeg2000 jpeg2000 and h 264 avc intra coding

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

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