singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

optimized compact support interpolation kernels in this paper we investigate the problem of designing compact support interpolation kernels for a given class of signals by using calculus of variations we simplify the optimization problem from an nonlinear infinite dimensional problem to a linear finite dimensional case and then find the optimum compact support function that best approximates a given filter in the least square sense l2 norm the benefit of compact support interpolants is the low computational complexity in the interpolation process while the optimum compact support interpolant guarantees the highest achievable signal to noise ratio snr our simulation results confirm the superior performance of the proposed kernel compared to other conventional compact support interpolants such as cubic spline

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

image denoising based on wavelets and multifractals for singularity detection this paper presents a very efficient algorithm for image denoising based on wavelets and multifractals for singularity detection a challenge of image denoising is how to preserve the edges of an image when reducing noise by modeling the intensity surface of a noisy image as statistically self similar multifractal processes and taking advantage of the multiresolution analysis with wavelet transform to exploit the local statistical self similarity at different scales the pointwise singularity strength value characterizing the local singularity at each scale was calculated by thresholding the singularity strength wavelet coefficients at each scale were classified into two categories the edge related and regular wavelet coefficients and the irregular coefficients the irregular coefficients were denoised using an approximate minimum mean squared error mmse estimation method while the edge related and regular wavelet coefficients were smoothed using the fuzzy weighted mean fwm filter aiming at preserving the edges and details when reducing noise furthermore to make the fwm based filtering more efficient for noise reduction at the lowest decomposition level the mmse based filtering was performed as the first pass of denoising followed by performing the fwm based filtering experimental results demonstrated that this algorithm could achieve both good visual quality and high psnr for the denoised images

multifractal characterization of stochastic resonance we use a multifractal formalism to study the effect of stochastic resonance in a noisy bistable system driven by various input signals to characterize the response of a stochastic bistable system we introduce a new measure based on the calculation of a singularity spectrum for a return time sequence we use wavelet transform modulus maxima method for the singularity spectrum computations it is shown that the degree of multifractality defined as a width of singularity spectrum can be successfully used as a measure of complexity both in the case of periodic and aperiodic stochastic or chaotic input signals we show that in the case of periodic driving force singularity spectrum can change its structure qualitatively becoming monofractal in the regime of stochastic synchronization this fact allows us to consider the degree of multifractality as a new measure of stochastic synchronization also moreover our calculations have shown that the effect of stochastic resonance can be catched by this measure even from a very short return time sequence we use also the proposed approach to characterize the noise enhanced dynamics of a coupled stochastic neurons model

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

parameter estimation based on scale dependent algebraic expressions and scale space fitting we present our results of applying wavelet theory to the classic problem of estimating the unknown parameters of a model function subject to noise the model function studied in this context is a generalization of the second order gaussian derivative of which the gaussian function is a special case for all five model parameters amplitude width location baseline undershoot size scale dependent algebraic expressions are derived based on this analytical framework our first method estimates all parameters by substituting into a given expression numerically obtained values such as the zero crossings of the multiscale decompositions of the noisy input signal using gaussian derivative wavelets our second method takes these estimates as starting values for iterative least squares optimization to fit our algebraic zero crossing model to observed numeric zero crossings in scale space for evaluation we apply our method together with three reference methods to the three parameter gaussian model function the results show that our method is on average 3 7 times more accurate than the respective best reference method for signal to noise ratios snr from text 10 to 70 db using a synthetic test scenario proposed by a competitor for our full five parameter model we investigate overall estimation error as well as per parameter error and per parameter uncertainty as a function of snr and various noise models including correlated noise to demonstrate practical effectiveness and relevance we apply our method to the well studied problem of qrs complex delineation in electrocardiography signals out of the box results show a performance comparable to the best algorithms known to date without relying on problem specific heuristic decision rules

on the long time stability of a temporal discretization scheme for the three dimensional viscous primitive equations in this article a semi discretized euler scheme to solve the three dimensional viscous primitive equations is studied based on suitable assumptions on the initial data and forcing terms the long time stability of the proposed scheme is proven by showing that the h 1 norm in space variables of the solutions is bounded at each time step when the time step satisfies certain smallness condition

skeletonization of ribbon like shapes based on a new wavelet function a wavelet based scheme to extract skeleton of ribbon like shape is proposed in this paper where a novel wavelet function plays a key role in this scheme which possesses three significant characteristics namely 1 the position of the local maximum moduli of the wavelet transform with respect to the ribbon like shape is independent of the gray levels of the image 2 when the appropriate scale of the wavelet transform is selected the local maximum moduli of the wavelet transform of the ribbon like shape produce two new parallel contours which are located symmetrically at two sides of the original one and have the same topological and geometric properties as that of the original shape 3 the distance between these two parallel contours equals to the scale of the wavelet transform which is independent of the width of the shape this new scheme consists of two phases 1 generation of wavelet skeleton based on the desirable properties of the new wavelet function symmetry analyses of the maximum moduli of the wavelet transform is described midpoints of all pairs of contour elements can be connected to generate a skeleton of the shape which is defined as wavelet skeleton 2 modification of the wavelet skeleton thereafter a set of techniques are utilized for modifying the artifacts of the primary wavelet skeleton the corresponding algorithm is also developed in this paper experimental results show that the proposed scheme is capable of extracting exactly the skeleton of the ribbon like shape with different width as well as different gray levels the skeleton representation is robust against noise and affine transformation

finite element solution of nonlinear elliptic problems the study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed dirichlet neumann boundary conditions is presented in the discretization variational crimes are commited approximation of the given domain by a polygonal one numerical integration with the assumption that the corresponding operator is strongly monotone and lipschitz continuous and that the exact solutionu h1 the convergence of the method is proved under the additional assumptionu h2 the rate of convergenceo h is derived without the use of green s theorem

arbitrary order hilbert spectral analysis for time series possessing scaling statistics comparison study with detrended fluctuation analysis and wavelet leaders in this paper we present an extended version of hilbert huang transform namely arbitrary order hilbert spectral analysis to characterize the scale invariant properties of a time series directly in an amplitude frequency space we first show numerically that due to a nonlinear distortion traditional methods require high order harmonic components to represent nonlinear processes except for the hilbert based method this will lead to an artificial energy flux from the low frequency large scale to the high frequency small scale part thus the power law if it exists is contaminated we then compare the hilbert method with structure functions sf detrended fluctuation analysis dfa and wavelet leader wl by analyzing fractional brownian motion and synthesized multifractal time series for the former simulation we find that all methods provide comparable results for the latter simulation we perform simulations with an intermittent parameter 0 15 we find that the sf underestimates scaling exponent when q 3 the hilbert method provides a slight underestimation when q 5 however both dfa and wl overestimate the scaling exponents when q 5 it seems that hilbert and dfa methods provide better singularity spectra than sf and wl we finally apply all methods to a passive scalar temperature data obtained from a jet experiment with a taylor s microscale reynolds number re 250 due to the presence of strong ramp cliff structures the sf fails to detect the power law behavior for the traditional method the ramp cliff structure causes a serious artificial energy flux from the low frequency large scale to the high frequency small scale part thus dfa and wl underestimate the scaling exponents however the hilbert method provides scaling exponents q quite close to the one for longitudinal velocity indicating a less intermittent passive scalar field than what was believed before 2011 american physical society

first order decoupled finite element method of the three dimensional primitive equations of the ocean this paper is concerned with a first order fully discrete decoupled method for solving the three dimensional 3d primitive equations of the ocean with the dirichlet boundary conditions on the side where a decoupled semi implicit scheme is used for the time discretization and the p 1 p 1 p 1 p 1 p 1 finite element for velocity pressure and density is used for the spatial discretization of these equations the h 1 l 2 h 1 optimal error estimates for the numerical solution u h n p h n theta h n and the l 2 optimal error estimate for u n h theta h n are established under the restriction of 0 h le beta 1 and 0 tau le beta 2 for some positive constants beta 1 and beta 2 moreover numerical investigations are provided to show that the first order decoupled method is of almost unconditional convergence with accuracy mathcal o h tau in the h 1 norm and mathcal o h 2 tau in the l 2 norm for solving the 3d primitive equations of the ocean numerical results are gi

wavelet analysis for brain function imaging the authors present a new algorithmic procedure for the analysis of brain images this procedure is specifically designed to image the activity and functional organization of the brain the authors results are tested on data collected and previously analyzed with the technique known as in vivo optical imaging of intrinsic signals the authors procedure enhances the applicability of this technique and facilitates the extension of the underlying ideas to other imaging problems e g functional mri the authors thrust is two fold first they give a systematic method to control the blood vessel artifacts which typically reduce the dynamic range of the image they propose a mathematical model for the vibrations in time of the veins and arteries and they design a new method for cleaning the images of the vessels with the highest time variations this procedure is based on the analysis of the singularities of the images the use of wavelet transform is of crucial importance in characterizing the singularities and reconstructing appropriate versions of the original images the second important component of the authors work is the analysis of the time evolution of the fine structure of the images they show that once the images have been cleaned of the blood vessel vibrations variations the principal component of the time evolutions of the signals is due to the functional activity following the stimuli the part of the brain where this function takes place can be localized and delineated with precision

reconstruction from projections under time frequency constraints low pass filtering computed tomography ct images to reduce noise may smooth or modify image features which are very important to the physician image features are often more easily identified and processed in the time frequency plane the authors use time frequency distributions for spatially varying filtering of noisy ct images constraining time frequency representation coefficients of the projection data or of the reconstructed image to be zero in certain regions of the time frequency plane the authors consider two different applications 1 filtering the projection data and then performing image reconstruction and 2 filtering the reconstructed image directly criteria minimized subject to constraints may be either a deterministic minimum weighted perturbation of the given projection data or a stochastic minimum mean square error in colored gaussian noise results show improvement over processing the image with a linear spatially invariant filter

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

change curve estimation via wavelets abstract the recently developed theory of wavelets has a remarkable ability to zoom in on very short lived frequency phenomena such as transients in signals and singularities in functions and hence provides an ideal tool to study localized changes this article proposes a wavelet method for estimating jump and sharp cusp curves of a function in the plane the method involves first computing wavelet transformation of data and then estimating jump and sharp cusp curves by wavelet transformation across fine scales asymptotic theory is established and simulations are carried out to lend some credence to the asymptotic theory the wavelet estimate is nearly optimal and can be computed by fast algorithms the method is applied to a real image

multiscale representation and analysis of spherical data by spherical wavelets classic wavelet methods were developed in the euclidean spaces for multiscale representation and analysis of regularly sampled signals time series and images this paper introduces a method of representing scattered spherical data by multiscale spherical wavelets the method extends the recent pioneering work of narcowich and ward appl comput harmon anal 3 1996 pp 324 336 by employing multiscale rather than single scale spherical basis functions and by introducing a bottom up procedure for network design and bandwidth selection decomposition and reconstruction algorithms are proposed for efficient computation an analytical investigation confirms the localization property of the resulting spherical wavelets the proposed method is illustrated by numerical examples it is also employed to analyze and compress a real data set consisting of the surface air temperatures observed on a global network of weather stations

global well posedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics in this paper we prove the global existence and uniqueness regularity of strong solutions to the three dimensional viscous primitive equations which model large scale ocean and atmosphere dynamics 1 introduction large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the navier stokes equations with rotation coupled to thermodynamics and salinity diffusion transport equations which account for the buoyancy forces and stratification effects under the boussinesq approximation moreover and due to the shallowness of the oceans and the atmosphere i e the depth of the fluid layer is very small in comparison to the radius of the earth the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one which in turn leads to modeling the vertical motion by the hydrostatic balance as a result

oscillating singularities on cantor sets a grand canonical multifractal formalism the singular behavior of functions is generally characterized by their holder exponent however we show that this exponent poorly characterizes oscillating singularities we thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform we then elaborate on a grand canonical multifractal formalism that describes statistically the fluctuations of both the holder and the oscillation exponents we prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities

understanding imaging methods for potential field data several noniterative imaging methods for potential field data have been proposed that provide an estimate of the 3d magnetization density distribution within the subsurface or that produce images of quantities related or proportional to such distributions they have been derived in various ways using generalized linear inversion wiener filtering wavelet and depth from extreme points dexp transformations crosscorrelation and migration we demonstrated that the resulting images from each of these approaches are equivalent to an upward continuation of the data weighted by a possibly depth dependent function source distributions or related quantities imaged by all of these methods are smeared diffuse versions of the true distributions but owing to the stability of upward continuation resolution may be substantially increased by coupling derivative and upward continuation operators these imaging techniques appeared most effective in the case of isolated compact and depth limited sources beca

a joint inter and intrascale statistical model for bayesian wavelet based image denoising this paper presents a new wavelet based image denoising method which extends a geometrical bayesian framework the new method combines three criteria for distinguishing supposedly useful coefficients from noise coefficient magnitudes their evolution across scales and spatial clustering of large coefficients near image edges these three criteria are combined in a bayesian framework the spatial clustering properties are expressed in a prior model the statistical properties concerning coefficient magnitudes and their evolution across scales are expressed in a joint conditional model the three main novelties with respect to related approaches are 1 the interscale ratios of wavelet coefficients are statistically characterized and different local criteria for distinguishing useful coefficients from noise are evaluated 2 a joint conditional model is introduced and 3 a novel anisotropic markov random field prior model is proposed the results demonstrate an improved denoising performance over related earlier techniques

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

remarks on decay of correlations and witten laplacians brascamp lieb inequalities and semiclassical limit as it appears in recent articles by helffer or sjostrand and naddaf spencer the analysis in the context of the statistical mechanics of measures of the type exp x dxis connected with the analysis of suitable witten laplacians on 1 forms for illustrating this point of view we present here remarks about the brascamp lieb inequalities and its extensions and prove the decay of the correlation in some cases when is weakly non convex

stable edge adaptive multiscale decompositions using updated normal offsets the normal offset decomposition is an adaptive nonlinear multiscale data transform constructed in a way similar to the lifting scheme for wavelet transforms the main difference lies in the fact that the detail coefficients are not wavelet coefficients corresponding to a wavelet basis function wavelet coefficients carry detail information situated at the location of the basis function besides detail information normal offset coefficients also contain directional geometrical information i e information on where to locate the details the normal offset decomposition is a jump adaptive transform leading to a sparse representation of edges in images and sharp reconstructions of these edges a major issue in this paper is the design of a new class of wavelet transforms that can be extended to normal offset decompositions the combination of normal offsets and a stable lifting scheme is nontrivial a necessary condition for an stable lifted wavelet transform is that the wavelet basis functions have a vanishing integral i e at least one vanishing moment although the normal offset decomposition is not a basis transform in the strict sense a similar criterion for stability can be established and instability is equally a problem in a naive implementation this paper constructs an ldquoupdatedrdquo normal offset decomposition that is stable and therefore appropriate for use in applications that sensitive to the numerical condition such as applications involving heavy noise the new scheme combines numerical stability fast computation and sharp edge representations

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

bandwidth selection for changepoint estimation in nonparametric regression nonparametric estimation of abrupt changes in a regression function involves choosing smoothing bandwidth parameters the performance of estimation procedures depends heavily on this choice so far little attention has been paid to the crucial issue of choosing appropriate bandwidth parameters in practice in this article we propose a bootstrap procedure for selecting the bandwidth parameters in a nonparametric two step estimation method this method results in a fully data driven procedure for estimating a finite but possibly unknown number of changepoints in a regression function we evaluate the performance of the data driven procedure via a simulation study which reveals that the fully automatic procedure performs quite well as an illustration we apply the procedure to some real data

wavelet thresholding techniques for power spectrum estimation estimation of the power spectrum s f of a stationary random process can be viewed as a nonparametric statistical estimation problem we introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown s f this approach offers the ability to capture statistically significant components of ln s f at different resolution levels and guarantees nonnegativity of the spectrum estimator the spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non gaussian noise we propose a wavelet thresholding technique to solve this problem under specified noise resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables the thresholds are computed using a saddle point approximation to the distribution of the noise coefficients

low temperature expansions for the gibbs states of weakly interacting quantum ising lattice systems low temperature expansions for the gibbs states of weakly interacting transverse ising like models are developed by conditioning the states on a sub algebra of observables the conditioned states have effective classical hamiltonians which are estimated by the solution to a simple implicit equation provided the interaction is sufficiently weak but fixed independent of temperature and the temperature is sufficiently low exponential clustering of the correlation functions holds

asymptotic analysis of the primitive equations under the small depth assumption abstract in this article we study the asymptotic behavior of solutions of the primitive equations pes as the depth of the domain goes to zero we prove that the solutions of the pes can be expanded as a sum of barotropic flow and baroclinic flow up to a uniformly bounded in time and space initial time layer the barotropic flow is solution of the 2d navier stokes equations with coriolis force coupled with density by employing a comparison theorem the baroclinic flow can be approximated by a quasi stationary nonlinear gfd stokes problem this article presents a mathematically rigorous justification that the barotropic flow dominates the baroclinic flow in the motion of the atmosphere and ocean

texture roughness analysis and synthesis via extended self similar ess model the 2d fractional brownian motion fbm model provides a useful tool to model textured surfaces whose roughness is scale invariant to represent textures whose roughness is scale dependent we generalize the fbm model to the extended self similar ess model in this research we present an estimation algorithm to extract the model parameters from real texture data furthermore a new incremental fourier synthesis algorithm is proposed to generate the 2d realizations of the ess model finally the estimation and rendering methods are combined to synthesize real textured surfaces

qrs complex detection based on multi wavelet packet decomposition abstract we present in this paper a wavelet packet based qrs complex detection algorithm our proposed algorithm consists of a particular combination of two vectors obtained by applying a designed routine of qrs detection process using haar and db10 wavelet functions respectively the qrs complex detection routine is based on the histogram approach where our key idea was to search for the node with highest number of histogram coefficients at center which we assume that they are related to the iso electric baseline whereas the remaining least number coefficients reflect the r waves peaks following a classical approach based of a calculated fixed threshold the possible qrs complexes will be determined the qrs detection complex algorithm has been applied to the whole mit bih arrhythmia database to assess its robustness the algorithm reported a global sensitivity of 98 68 positive predictive value of 97 24 and a percentage error of 04 12 eventhough the obtained global results are not as excellent as expected we have demonstrate that our designed qrs detection algorithm performs good on a partial selected high percentage of the whole database e g the partial results obtained when applying the algorithm on 85 01 of the whole mit bih arrhythmia database are 99 14 of sensitivity 98 94 of positive predictive value and 01 92 of percentage error

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

a two grid finite difference method for the primitive equations of the ocean abstract a two level method is proposed for the finite difference approximation of a class of dissipative systems including the primitive equations of the ocean the two level or multilevel method amounts to solving a nonlinear problem on the single coarsest grid subsequent approximations are generated on a succession of refined meshes by solving a linearized problem about the solution on the previous level the stability analysis for an euler type scheme shows that the multilevel method can allow larger time steps than the traditional one level methods some numerical simulations of the primitive equations of the ocean using this new multi scale algorithm are presented to illustrate the method

adaptive regularized constrained least squares image restoration in noisy environments a constrained least squares cls approach is presented to restore images blurred by a gaussian impulse response where instead of choosing a global regularization parameter each point in the signal has its own associated regularization parameter these parameters are found by constraining the weighted standard deviation of the wavelet transform coefficients on the finest scale of the inverse signal by a function r which is a local measure of the intensity variations around each point of the blurred and noisy observed signal border ringing in the inverse solution is proposed decreased by manipulating its wavelet transform coefficients on the finest scales close to the borders if the noise in the inverse solution is significant wavelet transform techniques are also applied to denoise the solution examples are given for images and the results are shown to outperform the optimum constrained least squares solution using a global regularization parameter both visually and in the mean squared error sense

global well posedness and finite dimensional global attractor for a 3 d planetary geostrophic viscous model in this paper we consider a three dimensional planetary geostrophic viscous model of the gyre scale mid latitude ocean we show the global existence and uniqueness of the weak and strong solutions to this model moreover we establish the existence of a nite dimensional global attractor to this dissipative evolution system c 2003 wiley periodicals inc

first order decoupled method of the primitive equations of the ocean i time discretization abstract in this article we study the time discretization of the first order decoupled semi implicit scheme of the 3d primitive equations of the ocean in the case of the non slip and non heat flux boundary conditions on the side the almost unconditional stability of the scheme and the optimal error estimates of the time discrete velocity pressure and density are provided

a second order decoupled implicit explicit method of the 3d primitive equations of ocean ii finite element spatial discretization summary a fully discrete second order decoupled implicit explicit method is proposed for solving 3d primitive equations of ocean in the case of dirichlet boundary conditions on the side where a second order decoupled implicit explicit scheme is used for time discretization and a finite element method based on the p1 p1 p1 p1 p1 elements for velocity pressure and density is used for spatial discretization of these primitive equations optimal h1 l2 h1 error estimates for numerical solution uhn phn hn and an optimal l2 error estimate for uhn hn are established under the convergence condition of 0 h 1 0 2 and 3h for some positive constants 1 2 and 3 furthermore numerical computations show that the h1 l2 h1 convergence rate for numerical solution uhn phn hn is of o h 2 and an l2 convergence rate for uhn hn is o h2 2 with the assumed convergence condition where h is a mesh size and is a time step size more practical calculations are performed as a further validation of the numerical method copyright 2016 john wiley sons ltd

wavelet based image denoising using a markov random field a priori model this paper describes a new method for the suppression of noise in images via the wavelet transform the method relies on two measures the first is a classic measure of smoothness of the image and is based on an approximation of the local holder exponent via the wavelet coefficients the second novel measure takes into account geometrical constraints which are generally valid for natural images the smoothness measure and the constraints are combined in a bayesian probabilistic formulation and are implemented as a markov random field mrf image model the manipulation of the wavelet coefficients is consequently based on the obtained probabilities a comparison of quantitative and qualitative results for test images demonstrates the improved noise suppression performance with respect to previous wavelet based image denoising methods

noise suppression by removing singularities a method is proposed for suppressing gaussian noise by extracting local singularities the method is based on truncating riemann series decomposition whose components naturally characterize different orders of holder regularity the approach yields a single step filtering technique whose performance is comparable to three step wavelet decomposition and thresholding techniques

low temperature expansions for the gibbs states of quantum ising lattice systems by conditioning on a subalgebra of observables we develop low temperature expansions for certain quantum ising models which include transverse ising models and anisotropic xy models these expansions are majorized by the series for functions solving certain nonlinear partial differential equations convergence for the majorizing series follows from cauchy kowalevski theory applied to the partial differential equations the low temperature expansions imply long range order

solution of inverse problems in image processing by wavelet expansion we describe a wavelet based approach to linear inverse problems in image processing in this approach both the images and the linear operator to be inverted are represented by wavelet expansions leading to a multiresolution sparse matrix representation of the inverse problem the constraints for a regularized solution are enforced through wavelet expansion coefficients a unique feature of the wavelet approach is a general and consistent scheme for representing an operator in different resolutions an important problem in multigrid multiresolution processing this and the sparseness of the representation induce a multigrid algorithm the proposed approach was tested on image restoration problems and produced good results

analytic structure and explicit solution of an important implicit equation the equationz 2g z expg z 1 and similar ones obtained from it by substitutions appears in connection with a variety of problems ranging from pure mathematics combinatorics some first order nonlinear differential equations over statistical thermodynamics to renormalization theory it is therefore of interest to solve this equation forg z explicitly it turns out after study of the complex structure of thez andg planes that an explicit integral representation ofg z can be given which may be directly used for numerical calculations of high precision

diverging moments and parameter estimation heavy tailed distributions are enjoying increased popularity and are becoming more readily applicable as the arsenal of analytical and numerical tools grows they play key roles in modeling approaches in networking finance and hydrology to name but a few areas the tail parameter is of central importance because it governs both the existence of moments of positive order and the thickness of the tails of the distribution some of the best known tail estimators such as those of koutrouvelis and hill are either parametric or show a lack of robustness or accuracy this article develops a shift and scale invariant nonparametric estimator for both upper and lower bounds for orders with finite moments the estimator builds on the equivalence between tail behavior and the regularity of the characteristic function at the origin and achieves its goal by deriving a simplified wavelet analysis that is particularly suited to characteristic functions

impulse noise suppression in speech using the stationary wavelet transform an approach for detecting and removing impulse noise from speech using the wavelet transform is proposed the approach utilizes the multi resolution property of the wavelet transform which provides finer time resolution at higher frequencies than the short time fourier transform to effectively identify and remove impulse noise the paper then describes how the impulse detection performance is dependent on certain wavelet features and their relationships with the impulse noise and the underlying speech signal performance comparisons carried out with an existing method show that the wavelet approach yields much better features for detecting the impulses to remove the impulses an algorithm that uses the stationary wavelet transform has been developed the algorithm uses a two step approach where the wavelet coefficients corresponding to the impulses are suppressed in the first step and then substituted by suitable coefficients located within the vicinity of the impulse in the second step performance evaluations with an existing method show that the proposed algorithm gives superior results

study of a robust feature the pointwise lipschitz regularity the aim of this paper is to highlight the relevance in computer vision of the pointwise lipschitz regularity the regularity gives a measure of the local regularity of the intensity function associated to an image known wavelet methods provide an efficient computation of at contour points of the image from a theoretical point of view we study the effect of geometric deformations and other specific transformations applied to the image showing invariance properties from a practical point of view we assess the robustness of the regularity when the image undergoes various transformations the results we obtain show the lipschitz regularity is a suitable feature for applications in computer vision

amplitude variation with angle behavior of self similar interfaces amplitude variation with angle ava analysis is generally based on the assumption that the medium parameters behave as step functions of the depth coordinate z at least in a finite region around the interface however outliers observed in well logs often behave quite differently from step functions in this paper outliers in the acoustic propagation velocity are parameterized by functions of the form c z c1 z z1 the wavelet transform of this function reveals properties similar to those of several outliers in real well logs moreover this function is self similar according to c z c z for 0 analytical expressions are derived for the acoustic normal incidence reflection and transmission coefficients for this type of velocity function for oblique incidence no explicit solutions are available however by exploiting the self similarity property of the velocity function it turns out that the acoustic angle dependent and frequency dependent reflection and transmission coefficients are sel

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

noise analysis of a new singularity index we analyze the noise sensitivity of a new singularity index that was designed to detect impulse singularities in signals of arbitrary dimensionality while rejecting step like singularities see muralidhar ieee signal process lett vol 20 no 1 pp 7 10 2013 and muralidhar proc ieee int conf image process 2012 for example the index responds strongly to curvilinear masses ridges in images while weakly to jump discontinuities edges we analyze the detection power of the index in the presence of noise our analysis is geared towards answering the following questions a in the presence of noise only what is the probability of falsely detecting an impulse given a threshold b given an impulse submerged in noise what is the probability of detecting it given a threshold and c since the index is designed to be edge suppressing what is the probability of incorrectly detecting an edge submerged in noise given a threshold we compare the detection power of the index with that of a nominal impulse detector the second derivative operator simulations and example applications in 1 d and 2 d reveal the efficacy of the new singularity index for correctly detecting impulses submerged in noise while suppressing edges a software version of the 2 d singularity index can be downloaded from http live ece utexas edu research singularityindex singularityindexcode zip

signal recovery from wavelet transform maxima the paper presents an iterative algorithm for signal recovery from discrete time wavelet transform maxima the signal recovery algorithm is developed by using the method of projections onto convex sets convergence of the algorithm is assured

an efficient and provably correct algorithm for the multiscale estimation of image contours by means of polygonal lines a large portion of image contours is characterized by local properties such as sharp variations of the image intensity across the contour the integration of local image descriptors estimated by using these local properties into curvilinear descriptors is a difficult problem from a theoretical viewpoint because of the combinatorially large number of possible curvilinear descriptors to deal with this difficulty the notion of compressible graphs is introduced and a contour data model is defined leading to an efficient linear time algorithm which provably recovers contours with an upper bound on the approximation error

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

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

progressive quantized projection approach to data hiding a new image data hiding technique is proposed the proposed approach modifies blocks of the image after projecting them onto certain directions by quantizing the projected blocks to even and odd values one can represent the hidden information properly the proposed algorithm performs the modification progressively to ensure successful data extraction without the need for the original image at the receiver side two techniques are also presented for correcting scaling and rotation attacks the first approach is an exhaustive search in nature which is based on a training sequence that is inserted as part of the hidden information the second approach uses wavelet maxima as image semantics for rotation and scaling estimation both algorithms have proved to be effective in correcting rotation and scaling distortion

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

wavelet based detection of singularities in acoustic impedances from surface seismic reflection data although the passage of singularity information from acoustic impedance to seismic traces is now well understood it remains unanswered how routine seismic processing mode conversions and multiple reflections can affect the singularity analysis of surface seismic data we make theoretical investigations on the transition of singularity behaviors from acoustic impedances to surface seismic data we also perform numerical wavelet based singularity analysis on an elastic synthetic data set that is processed through routine seismic processing steps such as stacking and migration and that contains mode conversions multiple reflections and other wave equation effects theoretically seismic traces can be approximated as proportional to a smoothed version of the n 1 th derivative of acoustic impedance where n is the vanishing moment of the seismic wavelet this theoretical approach forms the basis of linking singularity exponents holder exponents in acoustic impedance with those computable from seismi

multiridge analysis of potential fields geometric method and reduced euler deconvolution a new method based on 3d multiridge analysis of potential fields assumes a 3d subset in the harmonic region and studies the behavior of potential field ridges which are built by joining extreme points of the analyzed field computed at different altitudes three types of ridges are formed by searching for the zeros of the first horizontal and first vertical derivatives of the potential field types i and ii respectively and the zeros of the potential field itself type iii this method uses a redundant set of ridges called a multiridge set to determine source type and location for homogeneous potential fields generated by simple sources all of the ridges are straight lines converging to the source position this method analyzes the multiridges by using a geometric criterion to find the source position at the intersection of the multiridge set and by solving the three reduced euler equations associated with ridge types i ii and iii the reduced euler type i and ii equations are used to obtain the structural index and the vertical and horizontal source positions equation type iii estimates the horizontal and vertical source positions tests on synthetic as well as the bishop model field yield good results even with noise corrupted data results obtained using magnetic data collected over the wreck of a military ship in the tyrrhenian sea successfully determine its vertical and horizontal positions and the structural index

three dimensional wavelet based multifractal method the need for revisiting the multifractal description of turbulence dissipation data we generalize the wavelet transform modulus maxima wtmm method to multifractal analysis of 3d random fields this method is calibrated on synthetic 3d monofractal fractional brownian fields and on 3d multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration then we apply the 3d wtmm method to the dissipation field issued from 3d isotropic turbulence simulations we comment on the need to revisit previous box counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures

low temperature phase diagrams of quantum lattice systems i stability for quantum perturbations of classical systems with finitely many ground states starting from classical lattice systems ind 2 dimensions with a regular zerotemperature phase diagram involving a finite number of periodic ground states we prove that adding a small quantum perturbation and or increasing the temperature produce only smooth deformations of their phase diagrams the quantum perturbations can involve bosons or fermions and can be of infinite range but decaying exponentially fast with the size of the bonds for fermions the interactions must be given by monomials of even degree in creation and annihilation operators our methods can be applied to some anyonic systems as well our analysis is based on an extension of pirogov sinai theory to contour expansions ind 1 dimensions obtained by iteration of the duhamel formula

reconstruction of finite signal derivatives from multiscale extrema representations application to transient estimation and signal approximation in this correspondence we propose a novel approach to the reconstruction of finite signal derivatives from the extrema of a multiscale representation first introduced by berkner we show that this reconstruction procedure naturally leads to the definition of a signal characteristic scale we also focus on signal approximation from the multiscale extrema representation of one of its derivatives numerical applications involve the use of the signal characteristic scale for transient estimation and also the study of the new signal approximation method we propose

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

wavelet solutions for the dirichlet problem a modified classical penalty method for solving a dirichlet boundary value problem is presented this new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains it is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast approximate calculation of boundary integral the compact support and orthonormality of the basis are essential for representing the boundary measure numerically and therefore for implementing this methodology

wavelet analysis of transients in fractal superlattices fractal superlattices are designed by alternating dielectric layers according to an iterative fractal process the discrete self similarity of such stratified structures can be remotely detected from interrogation by an incident pulse in this paper the impulse response of one dimensional cantor superlattices is computed and the wavelet transform is applied to the reflected signal in order to explore its temporal distribution of singularities for a sufficiently narrow pulse the skeleton of the wavelet transform modulus maxima exhibits a hierarchical structure that makes apparent the iterative process governing the construction rule of the interrogated fractal superlattice and its geometry in the reflected signal such hierarchy reveals the existence of singularities that are distributed on the governing cantor set finally the similarity dimension is extracted from reflection data and a strategy for estimating the stage of growth is developed

feature based wavelet shrinkage algorithm for image denoising a selective wavelet shrinkage algorithm for digital image denoising is presented the performance of this method is an improvement upon other methods proposed in the literature and is algorithmically simple for large computational savings the improved performance and computational speed of the proposed wavelet shrinkage algorithm is presented and experimentally compared with established methods the denoising method incorporated in the proposed algorithm involves a two threshold validation process for real time selection of wavelet coefficients the two threshold criteria selects wavelet coefficients based on their absolute value spatial regularity and regularity across multiresolution scales the proposed algorithm takes image features into consideration in the selection process statistically most images have regular features resulting in connected subband coefficients therefore the resulting subbands of wavelet transformed images in large part do not contain isolated coefficients in the proposed algorithm coefficients are selected due to their magnitude and only a subset of those selected coefficients which exhibit a spatially regular behavior remain for image reconstruction therefore two thresholds are used in the coefficient selection process the first threshold is used to distinguish coefficients of large magnitude and the second is used to distinguish coefficients of spatial regularity the performance of the proposed wavelet denoising technique is an improvement upon several other established wavelet denoising techniques as well as being computationally efficient to facilitate real time image processing applications

a wavelet based method for solving discrete first kind fredholm equations with piecewise constant solutions the inverse problem of finding piecewise constant solutions to discrete fredholm integral equations of the first kind arises in many applied fields e g in geophysics this equation is usually an ill posed problem when it is considered in a hilbert space framework requiring regularization techniques to control arbitrary error amplifications and to get adequate solutions in this work we describe an iterative regularizing method for computing piecewise constant solutions to first kind discrete fredholm integral equations the algorithm involves two main steps at each iteration 1 approximating the solution using a new signal reconstruction algorithm from its wavelet maxima which involves a previous step of detecting discontinuities by estimation of its local holder exponents and 2 obtaining a regularized solution of the original equation using the a priori knowledge and the above approximation in order to check the behaviour of the proposed technique we have carried out a statistical study from a high number of simulations obtaining excellent results their comparisons with the results coming from using classical tikhonov regularization by the multiresolution support total variation tv regularization and piecewise polynomial truncated singular value decomposition pp tsvd algorithm serve to illustrate the advantages of the new method copyright 2003 john wiley sons ltd

wavelet based approach to character skeleton character skeleton plays a significant role in character recognition the strokes of a character may consist of two regions i e singular and regular regions the intersections and junctions of the strokes belong to singular region while the straight and smooth parts of the strokes are categorized to regular region therefore a skeletonization method requires two different processes to treat the skeletons in theses two different regions all traditional skeletonization algorithms are based on the symmetry analysis technique the major problems of these methods are as follows 1 the computation of the primary skeleton in the regular region is indirect so that its implementation is sophisticated and costly 2 the extracted skeleton cannot be exactly located on the central line of the stroke 3 the captured skeleton in the singular region may be distorted by artifacts and branches to overcome these problems a novel scheme of extracting the skeleton of character based on wavelet transform is presented in this paper this scheme consists of two main steps namely a extraction of primary skeleton in the regular region and b amendment processing of the primary skeletons and connection of them in the singular region a direct technique is used in the first step where a new wavelet based symmetry analysis is developed for finding the central line of the stroke directly a novel method called smooth interpolation is designed in the second step where a smooth operation is applied to the primary skeleton and thereafter the interpolation compensation technique is proposed to link the primary skeleton so that the skeleton in the singular region can be produced experiments are conducted and positive results are achieved which show that the proposed skeletonization scheme is applicable to not only binary image but also gray level image and the skeleton is robust against noise and affine transform

element analysis a wavelet based method for analysing time localized events in noisy time series a method is derived for the quantitative analysis of signals that are composed of superpositions of isolated time localized events here these events are taken to be well represented as rescaled and phase rotated versions of generalized morse wavelets a broad family of continuous analytic functions analysing a signal composed of replicates of such a function using another morse wavelet allows one to directly estimate the properties of events from the values of the wavelet transform at its own maxima the distribution of events in general power law noise is determined in order to establish significance based on an expected false detection rate finally an expression for an event s region of influence within the wavelet transform permits the formation of a criterion for rejecting spurious maxima due to numerical artefacts or other unsuitable events signals can then be reconstructed based on a small number of isolated points on the time scale plane this method termed element analysis is applied to the identification of long lived eddy structures in ocean currents as observed by along track measurements of sea surface elevation from satellite altimetry

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

modeling and simulation of an impact test using wavelets analytical solutions and finite elements an instrumented falling weight impact test is thoroughly investigated in two cases that of a rebound and that of a rupture in the sample first the force signal is analyzed with four methods the continuous wavelet transform with different wavelets derivatives of a gaussian and morlet s the gabor transform the wigner ville transform and the classical fourier analysis it is shown how a proper use of the time frequency methods allows to precisely detect the discontinuities in the signal the moment of rupture if any and the frequencies that are excited at the impact in a second part modal and stress analyses are carried out both by an analytical method and by finite element computations the results confirm those obtained in the first part and agree with experimental findings c 2001 elsevier science ltd all rights reserved

thresholding neural network for adaptive noise reduction in the paper a type of thresholding neural network tnn is developed for adaptive noise reduction new types of soft and hard thresholding functions are created to serve as the activation function of the tnn unlike the standard thresholding functions the new thresholding functions are infinitely differentiable by using the new thresholding functions some gradient based learning algorithms become possible or more effective the optimal solution of the tnn in a mean square error mse sense is discussed it is proved that there is at most one optimal solution for the soft thresholding tnn general optimal performances of both soft and hard thresholding tnns are analyzed and compared to the linear noise reduction method gradient based adaptive learning algorithms are presented to seek the optimal solution for noise reduction the algorithms include supervised and unsupervised batch learning as well as supervised and unsupervised stochastic learning it is indicated that the tnn with the stochastic learning algorithms can be used as a novel nonlinear adaptive filter it is proved that the stochastic learning algorithm is convergent in certain statistical sense in ideal conditions numerical results show that the tnn is very effective in finding the optimal solutions of thresholding methods in an mse sense and usually outperforms other noise reduction methods especially it is shown that the tnn based nonlinear adaptive filtering outperforms the conventional linear adaptive filtering in both optimal solution and learning performance

multiple depth layers and all in focus image generations by blurring and deblurring operations the depth map and all in focus generations from a single image are proposed the depth map bases on the characteristic curve of coc vs the depth characteristic curve of a camera all in focus image generation is from the estimated depth map a joint deblurring process is proposed to generate an all in focus image in this work we propose novel methods for the generation of multiple depth layers and all in focus images multiple depth layers are produced by applying a sequence of blurring and deblurring operations to determine the depth the proposed method makes no assumptions with regard to the properties of the scene in resolving depth ambiguity in complex images we also developed a method by which to generate all in focus images in which the sharpness is preserved and instabilities imposed by conventional deblurring operations largely eliminated this method involves updating outliers for which depth was over estimated while compensating for errors in the estimation of depth imposed by deblurring

wavelet simultaneous approximation from samples affected by noise the authors recently introduced a pth order wavelet regularization method together with the gcv criterion for approximating a function from a finite sample affected by noise convergence results of the method were proven for the l2 norm the present paper addresses the problem of simultaneous approximation which is of interest in applications where derivatives are useful for extracting several features from a signal it is proved that it is not needed to devise a special method to this purpose since the convergence of the method devised by the authors also works for the hq norm 0 q p

application of multifractal wavelet analysis to spontaneous fermentation processes abstract an algorithm is presented here to get more detailed information of mixed culture type based exclusively on the biomass concentration data for fermentation processes the analysis is performed with only the on line measurements of the redox potential being available it is a two step procedure which includes an artificial neural network ann that relates the redox potential to the biomass concentrations in the first step next a multifractal wavelet analysis is performed using the biomass estimates of the process in this context our results show that the redox potential is a valuable indicator of microorganism metabolic activity during the spontaneous fermentation in this paper the detailed design of the multifractal wavelet analysis is presented as well as its direct experimental application at the laboratory level

efficient moment computation over polygonal domains with an application to rapid wedgelet approximation many algorithms in image processing rely on the computation of sums of pixel values over a large variety of subsets of the image domain this includes the computation of image moments for pattern recognition purposes or adaptive smoothing and regression methods such as wedgelets in the first part of the paper we present a general method which allows the fast computation of sums over a large class of polygonal domains the approach relies on the idea of considering polygonal domains with a fixed angular resolution combined with an efficient implementation of a discrete version of green s theorem the second part deals with the application of the new methodology to a particular computational problem namely wedgelet approximation our technique results in a speedup of o 10 3 by comparison to preexisting implementations a further attractive feature of our implementation is the instantaneous access to the full scale of wedgelet minimizers we introduce a new scheme that replaces the locally constant regression underlying wedgelets by basically arbitrary local regression models due to the speedup obtained by the techniques explained in the first part this scheme is computationally efficient and at the same time much more flexible than previously suggested methods such as wedgelets or platelets in the final section we present numerical experiments showing the increase in speed and flexibility

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

the multifractal structure of contrast changes in natural images from sharp edges to textures we present a formalism that leads naturally to a hierarchical description of the different contrast structures in images providing precise definitions of sharp edges and other texture components within this formalism we achieve a decomposition of pixels of the image in sets the fractal components of the image such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood a crucial role in this description of images is played by the behavior of contrast differences under changes in scale contrary to naive scaling ideas where the image is thought to have uniform transformation properties field 1987 each of these fractal components has its own transformation law and scaling exponents a conjecture on their biological relevance is also given

crack identification in plates using wavelet analysis in this paper a method for crack identification in plates based on wavelet analysis is presented the case of an all over part through crack parallel to one edge of the plate is considered the vibration modes of the plate are analyzed using the continuous wavelet transform and both the location and depth of the crack are estimated the position of the crack is determined by the sudden change in the spatial variation of the transformed displacement response to estimate the depth of the crack an intensity factor is defined which relates the depth of the crack to the coefficients of the wavelet transform an intensity factor law is established which allows accurate prediction of crack depth the viability of the proposed approach is demonstrated using simulation examples in view of the obtained results the advantages and limitations of the proposed approach as well as suggestions for future work are presented and discussed

wavelet footprints theory algorithms and applications wavelet based algorithms have been successful in different signal processing tasks the wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients discontinuities often carry relevant signal information and therefore they represent a critical part to analyze we study the dependency across scales of the wavelet coefficients generated by discontinuities we start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual theorem 1 we then introduce the notion of footprints which are scale space vectors that model discontinuities in piecewise polynomial signals exactly we show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints this also leads to efficient approximation of piecewise smooth functions finally we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising compression and nonblind deconvolution in the case of compression we also prove that at high rates footprint based algorithms attain optimal performance theorem 3