adaptive bayesian wavelet shrinkage abstract when fitting wavelet based models shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data this article outlines a bayesian approach to shrinkage obtained by placing priors on the wavelet coefficients the prior for each coefficient consists of a mixture of two normal distributions with different standard deviations the simple and intuitive form of prior allows us to propose automatic choices of prior parameters these parameters are chosen adaptively according to the resolution level of the coefficients typically shrinking high resolution frequency coefficients more heavily assuming a good estimate of the background noise level we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients the latter may be used to assess uncertainty in the reconstruction several examples are used to illustrate the method and comparisons are made with other shrinkage methods

structural topology optimization with eigenvalues the paper considers different problem formulations of topology optimization of discrete or discretized structures with eigenvalues as constraints or as objective functions we study multiple load case formulations of minimum volume or weight minimum compliance problems and the problem of maximizing the minimal eigenvalue of the structure including the effect of nonstructural mass the paper discusses interrelations of the problems and in particular shows how solutions of one problem can be derived from solutions of the others moreover we present equivalent reformulations as semidefinite programming problems with the property that for the minimum volume and minimum compliance problem each local optimizer of these problems is also a global one this allows for the calculation of guaranteed global optimizers of the original problems through the use of modern solution techniques of semidefinite programming for the problem of maximization of the minimum eigenvalue we show how to verify the global optimality and present an algorithm for finding a tight approximation of a globally optimal solution numerical examples are provided for truss structures both academic and larger size examples illustrate the theoretical results achieved and demonstrate the practical use of this approach we conclude with an extension on multiple nonstructural mass conditions

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

semi definite programming for topology optimization of trusses under multiple eigenvalue constraints abstract topology optimization problem of trusses for specified eigenvalue of vibration is formulated as semi definite programming sdp and an algorithm is presented based on the semi definite programming algorithm sdpa which utilizes extensively the sparseness of the matrices since the sensitivity coefficients of the eigenvalues with respect to the design variables are not needed the sdpa is especially useful for the case where the optimal design has multiple fundamental eigenvalues global and local modes are defined and a procedure is presented for generating optimal topology from the practical point of view it is shown in the examples that sdpa has advantage over existing methods in view of computational efficiency and accuracy of the solutions and an optimal topology with five fold fundamental eigenvalue is found without any difficulty

on the equivalence of set theoretic and maxent map estimation we establish an equivalence between two conceptually different methods of signal estimation under modeling uncertainty viz set theoretic st estimation and maximum entropy maxent map estimation the first method assumes constraints on the signal to be estimated and the second assumes constraints on a probability distribution for the signal we provide broad conditions under which these two estimation paradigms produce the same signal estimate we also show how the maxent formalism can be used to provide solutions to three important problems how to select sizes of constraint sets in st estimation the analysis highlights the role of shrinkage how to choose the values of parameters in regularized restoration when using multiple regularization functionals how to trade off model complexity and goodness of fit in a model selection problem

geodesics on the manifold of multivariate generalized gaussian distributions with an application to multicomponent texture discrimination we consider the rao geodesic distance gd based on the fisher information as a similarity measure on the manifold of zero mean multivariate generalized gaussian distributions mggd the mggd is shown to be an adequate model for the heavy tailed wavelet statistics in multicomponent images such as color or multispectral images we discuss the estimation of mggd parameters using various methods we apply the gd between mggds to color texture discrimination in several classification experiments taking into account the correlation structure between the spectral bands in the wavelet domain we compare the performance both in terms of texture discrimination capability and computational load of the gd and the kullback leibler divergence kld likewise both uni and multivariate generalized gaussian models are evaluated characterized by a fixed or a variable shape parameter the modeling of the interband correlation significantly improves classification efficiency while the gd is shown to consistently outperform the kld as a similarity measure

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near ideal estimation performance in various asymptotic frameworks however image processing practice has shown that universal thresholding methods are outperformed by simple bayesian estimators assuming independent wavelet coefficients and heavy tailed priors such as generalized gaussian distributions ggds in this paper we investigate various connections between shrinkage methods and maximum a posteriori map estimation using such priors in particular we state a simple condition under which map estimates are sparse we also introduce a new family of complexity priors based upon rissanen s universal prior on integers one particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length mdl principle we develop analytical expressions for the shrinkage rules implied by ggd and complexity priors this allows us to show the equivalence between universal hard thresholding map estimation using a very heavy tailed ggd and mdl estimation using one of the new complexity priors theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis specifications of the prior a basic concern in image processing applications

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

an unbiased risk estimator for image denoising in the presence of mixed poisson gaussian noise the behavior and performance of denoising algorithms are governed by one or several parameters whose optimal settings depend on the content of the processed image and the characteristics of the noise and are generally designed to minimize the mean squared error mse between the denoised image returned by the algorithm and a virtual ground truth in this paper we introduce a new poisson gaussian unbiased risk estimator pg ure of the mse applicable to a mixed poisson gaussian noise model that unifies the widely used gaussian and poisson noise models in fluorescence bioimaging applications we propose a stochastic methodology to evaluate this estimator in the case when little is known about the internal machinery of the considered denoising algorithm and we analyze both theoretically and empirically the characteristics of the pg ure estimator finally we evaluate the pg ure driven parametrization for three standard denoising algorithms with and without variance stabilizing transforms and different characteristics of the poisson gaussian noise mixture

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

a support function based algorithm for optimization with eigenvalue constraints optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and hermitian matrix valued function we propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally the quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of hermitian matrices we establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration the convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue we illustrate the

llsure local linear sure based edge preserving image filtering in this paper we propose a novel approach for performing high quality edge preserving image filtering based on a local linear model and using the principle of stein s unbiased risk estimate as an estimator for the mean squared error from the noisy image only we derive a simple explicit image filter which can filter out noise while preserving edges and fine scale details moreover this filter has a fast and exact linear time algorithm whose computational complexity is independent of the filtering kernel size thus it can be applied to real time image processing tasks the experimental results demonstrate the effectiveness of the new filter for various computer vision applications including noise reduction detail smoothing and enhancement high dynamic range compression and flash no flash denoising

a new sure approach to image denoising interscale orthonormal wavelet thresholding this paper introduces a new approach to orthonormal wavelet image denoising instead of postulating a statistical model for the wavelet coefficients we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights we then minimize an estimate of the mean square error between the clean image and the denoised one the key point is that we have at our disposal a very accurate statistically unbiased mse estimate stein s unbiased risk estimate that depends on the noisy image alone not on the clean one like the mse this estimate is quadratic in the unknown weights and its minimization amounts to solving a linear system of equations the existence of this a priori estimate makes it unnecessary to devise a specific statistical model for the wavelet coefficients instead and contrary to the custom in the literature these coefficients are not considered random any more we describe an interscale orthonormal wavelet thresholding algorithm based on this new approach and show its near optimal performance both regarding quality and cpu requirement by comparing it with the results of three state of the art nonredundant denoising algorithms on a large set of test images an interesting fallout of this study is the development of a new group delay based parent child prediction in a wavelet dyadic tree

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

bayesian selection for the ell _2 potts model regularization parameter 1 d piecewise constant signal denoising piecewise constant denoising can be solved either by deterministic optimization approaches based on the potts model or by stochastic bayesian procedures the former lead to low computational time but require the selection of a regularization parameter whose value significantly impacts the achieved solution and whose automated selection remains an involved and challenging problem conversely fully bayesian formalisms encapsulate the regularization parameter selection into hierarchical models at the price of high computational costs this contribution proposes an operational strategy that combines hierarchical bayesian and potts model formulations with the double aim of automatically tuning the regularization parameter and maintaining computational efficiency the proposed procedure relies on formally connecting a bayesian framework to a ell 2 potts functional behaviors and performance for the proposed piecewise constant denoising and regularization parameter tuning techniques are studied qualitatively and assessed quantitatively and shown to compare favorably against those of a fully bayesian hierarchical procedure both in accuracy and computational load

a nonlinear stein based estimator for multichannel image denoising the use of multicomponent images has become widespread with the improvement of multisensor systems having increased spatial and spectral resolutions however the observed images are often corrupted by an additive gaussian noise in this paper we are interested in multichannel image denoising based on a multiscale representation of the images a multivariate statistical approach is adopted to take into account both the spatial and the intercomponent correlations existing between the different wavelet subbands more precisely we propose a new parametric nonlinear estimator which generalizes many reported denoising methods the derivation of the optimal parameters is achieved by applying stein s principle in the multivariate case experiments performed on multispectral remote sensing images clearly indicate that our method outperforms conventional wavelet denoising techniques

the sure let approach to image denoising we propose a new approach to image denoising based on the image domain minimization of an estimate of the mean squared error stein s unbiased risk estimate sure unlike most existing denoising algorithms using the sure makes it needless to hypothesize a statistical model for the noiseless image a key point of our approach is that although the nonlinear processing is performed in a transformed domain typically an undecimated discrete wavelet transform but we also address nonorthonormal transforms this minimization is performed in the image domain indeed we demonstrate that when the transform is a ldquotightrdquo frame an undecimated wavelet transform using orthonormal filters separate subband minimization yields substantially worse results in order for our approach to be viable we add another principle that the denoising process can be expressed as a linear combination of elementary denoising processes linear expansion of thresholds let armed with the sure and let principles we show that a denoising algorithm merely amounts to solving a linear system of equations which is obviously fast and efficient quite remarkably the very competitive results obtained by performing a simple threshold image domain sure optimized on the undecimated haar wavelet coefficients show that the sure let principle has a huge potential

wavelet denoising of multicomponent images using gaussian scale mixture models and a noise free image as priors in this paper a bayesian wavelet based denoising procedure for multicomponent images is proposed a denoising procedure is constructed that 1 fully accounts for the multicomponent image covariances 2 makes use of gaussian scale mixtures as prior models that approximate the marginal distributions of the wavelet coefficients well and 3 makes use of a noise free image as extra prior information it is shown that such prior information is available with specific multicomponent image data of e g remote sensing and biomedical imaging experiments are conducted in these two domains in both simulated and real noisy conditions

sufficient optimality conditions hold for almost all nonlinear semidefinite programs we derive a new genericity result for nonlinear semidefinite programming nlsdp namely almost all linear perturbations of a given nlsdp are shown to be nondegenerate here nondegeneracy for nlsdp refers to the transversality constraint qualification strict complementarity and second order sufficient condition due to the presence of the second order sufficient condition our result is a nontrivial extension of the corresponding results for linear semidefinite programs sdp from alizadeh et al math program 77 2 ser b 111 128 1997 the proof of the genericity result makes use of forsgren s derivation of optimality conditions for nlsdp in forsgren math program 88 1 ser a 105 128 2000 due to the latter approach the positive semidefiniteness of a symmetric matrix g x depending continuously on x is locally equivalent to the fact that a certain schur complement s x of g x is positive semidefinite this yields a reduced nlsdp by considering the new semidefinite constraint s x succeq 0 s x 0 instead of g x succeq 0 g x 0 while deriving optimality conditions for the reduced nlsdp the well known and often mentioned h term in the second order sufficient condition vanishes this allows us to access the proof of the genericity result for nlsdp

computational structural mechanics and optimal control the simulation of substructural chain theory to linear quadratic optimal control problems the theory of optimal control and the theory of a substructural chain in static structural analysis are mutually simulated issues from the minimum potential energy variational principle of the substructural chain the generalized variational principle with two kinds of variables is derived first by comparing that generalized variational principle with the variational principle in lq control theory the simulation relation is established based on that relation the potential energy and mixed energy formulation of the algebraic riccati equations are derived then iterative algorithms are proposed which give the upper and lower bounds to the solution matrix by using the solutions of the positive and negative co ordinate algebraic riccati equations the canonical transformation matrices for the eigenproblems of the substructural chain and lq control are constructed respectively which reduce the eigenproblem to half size the properties of the solutions are analysed which establishes the basis for expansion solutions

sure let multichannel image denoising interscale orthonormal wavelet thresholding we propose a vector matrix extension of our denoising algorithm initially developed for grayscale images in order to efficiently process multichannel e g color images this work follows our recently published sure let approach where the denoising algorithm is parameterized as a linear expansion of thresholds let and optimized using stein s unbiased risk estimate sure the proposed wavelet thresholding function is pointwise and depends on the coefficients of same location in the other channels as well as on their parents in the coarser wavelet subband a nonredundant orthonormal wavelet transform is first applied to the noisy data followed by the subband dependent vector valued thresholding of individual multichannel wavelet coefficients which are finally brought back to the image domain by inverse wavelet transform extensive comparisons with the state of the art multiresolution image denoising algorithms indicate that despite being nonredundant our algorithm matches the quality of the best redundant approaches while maintaining a high computational efficiency and a low cpu memory consumption an online java demo illustrates these assertions

near optimal compressed sensing without priors parametric sure approximate message passing both theoretical analysis and empirical evidence confirm that the approximate message passing amp algorithm can be interpreted as recursively solving a signal denoising problem at each amp iteration one observes a gaussian noise perturbed original signal retrieving the signal amounts to a successive noise cancellation until the noise variance decreases to a satisfactory level in this paper we incorporate the stein s unbiased risk estimate sure based parametric denoiser with the amp framework and propose the novel parametric sure amp algorithm at each parametric sure amp iteration the denoiser is adaptively optimized within the parametric class by minimizing sure which depends purely on the noisy observation in this manner the parametric sure amp is guaranteed with the best in class recovery and convergence rate if the parametric family includes the families of the mimimum mean squared error mmse estimators we are able to achieve the bayesian optimal amp performance without knowing the signal prior in the paper we resort to the linear parameterization of the sure based denoiser and propose three different kernel families as the base functions numerical simulations with the bernoulli gaussian k dense and student s t signals demonstrate that the parametric sure amp does not only achieve the state of the art recovery but also runs more than 20 times faster than the em gm gamp algorithm natural image simulations confirm the advantages of the parametric sure amp for signals without prior information

wavelet shrinkage using cross validation summary wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics this paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage a modified form of twofold cross validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2 the cross validation algorithm is then extended to data sets of any length and to multidimensional data sets the algorithms are compared with established threshold choosers by using simulation an application to a real data set arising from anaesthesia is presented

least squares estimation without priors or supervision selection of an optimal estimator typically relies on either supervised training samples pairs of measurements and their associated true values or a prior probability model for the true values here we consider the problem of obtaining a least squares estimator given a measurement process with known statistics i e a likelihood function and a set of unsupervised measurements each arising from a corresponding true value drawn randomly from an unknown distribution we develop a general expression for a nonparametric empirical bayes least squares nebls estimator which expresses the optimal least squares estimator in terms of the measurement density with no explicit reference to the unknown prior density we study the conditions under which such estimators exist and derive specific forms for a variety of different measurement processes we further show that each of these nebls estimators may be used to express the mean squared estimation error as an expectation over the measurement density alone thus generalizing stein s unbiased risk estimator sure which provides such an expression for the additive gaussian noise case this error expression may then be optimized over noisy measurement samples in the absence of supervised training data yielding a generalized sure optimized parametric least squares sure2pls estimator in the special case of a linear parameterization i e a sum of nonlinear kernel functions the objective function is quadratic and we derive an incremental form for learning this estimator from data we also show that combining the nebls form with its corresponding generalized sure expression produces a generalization of the score matching procedure for parametric density estimation finally we have implemented several examples of such estimators and we show that their performance is comparable to their optimal bayesian or supervised regression counterparts for moderate to large amounts of data

should penalized least squares regression be interpreted as maximum a posteriori estimation penalized least squares regression is often used for signal denoising and inverse problems and is commonly interpreted in a bayesian framework as a maximum a posteriori map estimator the penalty function being the negative logarithm of the prior for example the widely used quadratic program with an l1 penalty associated to the lasso basis pursuit denoising is very often considered as map estimation under a laplacian prior in the context of additive white gaussian noise awgn reduction this paper highlights the fact that while this is one possible bayesian interpretation there can be other equally acceptable bayesian interpretations therefore solving a penalized least squares regression problem with penalty x need not be interpreted as assuming a prior c exp x and using the map estimator in particular it is shown that for any prior px the minimum mean square error mmse estimator is the solution of a penalized least square problem with some penalty x which can be interpreted as the map estimator with the prior c exp x vice versa for certain penalties x the solution of the penalized least squares problem is indeed the mmse estimator with a certain prior px in general dpx x c exp x dx

generalized sure for exponential families applications to regularization stein s unbiased risk estimate sure was proposed by stein for the independent identically distributed i i d gaussian model in order to derive estimates that dominate least squares ls recently the sure criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean squared error mse however its use has been limited to the i i d case which precludes many important applications in this paper we begin by deriving a sure counterpart for general not necessarily i i d distributions from the exponential family this enables extending the sure design technique to a much broader class of problems based on this generalization we suggest a new method for choosing regularization parameters in penalized ls estimators we then demonstrate its superior performance over the conventional generalized cross validation and discrepancy approaches in the context of image deblurring and deconvolution the sure technique can also be used to design estimates without predefining their structure however allowing for too many free parameters impairs the estimate s performance to address this inherent tradeoff we propose a regularized sure objective and demonstrate its use in the context of wavelet denoising

discrete linear hamiltonian eigenvalue problems abstract this paper introduces general discrete linear hamiltonian eigenvalue problems and characterizes the eigenvalues assumptions are given among them the new notion of strict controllability of a discrete system that imply isolatedness and lower boundedness of the eigenvalues due to the quite general assumptions discrete sturm liouville eigenvalue problems of higher order are included in the presented theory

a sure approach for digital signal image deconvolution problems in this paper we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white gaussian noise the originality of the proposed approach is twofold first we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from stein s unbiased quadratic risk estimate secondly the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not new theoretical results concerning the calculation of the variance of the stein s risk estimate are also provided in this work simulations carried out on natural images show the good performance of our method with respect to conventional wavelet based restoration methods

bivariate shrinkage functions for wavelet based denoising exploiting interscale dependency most simple nonlinear thresholding rules for wavelet based denoising assume that the wavelet coefficients are independent however wavelet coefficients of natural images have significant dependencies we only consider the dependencies between the coefficients and their parents in detail for this purpose new non gaussian bivariate distributions are proposed and corresponding nonlinear threshold functions shrinkage functions are derived from the models using bayesian estimation theory the new shrinkage functions do not assume the independence of wavelet coefficients we show three image denoising examples in order to show the performance of these new bivariate shrinkage rules in the second example a simple subband dependent data driven image denoising system is described and compared with effective data driven techniques in the literature namely visushrink sureshrink bayesshrink and hidden markov models in the third example the same idea is applied to the dual tree complex wavelet coefficients

nonlinear wavelet shrinkage with bayes rules and bayes factors abstract wavelet shrinkage the method proposed by the seminal work of donoho and johnstone is a disarmingly simple and efficient way of denoising data shrinking wavelet coefficients was proposed from several optimality criteria in this article a wavelet shrinkage by coherent bayesian inference in the wavelet domain is proposed the methods are tested on standard donoho johnstone test functions

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

calibration and empirical bayes variable selection for the problem of variable selection for the normal linear model selection criteria such as aic c p bic and ric have fixed dimensionality penalties such criteria are shown to correspond to selection of maximum posterior models under implicit hyperparameter choices for a particular hierarchical bayes formulation based on this calibration we propose empirical bayes selection criteria that use hyperparameter estimates instead of fixed choices for obtaining these estimates both marginal and conditional maximum likelihood methods are considered as opposed to traditional fixed penalty criteria these empirical bayes criteria have dimensionality penalties that depend on the data their performance is seen to approximate adaptively the performance of the best fixed penalty criterion across a variety of orthogonal and nonorthogonal set ups including wavelet regression empirical bayes shrinkage estimators of the selected coefficients are also proposed

critical value computation for h riccati difference equations abstract in designing finite horizon discrete time h controllers the associated h riccati difference equations must be solved but the riccati equation has a non negative solution only when 2 is small enough so it is important to get the upper bound of the parameter i e the critical value that ensures the existence of the solution to the riccati equation the solution sequence of the riccati difference equation can be constructed by the conjoined basis of an associated linear hamiltonian difference system based on this expression and the hamiltonian difference system eigenvalue theorems the equivalence between the critical value and the first order eigenvalue of the linear hamiltonian difference system is presented since the critical value is also shown to be the fundamental eigenvalue of a generalized rayleigh quotient an extended form of wittrick williams algorithm is presented to search this value

multispectral image denoising with optimized vector bilateral filter vector bilateral filtering has been shown to provide good tradeoff between noise removal and edge degradation when applied to multispectral hyperspectral image denoising it has also been demonstrated to provide dynamic range enhancement of bands that have impaired signal to noise ratios snrs typical vector bilateral filtering described in the literature does not use parameters satisfying optimality criteria we introduce an approach for selection of the parameters of a vector bilateral filter through an optimization procedure rather than by ad hoc means the approach is based on posing the filtering problem as one of nonlinear estimation and minimization of the stein s unbiased risk estimate of this nonlinear estimator along the way we provide a plausibility argument through an analytical example as to why vector bilateral filtering outperforms band wise 2d bilateral filtering in enhancing snr experimental results show that the optimized vector bilateral filter provides improved denoising performance on multispectral images when compared with several other approaches

minimum h sub infinity norm regulation of linear discrete time systems and its relation to linear quadratic discrete games a solution is derived to the h sub infinity optimization problem that arises in multivariable discrete time regulation when the controller has full access to the state vector the solution method is based on the close relations that exist between linear quadratic differential game theory and h sub infinity optimization the existing theory of discrete time quadratic games is readily applied in order to derive the solution to a finite time horizon version of the h sub infinity optimization problem the solution of the infinite time horizon h sub infinity optimization problem is obtained by formally taking the limit of the number of stages to infinity

spatially adaptive wavelet thresholding with context modeling for image denoising the method of wavelet thresholding for removing noise or denoising has been researched extensively due to its effectiveness and simplicity much of the literature has focused on developing the best uniform threshold or best basis selection however not much has been done to make the threshold values adaptive to the spatially changing statistics of images such adaptivity can improve the wavelet thresholding performance because it allows additional local information of the image such as the identification of smooth or edge regions to be incorporated into the algorithm this work proposes a spatially adaptive wavelet thresholding method based on context modeling a common technique used in image compression to adapt the coder to changing image characteristics each wavelet coefficient is modeled as a random variable of a generalized gaussian distribution with an unknown parameter context modeling is used to estimate the parameter for each coefficient which is then used to adapt the thresholding strategy this spatially adaptive thresholding is extended to the overcomplete wavelet expansion which yields better results than the orthogonal transform experimental results show that spatially adaptive wavelet thresholding yields significantly superior image quality and lower mse than the best uniform thresholding with the original image assumed known

bounds for decoupled design and analysis discretizations in topology optimization summary topology optimization formulations using multiple design variables per finite element have been proposed to improve the design resolution this paper discusses the relation between the number of design variables per element and the order of the elements used for analysis we derive that beyond a maximum number of design variables certain sets of material distributions cannot be discriminated based on the corresponding analysis results this makes the design description inefficient and the solution of the optimization problem non unique to prevent this we establish bounds for the maximum number of design variables that can be used to describe the material distribution for any given finite element scheme without introducing non uniqueness copyright 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

wavelet transforms versus fourier transforms this note is a very basic introduction to wavelets it starts with an orthogonal basis of piecewise constant functions constructed by dilation and translation the wavelet transform maps each f x to its coefficients with respect to this basis the mathematics is simple and the transform is fast faster than the fast fourier transform which we briefly explain but approximation by piecewise constants is poor to improve this first wavelet we are led to dilation equations and their unusual solutions higher order wavelets are constructed and it is surprisingly quick to compute with them always indirectly and recursively

free material optimization with fundamental eigenfrequency constraints the goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure a natural formulation of this problem as a linear semidefinite program turns out to be numerically intractable as an alternative we propose a new approach which is based on a nonlinear semidefinite low rank approximation of the semidefinite dual we introduce an algorithm based on this approach and analyze its convergence properties the article is concluded by numerical experiments proving the effectiveness of the new approach

nonlinear multivariate image filtering techniques in this paper nonlinear multivariate image filtering techniques are proposed to handle color images corrupted by noise first we briefly review the principle of reduced ordering r ordering and then define three r orderings by selecting different central locations considering noise attenuation edge preservation and detail retention r ordering based multivariate filters are designed by combining the r ordering schemes to implement color image filtering more effectively we develop them into a locally adaptive version the output of the adaptive filter is the closest sample to a central location that is a weighted linear combination of the mean the marginal median and the center sample as a result we study an adaptive hybrid multivariate ahm filter consisting of the mean filter the marginal median filter and the identity filter the performance of the two adaptive filtering techniques is compared with that of some nonadaptive ones the examples of color image filtering show that the adaptive multivariate image filtering gives a rather good performance improvement

time invariant orthonormal wavelet representations a simple construction of an orthonormal basis starting with a so called mother wavelet together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications an orthonormal basis may not however always be a suitable representation of a signal particularly when time or space invariance is a required property the conventional way around this problem is to use a redundant decomposition we address the time invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions we show that it is possible to achieve time invariance and preserve the orthonormality we subsequently propose an efficient approach to obtain such a decomposition we demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation

nonparametric cepstrum estimation via optimal risk smoothing this paper proposes a new cepstrum estimation procedure that is capable of producing smoother and improved cepstrum estimates without the use of any parametric modeling this procedure consists of two main steps in the first step it applies a so called grid transformation to the empirical cepstral coefficients while in the second step it nonparametrically smooths the transformed coefficients with local linear regression the stein s unbiased risk estimation sure approach is adopted to select both the extent of the grid transformation and the amount of smoothing it is shown that the use of this sure selection method for the current problem is asymptotically optimal in a well defined sense lastly the good practical performance of the new cepstrum estimation procedure is demonstrated via numerical experiments

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

clustering based denoising with locally learned dictionaries in this paper we propose k lld a patch based locally adaptive denoising method based on clustering the given noisy image into regions of similar geometric structure in order to effectively perform such clustering we employ as features the local weight functions derived from our earlier work on steering kernel regression these weights are exceedingly informative and robust in conveying reliable local structural information about the image even in the presence of significant amounts of noise next we model each region or cluster which may not be spatially contiguous by ldquolearningrdquo a best basis describing the patches within that cluster using principal components analysis this learned basis or ldquodictionaryrdquo is then employed to optimally estimate the underlying pixel values using a kernel regression framework an iterated version of the proposed algorithm is also presented which leads to further performance enhancements we also introduce a novel mechanism for optimally choosing the local patch size for each cluster using stein s unbiased risk estimator sure we illustrate the overall algorithm s capabilities with several examples these indicate that the proposed method appears to be competitive with some of the most recently published state of the art denoising methods

on the selection of optimum savitzky golay filters savitzky golay s g filters are finite impulse response lowpass filters obtained while smoothing data using a local least squares ls polynomial approximation savitzky and golay proved in their hallmark paper that local ls fitting of polynomials and their evaluation at the mid point of the approximation interval is equivalent to filtering with a fixed impulse response the problem that we address here is how to choose a pointwise minimum mean squared error mmse s g filter length or order for smoothing while preserving the temporal structure of a time varying signal we solve the bias variance tradeoff involved in the mmse optimization using stein s unbiased risk estimator sure we observe that the 3 db cutoff frequency of the sure optimal s g filter is higher where the signal varies fast locally and vice versa essentially enabling us to suitably trade off the bias and variance thereby resulting in near mmse performance at low signal to noise ratios snrs it is seen that the adaptive filter length algorithm performance improves by incorporating a regularization term in the sure objective function we consider the algorithm performance on real world electrocardiogram ecg signals the results exhibit considerable snr improvement noise performance analysis shows that the proposed algorithms are comparable and in some cases better than some standard denoising techniques available in the literature

ℋ control in discrete time state feedback control and norm bounds h norm bounds are useful in control design for disturbance rejection for discrete systems h norm optimizing controls can be characterized via the solution of the discrete algebraic riccati equation dare in addition the dare can be used to characterize controls that guarantee a uniform h norm bound in this paper one identifies an entire class of state feedback controls that guarantee an h norm bound and provides new lower bounds for the h norm of stable systems as well as for the best achievable h norm using state feedback controls

monte carlo sure a black box optimization of regularization parameters for general denoising algorithms we consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white gaussian noise to achieve this we propose to minimize stein s unbiased risk estimate sure which provides a means of assessing the true mean squared error mse purely from the measured data without need for any knowledge about the noise free signal specifically we present a novel monte carlo technique which enables the user to calculate sure for an arbitrary denoising algorithm characterized by some specific parameter setting our method is a black box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form this therefore permits the use of sure for optimization of a wide variety of denoising algorithms we justify our claims by presenting experimental results for sure based optimization of a series of popular image denoising algorithms such as total variation denoising wavelet soft thresholding and wiener filtering smoothing splines in the process we also compare the performance of these methods we demonstrate numerically that sure computed using the new approach accurately predicts the true mse for all the considered algorithms we also show that sure uncovers the optimal values of the parameters in all cases

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

adaptive wavelet thresholding for image denoising and compression the first part of this paper proposes an adaptive data driven threshold for image denoising via wavelet soft thresholding the threshold is derived in a bayesian framework and the prior used on the wavelet coefficients is the generalized gaussian distribution ggd widely used in image processing applications the proposed threshold is simple and closed form and it is adaptive to each subband because it depends on data driven estimates of the parameters experimental results show that the proposed method called bayesshrink is typically within 5 of the mse of the best soft thresholding benchmark with the image assumed known it also outperforms sureshrink donoho and johnstone 1994 1995 donoho 1995 most of the time the second part of the paper attempts to further validate claims that lossy compression can be used for denoising the bayesshrink threshold can aid in the parameter selection of a coder designed with the intention of denoising and thus achieving simultaneous denoising and compression specifically the zero zone in the quantization step of compression is analogous to the threshold value in the thresholding function the remaining coder design parameters are chosen based on a criterion derived from rissanen s minimum description length mdl principle experiments show that this compression method does indeed remove noise significantly especially for large noise power however it introduces quantization noise and should be used only if bitrate were an additional concern to denoising

wavelet based statistical signal processing using hidden markov models wavelet based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly gaussian these models are unrealistic for many real world signals we develop a new framework for statistical signal processing based on wavelet domain hidden markov models hmms that concisely models the statistical dependencies and non gaussian statistics encountered in real world signals wavelet domain hmms are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models efficient expectation maximization algorithms are developed for fitting the hmms to observational signal data the new framework is suitable for a wide range of applications including signal estimation detection classification prediction and even synthesis to demonstrate the utility of wavelet domain hmms we develop novel algorithms for signal denoising classification and detection

noise reduction of cdna microarray images using complex wavelets noise reduction is an essential step of cdna microarray image analysis for obtaining better quality gene expression measurements wavelet based denoising methods have shown significant success in traditional image processing the complex wavelet transform cwt is preferred to the classical discrete wavelet transform for denoising of microarray images due to its improved directional selectivity for better representation of the circular edges of spots and near shift invariance property existing cwt based denoising methods are not efficient for microarray image processing because they fail to take into account the signal as well as noise correlations that exist between red and green channel images in this paper two bivariate estimators are developed for the cwt based denoising of microarray images using the standard maximum a posteriori and linear minimum mean squared error estimation criteria the proposed denoising methods are capable of taking into account both the interchannel signal and noise correlations significance of the proposed denoising methods is assessed by examining the effect of noise reduction on the estimation of the log intensity ratio extensive experimentations are carried out to show that the proposed methods provide better noise reduction of microarray images leading to more accurate estimation of the log intensity ratios as compared to the other cwt based denoising methods

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

on the homotopy method for perturbed symmetric generalized eigenvalue problems the generalized eigenvalue problem plays a significant role in many applications usually only a few smallest eigenpairs i e eigenvalues and their corresponding eigenvectors are desired a frequently encountered problem is to solve a system slightly perturbed from the original system if the perturbation is small the new system can be solved by using rayleigh quotient iteration rqi the initial ritz vectors are provided by the eigenvectors from the original system however if the perturbation is relatively large direct use of rqi will not be sufficient and in many cases will give inaccurate results such as missing some of the eigenvalues the homotopy method can be used to remedy this problem in this paper we first review the homotopy method and its theoretical background the approach employed here is based on perturbation theory we then discuss some algorithmic issues such as step size estimation and grouping clustered eigenvalues numerical examples are given to illustrate the potential applications of this method

optimal structural design under multiple eigenvalue constraints abstract with increasing complexity structures that are optimally designed subject to an eigenvalue constraint buckling load fundamental frequency etc are likely to display multiple coincident eigenvalues for example optimal statically determinate and simply indeterminate columns buckle into a single mode whereas fixed fixed columns may exhibit a dual buckling mode this phenomenon has been observed first by olhoff and rasmussen 1 even if the optimal design is in the interior of the admissible design space multiple eigenvalue solutions are not stationary but singular the nature of the singularity is the main topic of this investigation necessary and sufficient conditions for local and global optimality are explored and explicit optimality criteria are established for a double mode solution in that case the space of all admissible design changes is split into a two dimensional subspace in which the dual eigenvalues separate and a complementary subspace in which they remain coincident it is within this latter subspace that sequential approximations must take place the criteria developed here are applied first to a two degree of freedom system in addition an exact analytical solution is established for the fixed fixed column problem the accuracy of the numerical solution of this problem in 1 had been challenged but is now confirmed the paper ends with a discussion of the ring buckling problem the prismatic design is shown to be optimal without being stationary however unlike the cases discussed previously the optimal eigenvalue for the ring corresponds to a continuous spectrum of buckling modes

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

a transfer function approach to the problems of discrete time systems h sub infinity optimal linear control and filtering a solution is derived to the problems of h sub infinity optimal linear state regulation and filtering the solution method for both problems is based on a transfer function approach which applies standard spectral factorization return difference relations are given which are extensions of the relations associated with the linear quadratic problem

wavelet shrinkage and generalized cross validation for image denoising we present a denoising method based on wavelets and generalized cross validation and apply these methods to image denoising we describe the method of modified wavelet reconstruction and show that the related shrinkage parameter vector can be chosen without prior knowledge of the noise variance by using the method of generalized cross validation by doing so we obtain an estimate of the shrinkage parameter vector and hence the image which is very close to the best achievable mean squared error result that given by complete knowledge of the underlying clean image

bayesian tree structured image modeling using wavelet domain hidden markov models wavelet domain hidden markov models have proven to be useful tools for statistical signal and image processing the hidden markov tree hmt model captures the key features of the joint probability density of the wavelet coefficients of real world data one potential drawback to the hmt framework is the need for computationally expensive iterative training to fit an hmt model to a given data set e g using the expectation maximization algorithm we greatly simplify the hmt model by exploiting the inherent self similarity of real world images the simplified model specifies the hmt parameters with just nine meta parameters independent of the size of the image and the number of wavelet scales we also introduce a bayesian universal hmt uhmt that fixes these nine parameters the uhmt requires no training of any kind while extremely simple we show using a series of image estimation denoising experiments that these new models retain nearly all of the key image structure modeled by the full hmt finally we propose a fast shift invariant hmt estimation algorithm that outperforms other wavelet based estimators in the current literature both visually and in mean square error

building robust wavelet estimators for multicomponent images using stein s principle multichannel imaging systems provide several observations of the same scene which are often corrupted by noise in this paper we are interested in multispectral image denoising in the wavelet domain we adopt a multivariate statistical approach in order to exploit the correlations existing between the different spectral components our main contribution is the application of stein s principle to build a new estimator for arbitrary multichannel images embedded in additive gaussian noise simulation tests carried out on optical satellite images show that the proposed method outperforms conventional wavelet shrinkage techniques

optimal denoising in redundant representations image denoising methods are often designed to minimize mean squared error mse within the subbands of a multiscale decomposition however most high quality denoising results have been obtained with overcomplete representations for which minimization of mse in the subband domain does not guarantee optimal mse performance in the image domain we prove that despite this suboptimality the expected image domain mse resulting from applying estimators to subbands that are made redundant through spatial replication of basis functions e g cycle spinning is always less than or equal to that resulting from applying the same estimators to the original nonredundant representation in addition we show that it is possible to further exploit overcompleteness by jointly optimizing the subband estimators for image domain mse we develop an extended version of stein s unbiased risk estimate sure that allows us to perform this optimization adaptively for each observed noisy image we demonstrate this methodology using a new class of estimator formed from linear combinations of localized ldquobumprdquo functions that are applied either pointwise or on local neighborhoods of subband coefficients we show through simulations that the performance of these estimators applied to overcomplete subbands and optimized for image domain mse is substantially better than that obtained when they are optimized within each subband this performance is in turn substantially better than that obtained when they are optimized for use on a nonredundant representation

nonlocal means with dimensionality reduction and sure based parameter selection nonlocal means nlm is an effective denoising method that applies adaptive averaging based on similarity between neighborhoods in the image an attractive way to both improve and speed up nlm is by first performing a linear projection of the neighborhood one particular example is to use principal components analysis pca to perform dimensionality reduction here we derive stein s unbiased risk estimate sure for nlm with linear projection of the neighborhoods the sure can then be used to optimize the parameters by a search algorithm or we can consider a linear expansion of multiple nlms each with a fixed parameter set for which the optimal weights can be found by solving a linear system of equations the experimental results demonstrate the accuracy of the sure and its successful application to tune the parameters for nlm

on perturbations of matrix pencils with real spectra ii a well known result on spectral variation of a hermitian matrix due to mirsky is the following let a and a be two n x n hermitian matrices and let 1 n and 1 n be their eigenvalues arranged in ascending order then diag 1 1 n n a a for any unitarily invariant norm in this paper we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra the much studied case of definite pencils is included in this

eigenvalue problems of linear hamiltonian systems arising from h filtering abstract from the viewpoint of differential eigenvalue problem of hamiltonian system a linear finite time h filtering problem can be addressed by computing eigenvalue and solving riccati differential equation of an associated linear hamiltonian system this paper shows how a problem of determining the minimum induced norm of the h filter is formulated as a hamiltonian differential eigenvalue problem the h filters concerned here include central filter perturbed filter and decentralised filter the methods presented in the paper are based on characteristics of eigensolution of the corresponding hamiltonian system and riccati differential equation with eigensolution of the hamiltonian system arising from the central h filtering problem variational methods are proposed to compute eigenvalues of perturbed hamiltonian systems and large scale hamiltonian systems derived from perturbed h filters and decentralised h filters respectively then eigenvalues can be obtained by calculating stationary values of corresponding extended rayleigh s quotient with dual argument functions which is essentially different from the well known rayleigh s quotient of lagrange systems with only one independent argument function numerical examples are also presented to illustrate the variational approaches presented in this paper

function estimation via wavelet shrinkage for long memory data in this article we study function estimation via wavelet shrinkage for data with long range dependence we propose a fractional gaussian noise model to approximate nonparametric regression with long range dependence and establish asymptotics for minimax risks because of long range dependence the minimax risk and the minimax linear risk converge to 0 at rates that differ from those for data with independence or short range dependence wavelet estimates with best selection of resolution leveldependent threshold achieve minimax rates over a wide range of spaces cross validation for dependent data is proposed to select the optimal threshold the wavelet estimates significantly outperform linear estimates the key to proving the asymptotic results is a wavelet vaguelette decomposition which decorrelates fractional gaussian noise such wavelet vaguelette decomposition is also very useful in fractal signal processing

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

wavelet thresholding via a bayesian approach we discuss a bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression a prior distribution is imposed on the wavelet coefficients of the unknown response function designed to capture the sparseness of wavelet expansion that is common to most applications for the prior specified the posterior median yields a thresholding procedure our prior model for the underlying function can be adjusted to give functions falling in any specific besov space we establish a relationship between the hyperparameters of the prior model and the parameters of those besov spaces within which realizations from the prior will fall such a relationship gives insight into the meaning of the besov space parameters moreover the relationship established makes it possible in principle to incorporate prior knowledge about the function s regularity properties into the prior model for its wavelet coefficients however prior knowledge about a function s regularity properties might be difficult to elicit with this in mind we propose a standard choice of prior hyperparameters that works well in our examples several simulated examples are used to illustrate our method and comparisons are made with other thresholding methods we also present an application to a data set that was collected in an anaesthesiological study

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