spatially adaptive wavelet denoising using the minimum description length principle this paper presents a new spatially adaptive wavelet denoising method based on a doubly stochastic process model of wavelet coefficients the method gives a new threshold which varies spatially according to the variances of the coefficients using the minimum description length mdl principle the new threshold is not only easier to analyze since it is in a closed form but also provides more facility for future compression than several other methods almost without deteriorating mean square error risk

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

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

complexity analysis of hypergeometric orthogonal polynomials the complexity measures of the cramer rao fisher shannon and lmc lopez ruiz mancini and calvet types of the rakhmanov probability density n x x p n 2 x of the polynomials p n x orthogonal with respect to the weight function x x a b are used to quantify various two fold facets of the spreading of the hermite laguerre and jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways their explicit cramer rao and asymptotical fisher shannon lmc values are given for the three systems of orthogonal polynomials then these complexity type mathematical quantities are numerically examined in terms of the polynomial s degree n and the parameters which characterize the weight function finally several open problems about the generalized hypergeometric functions of lauricella and srivastava daoust types as well as on the asymptotics of weighted l q norms of laguerre and jacobi polynomials are pointed out

two fast nearest neighbor searching algorithms for image vector quantization in this paper two efficient codebook searching algorithms for vector quantization vq are presented the first fast search algorithm utilizes the compactness property of signal energy on transform domain and the geometrical relations between the input vector and every codevector to eliminate those codevectors that have no chance to be the closest codeword of the input vector it achieves a full search equivalent performance as compared with other fast methods of the same kind this algorithm requires the fewest multiplications and the least total times of distortion measurements then a suboptimal searching method which sacrifices the reconstructed signal quality to speed up the search of nearest neighbor is presented this algorithm performs the search process on predefined small subcodebooks instead of the whole codebook for the closest codevector experimental results show that this method not only needs less cpu time to encode an image but also encounters less loss of reconstructed signal quality than tree structured vq does

nonlinear elliptic systems with general growth we prove local lipschitz continuity and as a consequence c k and c regularity of weak solutions u for a class of nonlinear elliptic differential systems of the formn 1 xi a i du 0 1 2 m the growth conditions on the dependence of functions a i on the gradient du are so mild to allow us to embrace growths between the linear and the exponential cases and they are more general than those known in the literature

understanding waveshrink variance and bias estimation summary donoho johnstone s waveshrink procedure has proved valuable for function estimation and nonparametric regression waveshrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise waveshrink has very broad asymptotic near optimality properties and achieves the optimal risk to within a factor of log n in this paper we derive computationally efficient formulae for computing the exact bias variance and l2 risk of waveshrink estimates in finite sample situations we use these formulae to understand the behaviour of waveshrink estimators construct approximate confidence intervals and bias estimates for waveshrink and compute ideal thresholds for a given function we show that hard shrinkage has smaller bias but larger variance than soft shrinkage and that significantly smaller thresholds should be used for soft shrinkage we also compute minimax thresholds for waveshrink estimators and demonstrate that the minimax thresholds can nearly achieve the ideal rank for a range of functions

pattern generation using likelihood inference for cellular automata cellular automata are discrete dynamical systems which evolve on a discrete grid recent studies have shown that cellular automata with relatively simple rules can produce highly complex patterns we develop likelihood based methods for estimating rules of cellular automata aimed at the re generation of observed regular patterns under noisy data our approach is equivalent to estimating the local map of a stochastic cellular automaton direct computations of the maximum likelihood estimates are possible for regular binary patterns the likelihood formulation of the problem is congenial with the use of the minimum description length principle as a model selection tool we illustrate our method with a series of examples using binary images

regularity for minimizers of integrals with nonstandard growth abstract we deal with variational integrals f x d u x d x and we consider a minimizer u r n r among all functions that agree on the boundary with some fixed boundary value u we assume that the boundary datum u makes the density f x d u x more integrable and we prove that the minimizer u enjoy higher integrability

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

everywhere regularity for a class of vectorial functionals under subquadratic general growth conditions we consider the integral functional of the calculus of variations f du dx where f rnn r satisfies f z g z and g is an n function with subquadratic p q growth we prove that minimizers u rn rn of such a functional are locally lipschitz continuous provided g verifies some additional conditions

existence of parabolic minimizers on metric measure spaces abstract the objects of our studies are vector valued parabolic minimizers u associated to a convex caratheodory integrand f obeying a p growth assumption from below and a certain monotonicity condition in the gradient variable here the functions being considered are defined on a metric measure space x d for such parabolic minimizers that coincide with a time independent cauchy dirichlet datum u 0 on the parabolic boundary of a space time cylinder 0 t with an open subset x and t 0 we prove existence in the parabolic newtonian space l p 0 t n 1 p r n in this paper we generalize results from bogelein et al 2014 2015 to the metric setting and argue completely on a variational level

multiple wavelet threshold estimation by generalized cross validation for images with correlated noise denoising algorithms based on wavelet thresholding replace small wavelet coefficients by zero and keep or shrink the coefficients with absolute value above the threshold the optimal threshold minimizes the error of the result as compared to the unknown exact data to estimate this optimal threshold we use generalized cross validation this procedure does not require an estimation for the noise energy originally this method assumes uncorrelated noise in this paper we describe how we can extend it to images with correlated noise

morrey regularity results for asymptotically convex variational problems with p q growth abstract we prove some global morrey regularity results for almost minimizers of functionals of the form u f x u d x this regularity is valid up to the boundary provided the boundary data is sufficiently regular the main assumption on f is that for each x the function f x behaves asymptotically like a convex function with p q growth some discontinuous behavior in the first argument is allowed as a main application we establish analogous regularity results for a broad class of systems of nonhomogeneous partial differential equations

regularity for parabolic systems of uhlenbeck type with orlicz growth abstract we study the local regularity of p caloric functions or more generally of caloric functions in particular we study local solutions of non linear parabolic systems with homogeneous right hand side where the leading terms have uhlenbeck structure of orlicz type this paper closes the gap of 23 where liebermann proved that if the gradient of a solution is bounded it is holder continuous the crucial step is a novel local estimates for the gradient of the solutions which generalize and improve the pioneering estimates of dibenedetto and friedman 12 10 for the p laplace heat equation

pointwise bounds for minimizers of some anisotropic functionals abstract we consider variational integral functionals g x u x d u x d x where is a bounded open subset in r n and the integrand g x s f x b x s is not subjected to any growth condition from above neither satisfies any regularity assumption growing linearly with respect to the s variable their interest relies on the fact that they are strictly connected with optimal transport problems with congestion effects the aim of this paper is to find sufficient conditions on the boundary datum u in order to obtain global and explicit estimates for the solutions of the following minimization problem min g x u x d u x d x u u w 0 1 1

existence of evolutionary variational solutions via the calculus of variations abstract in this paper we introduce a purely variational approach to time dependent problems yielding the existence of global parabolic minimizers that is 0 t u t f x d u d x d t 0 t f x d u d d x d t whenever t 0 and c 0 0 t r n for the integrand f r n n 0 we merely assume convexity with respect to the gradient variable and coercivity these evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals in the simplest situation with some growth conditions on f the method provides the existence of global weak solutions to cauchy dirichlet problems of parabolic systems of the type t u div d f x d u 0 in 0

wavelet thresholding via mdl for natural images we study the application of rissanen s 1989 principle of minimum description length mdl to the problem of wavelet denoising and compression for natural images after making a connection between thresholding and model selection we derive an mdl criterion based on a laplacian model for noiseless wavelet coefficients we find that this approach leads to an adaptive thresholding rule while achieving mean squared error performance comparable with other popular thresholding schemes the mdl procedure tends to keep far fewer coefficients from this property we demonstrate that our method is an excellent tool for simultaneous denoising and compression we make this claim precise by analyzing mdl thresholding in two optimality frameworks one in which we measure rate and distortion based on quantized coefficients and one in which we do not quantize but instead record rate simply as the number of nonzero coefficients

splitting type variational problems with mixed linear superlinear growth conditions abstract splitting type variational problems with mixed linear superlinear growth conditions are considered in the two dimensional case the minimization problem is given by j w f 1 1 w f 2 2 w d x min with respect to a suitable class of comparison functions here f 1 is a convex energy density with linear growth and f 2 is a convex energy density with superlinear growth for instance given by an n function or just bounded from below by an n function one motivation for this kind of problem located between the well known splitting type problems of superlinear growth and the splitting type problems of linear growth recently considered by m bildhauer and m fuchs 2020 8 is the link to mathematical problems in plasticity see r temam 2018 29 here we prove results in the appropriate way of relaxation including approximation procedures duality and existence and uniqueness of solutions as well as some new higher integrability results

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

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

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

a variational approach to parabolic equations under general and p q growth conditions abstract we consider variational solutions to the cauchy dirichlet problem t u div d f x u d u d u f x u d u in t u u 0 on par t where the function f f x u f r n r n r n n 0 is convex with respect to u and coercive in r n n but it not necessarily satisfies a growth condition from above a motivation to consider a class of such energy functions f can be also easily found in the stationary case where a large literature in the calculus of variations is devoted to the minimization of p q growth problems 45 and to double phase problems 23 24 4 5 6 in the parabolic context the notion of variational solution see the references from 8 to 15 is compatible with the lack of the same polynomial growth from below and from above

minimum description length induction bayesianism and kolmogorov complexity the relationship between the bayesian approach and the minimum description length approach is established we sharpen and clarify the general modeling principles minimum description length mdl and minimum message length mml abstracted as the ideal mdl principle and defined from bayes s rule by means of kolmogorov complexity the basic condition under which the ideal principle should be applied is encapsulated as the fundamental inequality which in broad terms states that the principle is valid when the data are random relative to every contemplated hypothesis and also these hypotheses are random relative to the universal prior the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized if we restrict the model class to finite sets then application of the ideal principle turns into kolmogorov s minimal sufficient statistic in general we show that data compression is almost always the best strategy both in model selection and prediction

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

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

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

regularity of minimizers of vectorial integrals with p q growth local lipschitz continuity of local minimizers of vectorial integrals f x du x dx is proved when f satisfies p q growth condition and f x is convex the uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

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

parabolic systems with p q growth a variational approach we consider the evolution problem associated with a convex integrand f mathbb r nn to 0 infty satisfying a non standard p q growth assumption to establish the existence of solutions we introduce the concept of variational solutions in contrast to weak solutions that is mappings u colon omega t to mathbb r n which solve partial tu rm div df du 0 weakly in omega t variational solutions exist under a much weaker assumption on the gap q p here we prove the existence of variational solutions provided the integrand f is strictly convex and frac 2n n 2 p le q p 1 these variational solutions turn out to be unique under certain mild additional assumptions on the data moreover if the gap satisfies the natural stronger assumption 2 le p le q p rm min big 1 frac 4 n big we show that variational solutions are actually weak solutions this means that solutions u admit the necessary higher integrability of the spatial derivative du to satisfy the parabolic system in the weak sense that is we prove that u in l q rm loc big 0 t w 1 q rm loc omega mathbb r n big

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

minimax description length for signal denoising and optimized representation approaches to wavelet based denoising or signal enhancement have generally relied on the assumption of normally distributed perturbations to relax this assumption which is often violated in practice we derive a robust wavelet thresholding technique based on the minimax description length mmdl principle we first determine the least favorable distribution in the spl epsiv contaminated normal family as the member that maximizes the entropy we show that this distribution and the best estimate based upon it namely the maximum likelihood estimate together constitute a saddle point the mmdl approach results in a thresholding scheme that is resistant to heavy tailed noise we further extend this framework and propose a novel approach to selecting an adapted or best basis bb that results in optimal signal reconstruction finally we address the practical case where the underlying signal is known to be bounded and derive a two sided thresholding technique that is resistant to outliers and has bounded error

a general regularity theorem for functionals with φ growth abstract we prove c 1 regularity for local minimizers of functionals u d x where is a young function in order to generalize the results of fuchs 2011 9 and diening et al 2009 6 we assume for the function only the 2 condition and t t t for a positive constant and of course a holder condition for the second derivatives