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

tree approximations to markov random fields methods for approximately computing the marginal probability mass functions and means of a markov random field mrf by approximating the lattice by a tree are described applied to the a posteriori mrf these methods solve bayesian spatial pattern classification and image restoration problems the methods are described several theoretical results concerning fixed point problems are proven and four numerical examples are presented including comparison with optimal estimators and the iterated conditional mode estimator and including two agricultural optical remote sensing problems

statistical wavelet subband characterization based on generalized gamma density and its application in texture retrieval the modeling of image data by a general parametric family of statistical distributions plays an important role in many applications in this paper we propose to adopt the three parameter generalized gamma density g d for modeling wavelet detail subband histograms and for texture image retrieval the advantage of g d over the existing generalized gaussian density ggd is that it provides more flexibility to control the shape of model which is critical for practical histogram based applications to measure the discrepancy between g ds we use the symmetrized kullback leibler distance skld and derive a closed form for the skld between g ds such a distance can be computed directly and effectively via the model parameters making our proposed scheme particularly suitable for image retrieval systems with large image database experimental results on the well known databases reveal the superior performance of our proposed method compared with the current existing approaches

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

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

computational methods for hidden markov tree models an application to wavelet trees the hidden markov tree models were introduced by crouse et al in 1998 for modeling nonindependent non gaussian wavelet transform coefficients in their paper they developed the equivalent of the forward backward algorithm for hidden markov tree models and called it the upward downward algorithm this algorithm is subject to the same numerical limitations as the forward backward algorithm for hidden markov chains hmcs in this paper adapting the ideas of devijver from 1985 we propose a new upward downward algorithm which is a true smoothing algorithm and is immune to numerical underflow furthermore we propose a viterbi like algorithm for global restoration of the hidden state tree the contribution of those algorithms as diagnosis tools is illustrated through the modeling of statistical dependencies between wavelet coefficients with a special emphasis on local regularity changes

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

regularity for a local nonlocal transmission problem we formulate and study an elliptic transmission like problem combining local and nonlocal elements let mathbb r n be separated into two components by a smooth hypersurface on one side of a function satisfies a local second order elliptic equation on the other it satisfies a nonlocal one of lower order in addition a nonlocal transmission condition is imposed across the interface which has a natural variational interpretation we deduce the existence of solutions to the corresponding dirichlet problem and show that under mild assumptions they are holder continuous following the method of de giorgi the principal difficulty stems from the lack of scale invariance near which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter we then turn to the question of optimal regularity and qualitative properties of solutions and show that in the case of constant coefficients and flat they satisfy a kind of transmission condition where the ratio of fractional conormal derivatives on the two sides of the interface is fixed a perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface throughout we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution which has an interpretation in the context of quasigeostrophic dynamics

the quadrilateral mini finite element for the stokes problem abstract we propose two mixed finite element methods for the stokes equations in two space dimensions starting from a q 1 q 1 combination for the velocity and pressure it is studied how to introduce internal degrees of freedom into the velocity space in order to obtain a lbb stable combination the lbb stability conditions of the two proposed methods are checked by using the macroelement technique

multiresolution gauss markov random field models for texture segmentation this paper presents multiresolution models for gauss markov random fields gmrfs with applications to texture segmentation coarser resolution sample fields are obtained by subsampling the sample field at fine resolution although the markov property is lost under such resolution transformation coarse resolution non markov random fields can be effectively approximated by markov fields we present two techniques to estimate the gmrf parameters at coarser resolutions from the fine resolution parameters one by minimizing the kullback leibler distance and another based on local conditional distribution invariance we also allude to the fact that different gmrf parameters at the fine resolution can result in the same probability measure after subsampling and present the results for first and second order cases we apply this multiresolution model to texture segmentation different texture regions in an image are modeled by gmrfs and the associated parameters are assumed to be known parameters at lower resolutions are estimated from the fine resolution parameters the coarsest resolution data is first segmented and the segmentation results are propagated upward to the finer resolution we use the iterated conditional mode icm minimization at all resolutions our experiments with synthetic brodatz texture and real satellite images show that the multiresolution technique results in a better segmentation and requires lesser computation than the single resolution algorithm

global optimization of wavelet domain hidden markov tree for image segmentation this work presents a global energy minimization method for multiscale image segmentation using convex optimization theory the construction of energy function is motivated by the intuition that the larger the entropy the less a priori information one has on the value of the random variables first we represent the wavelet domain hidden markov tree whmt model of the original image as a structured energy function which is proved convex in marginal distributions next we derive the maximum lower bound of the energy function through lagrange dual transform for the purpose of incorporating marginal constraints into optimization finally a modified belief propagation optimization algorithm is used to perform global energy minimization of the dual convex energy function experiments on real image segmentation problems demonstrate the superior performance of this new algorithm when compared with nonconvex ones

fuzzy model based clustering and its application in image segmentation a fuzzy generalized gaussian density ggd segmentation model is proposed a ggd based agglomerative fuzzy algorithm is developed a two stage fuzzy ggd segmentation algorithm is proposed this paper proposes a novel fuzzy model based unsupervised learning algorithm with boundary correction for image segmentation we propose a fuzzy generalized gaussian density ggd segmentation model and the ggd based agglomerative fuzzy algorithm for grouping image pixels the merits of algorithm are that it is not sensitive to initial parameters and that the number of groups can be estimated via the validation technique to minimize the objective function of the model we define a dissimilarity measure based on the kullbackleibler divergence of the ggds that computes the discrepancy between ggds in the space of generalized probability distributions to effectively segment images with various textures we propose a two stage fuzzy ggd segmentation algorithm the first stage adopts the proposed fuzzy algorithm to obtain initial segmentation and the second stage improves initial segmentation by image boundary correction experimental results show that our proposed method has a promising performance compared with existing approaches

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

classification of textures using gaussian markov random fields the problem of texture classification arises in several disciplines such as remote sensing computer vision and image analysis in this paper we present two feature extraction methods for the classification of textures using two dimensional 2 d markov random field mrf models it is assumed that the given m m texture is generated by a gaussian mrf model in the first method the least square ls estimates of model parameters are used as features in the second method using the notion of sufficient statistics it is shown that the sample correlations over a symmetric window including the origin are optimal features for classification simple minimum distance classifiers using these two feature sets yield good classification accuracies for a seven class problem

a sequence based extension of mean field annealing using the forward backward algorithm application to image segmentation optimization based on mean field theory including mean field annealing mfa is widely used for discrete optimization label assignment problems defined on the pixel sites of an image one formulation of mfa is via maximum entropy where one seeks the joint distribution over the random assignments subject to an average level of cost mfa is obtained by assuming the assignments at each pixel are statistically independent given the observed image alternatively we make the less restrictive assumption of independent row labelings the independence assumption means that at each step mfa optimizes over only one pixel whereas our method jointly optimizes over an entire row i e our method is less greedy in principle an mfa extension could be developed that explicitly re estimates the row labeling distributions but such an approach is in practice infeasible even so we can indirectly implement this re estimation by re estimating quantities that determine the row labeling distributions these quantities are the a posteriori site probabilities re estimable via the well known forward backward f b algorithm thus our algorithm which descends in the me lagrangian free energy consists of iterative application of f b to the image rows columns at convergence maximum a posteriori site labeling is performed our method was applied to segmentation of both real and synthetic noise corrupted images it achieved lower markov random field model potentials and better segmentations compared with other methods and in high noise standard mfa

multiresolution image classification by hierarchical modeling with two dimensional hidden markov models this paper treats a multiresolution hidden markov model for classifying images each image is represented by feature vectors at several resolutions which are statistically dependent as modeled by the underlying state process a multiscale markov mesh unknowns in the model are estimated by maximum likelihood in particular by employing the expectation maximization algorithm an image is classified by finding the optimal set of states with maximum a posteriori probability states are then mapped into classes the multiresolution model enables multiscale information about context to be incorporated into classification suboptimal algorithms based on the model provide progressive classification that is much faster than the algorithm based on single resolution hidden markov models

a mixed finite element method for non linear and nearly incompressible elasticity based on biorthogonal systems we present a finite element method for non linear and nearly incompressible elasticity the formulation is based on petrov galerkin discretization for the pressure and is closely related to the average nodal pressure formulation presented earlier in the context of incompressible and nearly incompressible dynamic explicit applications commun numer meth engng 1998 14 437 449 some numerical examples are presented to show the efficiency of the approach copyright 2009 john wiley sons ltd

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

local projection stabilization of equal order interpolation applied to the stokes problem the local projection stabilization allows us to circumvent the babuska brezzi condition and to use equal order interpolation for discretizing the stokes problem the projection is usually done in a two level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements we propose a new local projection stabilization method based on possibly enriched finite element spaces and discontinuous projection spaces defined on the same mesh optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf sup condition examples are enriched simplicial finite elements and standard quadrilateral hexahedral elements the new approach overcomes the problem of an increasing discretization stencil and thus is simple to implement in existing computer codes numerical tests confirm the theoretical convergence results which are robust with respect to the user chosen stabilization parameter

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

holder regularity up to the boundary for critical sqg on bounded domains we consider the dissipative sqg equation in bounded domains first introduced by constantin and ignatova in 2016 we show global holder regularity up to the boundary of the solution with a method based on the de giorgi techniques the boundary introduces several difficulties in particular the dirichlet laplacian is not translation invariant near the boundary which leads to complications involving the riesz transform

multiscale image segmentation using wavelet domain hidden markov models we introduce a new image texture segmentation algorithm hmtseg based on wavelets and the hidden markov tree hmt model the hmt is a tree structured probabilistic graph that captures the statistical properties of the coefficients of the wavelet transform since the hmt is particularly well suited to images containing singularities edges and ridges it provides a good classifier for distinguishing between textures utilizing the inherent tree structure of the wavelet hmt and its fast training and likelihood computation algorithms we perform texture classification at a range of different scales we then fuse these multiscale classifications using a bayesian probabilistic graph to obtain reliable final segmentations since hmtseg works on the wavelet transform of the image it can directly segment wavelet compressed images without the need for decompression into the space domain we demonstrate the performance of hmtseg with synthetic aerial photo and document image segmentations

image segmentation using fuzzy region competition and spatial frequency information this paper presents a multiphase fuzzy region competition model that takes into account spatial and frequency information for image segmentation in the proposed energy functional each region is represented by a fuzzy membership function and a data fidelity term that measures the conformity of spatial and frequency data within each region to generalized gaussian densities whose parameters are determined jointly with the segmentation process compared with the classical region competition model our approach gives soft segmentation results via the fuzzy membership functions and moreover the use of frequency data provides additional region information that can improve the overall segmentation result to efficiently solve the minimization of the energy functional we adopt an alternate minimization procedure and make use of chambolle s fast duality projection algorithm we apply the proposed method to synthetic and natural textures as well as real world natural images experimental results show that our proposed method has very promising segmentation performance compared with the current state of the art approaches

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

a mixed finite element method for nearly incompressible elasticity and stokes equations using primal and dual meshes with quadrilateral and hexahedral grids we consider a mixed finite element method for approximating the solution of nearly incompressible elasticity and stokes equations the finite element method is based on quadrilateral and hexahedral triangulation using primal and dual meshes we use the standard bilinear and trilinear finite element space enriched with element wise defined bubble functions with respect to the primal mesh for the displacement or velocity whereas the pressure space is discretized by using a piecewise constant finite element space with respect to the dual mesh

quasi nonlocal coupling of nonlocal diffusions we developed a new self adjoint consistent and stable coupling strategy for nonlocal diffusion models inspired by the quasi nonlocal atomistic to continuum method for crystalline solids the proposed coupling model is coercive with respect to the energy norms induced by the nonlocal diffusion kernels as well as the l 2 norm and it satisfies the maximum principle a finite difference approximation is used to discretize the coupled system which inherits the property from the continuous formulation furthermore we design a numerical example that shows the discrepancy between the fully nonlocal and fully local diffusions whereas the result of the coupled diffusion agrees with that of the fully nonlocal diffusion

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

stable nonconforming quadrilateral finite elements for the stokes problem stability result is obtained for the approximation of the stationary stokes problem with nonconforming elements proposed by douglas et al j douglas jr j e santos d s xiu ye nonconforming galerkin methods based on quadrilateral elements for second order elliptic problems preprints added by conforming bubbles to the velocity and discontinuous piecewise linear to the pressure on quadrilateral elements optimal order error estimates derived

sar image segmentation based on mixture context and wavelet hidden class label markov random field in order to suppress the effect of multiplicative speckle noise on synthetic aperture radar sar image segmentation a new sar image segmentation algorithm is proposed based on the mixture context and the wavelet hidden class label markov random field mrf in our paper a wavelet mixture heavy tailed model is constructed and the hidden class label mrf is extended to the wavelet domain to suppress the effect of speckle noise the multiscale segmentation with overlapping window is presented here to segment the finest scale of stationary wavelet transform swt domain and the classical segmentation method is still utilized at the coarse scales of discrete wavelet transform dwt domain moreover a mixture context model is proposed to combine the two different segmentation methods finally a new maximum a posteriori map classification is obtained the experimental results demonstrate that our segmentation method outperforms several other segmentation methods

wavelet based bayesian image estimation from marginal and bivariate prior models to multivariate prior models prior models play an important role in the wavelet based bayesian image estimation problem although it is well known that a residual dependency structure always remains among natural image wavelet coefficients only few multivariate prior models with a closed parametric form are available in the literature in this paper we develop new multivariate prior models that not only match well with the observed statistics of the wavelet coefficients of natural images but also have a simple parametric form these prior models are very effective for bayesian image estimation and lead to an improved estimation performance over related earlier techniques

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

censored stable processes we present several constructions of a censored stable process in an open set d rn i e a symmetric stable process which is not allowed to jump outside d we address the question of whether the process will approach the boundary of d in a finite time we give sharp conditions for such approach in terms of the stability index and the thickness of the boundary as a corollary new results are obtained concerning besov spaces on non smooth domains including the critical exponent case we also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary we derive a boundary harnack principle in c1 1 open sets