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

image denoising in steerable pyramid domain based on a local laplace prior this paper presents a new image denoising algorithm based on the modeling of coefficients in each subband of steerable pyramid employing a laplacian probability density function pdf with local variance this pdf is able to model the heavy tailed nature of steerable pyramid coefficients and the empirically observed correlation between the coefficient amplitudes within this framework we describe a novel method for image denoising based on designing both maximum a posteriori map and minimum mean squared error mmse estimators which relies on the zero mean laplacian random variables with high local correlation despite the simplicity of our spatially adaptive denoising method both in its concern and implementation our denoising results achieves better performance than several published methods such as bayes least squared gaussian scale mixture bls gsm technique that is a state of the art denoising technique

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

regularization in tomographic reconstruction using thresholding estimators in tomographic medical devices such as single photon emission computed tomography or positron emission tomography cameras image reconstruction is an unstable inverse problem due to the presence of additive noise a new family of regularization methods for reconstruction based on a thresholding procedure in wavelet and wavelet packet wp decompositions is studied this approach is based on the fact that the decompositions provide a near diagonalization of the inverse radon transform and of prior information in medical images a wp decomposition is adaptively chosen for the specific image to be restored corresponding algorithms have been developed for both two dimensional and full three dimensional reconstruction these procedures are fast noniterative and flexible numerical results suggest that they outperform filtered back projection and iterative procedures such as ordered subset expectation maximization

the contourlet transform an efficient directional multiresolution image representation the limitations of commonly used separable extensions of one dimensional transforms such as the fourier and wavelet transforms in capturing the geometry of image edges are well known in this paper we pursue a true two dimensional transform that can capture the intrinsic geometrical structure that is key in visual information the main challenge in exploring geometry in images comes from the discrete nature of the data thus unlike other approaches such as curvelets that first develop a transform in the continuous domain and then discretize for sampled data our approach starts with a discrete domain construction and then studies its convergence to an expansion in the continuous domain specifically we construct a discrete domain multiresolution and multidirection expansion using nonseparable filter banks in much the same way that wavelets were derived from filter banks this construction results in a flexible multiresolution local and directional image expansion using contour segments and thus it is named the contourlet transform the discrete contourlet transform has a fast iterated filter bank algorithm that requires an order n operations for n pixel images furthermore we establish a precise link between the developed filter bank and the associated continuous domain contourlet expansion via a directional multiresolution analysis framework we show that with parabolic scaling and sufficient directional vanishing moments contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves finally we show some numerical experiments demonstrating the potential of contourlets in several image processing applications

pointwise shape adaptive dct for high quality denoising and deblocking of grayscale and color images the shape adaptive discrete cosine transform sa dct transform can be computed on a support of arbitrary shape but retains a computational complexity comparable to that of the usual separable block dct b dct despite the near optimal decorrelation and energy compaction properties application of the sa dct has been rather limited targeted nearly exclusively to video compression in this paper we present a novel approach to image filtering based on the sa dct we use the sa dct in conjunction with the anisotropic local polynomial approximation intersection of confidence intervals technique which defines the shape of the transform s support in a pointwise adaptive manner the thresholded or attenuated sa dct coefficients are used to reconstruct a local estimate of the signal within the adaptive shape support since supports corresponding to different points are in general overlapping the local estimates are averaged together using adaptive weights that depend on the region s statistics this approach can be used for various image processing tasks in this paper we consider in particular image denoising and image deblocking and deringing from block dct compression a special structural constraint in luminance chrominance space is also proposed to enable an accurate filtering of color images simulation experiments show a state of the art quality of the final estimate both in terms of objective criteria and visual appearance thanks to the adaptive support reconstructed edges are clean and no unpleasant ringing artifacts are introduced by the fitted transform

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

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

experimental evaluation of variance reduction techniques for queueing simulation using generalized concomitant variables in a companion paper we developed a unified scheme for using poststratified sampling and control variables to improve the efficiency of regenerative queueing simulations we adapted these variance reduction techniques to the estimation methods of replication analysis and regenerative analysis by exploiting the asymptotic joint normality of certain standardized concomitant variables that are defined on each input process sampled during a queueing simulation in this paper we present an experimental evaluation of the reductions in point estimator variance and confidence interval width that can be achieved with each procedure in several closed and mixed machine repair systems for the analytically tractable model nominal and actual confidence interval coverage probabilities are also compared poststratification generally produced variance reductions ranging from 10 to 40 and confidence interval reductions between 1 and 20 the control variates schemes yielded variance reductions ranging from 20 to 90 and confidence interval reductions between 10 and 70 in some instances where small sample sizes were used the poststratification schemes produced confidence interval width increases between 1 and 10 with a small number of regenerative cycles some loss of confidence interval coverage was observed with both post stratified and controlled regenerative analysis when larger sample sizes were used all of the estimation schemes yielded fairly consistent efficiency gains

the curvelet transform for image denoising we describe approximate digital implementations of two new mathematical transforms namely the ridgelet transform and the curvelet transform our implementations offer exact reconstruction stability against perturbations ease of implementation and low computational complexity a central tool is fourier domain computation of an approximate digital radon transform we introduce a very simple interpolation in the fourier space which takes cartesian samples and yields samples on a rectopolar grid which is a pseudo polar sampling set based on a concentric squares geometry despite the crudeness of our interpolation the visual performance is surprisingly good our ridgelet transform applies to the radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain our curvelet transform uses our ridgelet transform as a component step and implements curvelet subbands using a filter bank of a spl grave trous wavelet filters our philosophy throughout is that transforms should be overcomplete rather than critically sampled we apply these digital transforms to the denoising of some standard images embedded in white noise in the tests reported here simple thresholding of the curvelet coefficients is very competitive with state of the art techniques based on wavelets including thresholding of decimated or undecimated wavelet transforms and also including tree based bayesian posterior mean methods moreover the curvelet reconstructions exhibit higher perceptual quality than wavelet based reconstructions offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curvilinear features existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems the empirical results reported here are in encouraging agreement

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

signal analysis using a multiresolution form of the singular value decomposition this paper proposes a multiresolution form of the singular value decomposition svd and shows how it may be used for signal analysis and approximation it is well known that the svd has optimal decorrelation and subrank approximation properties the multiresolution form of svd proposed here retains those properties and moreover has linear computational complexity by using the multiresolution svd the following important characteristics of a signal may he measured at each of several levels of resolution isotropy sphericity of principal components self similarity under scaling and resolution of mean squared error into meaningful components theoretical calculations are provided for simple statistical models to show what might be expected results are provided with real images to show the usefulness of the svd decomposition

optimal spatial adaptation for patch based image denoising a novel adaptive and patch based approach is proposed for image denoising and representation the method is based on a pointwise selection of small image patches of fixed size in the variable neighborhood of each pixel our contribution is to associate with each pixel the weighted sum of data points within an adaptive neighborhood in a manner that it balances the accuracy of approximation and the stochastic error at each spatial position this method is general and can be applied under the assumption that there exists repetitive patterns in a local neighborhood of a point by introducing spatial adaptivity we extend the work earlier described by buades et al which can be considered as an extension of bilateral filtering to image patches finally we propose a nearly parameter free algorithm for image denoising the method is applied to both artificially corrupted white gaussian noise and real images and the performance is very close to and in some cases even surpasses that of the already published denoising methods

enhancing synchronization by directionality in complex networks we propose a method called the residual edge betweenness gradient rebg to enhance the synchronizability of networks by assigning the link direction while keeping the topology and link weights unchanged direction assignment has been shown to improve the synchronizability of undirected networks in general but we find that in some cases incommunicable components emerge and networks fail to synchronize we show that the rebg method improves the residual degree gradient rdg method by effectively avoiding the synchronization failure further experiments show that the rebg method enhances the synchronizability in networks with a community structure compared with the rdg method

image restoration using space variant gaussian scale mixtures in overcomplete pyramids in recent years bayes least squares gaussian scale mixtures bls gsm has emerged as one of the most powerful methods for image restoration its strength relies on providing a simple and yet very effective local statistical description of oriented pyramid coefficient neighborhoods via a gsm vector this can be viewed as a fine adaptation of the model to the signal variance at each scale orientation and spatial location here we present an enhancement of the model by introducing a coarser adaptation level where a larger neighborhood is used to estimate the local signal covariance within every subband we formulate our model as a bls estimator using space variant gsm the model can be also applied to image deconvolution by first performing a global blur compensation and then doing local adaptive denoising we demonstrate through simulations that the proposed method besides being model based and noniterative it is also robust and efficient its performance measured visually and in l2 norm terms is significantly higher than the original bls gsm method both for denoising and deconvolution

penalized weighted least squares approach to sinogram noise reduction and image reconstruction for low dose x ray computed tomography reconstructing low dose x ray computed tomography ct images is a noise problem this work investigated a penalized weighted least squares pwls approach to address this problem in two dimensions where the wls considers first and second order noise moments and the penalty models signal spatial correlations three different implementations were studied for the pwls minimization one utilizes a markov random field mrf gibbs functional to consider spatial correlations among nearby detector bins and projection views in sinogram space and minimizes the pwls cost function by iterative gauss seidel algorithm another employs karhunen loeve kl transform to de correlate data signals among nearby views and minimizes the pwls adaptively to each kl component by analytical calculation where the spatial correlation among nearby bins is modeled by the same gibbs functional the third one models the spatial correlations among image pixels in image domain also by a mrf gibbs functional and minimizes the pwls by iterative successive over relaxation algorithm in these three implementations a quadratic functional regularization was chosen for the mrf model phantom experiments showed a comparable performance of these three pwls based methods in terms of suppressing noise induced streak artifacts and preserving resolution in the reconstructed images computer simulations concurred with the phantom experiments in terms of noise resolution tradeoff and detectability in low contrast environment the kl pwls implementation may have the advantage in terms of computation for high resolution dynamic low dose ct imaging

image denoising with neighbour dependency and customized wavelet and threshold image denoising by means of wavelet transforms has been an active research topic for many years for a given noisy image which kind of wavelet and what threshold we use should have significant impact on the quality of the denoised image in this paper we use simulated annealing to find the customized wavelet filters and the customized threshold corresponding to the given noisy image at the same time also we propose to consider a small neighbourhood around the customized wavelet coefficient to be thresholded for image denoising experimental results show that our approach is better than visushrink our neighshrink with fixed wavelet and the wiener2 filter that is available in matlab image processing toolbox in addition our neighshrink with fixed wavelet already outperforms visushrink for all the experiments

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

image denoising by sparse 3 d transform domain collaborative filtering we propose a novel image denoising strategy based on an enhanced sparse representation in transform domain the enhancement of the sparsity is achieved by grouping similar 2d image fragments e g blocks into 3d data arrays which we call groups collaborative altering is a special procedure developed to deal with these 3d groups we realize it using the three successive steps 3d transformation of a group shrinkage of the transform spectrum and inverse 3d transformation the result is a 3d estimate that consists of the jointly filtered grouped image blocks by attenuating the noise the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block the filtered blocks are then returned to their original positions because these blocks are overlapping for each pixel we obtain many different estimates which need to be combined aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy a significant improvement is obtained by a specially developed collaborative wiener filtering an algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail an extension to color image denoising is also developed the experimental results demonstrate that this computationally scalable algorithm achieves state of the art denoising performance in terms of both peak signal to noise ratio and subjective visual quality

shiftable multiscale transforms one of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance the content of wavelet subbands is unstable under translations of the input signal wavelet transforms are also unstable with respect to dilations of the input signal and in two dimensions rotations of the input signal the authors formalize these problems by defining a type of translation invariance called shiftability in the spatial domain shiftability corresponds to a lack of aliasing thus the conditions under which the property holds are specified by the sampling theorem shiftability may also be applied in the context of other domains particularly orientation and scale jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored two examples of jointly shiftable transforms are designed and implemented a 1 d transform that is jointly shiftable in position and scale and a 2 d transform that is jointly shiftable in position and orientation the usefulness of these image representations for scale space analysis stereo disparity measurement and image enhancement is demonstrated

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

image compression via joint statistical characterization in the wavelet domain we develop a probability model for natural images based on empirical observation of their statistics in the wavelet transform domain pairs of wavelet coefficients corresponding to basis functions at adjacent spatial locations orientations and scales are found to be non gaussian in both their marginal and joint statistical properties specifically their marginals are heavy tailed and although they are typically decorrelated their magnitudes are highly correlated we propose a markov model that explains these dependencies using a linear predictor for magnitude coupled with both multiplicative and additive uncertainties and show that it accounts for the statistics of a wide variety of images including photographic images graphical images and medical images in order to directly demonstrate the power of the model we construct an image coder called epwic embedded predictive wavelet image coder in which subband coefficients are encoded one bitplane at a time using a nonadaptive arithmetic encoder that utilizes conditional probabilities calculated from the model bitplanes are ordered using a greedy algorithm that considers the mse reduction per encoded bit the decoder uses the statistical model to predict coefficient values based on the bits it has received despite the simplicity of the model the rate distortion performance of the coder is roughly comparable to the best image coders in the literature

analytical form for a bayesian wavelet estimator of images using the bessel k form densities a novel bayesian nonparametric estimator in the wavelet domain is presented in this approach a prior model is imposed on the wavelet coefficients designed to capture the sparseness of the wavelet expansion seeking probability models for the marginal densities of the wavelet coefficients the new family of bessel k forms bkf densities are shown to fit very well to the observed histograms exploiting this prior we designed a bayesian nonlinear denoiser and we derived a closed form for its expression we then compared it to other priors that have been introduced in the literature such as the generalized gaussian density ggd or the spl alpha stable models where no analytical form is available for the corresponding bayesian denoisers specifically the bkf model turns out to be a good compromise between these two extreme cases hyperbolic tails for the spl alpha stable and exponential tails for the ggd moreover we demonstrate a high degree of match between observed and estimated prior densities using the bkf model finally a comparative study is carried out to show the effectiveness of our denoiser which clearly outperforms the classical shrinkage or thresholding wavelet based techniques

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