nonlocal discrete regularization on weighted graphs a framework for image and manifold processing we introduce a nonlocal discrete regularization framework on weighted graphs of the arbitrary topologies for image and manifold processing the approach considers the problem as a variational one which consists of minimizing a weighted sum of two energy terms a regularization one that uses a discrete weighted dirichlet energy and an approximation one this is the discrete analogue of recent continuous euclidean nonlocal regularization functionals the proposed formulation leads to a family of simple and fast nonlinear processing methods based on the weighted laplace operator parameterized by the degree of regularity the graph structure and the graph weight function these discrete processing methods provide a graph based version of recently proposed semi local or nonlocal processing methods used in image and mesh processing such as the bilateral filter the tv digital filter or the nonlocal means filter it works with equal ease on regular 2 d and 3 d images manifolds or any data we illustrate the abilities of the approach by applying it to various types of images meshes manifolds and data represented as graphs

nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising part ii adaptive algorithms we combine the main ideas introduced in part i with adaptive techniques to arrive at a powerful algorithm that estimates missing data in nonstationary signals the proposed approach operates automatically based on a chosen linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coefficients over missing regions are zero or close to zero unlike prevalent algorithms our method does not necessitate any complex preconditioning segmentation or edge detection steps and it can be written as a progression of denoising operations we show that constructing estimates based on nonlinear approximants is fundamentally a nonconvex problem and we propose a progressive algorithm that is designed to deal with this issue directly the algorithm is applied to images through an extensive set of simulation examples primarily on missing regions containing textures edges and other image features that are not readily handled by established estimation and recovery methods we discuss the properties required of good transforms and in conjunction show the types of regions over which well known transforms provide good predictors we further discuss extensions of the algorithm where the utilized transforms are also chosen adaptively where unpredictable signal components in the progressions are identified and not predicted and where the prediction scenario is more general

robust compressive sensing of multichannel eeg signals in the presence of impulsive noise abstract latest studies have shown that combining cosparsity and low rank property usually results in efficient compressive sensing cs recovery approaches for multichannel electroencephalogram eeg signals however existing methods rarely consider noise or consider only the influence of gaussian noise generated during transmission when the measurement is corrupted by impulsive noise the performance of these cs approaches will degrade in this study a new robust cs approach is proposed to accurately recover multichannel eeg signals from noisy measurements in the presence of impulsive noise we first employ welsch estimator to depress the influence of impulsive noise in cs reconstruction we further develop an efficient iterative scheme based on half quadratic theory and alternating direction method of multipliers to solve the resulting nonconvex optimization problem experimental results show that our approach can obtain better reconstruction results than those of existing state of the art cs methods when measurements are corrupted by impulsive noise

compressive sensing via nonlocal low rank regularization sparsity has been widely exploited for exact reconstruction of a signal from a small number of random measurements recent advances have suggested that structured or group sparsity often leads to more powerful signal reconstruction techniques in various compressed sensing cs studies in this paper we propose a nonlocal low rank regularization nlr approach toward exploiting structured sparsity and explore its application into cs of both photographic and mri images we also propose the use of a nonconvex log det x as a smooth surrogate function for the rank instead of the convex nuclear norm and justify the benefit of such a strategy using extensive experiments to further improve the computational efficiency of the proposed algorithm we have developed a fast implementation using the alternative direction multiplier method technique experimental results have shown that the proposed nlr cs algorithm can significantly outperform existing state of the art cs techniques for image recovery

solution of the helmholtz eigenvalue problem via the boundary element method the numerical solution of the helmholtz eigenvalue problem is considered the application of the boundary element method reduces it to that of a non linear eigenvalue problem through a polynomial approximation with respect to the wavenumber the non linear eigenvalue problem is reduced to a standard generalized eigenvalue problem the method is applied to the test problems of a three dimensional sphere with an axisymmetric boundary condition and a two dimensional square

model assisted adaptive recovery of compressed sensing with imaging applications in compressive sensing cs a challenge is to find a space in which the signal is sparse and hence faithfully recoverable since many natural signals such as images have locally varying statistics the sparse space varies in time spatial domain as such cs recovery should be conducted in locally adaptive signal dependent spaces to counter the fact that the cs measurements are global and irrespective of signal structures on the contrary existing cs reconstruction methods use a fixed set of bases e g wavelets dct and gradient spaces for the entirety of a signal to rectify this problem we propose a new framework for model guided adaptive recovery of compressive sensing marx and show how a 2 d piecewise autoregressive model can be integrated into the marx framework to make cs recovery adaptive to spatially varying second order statistics of an image in addition marx offers a mechanism of characterizing and exploiting structured sparsities of natural images greatly restricting the cs solution space simulation results over a wide range of natural images show that the proposed marx technique can improve the reconstruction quality of existing cs methods by 2 7 db

on the early history of the singular value decomposition this paper surveys the contributions of five mathematicians eugenio beltrami 1835 1899 camille jordan 1838 1921 james joseph sylvester 1814 1897 erhard schmidt 1876 1959 and hermann weyl 1885 1955 who were responsible for establishing the existence of the singular value decomposition and developing its theory

a douglas rachford splitting approach to compressed sensing image recovery using low rank regularization in this paper we study the compressed sensing cs image recovery problem the traditional method divides the image into blocks and treats each block as an independent sub cs recovery task this often results in losing global structure of an image in order to improve the cs recovery result we propose a nonlocal nl estimation step after the initial cs recovery for denoising purpose the nl estimation is based on the well known nl means filtering that takes an advantage of self similarity in images we formulate the nl estimation as the low rank matrix approximation problem where the low rank matrix is formed by the nl similarity patches an efficient algorithm nonlocal douglas rachford nldr based on douglas rachford splitting is developed to solve this low rank optimization problem constrained by the cs measurements experimental results demonstrate that the proposed nldr algorithm achieves significant performance improvements over the state of the art in cs image recovery

mixed noise removal via laplacian scale mixture modeling and nonlocal low rank approximation recovering the image corrupted by additive white gaussian noise awgn and impulse noise is a challenging problem due to its difficulties in an accurate modeling of the distributions of the mixture noise many efforts have been made to first detect the locations of the impulse noise and then recover the clean image with image in painting techniques from an incomplete image corrupted by awgn however it is quite challenging to accurately detect the locations of the impulse noise when the mixture noise is strong in this paper we propose an effective mixture noise removal method based on laplacian scale mixture lsm modeling and nonlocal low rank regularization the impulse noise is modeled with lsm distributions and both the hidden scale parameters and the impulse noise are jointly estimated to adaptively characterize the real noise to exploit the nonlocal self similarity and low rank nature of natural image a nonlocal low rank regularization is adopted to regularize the denoising process experimental results on synthetic noisy images show that the proposed method outperforms existing mixture noise removal methods

streamflow simulation a nonparametric approach in this paper kernel estimates of the joint and conditional probability density functions are used to generate synthetic streamflow sequences streamflow is assumed to be a markov process with time dependence characterized by a multivariate probability density function kernel methods are used to estimate this multivariate density function simulation proceeds by sequentially resampling from the conditional density function derived from the kernel estimate of the underlying multivariate probability density function this is a nonparametric method for the synthesis of streamflow that is data driven and avoids prior assumptions as to the form of dependence e g linear or nonlinear and the form of the probability density functions e g gaussian we show using synthetic examples with known underlying models that the nonparametric method presented is more flexible than the conventional models used in stochastic hydrology and is capable of reproducing both linear and nonlinear dependence the effectiveness of this model is illustrated through its application to simulation of monthly streamflow from the beaver river in utah

image denoising via sparse and redundant representations over learned dictionaries we address the image denoising problem where zero mean white and homogeneous gaussian additive noise is to be removed from a given image the approach taken is based on sparse and redundant representations over trained dictionaries using the k svd algorithm we obtain a dictionary that describes the image content effectively two training options are considered using the corrupted image itself or training on a corpus of high quality image database since the k svd is limited in handling small image patches we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image we show how such bayesian treatment leads to a simple and effective denoising algorithm this leads to a state of the art denoising performance equivalent and sometimes surpassing recently published leading alternative denoising methods

solution of discontinuous interior helmholtz problems by the boundary and shell element method the helmholtz equation governing an interior domain with shell discontinuities is not efficiently solvable by the traditional boundary element method in this paper it is shown how the helmholtz equation can be recast as an integral equation known as the boundary and shell integral equation the application of collocation to the integral equation gives rise to a method termed the boundary and shell element method the associated problem of finding the eigenvalues and eigenfunctions of the helmholtz equation in a discontinuous domain via the same method is also considered this leads to a non linear eigenvalue problem such a problem may be solved through polynomial interpolation of the matrix components in this paper methods for solving the helmholtz equation and the associated eigenvalue problem are implemented and applied to a test problem

physical basis of stochastic models of annual flows the main objective of this paper is to demonstrate that annual streamflow time series can be represented by autoregressive and moving average arma processes the method follows the conceptual representation of a watershed given by fiering 1967

eigenanalysis for membranes with stringers using conventional bem in conjunction with svd technique it is well known that either the multi domain boundary element method bem or the dual bem can solve boundary value problems with degenerate boundaries where double values are defined i e two portions of boundary meet together with a zero thickness in this paper the eigensolutions for membranes with stringers are obtained in a single domain by using the conventional bem in conjunction with the svd technique by adopting the svd technique for rank revealing the nontrivial boundary mode can be detected by the successive zero singular values which are not due to the degeneracy of degenerate boundary the boundary modes are obtained according to the right unitary vectors with respect to the zero singular values in the svd three examples a single edge stringer a double edge stringer and a central stringer in a circular membrane are considered the results of the present method are compared with those of the multi domain bem the dual bem the dtn method dirichlet to neumann the fem abaqus and analytical solutions if available good agreement is obtained the goal to deal with the eigenproblem in a single domain without hypersingularity is achieved

image inpainting using reproducing kernel hilbert space and heaviside functions image inpainting a technique of repairing damaged images is an important topic in image processing in this paper we solve the problem from an intensity function estimation perspective we assume the underlying image is defined on a continuous domain and belongs to a space spanned by a basis of a reproducing kernel hilbert space and some variations of the heaviside function the reproducing kernel hilbert space is used to model the smooth component of the image while heaviside function variations are used to model the edges the coefficients of the redundant basis are computed by the discrete intensity at undamaged domain we test the proposed model through various images numerical experiments show the effectiveness of the proposed method especially in recovering edges

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

low rank matrix completion in the presence of high coherence prevalent matrix completion methods capture only the low rank property which gives merely a constraint that the data points lie on some low dimensional subspace but generally ignore the extra structures beyond low rank that specify in more detail how the data points lie on the subspace whenever the data points are not uniformly distributed on the low dimensional subspace the row coherence of the target matrix to recover could be considerably high and accordingly prevalent methods might fail even if the target matrix is fairly low rank to relieve this challenge we suggest to consider a model termed low rank factor decomposition lrfd which imposes an additional restriction that the data points must be represented as linear compressive combinations of the bases in a given dictionary we show that lrfd can effectively mitigate the challenges of high row coherence provided that its dictionary is configured properly namely it is mathematically proven that if the dictionary is well conditioned and low rank then lrfd can weaken the dependence on the row coherence in particular if the dictionary itself is low rank then the dependence on the row coherence can be entirely removed subsequently we devise two practical algorithms to obtain proper dictionaries in unsupervised environments one uses the existing matrix completion methods to construct the dictionary in lrfd and the other tries to learn a proper dictionary from the data given experiments on randomly generated matrices and motion datasets show superior performance of our proposed algorithms

image interpolation by adaptive 2 d autoregressive modeling and soft decision estimation the challenge of image interpolation is to preserve spatial details we propose a soft decision interpolation technique that estimates missing pixels in groups rather than one at a time the new technique learns and adapts to varying scene structures using a 2 d piecewise autoregressive model the model parameters are estimated in a moving window in the input low resolution image the pixel structure dictated by the learnt model is enforced by the soft decision estimation process onto a block of pixels including both observed and estimated the result is equivalent to that of a high order adaptive nonseparable 2 d interpolation filter this new image interpolation approach preserves spatial coherence of interpolated images better than the existing methods and it produces the best results so far over a wide range of scenes in both psnr measure and subjective visual quality edges and textures are well preserved and common interpolation artifacts blurring ringing jaggies zippering etc are greatly reduced

segmentation of multivariate mixed data via lossy data coding and compression in this paper based on ideas from lossy data coding and compression we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of gaussian distributions which are allowed to be almost degenerate the goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data subject to a given distortion by analyzing the coding length rate of mixed data we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory we show that a deterministic segmentation is approximately the asymptotically optimal solution for compressing mixed data we propose a very simple and effective algorithm that depends on a single parameter the allowable distortion at any given distortion the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation simulation results reveal intriguing phase transition like behaviors of the number of segments when changing the level of distortion or the amount of outliers finally we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data

group based image decomposition using 3 d cartoon and texture priors we propose a novel image decomposition method to decompose an image into its cartoon and texture components to exploit the nonlocal self similarity of cartoon plus texture images we construct groups by stacking together similar image patches into 3 d arrays and consider group as the basic unit of decomposition we decompose each group via a convex optimization model consisting of 3 d cartoon and texture priors these priors characterize the local properties of the cartoon and texture components and the nonlocal similarity within each component in a unified and natural manner we develop the alternating direction method of multipliers admm to efficiently solve the proposed model for further improvement we investigate an adaptive rule for the estimation of the regularization parameter the proposed method is also extended to tackle noisy images numerical experiments confirm that the performance of the proposed method is competitive with some of the state of the art schemes

a stochastic streamflow model based on physical principles based on hydrologic arguments a stochastic model for discretized streamflow from rainfall is proposed in the form of an averaged filtered poisson process fpp using a two component description with excess precipitation and drainage as inputs to the surface and subsurface flow systems simplified models are presented for the physical processes the mean variance and covariance function of discretized streamflow are computed in terms of the first and second moments of excess precipitation and drainage which are derived analytically the behavior of the stochastic properties of streamflow is evaluated for various soil types basin travel times and effective initial soil water contents where there is an intuitive feeling for the interaction of physical characteristics and stochastic properties such as the mean the model appears to produce qualitatively correct results prediction of stochastic properties is limited by the selection of an effective initial soil moisture value

iterative regularization and nonlinear inverse scale space applied to wavelet based denoising in this paper we generalize the iterative regularization method and the inverse scale space method recently developed for total variation tv based image restoration to wavelet based image restoration this continues our earlier joint work with others where we applied these techniques to variational based image restoration obtaining significant improvement over the rudin osher fatemi tv based restoration here we apply these techniques to soft shrinkage and obtain the somewhat surprising result that a the iterative procedure applied to soft shrinkage gives firm shrinkage and converges to hard shrinkage and b that these procedures enhance the noise removal capability both theoretically in the sense of generalized bregman distance and for some examples experimentally in terms of the signal to noise ratio leaving less signal in the residual

nonlocal image restoration with bilateral variance estimation a low rank approach simultaneous sparse coding ssc or nonlocal image representation has shown great potential in various low level vision tasks leading to several state of the art image restoration techniques including bm3d and lssc however it still lacks a physically plausible explanation about why ssc is a better model than conventional sparse coding for the class of natural images meanwhile the problem of sparsity optimization especially when tangled with dictionary learning is computationally difficult to solve in this paper we take a low rank approach toward ssc and provide a conceptually simple interpretation from a bilateral variance estimation perspective namely that singular value decomposition of similar packed patches can be viewed as pooling both local and nonlocal information for estimating signal variances such perspective inspires us to develop a new class of image restoration algorithms called spatially adaptive iterative singular value thresholding saist for noise data saist generalizes the celebrated bayesshrink from local to nonlocal models for incomplete data saist extends previous deterministic annealing based solution to sparsity optimization through incorporating the idea of dictionary learning in addition to conceptual simplicity and computational efficiency saist has achieved highly competent often better objective performance compared to several state of the art methods in image denoising and completion experiments our subjective quality results compare favorably with those obtained by existing techniques especially at high noise levels and with a large amount of missing data

conceptual stochastic modeling of seasonal runoff using autoregressive moving average models and different scales of aggregation the statistical and phenomenological aspects of the runoff process observed on different scales of aggregation are taken as a priori information for the conceptually based stochastic modeling of seasonal runoff runoff is considered as the sum of two groundwater components with over year and subannual response lag and of a purely random component representing the direct runoff this scheme is equivalent to a linear system with two parallel linear reservoirs plus a zero lag linear channel the system output is the runoff and the input is the effective rainfall considered proportional to the direct runoff assuming the effective rainfall as a non gaussian periodic independent process and considering nonseasonal groundwater parameters this conceptualization leads to an autoregressive and moving average 2 2 stochastic process with periodic independent residual stochastic model parameters are directly related to the linear system coefficients and the effective rainfall structure can be determined from the estimated model residual in order to obtain parameter estimates consistent with the conceptual constraints two estimation stages on an annual and a seasonal basis and an iterative procedure are needed the model was applied to a number of time series of monthly streamflows in the apennine regions of italy with promising results

nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising part i theory we study the robust estimation of missing regions in images and video using adaptive sparse reconstructions our primary application is on missing regions of pixels containing textures edges and other image features that are not readily handled by prevalent estimation and recovery algorithms we assume that we are given a linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coefficients over missing regions are zero or close to zero we adaptively determine these small magnitude coefficients through thresholding establish sparsity constraints and estimate missing regions in images using information surrounding these regions unlike prevalent algorithms our approach does not necessitate any complex preconditioning segmentation or edge detection steps and it can be written as a sequence of denoising operations we show that the region types we can effectively estimate in a mean squared error sense are those for which the given transform provides a close approximation using sparse nonlinear approximants we show the nature of the constructed estimators and how these estimators relate to the utilized transform and its sparsity over regions of interest the developed estimation framework is general and can readily be applied to other nonstationary signals with a suitable choice of linear transforms part i discusses fundamental issues and part ii is devoted to adaptive algorithms with extensive simulation examples that demonstrate the power of the proposed techniques

a new boundary finite element method for fluid structure interaction problems in order to study problems on fluid structure interaction we have used a mixed formulation which couples the classical functional of the structure with a new variational formulation by integral equations for the fluid this formulation has the advantage over the finite element methods of avoiding the discretization of the fluid domain furthermore unlike collocation methods the explicit calculation of the hadamard finite part of the singular integrals is avoided this leads after discretization by boundary finite elements to a small and symmetrical algebraic system typical examples are presented that demonstrate the efficiency of this variational formulation by studying the sound transmission through a baffled plane structure and through a flexible panel backed by a rigid cavity these include the calculation of the transmission loss factor and the determination of which modes dominate the noise transmission good agreement is obtained between numerical results and analytical results found in the literature

robust pca via ell _ 0 ell _ 1 regularization we study the problem of low rank and sparse decomposition from possibly noisy observations known as robust pca where the sparse component can be seen as outliers we first propose a modified objective function where the nuclear norm captures the low rank term ell 0 norm addresses the sparse outlier term and an ell 1 norm to deal with the additive noise term the associated algorithm termed sparsity regularized principal component pursuit srpcp is shown to converge under certain model and algorithm parameter settings it is shown that srpcp can recover the low rank component and sparse component exactly in the noiseless case in the noisy case we first prove that the widely used principal component pursuit pcp method which was designed for the noiseless case is actually stable to dense noise then we show that srpcp has smaller estimation error bound and can identify outlier entries without any false alarm another important byproduct of our analysis is the result that pcp with missing entries is also stable to dense noise we further propose another objective function which replaces the above nuclear norm by the log determinant the proposed algorithm termed iterative reweighted sparsity regularized principal component pursuit is also shown to converge in each iteration it solves a weighted nuclear norm regularized robust matrix completion problem we propose an alternating direction method of multipliers algorithm to solve this nonconvex subproblem which also converges empirical studies demonstrate the efficacy of the proposed ell 0 ell 1 regularization framework to deal with the outliers as well as its advantage over the existing state of the art methods

compressive sensing image restoration using adaptive curvelet thresholding and nonlocal sparse regularization compressive sensing cs is a recently emerging technique and an extensively studied problem in signal and image processing which suggests a new framework for the simultaneous sampling and compression of sparse or compressible signals at a rate significantly below the nyquist rate maybe designing an effective regularization term reflecting the image sparse prior information plays a critical role in cs image restoration recently both local smoothness and nonlocal self similarity have led to superior sparsity prior for cs image restoration in this paper first an adaptive curvelet thresholding criterion is developed trying to adaptively remove the perturbations appeared in recovered images during cs recovery process imposing sparsity furthermore a new sparsity measure called joint adaptive sparsity regularization jasr is established which enforces both local sparsity and nonlocal 3 d sparsity in transform domain simultaneously then a novel technique for high fidelity cs image recovery via jasr is proposed cs jasr to efficiently solve the proposed corresponding optimization problem we employ the split bregman iterations extensive experimental results are reported to attest the adequacy and effectiveness of the proposed method comparing with the current state of the art methods in cs image restoration

nonparametric bayesian dictionary learning for analysis of noisy and incomplete images nonparametric bayesian methods are considered for recovery of imagery based upon compressive incomplete and or noisy measurements a truncated beta bernoulli process is employed to infer an appropriate dictionary for the data under test and also for image recovery in the context of compressive sensing significant improvements in image recovery are manifested using learned dictionaries relative to using standard orthonormal image expansions the compressive measurement projections are also optimized for the learned dictionary additionally we consider simpler incomplete measurements defined by measuring a subset of image pixels uniformly selected at random spatial interrelationships within imagery are exploited through use of the dirichlet and probit stick breaking processes several example results are presented with comparisons to other methods in the literature

physical basis of time series models for water table depths the relationship between precipitation excess and water table depth can be described by empirical time series models such as transfer function noise models autoregressive exogenous variable models arx and threshold autoregressive self exciting open loop models in this paper these models are interpreted in terms of the water balance of a soil column a physically based arx model is used to predict the effect of an intervention on the water table dynamics at two locations it is shown that the physically based arx model predicts the effect of interventions reasonably well

statistics of natural images scaling in the woods we study the statistics of an ensemble of images taken in the woods distributions of local quantities such as contrast are scale invariant and have nearly exponential tails power spectra exhibit scaling with a nontrivial exponent these data limit the information content of natural images and point to the importance of gain control strategies in visual processing

rank minimization with applications to image noise removal abstract rank minimization problem has a wide range of applications in different areas however since this problem is np hard and non convex the frequently used method is to replace the matrix rank minimization with nuclear norm minimization nuclear norm is the convex envelope of the matrix rank and it is more computationally tractable matrix completion is a special case of rank minimization problem in this paper we consider directly using matrix rank as the regularization term instead of nuclear norm in the cost function for matrix completion problem the solution is analyzed and obtained by a hard thresholding operation on the singular values of the observed matrix then by exploiting patch based nonlocal self similarity scheme we apply the proposed rank minimization algorithm to remove white gaussian additive noise in images gamma multiplicative noise is also removed in logarithm domain the experimental results illustrate that the proposed algorithm can remove noises in images more efficiently than nuclear norm can do and the results are also competitive with those obtained by using the existing state of the art noise removal methods in the literature

bayesian robust principal component analysis a hierarchical bayesian model is considered for decomposing a matrix into low rank and sparse components assuming the observed matrix is a superposition of the two the matrix is assumed noisy with unknown and possibly non stationary noise statistics the bayesian framework infers an approximate representation for the noise statistics while simultaneously inferring the low rank and sparse outlier contributions the model is robust to a broad range of noise levels without having to change model hyperparameter settings in addition the bayesian framework allows exploitation of additional structure in the matrix for example in video applications each row or column corresponds to a video frame and we introduce a markov dependency between consecutive rows in the matrix corresponding to consecutive frames in the video the properties of this markov process are also inferred based on the observed matrix while simultaneously denoising and recovering the low rank and sparse components we compare the bayesian model to a state of the art optimization based implementation of robust pca considering several examples we demonstrate competitive performance of the proposed model

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

image deblurring with an inaccurate blur kernel using a group based low rank image prior abstract we address the problem of restoring an original image from its blurry and noisy observation together with inaccurate information of the blurring process for this purpose we propose an enhanced regularized structured total least squares rstls model that can estimate the latent image and blur kernel simultaneously in the proposed model both the image and the blur kernel are characterized by a group based low rank prior which assumes that a group of vectorized similar data patches can be well approximated by a low rank matrix we develop an alternating minimization algorithm to solve the proposed model efficiently numerical experiments demonstrate the effectiveness of our method in terms of both quantitative measures and visual quality

weighted nuclear norm minimization and its applications to low level vision as a convex relaxation of the rank minimization model the nuclear norm minimization nnm problem has been attracting significant research interest in recent years the standard nnm regularizes each singular value equally composing an easily calculated convex norm however this restricts its capability and flexibility in dealing with many practical problems where the singular values have clear physical meanings and should be treated differently in this paper we study the weighted nuclear norm minimization wnnm problem which adaptively assigns weights on different singular values as the key step of solving general wnnm models the theoretical properties of the weighted nuclear norm proximal wnnp operator are investigated albeit nonconvex we prove that wnnp is equivalent to a standard quadratic programming problem with linear constrains which facilitates solving the original problem with off the shelf convex optimization solvers in particular when the weights are sorted in a non descending order its optimal solution can be easily obtained in closed form with wnnp the solving strategies for multiple extensions of wnnm including robust pca and matrix completion can be readily constructed under the alternating direction method of multipliers paradigm furthermore inspired by the reweighted sparse coding scheme we present an automatic weight setting method which greatly facilitates the practical implementation of wnnm the proposed wnnm methods achieve state of the art performance in typical low level vision tasks including image denoising background subtraction and image inpainting

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

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

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

local adaptivity to variable smoothness for exemplar based image regularization and representation a novel adaptive and exemplar based approach is proposed for image restoration denoising and representation the method is based on a pointwise selection of similar image patches of fixed size in the variable neighborhood of each pixel the main idea is to associate with each pixel the weighted sum of data points within an adaptive neighborhood we use small image patches e g 7 7 or 9 9 patches to compute these weights since they are able to capture local geometric patterns and texels seen in images in this paper we mainly focus on the problem of adaptive neighborhood selection in a manner that balances the accuracy of approximation and the stochastic error at each spatial position the proposed pointwise estimator is then iterative and automatically adapts to the degree of underlying smoothness with minimal a priori assumptions on the function to be recovered the method is applied to artificially corrupted real images and the performance is very close and in some cases even surpasses to that of the already published denoising methods the proposed algorithm is demonstrated on real images corrupted by non gaussian noise and is used for applications in bio imaging

total variation regularized tensor rpca for background subtraction from compressive measurements background subtraction has been a fundamental and widely studied task in video analysis with a wide range of applications in video surveillance teleconferencing and 3d modeling recently motivated by compressive imaging background subtraction from compressive measurements bscm is becoming an active research task in video surveillance in this paper we propose a novel tensor based robust principal component analysis tenrpca approach for bscm by decomposing video frames into backgrounds with spatial temporal correlations and foregrounds with spatio temporal continuity in a tensor framework in this approach we use 3d total variation to enhance the spatio temporal continuity of foregrounds and tucker decomposition to model the spatio temporal correlations of video background based on this idea we design a basic tensor rpca model over the video frames dubbed as the holistic tenrpca model to characterize the correlations among the groups of similar 3d patches of video background we further design a patch group based tensor rpca model by joint tensor tucker decompositions of 3d patch groups for modeling the video background efficient algorithms using the alternating direction method of multipliers are developed to solve the proposed models extensive experiments on simulated and real world videos demonstrate the superiority of the proposed approaches over the existing state of the art approaches

a truncated nuclear norm regularization method based on weighted residual error for matrix completion low rank matrix completion aims to recover a matrix from a small subset of its entries and has received much attention in the field of computer vision most existing methods formulate the task as a low rank matrix approximation problem a truncated nuclear norm has recently been proposed as a better approximation to the rank of matrix than a nuclear norm the corresponding optimization method truncated nuclear norm regularization tnnr converges better than the nuclear norm minimization based methods however it is not robust to the number of subtracted singular values and requires a large number of iterations to converge in this paper a tnnr method based on weighted residual error tnnr wre for matrix completion and its extension model etnnr wre are proposed tnnr wre assigns different weights to the rows of the residual error matrix in an augmented lagrange function to accelerate the convergence of the tnnr method the etnnr wre is much more robust to the number of subtracted singular values than the tnnr wre tnnr alternating direction method of multipliers and tnnr accelerated proximal gradient with line search methods experimental results using both synthetic and real visual data sets show that the proposed tnnr wre and etnnr wre methods perform better than tnnr and iteratively reweighted nuclear norm irnn methods

robust low rank tensor recovery via nonconvex singular value minimization tensor robust principal component analysis via tensor nuclear norm tnn minimization has been recently proposed to recover the low rank tensor corrupted with sparse noise outliers tnn is demonstrated to be a convex surrogate of rank however it tends to over penalize large singular values and thus usually results in biased solutions to handle this issue we propose a new definition of tensor logarithmic norm tln as the nonconvex surrogate of rank which can decrease the penalization on larger singular values and increase that on smaller ones simultaneously to preserve the low rank structure of a tensor then the strategy of tensor factorization is combined into the minimization of tln to improve computational performance to handle impulsive scenarios we propose a nonconvex ell p ball projection scheme with 0 instead of the conventional convex scheme with p 1 which enhances the robustness against outliers by incorporating the tln minimization and the ell p ball projection we finally propose two low rank recovery algorithms whose resulting optimization problems are efficiently solved by the alternating direction method of multipliers admm with convergence guarantees the proposed algorithms are applied to the synthetic data recovery and image and video restorations in real world experimental results demonstrate the superior performance of the proposed methods over several state of the art algorithms in terms of tensor recovery accuracy and computational efficiency