injectivity of compressing maps on primitive sequences over bbz p e let zopf pe be the integer residue ring with odd prime p and integer eges2 for a sequence a over zopf pe one has a unique p adic expansion a a 0 a 1 p a e 1 pe 1 where a i can be regarded as a sequence over zopf p for 0lesilese 1 let f x be a strongly primitive polynomial over zopf pe and g f x pe be the set of all primitive sequences generated f x by over zopf pe recently the authors xuan yong zhu and wen feng qi have proved that for a function phi x0 xe 1 g xe 1 eta x0 xe 2 over zopf p and a b ising f x pe where 2lesdeg glesp 1 phi a 0 a 1 a e 1 phi b 0 b 1 b e 1 if and only if a b to further complete their work we show that such injectivity also holds for deg g 1 that is for a function phi x0 xe 1 xe 1 eta x0 xe 2 over zopf p and a b ising f x pe phi a 0 a 1 a e 1 phi b 0 b 1 b e 1 if and only if a b

aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

an efficient two phase rm l 1 tv method for restoring blurred images with impulse noise a two phase image restoration method based upon total variation regularization combined with an l1 data fitting term for impulse noise removal and deblurring is proposed in the first phase suitable noise detectors are used for identifying image pixels contaminated by noise then in the second phase based upon the information on the location of noise free pixels images are deblurred and denoised simultaneously for efficiency reasons in the second phase a superlinearly convergent algorithm based upon fenchel duality and inexact semismooth newton techniques is utilized for solving the associated variational problem numerical results prove the new method to be a significantly advance over several state of the art techniques with respect to restoration capability and computational efficiency

distribution properties of compressing sequences derived from primitive sequences over bbz p e let z pe be the integer residue ring with odd prime p and integer e 2 any sequence a over z pe has a unique p adic expansion a a0 a1 p ae 1 pe 1 where ai can be regarded as a sequence over z p for 0 i e 1 let f x be a strongly primitive polynomial over z pe and a b be two primitive sequences generated by f x over z pe assume x0 xe 1 xe 1 x0 xe 2 is an e variable function over z p with the monomial p 1 2 xe 2 p 1 x1 p 1 not pearing in the expression of x0 x1 xe 2 it is shown that if there exists an s z p such that a0 t ae 1 t s if and only if b0 t be 1 t s for all nonnegative t with i 0 where is an m sequence determined by f x and a0 then a b this implies that for compressing sequences derived from primitive sequences generated by f x over z pe single element distribution is unique on all positions t with t 0 in particular when x0 x1 xe 2 0 it is a completion of the former result on the uniqueness of distribution of element 0 in highest level sequences

restoration of blurred color images with impulse noise restoration of images degraded by blurring and impulse noise has received considerable attention recently guo et al 2009 proposed a fast l 1 total variation algorithm for grayscale image restoration with impulse noise in this paper we extend their idea for deblurring color images with impulse noise an alternating iteration scheme is adopted for solving the corresponding problem more importantly we employ the five point property to analyze the convergence of the proposed alternating algorithm numerical experiments demonstrate that the proposed method could deblur color images with good quality

analysis and application of a novel fast algorithm for 2 d arma model parameter estimation abstract in this paper we analyze a novel algorithm for 2 d arma model parameter estimation in the presence of noise and then develop a fast and efficient blind image restoration algorithm we show that the novel algorithm can minimize a quadratic convex optimization problem and has a lower computational complexity than the conventional algorithms as a result the novel algorithm involves no convergence and local minimum issue moreover the proposed blind image restoration algorithm can overcome the local minimization problem computed results confirm that the novel algorithm can more quickly obtain more accurate estimates than the conventional algorithms in the presence of noise

factorized banded inverse preconditioners for matrices with toeplitz structure in this paper we study factorized banded inverse preconditioners for matrices with toeplitz structure we show that if a toeplitz matrix t has certain off diagonal decay property then the factorized banded inverse preconditioner approximates t 1 accurately and the spectra of these preconditioned matrices are clustered around 1 in nonlinear image restoration applications toeplitz related systems of the form i t d t are required to solve where d is a positive nonconstant diagonal matrix we construct inverse preconditioners for such matrices numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners a two dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner

lcd motion blur modeling analysis and algorithm liquid crystal display lcd devices are well known for their slow responses due to the physical limitations of liquid crystals therefore fast moving objects in a scene are often perceived as blurred this effect is known as the lcd motion blur in order to reduce lcd motion blur an accurate lcd model and an efficient deblurring algorithm are needed however existing lcd motion blur models are insufficient to reflect the limitation of human eye tracking system also the spatiotemporal equivalence in lcd motion blur models has not been proven directly in the discrete 2 d spatial domain although it is widely used there are three main contributions of this paper modeling analysis and algorithm first a comprehensive lcd motion blur model is presented in which human eye tracking limits are taken into consideration second a complete analysis of spatiotemporal equivalence is provided and verified using real video sequences third an lcd motion blur reduction algorithm is proposed the proposed algorithm solves an l1 norm regularized least squares minimization problem using a subgradient projection method numerical results show that the proposed algorithm gives higher peak snr lower temporal error and lower spatial error than motion compensated inverse filtering and lucy richardson deconvolution algorithm which are two state of the art lcd deblurring algorithms

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

semi supervised cross entropy clustering with information bottleneck constraint abstract in this paper we propose a semi supervised clustering method cec ib that models data with a set of gaussian distributions and that retrieves clusters based on a partial labeling provided by the user partition level side information by combining the ideas from cross entropy clustering cec with those from the information bottleneck method ib our method trades between three conflicting goals the accuracy with which the data set is modeled the simplicity of the model and the consistency of the clustering with side information experiments demonstrate that cec ib performs similar as gaussian mixture models in a classical semi supervised scenario but is faster more robust to noisy labels automatically determines the optimal number of clusters and performs well when not all classes are present in the side information moreover in contrast to many other semi supervised models it can be successfully applied in discovering natural subgroups if the partition level side information is derived from the top levels of a hierarchical clustering

gradient estimates for dirichlet parabolic problems in unbounded domains we consider autonomous parabolic dirichlet problems in a regular unbounded open set rn involving second order operator a with possibly unbounded coefficients we determine new conditions on the coefficients of a yielding global gradient estimates for the bounded classical solution

a nonconvex l1 l1 l2 model for image restoration with impulse noise abstract in this paper we propose a model for image restoration with blur and impulse noise it is composed of l 1 data fitting term and a nonconvex regularization term which is the weighted difference of l 1 norm and l 2 norm based on wavelet frame the combined model is difficult to solve by the corresponding euler lagrangian equations here we solve it by alternating direction method of multipliers admm we describe the detailed process of the algorithm and establish the convergence of the algorithm the experimental outcomes on different blurs and different impulse noises demonstrate that the proposed approach is better than those existing methods in terms of standard signal to noise ratio psnr relative error reerr and visual quality

application of the em algorithm for the multitarget multisensor tracking problem an important problem in surveillance and reconnaissance systems is the tracking of multiple moving targets in cluttered noise environments using outputs from a number of sensors possessing wide variations in individual characteristics and accuracies a number of approaches have been proposed for this multitarget multisensor tracking problem ranging from reasonably simple though ad hoc schemes to fairly complex but theoretically optimum approaches in this paper we describe an iterative procedure for time recursive multitarget multisensor tracking based on use of the expectation maximization em algorithm more specifically we pose the multitarget multisensor tracking problem as an incomplete data problem with the observable sensor outputs representing the incomplete data whereas the target associated sensor outputs constitute the complete data target updates at each time use an em based approach that calculates the maximum a posteriori map estimate of the target states under the assumption of appropriate motion models based on the outputs of disparate sensors the approach uses a markov random field mrf model of the associations between observations and targets and allows for estimation of joint association probabilities without explicit enumeration the advantage of this em based approach is that it provides a computationally efficient means for approaching the performance offered by theoretically optimum approaches that use explicit enumeration of the joint association probabilities we provide selected results illustrating the performance complexity characteristics of this em based approach compared with competing schemes

parameter selection for total variation based image restoration using discrepancy principle there are two key issues in successfully solving the image restoration problem 1 estimation of the regularization parameter that balances data fidelity with the regularity of the solution and 2 development of efficient numerical techniques for computing the solution in this paper we derive a fast algorithm that simultaneously estimates the regularization parameter and restores the image the new approach is based on the total variation tv regularized strategy and morozov s discrepancy principle the tv norm is represented by the dual formulation that changes the minimization problem into a minimax problem a proximal point method is developed to compute the saddle point of the minimax problem by adjusting the regularization parameter adaptively in each iteration the solution is guaranteed to satisfy the discrepancy principle we will give the convergence proof of our algorithm and numerically show that it is better than some state of the art methods in terms of both speed and accuracy

feature selection based on the approximation of class densities by finite mixtures of special type abstract a new method of feature selection based on the approximation of class conditional densities by a mixture of parameterized densities of a special type suitable especially for multimodal data is presented no search procedure is needed when using the proposed method its performance is tested both on real simulated data

a gradient estimate on a manifold with convex boundary we present a simple probability approach for establishing a gradient estimate for a solution of an elliptic equation on a compact riemannian manifold with convex boundary or on a noncompact complete manifold our method can also be applied to derive a similar gradient estimate for a nonlinear parabolic equation and an abstract gradient estimate for a markov semigroup

learning a coupled linearized method in online setting based on the alternating direction method of multipliers in this paper we propose analyze and test a coupled linearized method which aims to minimize an unconstrained problem consisting of a loss term and a regularization term in an online setting to solve this problem we first transform it into an equivalent constrained minimization problem with a separable structure then we split the corresponding augmented lagrangian function and minimize the resulting subproblems distributedly with one variable by fixing another one this method is easy to execute without calculating matrix inversion by implementing three linearized operations per iteration and at each iteration we can obtain a closed form solution in particular our update rule contains the well known soft thresholding operator as a special case moreover upper bound on the regret of the proposed method is analyzed under some mild conditions it can achieve o 1 sqrt t convergence rate for convex learning problems and o rm log t t for strongly convex learning numerical experiments and comparisons with several state of the art methods are reported which demonstrate the efficiency and effectiveness of our approach

compression mappings on primitive sequences over z p sup e let z p sup e be the integer residue ring with odd prime p spl ges 5 and integer e spl ges 2 for a sequence a over z p sup e there is a unique p adic expansion a a sub 0 a spl middot p a sub e 1 spl middot p sup e 1 where each a sub i is a sequence over 0 1 p 1 and can be regarded as a sequence over the finite field gf p naturally let f x be a primitive polynomial over z p sup e and g f x p sup e the set of all primitive sequences generated by f x over z p sup e set spl phi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 1 x sub 0 x sub 1 x sub e 2 spl psi sub e 1 x sub 0 x sub e 1 x sub e 1 sup k spl eta sub e 2 2 x sub 0 x sub 1 x sub e 2 where spl eta sub e 2 1 and spl eta sub e 2 2 are arbitrary functions of e 1 variables over gf p and 2 spl les k spl les p 1 then the compression mapping spl phi sub e 1 g f x p sup e spl rarr gf p sup spl infin a spl rarr spl phi sub e 1 a sub 0 a sub e 1 is injective that is a b if and only if spl phi sub e 1 a sub 0 a sub e 1 spl phi sub e 1 b sub 0 b sub e 1 for a b spl isin g f x p sup e furthermore if f x is a strongly primitive polynomial over z p sup e then spl phi sub e 1 a sub 0 a sub e 1 spl psi sub e 1 b sub 0 b sub e 1 if and only if a b and spl phi sub e 1 x sub 0 x sub e 1 spl psi sub e 1 x sub 0 x sub e 1 for a b spl isin g f x p sup e

the most significant bit of maximum length sequences over spl zopf 2 sup l autocorrelation and imbalance the imbalance and the autocorrelation of the binary sequences in the title are explored by combining the local weil bound with spectral analysis recent estimates on these quantities are improved by a factor of order 2 sup l 2 for large l

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

convergence rates for exponentially ill posed inverse problems with impulsive noise this paper is concerned with exponentially ill posed operator equations with additive impulsive noise on the right hand side i e the noise is large on a small part of the domain and small or zero outside it is well known that tikhonov regularization with an l 1 data fidelity term outperforms tikhonov regularization with an l 2 fidelity term in this case this effect has recently been explained and quantified for the case of finitely smoothing operators here we extend this analysis to the case of infinitely smoothing forward operators under standard sobolev smoothness assumptions on the solution i e exponentially ill posed inverse problems it turns out that high order polynomial rates of convergence in the size of the support of large noise can be achieved rather than the poor logarithmic convergence rates typical for exponentially ill posed problems the main tools of our analysis are banach spaces of analytic functions and interpolation type inequalities for such spaces we discuss two examp

an adaptive method for recovering image from mixed noisy data in this paper we present a new version of the famous rudin osher fatemi rof model to restore image the key point of the model is that it could reconstruct images with blur and non uniformly distributed noise we develop this approach by adding several statistical control parameters to the cost functional and these parameters could be adaptively determined by the given observed image in this way we could adaptively balance the performance of the fit to data term and the regularization term the numerical experiments have demonstrated the significant effectiveness and robustness of our model in restoring blurred images with mixed gaussian noise or salt and pepper noise

on the distinctness of binary sequences derived from primitive sequences modulo square free odd integers let m be a square free odd integer and z m the integer residue ring modulo m this paper studies the distinctness of primitive sequences over z m modulo 2 recently for the case of m pq a product of two distinct prime numbers p and q the problem has been almost completely solved as for the case that m is a product of more prime numbers the problem has been quite resistant to proof in this paper a partial proof is given by showing that a class of primitive sequences of order 2n 1 over z m is distinct modulo 2 where n is a positive integer besides as an independent interest this paper also involves two distribution properties of primitive sequences over z m which are related closely to our main results

an efficient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise we extend the alternating minimization algorithm recently proposed in y wang j yang w yin and y zhang siam j imag sci 1 2008 pp 248 272 j yang w yin y zhang and y wang siam j imag sci 2 2009 pp 569 592 to the case of recovering blurry multichannel color images corrupted by impulsive rather than gaussian noise the algorithm minimizes the sum of a multichannel extension of total variation and a data fidelity term measured in the ell 1 norm and is applicable to both salt and pepper and random valued impulsive noise we derive the algorithm by applying the well known quadratic penalty function technique and prove attractive convergence properties including finite convergence for some variables and q linear convergence rate under periodic boundary conditions the main computational requirements of the algorithm are fast fourier transforms and a low complexity gaussian elimination procedure numerical results on images with different blurs and impulsive noise are presented to demonstrate the efficiency of the algorithm in addition it is numerically compared to the least absolute deviation method h y fu m k ng m nikolova and j l barlow siam j sci comput 27 2006 pp 1881 1902 and the two phase method j f cai r chan and m nikolova aims j inverse problems and imaging 2 2008 pp 187 204 for recovering grayscale images we also present results of recovering multichannel images

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

a weighted dictionary learning model for denoising images corrupted by mixed noise this paper proposes a general weighted l2 l0 norms energy minimization model to remove mixed noise such as gaussian gaussian mixture impulse noise and gaussian impulse noise from the images the approach is built upon maximum likelihood estimation framework and sparse representations over a trained dictionary rather than optimizing the likelihood functional derived from a mixture distribution we present a new weighting data fidelity function which has the same minimizer as the original likelihood functional but is much easier to optimize the weighting function in the model can be determined by the algorithm itself and it plays a role of noise detection in terms of the different estimated noise parameters by incorporating the sparse regularization of small image patches the proposed method can efficiently remove a variety of mixed or single noise while preserving the image textures well in addition a modified k svd algorithm is designed to address the weighted rank one approximation the experimental results demonstrate its better performance compared with some existing methods

digital filters as absolute norm regularizers a variational formulation of the digital filtering problem based on a certain regularizing functional is presented in particular the special case where the regularizing functional is defined by means of l sub 1 norm is considered and a characterization of its minima is presented it is shown that for a certain class of signals one solution to the regularization problem can be obtained by means of a suitable recursive median filter some examples and a possible extension of the variational approach are also discussed

an adaptive gaussian model for satellite image deblurring the deconvolution of blurred and noisy satellite images is an ill posed inverse problem which can be regularized within a bayesian context by using an a priori model of the reconstructed solution since real satellite data show spatially variant characteristics we propose here to use an inhomogeneous model we use the maximum likelihood estimator mle to estimate its parameters and we show that the mle computed on the corrupted image is not suitable for image deconvolution because it is not robust to noise we then show that the estimation is correct only if it is made from the original image since this image is unknown we need to compute an approximation of sufficiently good quality to provide useful estimation results such an approximation is provided by a wavelet based deconvolution algorithm thus a hybrid method is first used to estimate the space variant parameters from this image and then to compute the regularized solution the obtained results on high resolution satellite images simultaneously exhibit sharp edges correctly restored textures and a high snr in homogeneous areas since the proposed technique adapts to the local characteristics of the data

on the distinctness of modular reductions of maximal length sequences modulo odd prime powers we discuss the distinctness problem of the reductions modulo m of maximal length sequences modulo powers of an odd prime p where the integer m has a prime factor different from p for any two different maximal length sequences generated by the same polynomial we prove that their reductions modulo m are distinct in other words the reduction modulo m of a maximal length sequence is proved to contain all the information of the original sequence

deblurring of color images corrupted by impulsive noise we consider the problem of restoring a multichannel image corrupted by blur and impulsive noise e g salt and pepper noise using the variational framework we consider the l1 fidelity term and several possible regularizers in particular we use generalizations of the mumford shah ms functional to color images and gamma convergence approximations to unify deblurring and denoising experimental comparisons show that the ms stabilizer yields better results with respect to beltrami and total variation regularizers color edge detection is a beneficial by product of our methods

salt and pepper noise removal by median type noise detectors and detail preserving regularization this paper proposes a two phase scheme for removing salt and pepper impulse noise in the first phase an adaptive median filter is used to identify pixels which are likely to be contaminated by noise noise candidates in the second phase the image is restored using a specialized regularization method that applies only to those selected noise candidates in terms of edge preservation and noise suppression our restored images show a significant improvement compared to those restored by using just nonlinear filters or regularization methods only our scheme can remove salt and pepper noise with a noise level as high as 90

a duality based splitting method for ell 1 tv image restoration with automatic regularization parameter choice a novel splitting method is presented for the ell 1 tv restoration of degraded images subject to impulsive noise the functional is split into an ell 2 tv denoising part and an ell 1 ell 2 deblurring part the dual problem of the relaxed functional is smooth with convex constraints and can be solved efficiently by applying an arrow hurwicz type algorithm to the augmented lagrangian formulation the regularization parameter is chosen automatically based on a balancing principle the accuracy the fast convergence and the robustness of the algorithm and the use of the parameter choice rule are illustrated on some benchmark images and compared with an existing method

color image enhancement via chromaticity diffusion a novel approach for color image denoising is proposed in this paper the algorithm is based on separating the color data into chromaticity and brightness and then processing each one of these components with partial differential equations or diffusion flows in the proposed algorithm each color pixel is considered as an n dimensional vector the vectors direction a unit vector gives the chromaticity while the magnitude represents the pixel brightness the chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data such as chromaticity both isotropic and anisotropic diffusion flows are presented for this n dimensional chromaticity diffusion flow the brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement we present the underlying theory a number of examples and briefly compare with the current literature

properties of l1 tgv2 the one dimensional case abstract we study properties of solutions to second order total generalized variation tgv 2 regularized l 1 fitting problems in dimension one special attention is paid to the analysis of the structure of the solutions their regularity and the effect of the regularization parameters

least squares restoration of multichannel images multichannel restoration using both within and between channel deterministic information is considered a multichannel image is a set of image planes that exhibit cross plane similarity existing optimal restoration filters for single plane images yield suboptimal results when applied to multichannel images since between channel information is not utilized multichannel least squares restoration filters are developed using the set theoretic and the constrained optimization approaches a geometric interpretation of the estimates of both filters is given color images three channel imagery with red green and blue components are considered constraints that capture the within and between channel properties of color images are developed issues associated with the computation of the two estimates are addressed a spatially adaptive multichannel least squares filter that utilizes local within and between channel image properties is proposed experiments using color images are described

adaptive variational method for restoring color images with high density impulse noise in this paper a new variational framework of restoring color images with impulse noise is presented the novelty of this work is the introduction of an adaptively weighting data fidelity term in the cost functional the fidelity term is derived from statistical methods and contains two weighting functions as well as some statistical control parameters of noise this method is based on the fact that impulse noise can be approximated as an additive noise with probability density function pdf being the finite mixture model a bayesian framework is then formulated in which likelihood functions are given by the mixture model inspired by the expectation maximization em algorithm we present two models with variational framework in this study the superiority of the proposed models is that the weighting functions can effectively detect the noise in the image with the noise information the proposed algorithm can automatically balance the regularity of the restored image and the fidelity term by updating the weighting functions and the control parameters these two steps ensure that one can obtain a good restoration even though the degraded color image is contaminated by impulse noise with large ration 90 or more in addition the numerical implementation of this algorithm is very fast by using a split algorithm some numerical experimental results and comparisons with other methods are provided to show the significant effectiveness of our approach

sar amplitude probability density function estimation based on a generalized gaussian model in the context of remotely sensed data analysis an important problem is the development of accurate models for the statistics of the pixel intensities focusing on synthetic aperture radar sar data this modeling process turns out to be a crucial task for instance for classification or for denoising purposes in this paper an innovative parametric estimation methodology for sar amplitude data is proposed that adopts a generalized gaussian gg model for the complex sar backscattered signal a closed form expression for the corresponding amplitude probability density function pdf is derived and a specific parameter estimation algorithm is developed in order to deal with the proposed model specifically the recently proposed method of log cumulants molc is applied which stems from the adoption of the mellin transform instead of the usual fourier transform in the computation of characteristic functions and from the corresponding generalization of the concepts of moment and cumulant for the developed gg based amplitude model the resulting molc estimates turn out to be numerically feasible and are also analytically proved to be consistent the proposed parametric approach was validated by using several real ers 1 xsar e sar and nasa jpl airborne sar images and the experimental results prove that the method models the amplitude pdf better than several previously proposed parametric models for backscattering phenomena

a multiplicative iterative algorithm for box constrained penalized likelihood image restoration image restoration is a computationally intensive problem as a large number of pixel values have to be determined since the pixel values of digital images can attain only a finite number of values e g 8 bit images can have only 256 gray levels one would like to recover an image within some dynamic range this leads to the imposition of box constraints on the pixel values the traditional gradient projection methods for constrained optimization can be used to impose box constraints but they may suffer from either slow convergence or repeated searching for active sets in each iteration in this paper we develop a new box constrained multiplicative iterative bcmi algorithm for box constrained image restoration the bcmi algorithm just requires pixelwise updates in each iteration and there is no need to invert any matrices we give the convergence proof of this algorithm and apply it to total variation image restoration problems where the observed blurry images contain poisson gaussian or salt and pepper noises

parameter estimation and tissue segmentation from multispectral mr images a statistical method is developed to classify tissue types and to segment the corresponding tissue regions from relaxation time t sub 1 t sub 2 and proton density p sub d weighted magnetic resonance images the method assumes that the distribution of image intensities associated with each tissue type can be expressed as a multivariate likelihood function of three weighted signal intensity values t sub 1 t sub 2 p sub d at each location within that tissue regions the method further assumes that the underlying tissue regions are piecewise contiguous and can be characterized by a markov random field prior in classifying the tissue types the method models the likelihood of realizing the images as a finite multivariate mixture function the class parameters associated with the tissue types i e the weighted intensity means variances and correlation coefficients of the multivariate function as well as the number of voxels within regions of the tissue types of are estimated by maximum likelihood the estimation fits the class parameters to the image data via the expectation maximization algorithm the number of classes associated with the tissue types is determined by the information criterion of minimum description length the method segments the tissue regions given the estimated class parameters by maximum a posteriori probability the prior is constructed by the tissue region membership of the first and second order neighborhood the method is tested by a few sets of t sub 1 t sub 2 and p sub d weighted images of the brain acquired with a 1 5 tesla whole body scanner the number of classes and the associated class parameters are automatically estimated the regions of different brain tissues are satisfactorily segmented

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

gcsdecolor gradient correlation similarity for efficient contrast preserving decolorization this paper presents a novel gradient correlation similarity gcs measure based decolorization model for faithfully preserving the appearance of the original color image contrary to the conventional data fidelity term consisting of gradient error norm based measures the newly defined gcs measure calculates the summation of the gradient correlation between each channel of the color image and the transformed grayscale image two efficient algorithms are developed to solve the proposed model on one hand due to the highly nonlinear nature of gcs measure a solver consisting of the augmented lagrangian and alternating direction method is adopted to deal with its approximated linear parametric model the presented algorithm exhibits excellent iterative convergence and attains superior performance on the other hand a discrete searching solver is proposed by determining the solution with the minimum function value from the linear parametric model induced candidate images the non iterative solver has advantages in simplicity and speed with only several simple arithmetic operations leading to real time computational speed in addition it is very robust with respect to the parameter and candidates extensive experiments under a variety of test images and a comprehensive evaluation against existing state of the art methods consistently demonstrate the potential of the proposed model and algorithms

noise adaptive soft switching median filter existing state of the art switching based median filters are commonly found to be nonadaptive to noise density variations and prone to misclassifying pixel characteristics at high noise density interference this reveals the critical need of having a sophisticated switching scheme and an adaptive weighted median filter we propose a novel switching based median filter with incorporation of fuzzy set concept called the noise adaptive soft switching median nasm filter to achieve much improved filtering performance in terms of effectiveness in removing impulse noise while preserving signal details and robustness in combating noise density variations the proposed nasm filter consists of two stages a soft switching noise detection scheme is developed to classify each pixel to be uncorrupted pixel isolated impulse noise nonisolated impulse noise or image object s edge pixel no filtering or identity filter standard median sm filter or our developed fuzzy weighted median fwm filter will then be employed according to the respective characteristic type identified experimental results show that our nasm filter impressively outperforms other techniques by achieving fairly close performance to that of ideal switching median filter across a wide range of noise densities ranging from 10 to 70

further result of compressing maps on primitive sequences modulo odd prime powers let be the integer residue ring with odd prime and integer for a sequence over there is an unique adic expansion where each is a sequence over and can be regarded as a sequence over the prime field gf naturally let be a strongly primitive polynomial over and the set of all primitive sequences generated by over suppose that gf gf it is shown that any function in is an injective map from to gf and the derived sequences of different functions are also different that is if and only if and for and these injective functions in can be considered as good candidates for the keys of a stream cipher

further result on distribution properties of compressing sequences derived from primitive sequences over bf z p e let p be an odd prime number e an integer greater than 1 and z pe the integer residue ring modulo pe in this paper we obtain an improved result of the previous paper ieee trans inf theory 56 1 2010 555 563 on distribution properties of compressing sequences derived from primitive sequences over z pe it is shown that two primitive sequences and b generated by a strongly primitive polynomial f x over z pe are the same if there exist s z p and k z p such that the distribution of in their compressing sequences ae 1 a0 ae 2 and be 1 b0 be 2 is coincident at the positions t with t k where x0 xe 2 is an e variable polynomial over z p with the coefficient of xe 2p 1 x1p 1x0p 1 not equal to 1 e p 1 2 and is an m sequence over z p determined by f x and a compared with the previous result this gives a more precise characterization on the positions of a compressing sequence i e of the form ae 1 a0 ae 2 derived from a primitive sequence a over z pe that completely determines a in particular the result is also true for the highest level sequence ae 1 by taking x0 xe 2 0

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

efficient minimization methods of mixed l 2 l 1 and l 1 l 1 norms for image restoration image restoration problems are often solved by finding the minimizer of a suitable objective function usually this function consists of a data fitting term and a regularization term for the least squares solution both the data fitting and the regularization terms are in the ell 2 norm in this paper we consider the least absolute deviation lad solution and the least mixed norm lmn solution for the lad solution both the data fitting and the regularization terms are in the ell 1 norm for the lmn solution the regularization term is in the ell 1 norm but the data fitting term is in the ell 2 norm since images often have nonnegative intensity values the proposed algorithms provide the option of taking into account the nonnegativity constraint the lmn and lad solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods at each iteration of the interior point method a structured linear system must be solved the preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems experimental results are used to demonstrate the effectiveness of our approach we also show the quality of the restored images using the minimization of mixed ell 2 ell 1 and ell1 ell 1 norms is better than that using only the ell 2 norm

selective removal of impulse noise based on homogeneity level information we propose a decision based signal adaptive median filtering algorithm for removal of impulse noise our algorithm achieves accurate noise detection and high snr measures without smearing the fine details and edges in the image the notion of homogeneity level is defined for pixel values based on their global and local statistical properties the cooccurrence matrix technique is used to represent the correlations between a pixel and its neighbors and to derive the upper and lower bound of the homogeneity level noise detection is performed at two stages noise candidates are first selected using the homogeneity level and then a refining process follows to eliminate false detections the noise detection scheme does not use a quantitative decision measure but uses qualitative structural information and it is not subject to burdensome computations for optimization of the threshold values empirical results indicate that our scheme performs significantly better than other median filters in terms of noise suppression and detail preservation

explicit ringing removal in image deblurring in this paper we present a simple yet effective image deblurring method to produce ringing free deblurred images our work is inspired by the observation that large scale deblurring ringing artifacts are measurable through a multi resolution pyramid of low pass filtering of the blurred deblurred image pair we propose to model such a quantification as a convex cost function and minimize it directly in the deblurring process in order to reduce ringing regardless of its cause an efficient primal dual algorithm is proposed as a solution to this optimization problem since the regularization is more biased toward ringing patterns the details of the reconstructed image are prevented from over smoothing an inevitable source of ringing is sensor saturation which can be detected costlessly contrary to most other sources of ringing however dealing with the saturation effect in deblurring introduces a non linear operator in optimization problem in this paper we also introduce a linear approximation as a solution to handling saturation in the proposed deblurring method as a result of these steps we significantly enhance the quality of the deblurred images experimental results and quantitative evaluations demonstrate that the proposed method performs favorably against state of the art image deblurring methods

random properties of the highest level sequences of primitive sequences over z 2 sup e using the estimates of the exponential sums over galois rings we discuss the random properties of the highest level sequences spl alpha sub e 1 of primitive sequences generated by a primitive polynomial of degree n over z 2 sup e first we obtain an estimate of 0 1 distribution in one period of spl alpha sub e 1 on the other hand we give an estimate of the absolute value of the autocorrelation function c sub n h of spl alpha sub e 1 which is less than 2 sup e 1 2 sup e 1 1 spl radic 3 2 sup 2e 1 2 sup n 2 2 sup e 1 for h spl ne 0 both results show that the larger n is the more random spl alpha sub e 1 will be

further results on the distinctness of binary sequences derived from primitive sequences modulo square free odd integers this paper studies the distinctness of primitive sequences over z m modulo 2 where m is an odd integer that is composite and square free and z m is the integer residue ring modulo m a new sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial f x over z m are pairwise distinct modulo 2 such result improves a recent result obtained in our previous paper and consequently the set of primitive sequences over z m that can be proven to be distinct modulo 2 is greatly enlarged

convergence rates for inverse problems with impulsive noise we study inverse problems f f g with perturbed right hand side g rm obs corrupted by so called impulsive noise i e noise which is concentrated on a small subset of the domain of definition of g it is well known that tikhonov type regularization with an mathbf l 1 data fidelity term yields significantly more accurate results than tikhonov regularization with classical mathbf l 2 data fidelity terms for this type of noise the purpose of this paper is to provide a convergence analysis explaining this remarkable difference in accuracy our error estimates significantly improve previous error estimates for tikhonov regularization with mathbf l 1 fidelity term in the case of impulsive noise we present numerical results which are in good agreement with the predictions of our analysis

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

structured total least squares for color image restoration the problem of 3 x 3 color mixing image restoration is considered the blurring matrices as well as the observed image are contaminated by noise therefore the total least squares tls method is employed to restore the original image since the blurring matrices are also structured we apply the structured total least squares stls method j b rosen h park and j glick siam j matrix anal appl 17 1996 pp 110 126 the blurring matrices are generally ill conditioned thus tikhonov s regularization is used to stabilize the solution since neumann boundary conditions are used in the restoration process the discrete cosine transform dct based preconditioner is effective for the linear systems encountered in our stls algorithm

distribution properties of compressing sequences derived from primitive sequences modulo odd prime powers let a and b be primitive sequences over z pe with odd prime p and e 2 for certain compressing maps we consider the distribution properties of compressing sequences of a and b and prove that a b if the compressing sequences are equal at the times t such that t k where is a sequence related to a we also discuss the s uniform distribution property of compressing sequences for some compressing maps we have that there exist different primitive sequences such that the compressing sequences are s uniform we also discuss that compressing sequences can be s uniform for how many elements s

a fast segmentation method based on constraint optimization and its applications intensity inhomogeneity and texture segmentation we propose a new constraint optimization energy and an iteration scheme for image segmentation which is connected to edge weighted centroidal voronoi tessellation ewcvt we show that the characteristic functions of the edge weighted voronoi regions are the minimizers may not unique of the proposed energy at each iteration we propose a narrow banding algorithm to accelerate the implementation which makes the proposed method very fast we generalize the cvt segmentation to hand intensity inhomogeneous and texture segmentation by incorporating the global and local image information into the energy functional compared with other approaches such as level set method the experimental results in this paper have shown that our approach greatly improves the calculation efficiency without losing segmentation accuracy

the l 1 potts functional for robust jump sparse reconstruction we investigate the nonsmooth and nonconvex l 1 potts functional in discrete and continuous time we show gamma convergence of discrete l 1 potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer for the discrete l 1 potts problem we introduce an o n 2 time and o n space algorithm to compute an exact minimizer we apply l 1 potts minimization to the problem of recovering piecewise constant signals from noisy measurements f it turns out that the l 1 potts functional has a quite interesting blind deconvolution property in fact we show that mildly blurred jump sparse signals are reconstructed by minimizing the l 1 potts functional furthermore for strongly blurred signals and a known blurring operator we derive an iterative reconstruction algorithm

asymptotic convergence rate of the em algorithm for gaussian mixtures it is well known that the convergence rate of the expectation maximization em algorithm can be faster than those of convention first order iterative algorithms when the overlap in the given mixture is small but this argument has not been mathematically proved yet this article studies this problem asymptotically in the setting of gaussian mixtures under the theoretical framework of xu and jordan 1996 it has been proved that the asymptotic convergence rate of the em algorithm for gaussian mixtures locally around the true solution is o e0 5 e where e 0 is an arbitrarily small number o x means that it is a higher order infinitesimal as x 0 and e is a measure of the average overlap of gaussians in the mixture in other words the large sample local convergence rate for the em algorithm tends to be asymptotically superlinear when e tends to zero

image deblurring in the presence of impulsive noise consider the problem of image deblurring in the presence of impulsive noise standard image deconvolution methods rely on the gaussian noise model and do not perform well with impulsive noise the main challenge is to deblur the image recover its discontinuities and at the same time remove the impulse noise median based approaches are inadequate because at high noise levels they induce nonlinear distortion that hampers the deblurring process distinguishing outliers from edge elements is difficult in current gradient based edge preserving restoration methods the suggested approach integrates and extends the robust statistics line process half quadratic and anisotropic diffusion points of view we present a unified variational approach to image deblurring and impulse noise removal the objective functional consists of a fidelity term and a regularizer data fidelity is quantified using the robust modified l 1 norm and elements from the mumford shah functional are used for regularization we show that the mumford shah regularizer can be viewed as an extended line process it reflects spatial organization properties of the image edges that do not appear in the common line process or anisotropic diffusion this allows to distinguish outliers from edges and leads to superior experimental results

recovering low rank matrices from corrupted observations via the linear conjugate gradient algorithm the matrix nuclear norm minimization problem has received much attention in recent years largely because its highly related to the matrix rank minimization problem arising from controller design signal processing and model reduction the alternating direction method is a very popular way to solve this problem due to its simplicity low storage practical computation efficiency and nice convergence properties in this paper we propose an alternating direction method where one variable is determined explicitly and the other variable is computed by a linear conjugate gradient algorithm at each iteration the method involves a singular value thresholding and its convergence result is guaranteed in this literature extensive experiments illustrate that the proposed algorithm compares favorable with the state of the art algorithms fpca and iadm bb which were specifically designed in recent years

parametric maximum flow algorithms for fast total variation minimization this report studies the global minimization of anisotropically discretized total variation tv energies with an l p in particular l 1 and l 2 fidelity term using parametric maximum flow algorithms to minimize s t cut representations of these energies the tv l 2 model also known as the rudin osher fatemi rof model is suitable for restoring images contaminated by gaussian noise while the tv l 1 model is able to remove impulsive noise from grayscale images and perform multiscale decompositions of them preliminary numerical results on large scale two dimensional ct and three dimensional brain mr images are presented to illustrate the effectiveness of these approaches

a dictionary learning approach for poisson image deblurring the restoration of images corrupted by blur and poisson noise is a key issue in medical and biological image processing while most existing methods are based on variational models generally derived from a maximum a posteriori map formulation recently sparse representations of images have shown to be efficient approaches for image recovery following this idea we propose in this paper a model containing three terms a patch based sparse representation prior over a learned dictionary the pixel based total variation regularization term and a data fidelity term capturing the statistics of poisson noise the resulting optimization problem can be solved by an alternating minimization technique combined with variable splitting extensive experimental results suggest that in terms of visual quality peak signal to noise ratio value and the method noise the proposed algorithm outperforms state of the art methods

gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities by using probabilistic approaches some uniform gradient estimates are obtained for dirichlet heat semigroups on a riemannian manifold with boundary as an application lower bound estimates of isoperimetric constants are presented in terms of functional inequalities