the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

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

gauge optimization and duality gauge functions significantly generalize the notion of a norm and gauge optimization as defined by r m freund math programming 38 1987 pp 47 67 seeks the element of a convex set that is minimal with respect to a gauge function this conceptually simple problem can be used to model a remarkable array of useful problems including a special case of conic optimization and related problems that arise in machine learning and signal processing the gauge structure of these problems allows for a special kind of duality framework this paper explores the duality framework proposed by freund and proposes a particular form of the problem that exposes some useful properties of the gauge optimization framework such as the variational properties of its value function and yet maintains most of the generality of the abstract form of gauge optimization

a class of accelerated uzawa algorithms for saddle point problems in this paper we establish a class of accelerated uzawa au algorithms for solving the large sparse nonsingular saddle point problems by making use of the extrapolation technique this extrapolation technique is based on the eigenvalues of the iterative matrix these au algorithms involve two iteration parameters whose special choices can cover the known classical uzawa method as well as yield new ones firstly the accelerated model for the uzawa algorithm is established and the detail algorithm description of au method is presented then the convergence analyse of the au method is given moreover theoretical analyses show that the au algorithm converges faster than some uzawa type methods the uzawa method is also included in when the eigenvalues of the iterative matrix and the parameter satisfy some conditions numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the au method

preconditioning multichannel adaptive filtering algorithms using evd and svd based signal prewhitening and system decoupling abstract it is well known that the convergence rate of multichannel lms based algorithms is limited by the correlation properties of the reference signals and the cross coupling within the plant dynamics these factors give rise to excessive eigenvalue spread and slow convergence rate of a gradient descent algorithm a preconditioning technique is developed in this study for the multichannel lms algorithm so as to improve its convergence rate signal prewhitening and system decoupling are the two key elements of the proposed techniques preconditioning filters are first formulated in the frequency domain by using eigenvalue decomposition and singular value decomposition these filters are then transformed into the time domain with causality taken into account the preconditioning filters are incorporated into a multichannel lms algorithm where the reference signals are prewhitened and the plants are decoupled prior to the adaptation process simulations for a two channel one listener cross talk cancellation problem illustrate the effectiveness of the preconditioning technique in improving the convergence rate of the adaptive algorithms

a multilevel algorithm for simultaneously denoising and deblurring images in this paper we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the total variation regularization although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved fast algorithms for the combined denoising and deblurring model remain to be a challenge recently several successful studies of approximating this model and subsequently finding fast algorithms were conducted which have partially solved this problem the aim of this paper is to generalize a fast multilevel denoising method to solving the minimization model for simultaneously denoising and deblurring our new idea is to overcome the complexity issue by a detailed study of the structured matrices that are associated with the blurring operator a fast algorithm can then be obtained for directly solving the variational model supporting numerical experiments on gray scale images are presented

total variation stokes strategy for sparse view x ray ct image reconstruction previous studies have shown that by minimizing the total variation tv of the to be estimated image with some data and or other constraints a piecewise smooth x ray computed tomography image can be reconstructed from sparse view projection data however due to the piecewise constant assumption for the tv model the reconstructed images are frequently reported to suffer from the blocky or patchy artifacts to eliminate this drawback we present a total variation stokes projection onto convex sets tvs pocs reconstruction method in this paper the tvs model is derived by introducing isophote directions for the purpose of recovering possible missing information in the sparse view data situation thus the desired consistencies along both the normal and the tangent directions are preserved in the resulting images compared to the previous tv based image reconstruction algorithms the preserved consistencies by the tvs pocs method are expected to generate noticeable gains in terms of eliminating the patchy artifacts and preserving subtle structures to evaluate the presented tvs pocs method both qualitative and quantitative studies were performed using digital phantom physical phantom and clinical data experiments the results reveal that the presented method can yield images with several noticeable gains measured by the universal quality index and the full width at half maximum merit as compared to its corresponding tv based algorithms in addition the results further indicate that the tvs pocs method approaches to the gold standard result of the filtered back projection reconstruction in the full view data case as theoretically expected while most previous iterative methods may fail in the full view case because of their artificial textures in the results

total variation projection with first order schemes this article proposes a new algorithm to compute the projection on the set of images whose total variation is bounded by a constant the projection is computed through a dual formulation that is solved by first order non smooth optimization methods this yields an iterative algorithm that applies iterative soft thresholding to the dual vector field and for which we establish convergence rate on the primal iterates this projection algorithm can then be used as a building block in a variety of applications such as solving inverse problems under a total variation constraint or for texture synthesis numerical results are reported to illustrate the usefulness and potential applicability of our tv projection algorithm on various examples including denoising texture synthesis inpainting deconvolution and tomography problems we also show that our projection algorithm competes favorably with state of the art tv projection methods in terms of convergence speed

network growth with preferential attachment for high indegree and low outdegree abstract we study the growth of a directed transportation network such as a food web in which links carry resources we propose a growth process in which new nodes or species preferentially attach to existing nodes with high indegree in food web language number of prey and low outdegree or number of predators this scheme which we call inverse preferential attachment is intended to maximize the amount of resources available to each new node we show that the outdegree predator distribution decays at least exponentially fast for large outdegree and is continuously tunable between an exponential distribution and a delta function the indegree prey distribution is poissonian in the large network limit

total bounded variation regularization as a bilaterally constrained optimization problem it is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem existence of a lagrange multiplier and an optimality system are established this allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations here in particular the primal dual active set method considered as a semismooth newton method is analyzed and superlinear convergence is proved as a by product we obtain that the lagrange multiplier associated with the box constraints acts as an edge detector numerical results for image denoising and zooming resizing show the efficiency of the new approach

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

the lms algorithm with delayed coefficient adaptation the behavior of the delayed least mean square dlms algorithm is studied it is found that the step size in the coefficient update plays a key role in the convergence and stability of the algorithm an upper bound for the step size is derived that ensures the stability of the dlms the relationship between the step size and the convergence speed and the effect of the delay on the convergence speed are also studied the analytical results are supported by computer simulations

model selection for convolutive ica with an application to spatiotemporal analysis of eeg we present a new algorithm for maximum likelihood convolutive independent component analysis ica in which components are unmixed using stable autoregressive filters determined implicitly by estimating a convolutive model of the mixing process by introducing a convolutive mixing model for the components we show how the order of the filters in the model can be correctly detected using bayesian model selection we demonstrate a framework for deconvolving a subspace of independent components in electroencephalography eeg initial results suggest that in some cases convolutive mixing may be a more realistic model for eeg signals than the instantaneous ica model

computational acceleration for mr image reconstruction in partially parallel imaging in this paper we present a fast numerical algorithm for solving total variation and l1 tvl1 based image reconstruction with application in partially parallel magnetic resonance imaging our algorithm uses variable splitting method to reduce computational cost moreover the barzilai borwein step size selection method is adopted in our algorithm for much faster convergence experimental results on clinical partially parallel imaging data demonstrate that the proposed algorithm requires much fewer iterations and or less computational cost than recently developed operator splitting and bregman operator splitting methods which can deal with a general sensing matrix in reconstruction framework to get similar or even better quality of reconstructed images

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

an efficient algorithm for image segmentation markov random fields and related problems problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements or order constraints in such problems the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modify value in markov random field problems there is also a pairwise relationship between the objects the objective in markov random field problem is to minimize the sum of the deviation cost function and a penalty function that grows with the distance between the values of related pairs separation function we discuss markov random fields problems in the context of a representative application the image segmentation problem in this problem the goal is to modify color shades assigned to pixels of an image so that the penalty function consisting of one term due to the deviation from the initial color shade and a second term that penalizes differences in assigned values to neighboring pixels is minimized we present here an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear and in strongly polynomial time when the deviation cost function is linear quadratic or piecewise linear convex with few pieces where few means a number exponential in a polynomial function of the number of variables and constraints the complexity of the algorithm for a problem on n pixels or variables m adjacency relations or constraints and range of variable values colors u is o t n m n log u where t n m is the complexity of solving the minimum s t cut problem on a graph with n nodes and m arcs furthermore other algorithms are shown to solve the problem with convex deviation and convex separation in running time o mn log n log nu and the problem with nonconvex deviation and convex separation in running time o t nu mu the nonconvex separation problem is np hard even for fixed value of u for the family of problems with convex deviation functions and linear separation function the algorithm described here runs in polynomial time which is demonstrated to be fastest possible

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

blind image restoration by anisotropic regularization this paper presents anisotropic regularization techniques to exploit the piecewise smoothness of the image and the point spread function psf in order to mitigate the severe lack of information encountered in blind restoration of shift invariantly and shift variantly blurred images the new techniques which are derived from anisotropic diffusion adapt both the degree and direction of regularization to the spatial activities and orientations of the image and the psf this matches the piecewise smoothness of the image and the psf which may be characterized by sharp transitions in magnitude and by the anisotropic nature of these transitions for shift variantly blurred images whose underlying psfs may differ from one pixel to another we parameterize the psf and then apply the anisotropic regularization techniques this is demonstrated for linear motion blur and out of focus blur alternating minimization is used to reduce the computational load and algorithmic complexity

robustness of the filtered x lms algorithm part i necessary conditions for convergence and the asymptotic pseudospectrum of toeplitz matrices errors in the secondary path model of the filtered x lms fxlms algorithm will lead to its divergence when the eigenvalues of the cross correlation matrix between the estimated filtered reference and the true filtered reference signals are not all located in the right half plane this cross correlation matrix has a block toeplitz structure whose dimension is determined by the number of adaptive filter coefficients using results on the asymptotic pseudospectrum of toeplitz matrices a frequency domain condition on the model is derived to ensure stability the condition is sufficient and necessary for a large number of filter coefficients a transient analysis shows that the sufficient condition given by wang and ren 1999 is only necessary to prevent an initial increase of the error in the adaptive filter coefficients critical behavior

quasi newton approach to nonnegative image restorations abstract image restoration or deblurring is the process of attempting to correct for degradation in a recorded image typically the blurring system is assumed to be linear and spatially invariant and fast fourier transform fft based schemes result in efficient computational image restoration methods however real images have properties that cannot always be handled by linear methods in particular an image consists of positive light intensities and thus a nonnegativity constraint should be enforced this constraint and other ways of incorporating a priori information have been suggested in various applications and can lead to substantial improvements in the reconstructions nevertheless such constraints are rarely implemented because they lead to nonlinear problems which require demanding computations we suggest efficient implementations for three nonnegatively constrained restorations schemes constrained least squares maximum likelihood and maximum entropy we show that with a certain parameterization and using a quasi newton scheme these methods are very similar in addition our formulation reveals a connection between our approach for maximum likelihood and the expectation maximization em method used extensively by astronomers numerical experiments illustrate that our approach is superior to em both in terms of accuracy and efficiency

frequency domain adaptation of causal digital filters the adaptation of causal fir digital filters in the discrete frequency domain is considered and it is shown how the bin normalized form of the lms algorithm can converge to a biased solution for problems such as linear prediction a discrete frequency domain version of newton s algorithm is derived and it is demonstrated how this can converge to the optimal causal solution even for linear prediction problems the algorithm employs a spectral factorization of the estimated power spectral density of the reference signal the entirely noncausal part of which is used before the causality constraint in the adaptation algorithm and the entirely causal part is applied after the causality constraint the spectral factors can be calculated online from a recursive estimate of the power spectral density without too great a loss of convergence speed the extension of the algorithm to the adaptation of feedforward controllers is also described in which case the spectral factors of the reference signals filtered by the plant response are required and these are shown to be equal to the spectral factors of the reference signal multiplied by the minimum phase part or the plant frequency response

optimal controllers and adaptive controllers for multichannel feedforward control of stochastic disturbances a time domain formulation for the multichannel feedforward control problem is used to derive an optimally condition for the least squares controller it is also used to motivate an instantaneous steepest descent algorithm for the adaptation of the controller which is known as the filtered error lms algorithm the convergence rate of this algorithm is limited by the correlation properties of each reference signal their cross correlation properties and the dynamics and coupling within the plant response an expression is then derived for the transfer function of the optimum least squares controller this suggests a new architecture for the adaptive controller whose convergence rate is not limited by the factors mentioned above a set of white and uncorrelated reference signals are generated to drive a modified matrix of control filters whose outputs are multiplied by the inverse of the minimum phase part of the plant response matrix before being fed to the physical plant the error signals from the plant are then used to update this controller after being fed through the time reverse transpose of the all pass part of the plant matrix the relationship is also discussed between this architecture and that using a singular value decomposition of the plant response which is used to control tonal disturbances

an efficient primal dual interior point method for minimizing a sum of euclidean norms the problem of minimizing a sum of euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature this nonsmooth optimization problem arises in many different kinds of modern scientific applications we derive a primal dual interior point algorithm for the problem by applying newton s method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition the main work at each step consists of solving a system of linear equations the schur complement equations this schur complement matrix is not symmetric unlike in linear programming we incorporate a mehrotra type predictor corrector scheme and present some experimental results comparing several variations of the algorithm including as one option explicit symmetrization of the schur complement with a skew corrector term we also present results obtained from a code implemented to solve large sparse problems using a symmetrized schur complement this has been applied to problems arising in plastic collapse analysis with hundreds of thousands of variables and millions of nonzeros in the constraint matrix the algorithm typically finds accurate solutions in less than 50 iterations and determines physically meaningful solutions previously unobtainable

non lipschitz ell_ p regularization and box constrained model for image restoration nonsmooth nonconvex regularization has remarkable advantages for the restoration of piecewise constant images constrained optimization can improve the image restoration using a priori information in this paper we study regularized nonsmooth nonconvex minimization with box constraints for image restoration we present a computable positive constant for using nonconvex nonsmooth regularization and show that the difference between each pixel and its four adjacent neighbors is either 0 or larger than in the recovered image moreover we give an explicit form of for the box constrained image restoration model with the non lipschitz nonconvex lp norm 0 p 1 regularization our theoretical results show that any local minimizer of this imaging restoration problem is composed of constant regions surrounded by closed contours and edges numerical examples are presented to validate the theoretical results and show that the proposed model can recover image restoration results very well

an algorithm for image removals and decompositions without inverse matrices partial differential equation pde based methods in image processing have been actively studied in the past few years one of the effective methods is the method based on a total variation introduced by rudin oshera and fatemi rof l i rudin s osher e fatemi nonlinear total variation based noise removal algorithms physica d 60 1992 259 268 this method is a well known edge preserving model and an useful tool for image removals and decompositions unfortunately this method has a nonlinear term in the equation which may yield an inaccurate numerical solution to overcome the nonlinearity a fixed point iteration method has been widely used the nonlinear system based on the total variation is induced from the rof model and the fixed point iteration method to solve the rof model is introduced by dobson and vogel d c dobson c r vogel convergence of an iterative method for total variation denoising siam j numer anal 34 5 1997 1779 1791 however some methods had to compute inverse matrices which led to roundoff error to address this problem we developed an efficient method for solving the rof model we make a sequence like richardson s method by using a fixed point iteration to evade the nonlinear equation this approach does not require the computation of inverse matrices the main idea is to make a direction vector for reducing the error at each iteration step in other words we make the next iteration to reduce the error from the computed error and the direction vector we describe that our method works well in theory in numerical experiments we show the results of the proposed method and compare them with the results by d dobson and c vogel and then we confirm the superiority of our method

a monolithic conservative level set method with built in redistancing abstract we introduce a new level set method for representing evolving interfaces in the case of divergence free velocity fields the new method satisfies a conservation principle conservation is important for many applications such as modeling two phase incompressible flow in the present implementation the conserved quantity is defined as the integral of a smoothed characteristic function the new approach embeds level sets into a volume of fluid formulation the evolution of an approximate signed distance function is governed by a conservation law for its smoothed sign the non linear level set transport equation is regularized by adding a flux correction term that assures a non singular jacobian and penalizes deviations from a distance function the result is a locally conservative level set method with built in elliptic redistancing the continuous model is monolithic in the sense that the level set transport model the volume of fluid law of mass conservation and the minimization problem that preserves the approximate distance function property are incorporated into a single equation there is no need for any extra stabilization artificial compression flux limiting redistancing mass correction and other numerical fixes which are commonly used in level set or volume of fluid methods in addition there is just one free parameter that controls the strength of regularization and penalization in the model the accuracy and conservation properties of the monolithic finite element level set method are illustrated by the results of numerical studies for passive advection of free interfaces

individual based predator prey model for biological coevolution fluctuations stability and community structure we study an individual based predator prey model of biological coevolution using linear stability analysis and large scale kinetic monte carlo simulations the model exhibits approximate 1 f noise in diversity and population size fluctuations and it generates a sequence of quasisteady communities in the form of simple food webs these communities are quite resilient toward the loss of one or a few species which is reflected in different power law exponents for the durations of communities and the lifetimes of species the exponent for the former is near ensuremath 1 while the latter is close to ensuremath 2 statistical characteristics of the evolving communities including degree predator and prey distributions and proportions of basal intermediate and top species compare reasonably with data for real food webs

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

preconditioning for allen cahn variational inequalities with non local constraints the solution of allen cahn variational inequalities with mass constraints is of interest in many applications this problem can be solved both in its scalar and vector valued form as a pde constrained optimization problem by means of a primal dual active set method at the heart of this method lies the solution of linear systems in saddle point form in this paper we propose the use of krylov subspace solvers and suitable preconditioners for the saddle point systems numerical results illustrate the competitiveness of this approach

effects of preference for attachment to low degree nodes on the degree distributions of a growing directed network and a simple food web model we study the growth of a directed network in which the growth is constrained by the cost of adding links to the existing nodes we propose a preferential attachment scheme in which a new node attaches to an existing node i with probability ensuremath pi k i ensuremath propto k i ensuremath 1 where k i is the number of outgoing links at i we calculate the degree distribution for the outgoing links in the asymptotic regime t ensuremath rightarrow ensuremath infty n k both analytically and by monte carlo simulations the distribution decays like k ensuremath mu k ensuremath gamma k for large k where ensuremath mu is a constant we investigate the effect of this preferential attachment scheme by comparing the results to an equivalent growth model with a degree independent probability of attachment which gives an exponential outdegree distribution also we relate this mechanism to simple food web models by implementing it in the cascade model we show that the low degree preferential attachment mechanism breaks the symmetry between in and outdegree distributions in the cascade model it also causes a faster decay in the tails of the outdegree distributions for both our network growth model and the cascade model

anisotropic second and fourth order diffusion models based on convolutional virtual electric field for image denoising this paper proposes novel second order and fourth order anisotropic diffusion models for image denoising these two models are based on convolutional virtual electric field convef the proposed second order anisotropic diffusion model was introduced by introducing the convef into the perona malik p m equation and the fourth order one is coined by combining convef and the you kaveh y k equation the employment of the convef model allows to improve the estimation of high order derivatives and makes the proposed methods more robust to noise in addition the proposed convef based y k model is an anisotropic diffusion equation which can avoid the over smoothness of edges and provide better edge protection capability except for better denoising some experiments are conducted to demonstrate the efficiency of the proposed methods

planar crack identification in 3d linear elasticity by the reciprocity gap method abstract we study the reciprocity gap method andrieux et al 1999 for the identification of planar cracks in the framework of three dimensional linear elasticity in order to achieve better accuracy on thick domains we propose to use polynomial test functions instead of fourier series we also propose to improve the solution by various regularization techniques that are tested and compared numerical assessments are provided in 2d and 3d

regularization operators for natural images based on nonlinear perception models image restoration requires some a priori knowledge of the solution some of the conventional regularization techniques are based on the estimation of the power spectrum density simple statistical models for spectral estimation just take into account second order relations between the pixels of the image however natural images exhibit additional features such as particular relationships between local fourier or wavelet transform coefficients biological visual systems have evolved to capture these relations we propose the use of this biological behavior to build regularization operators as an alternative to simple statistical models the results suggest that if the penalty operator takes these additional features in natural images into account it will be more robust and the choice of the regularization parameter is less critical

brief regularization networks for inverse problems a state space approach linear inverse problems with discrete data are equivalent to the estimation of the continuous time input of a linear dynamical system from samples of its output the solution obtained by means of regularization theory has the structure of a neural network similar to classical rbf networks however the basis functions depend in a nontrivial way on the specific linear operator to be inverted and the adopted regularization strategy by resorting to the bayesian interpretation of regularization we show that such networks can be implemented rigorously and efficiently whenever the linear operator admits a state space representation an analytic expression is provided for the basis functions as well as for the entries of the matrix of the linear system used to compute the weights the results are illustrated through a deconvolution problem where the spontaneous secretory rate of luteinizing hormone lh of the hypophisis is reconstructed from measurements of plasma lh concentrations

an infeasible primal dual algorithm for total bounded variation based inf convolution type image restoration in this paper a primal dual algorithm for total bounded variation tv type image restoration is analyzed and tested analytically it turns out that employing a global boldsymbol l s regularization with 1 s leq 2 in the dual problem results in a local smoothing of the tv regularization term in the primal problem the local smoothing can alternatively be obtained as the infimal convolution of the ell r norm with r 1 s 1 1 and a smooth function in the case r s 2 this results in gauss tv type image restoration the globalized primal dual algorithm introduced in this paper works with generalized derivatives converges locally at a superlinear rate and is stable with respect to noise in the data in addition it utilizes a projection technique which reduces the size of the linear system that has to be solved per iteration a comprehensive numerical study ends the paper

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

giant strongly connected component of directed networks we describe how to calculate the sizes of all giant connected components of a directed graph including the strongly connected one in particular the world wide web is a directed network the results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in and out degree distribution p k i k o we show that if p k i k o does not factorize the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in and out components the calculations of the relative sizes of all the giant components are demonstrated using the simplest examples we explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

multichannel blind deconvolution of spatially misaligned images existing multichannel blind restoration techniques assume perfect spatial alignment of channels correct estimation of blur size and are prone to noise we developed an alternating minimization scheme based on a maximum a posteriori estimation with a priori distribution of blurs derived from the multichannel framework and a priori distribution of original images defined by the variational integral this stochastic approach enables us to recover the blurs and the original image from channels severely corrupted by noise we observe that the exact knowledge of the blur size is not necessary and we prove that translation misregistration up to a certain extent can be automatically removed in the restoration process

an effective method for solving nonlinear equations and its application the linearized partial differential equation from the nonlinear partial differential equation which was proposed by rudin osher and fatemi l i rudin s osher and e fatemi nonlinear total variation based noise removal algorithms for solving image decomposition was introduced by chambolle a chambolle an algorithm for total variation minimization and applications and r acar and c r vogel r acar and c r vogel analysis of bounded variation penalty methods for ill posed problems in this paper we propose a method for solving the linearized partial differential equation and we show numerical results for denoising which demonstrate a significant improvement over other previous works

self whitening algorithms for adaptive equalization and deconvolution in equalization and deconvolution tasks the correlated nature of the input signal slows the convergence speeds of stochastic gradient adaptive filters prewhitening techniques have been proposed to improve the convergence performance but the additional coefficient memory and updates for the prewhitening filter can be prohibitive in some applications we present two simple algorithms that employ the equalizer as a prewhitening filter within the gradient updates these self whitening algorithms provide quasi newton convergence locally about the optimum coefficient solution for deconvolution and equalization tasks multichannel extensions of the techniques are also described

on a class of ill posed minimization problems in image processing in this paper we show that minimization problems involving sublinear regularizing terms are ill posed in general although numerical experiments in image processing give very good results the energies studied here are inspired by image restoration and image decomposition rewriting the nonconvex sublinear regularizing terms as weighted total variations we give a new approach to perform minimization via the well known chambolle s algorithm the approach developed here provides an alternative to the well known half quadratic minimization one

image restoration based on the minimized surface regularization abstract we propose an image restoration model based on the minimized surface regularization the proposed model closely relates to the classical smoothing rof model rudin et al 1992 we deduce two different conjugation forms via coupling the gradient operator with the smoothing parameter or not and then provide the existence of the minimizer in the continuous setting in order to efficiently solving the proposed model we employ the primal dual method by reformulating the proposed model as a min max problem relying on the convex conjugation the convergence of the algorithm is provided as well numerical implementations mainly emphasize the effectiveness of the proposed method by comparing it to other two well known methods in terms of the cpu time and restoration quality

shapes from pixels continuous domain visual signals are usually captured as discrete digital images this operation is not invertible in general in the sense that the continuous domain signal cannot be exactly reconstructed based on the discrete image unless it satisfies certain constraints e g bandlimitedness in this paper we study the problem of recovering shape images with smooth boundaries from a set of samples thus the reconstructed image is constrained to regenerate the same samples consistency as well as forming a shape bilevel image we initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes next we relax the non convex shape constraint to transform the problem into minimizing the total variation over consistent non negative valued images we also introduce a requirement called reducibility that guarantees equivalence between the two problems we illustrate that the reducibility property effectively sets a requirement on the minimum sampling density we also evaluate the performance of the relaxed alternative in various numerical experiments

a multigrid method for a fourth order diffusion equation with application to image processing in this paper we are concerned with the efficient solution of the elliptic system alpha l nobreak i u nobreak f where l is the biharmonic operator supplemented by higher order boundary conditions this is a core task in image processing arising from the minimization of rotation invariant energy functionals for the purpose of numerical solution we approximate the partial differential equation by finite differences in order to obtain efficient and robust solution methods we discuss how the system can be treated efficiently by multigrid methods finally experimental results are reported that demonstrate the effectiveness of the multigrid approach

area and length preserving geometric invariant scale spaces in this paper area preserving multi scale representations of planar curves are described this allows smoothing without shrinkage at the same time preserving all the scale space properties the representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant when the euclidean geometric heat flow is used the resulting representation is euclidean invariant and similarly it is affine invariant when the affine one is used the flows are geometrically intrinsic to the curve and exactly satisfy all the basic requirements of scale space representations in the case of the euclidean heat flow it is completely local as well the same approach is used to define length preserving geometric flows a similarity scale invariant geometric heat flow is studied as well in this work

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

noise removal using smoothed normals and surface fitting in this work we use partial differential equation techniques to remove noise from digital images the removal is done in two steps we first use a total variation filter to smooth the normal vectors of the level curves of a noise image after this we try to find a surface to fit the smoothed normal vectors for each of these two stages the problem is reduced to a nonlinear partial differential equation finite difference schemes are used to solve these equations a broad range of numerical examples are given in the paper

analytical solution of a model for complex food webs we investigate numerically and analytically a recently proposed model for food webs nature 404 180 2000 in the limit of large web sizes and sparse interaction matrices we obtain analytical expressions for several quantities with ecological interest in particular the probability distributions for the number of prey and the number of predators we find that these distributions have fast decaying exponential and gaussian tails respectively we also find that our analytical expressions are robust to changes in the details of the model

on the local and semilocal convergence of a parameterized multi step newton method abstract this paper is devoted to a family of newton like methods with frozen derivatives used to approximate a locally unique solution of an equation we perform a convergence study and an analysis of the efficiency this analysis gives us the opportunity to select the most efficient method in the family without the necessity of their implementation the method can be applied to many type of problems including the discretization of ordinary differential equations integral equations integro differential equations or partial differential equations moreover multi step iterative methods are computationally attractive

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding

a primal dual active set method for non negativity constrained total variation deblurring problems this paper studies image deblurring problems using a total variation based model with a non negativity constraint the addition of the non negativity constraint improves the quality of the solutions but makes the solution process a difficult one the contribution of our work is a fast and robust numerical algorithm to solve the non negatively constrained problem to overcome the nondifferentiability of the total variation norm we formulate the constrained deblurring problem as a primal dual program which is a variant of the formulation proposed by chan golub and mulet for unconstrained problems here dual refers to a combination of the lagrangian and fenchel duals to solve the constrained primal dual program we use a semi smooth newton s method we exploit the relationship between the semi smooth newton s method and the primal dual active set method to achieve considerable simplification of the computations the main advantages of our proposed scheme are no parameters need significant adjustment a standard inverse preconditioner works very well quadratic rate of local convergence theoretical and numerical numerical evidence of global convergence and high accuracy of solving the optimality system the scheme shows robustness of performance over a wide range of parameters a comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme