efficient algorithm for nonconvex minimization and its application to pm regularization in image processing nonconvex regularization has the ability to smooth homogeneous regions and sharpen edges but leads to challenging computation we propose some iterative schemes to minimize the energy function with nonconvex edge preserving potential the schemes are derived from the duality based algorithm proposed by bermudez and moreno and the fixed point iteration the convergence is proved for the convex energy function with nonconvex potential and the linear convergence rate is given applying the proposed schemes to perona and malik s nonconvex regularization we present some efficient algorithms based on our schemes and show the approximate convergence behavior for nonconvex energy function experimental results are presented which show the efficiency of our algorithms including better denoised performance of nonconvex regularization faster convergence speed higher calculation precision lower calculation cost under the same number of iterations and less implementation time under the same peak signal noise ratio level

the qr decomposition and the singular value decomposition in the symmetrized max plus algebra revisited this paper is an updated and extended version of the paper the qr decomposition and the singular value decomposition in the symmetrized max plus algebra b de schutter and b de moor siam j matrix anal appl 19 1998 pp 378 406 the max plus algebra which has maximization and addition as its basic operations can be used to describe and analyze certain classes of discrete event systems such as flexible manufacturing systems railway networks and parallel processor systems in contrast to conventional algebra and conventional linear system theory the max plus algebra and the max plus algebraic system theory for discrete event systems are at present far from fully developed and many fundamental problems still have to be solved currently much research is going on to deal with these problems and to further extend the max plus algebra and to develop a complete max plus algebraic system theory for discrete event systems in this paper we address one of the remaining gaps in the max plus algebra by considering matrix decompositions in the symmetrized max plus algebra the symmetrized max plus algebra is an extension of the max plus algebra obtained by introducing a max plus algebraic analogue of the operator we show that we can use well known linear algebra algorithms to prove the existence of max plus algebraic analogues of basic matrix decompositions from linear algebra such as the qr decomposition the singular value decomposition the hessenberg decomposition the lu decomposition and so on these max plus algebraic matrix decompositions could play an important role in the max plus algebraic system theory for discrete event systems

restoration of images based on subspace optimization accelerating augmented lagrangian approach we propose a new fast algorithm for solving a tv based image restoration problem our approach is based on merging subspace optimization methods into an augmented lagrangian method the proposed algorithm can be seen as a variant of the alm augmented lagrangian method and the convergence properties are analyzed from a drs douglas rachford splitting viewpoint experiments on a set of image restoration benchmark problems show that the proposed algorithm is a strong contender for the current state of the art methods

on the convergence rate of some nonlocal energies abstract we study the rate of convergence of some nonlocal functionals recently considered by bourgain brezis and mironescu and after a suitable rescaling we establish the convergence of the corresponding rate functionals to a limit functional of second order

characterization of sobolev and bv spaces abstract the main results of this paper are new characterizations of w 1 p 1 p and bv for r n an arbitrary open set using these results we answer some open questions of brezis 2002 10 and ponce 2004 25

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

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

a class of high order nonlocal operators we study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators we present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of sobolev inequalities such as the nonlocal poincare and nonlocal gagliardo nirenberg inequalities nonlocal characterizations of high order sobolev spaces in the spirit of bourgain brezis mironescu are provided applications of nonlocal calculus of variations to the well posedness of linear nonlocal models of elastic beams and plates are also considered

an iterative shrinkage approach to total variation image restoration the problem of restoration of digital images from their degraded measurements plays a central role in a multitude of practically important applications a particularly challenging instance of this problem occurs in the case when the degradation phenomenon is modeled by an ill conditioned operator in such a situation the presence of noise makes it impossible to recover a valuable approximation of the image of interest without using some a priori information about its properties such a priori information commonly referred to as simply priors is essential for image restoration rendering it stable and robust to noise moreover using the priors makes the recovered images exhibit some plausible features of their original counterpart particularly if the original image is known to be a piecewise smooth function one of the standard priors used in this case is defined by the rudin osher fatemi model which results in total variation tv based image restoration the current arsenal of algorithms for tv based image restoration is vast in this present paper a different approach to the solution of the problem is proposed based upon the method of iterative shrinkage aka iterated thresholding in the proposed method the tv based image restoration is performed through a recursive application of two simple procedures viz linear filtering and soft thresholding therefore the method can be identified as belonging to the group of first order algorithms which are efficient in dealing with images of relatively large sizes another valuable feature of the proposed method consists in its working directly with the tv functional rather then with its smoothed versions moreover the method provides a single solution for both isotropic and anisotropic definitions of the tv functional thereby establishing a useful connection between the two formulae finally a number of standard examples of image deblurring are demonstrated in which the proposed method can provide restoration results of superior quality as compared to the case of sparse wavelet deconvolution

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

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

split bregman iteration algorithm for total bounded variation regularization based image deblurring many existing algorithms taking the seminorm in bv for regularization have achieved great success in image processing however this paper considers the total bounded variation regularization based approach to perform image deblurring based on this novel model we introduce an extended split bregman iteration to obtain the optimum solution quickly we also provide the rigorous convergence analysis of the iterative algorithm here compared with the results of the rof method numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm

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

multiplicative noise removal using variable splitting and constrained optimization multiplicative noise also known as speckle noise models are central to the study of coherent imaging systems such as synthetic aperture radar and sonar and ultrasound and laser imaging these models introduce two additional layers of difficulties with respect to the standard gaussian additive noise scenario 1 the noise is multiplied by rather than added to the original image 2 the noise is not gaussian with rayleigh and gamma being commonly used densities these two features of multiplicative noise models preclude the direct application of most state of the art algorithms which are designed for solving unconstrained optimization problems where the objective has two terms a quadratic data term log likelihood reflecting the additive and gaussian nature of the noise plus a convex possibly nonsmooth regularizer e g a total variation or wavelet based regularizer prior in this paper we address these difficulties by 1 converting the multiplicative model into an additive one by taking logarithms as proposed by some other authors 2 using variable splitting to obtain an equivalent constrained problem and 3 dealing with this optimization problem using the augmented lagrangian framework a set of experiments shows that the proposed method which we name midal multiplicative image denoising by augmented lagrangian yields state of the art results both in terms of speed and denoising performance

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

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

the bond based peridynamic system with dirichlet type volume constraint in this paper the bond based peridynamic system is analysed as a non local boundary value problem with volume constraint the study extends earlier works in the literature on non local diffusion and non local peridynamic models to include non positive definite kernels we prove the well posedness of both linear and nonlinear variational problems with volume constraints the analysis is based on some non local poincare type inequalities and the compactness of the associated non local operators it also offers careful characterizations of the associated solution spaces such as compact embedding separability and completeness in the limit of vanishing non locality the convergence of the peridynamic system to the classical navier equations of elasticity with poisson ratio is demonstrated

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

efficient schemes for total variation minimization under constraints in image processing this paper presents new fast algorithms to minimize total variation and more generally l 1 norms under a general convex constraint such problems are standards of image processing the algorithms are based on a recent advance in convex optimization proposed by yurii nesterov depending on the regularity of the data fidelity term we solve either a primal problem or a dual problem first we show that standard first order schemes allow one to get solutions of precision epsilon in o frac 1 epsilon 2 iterations at worst we propose a scheme that allows one to obtain a solution of precision epsilon in o frac 1 epsilon iterations for a general convex constraint for a strongly convex constraint we solve a dual problem with a scheme that requires o frac 1 sqrt epsilon iterations to get a solution of precision epsilon finally we perform some numerical experiments which confirm the theoretical results on various problems of image processing

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

two subtle convex nonlocal approximations of the bv norm abstract inspired by the bbm formula and by work of g leoni and d spector we analyze the asymptotic behavior of two sequences of convex nonlocal functionals n u and n u which converge formally to the bv norm of u we show that pointwise convergence when u is not smooth can be delicate by contrast convergence to the bv norm is a robust and very useful mode of convergence

a posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models abstract in this paper we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond based peridynamic models in particular we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems a posteriori error estimators are consequently prompted the reliability and efficiency of the estimators are proved connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space numerical experiments 1d are also given to test the theoretical conclusions

polynomial convolutions in max plus algebra abstract recently in a work that grew out of their exploration of interlacing polynomials marcus spielman and srivastava 21 and marcus 20 studied certain combinatorial polynomial convolutions these convolutions preserve real rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices thus providing a link to free probability we explore analogues of these types of convolutions in the setting of max plus algebra in this setting the max permanent replaces the determinant the maximum is the analogue of the expected value and real rootedness is replaced by full canonical form our results resemble those of marcus et al however in contrast to the classical setting we obtain an exact and simple description of all roots of the convolution of p x and q x in terms of the roots of p x and q x

convergence of a fixed point iteration method for the osv model in image processing image denoising and texture extraction are important problems in which many new methods recently have been developed one of the most important models is the osv model s osher a sole l vese image decomposition and restoration using total variation minimization and the h 1 norm multiscale model simul a siam interdisciplinary j 1 3 2003 349 370 which is constructed by the total variation and h 1 norm this paper proves the existence of the minimizer of the functional from the osv model and analyzes the convergence of an iterative method for solving the problems our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated euler lagrange equation and the limit of our iterations satisfies the minimizer of the functional from the osv model in numerical experiments we compare the numerical results of our works with those of the rof model l i rudin s osher e fatemi nonlinear total variation based noise removal algorithms phys d 60 1992 259 268

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

a lattice boltzmann method for image denoising in this paper we construct a lattice boltzmann scheme to simulate the well known total variation based restoration model that is rof model the advantages of the lattice boltzmann method include the fast computational speed and the easily implemented fully parallel algorithm a conservative property of the lb method is discussed the macroscopic pde associated with the lb algorithm is derived which is just the rof model moreover the linearized stability of the method is analyzed the numerical computations demonstrate that the lb algorithm is efficient and robust even though the quality of the restored images is slightly lower than those by using the rof model the restored images of the lb method are satisfactory furthermore computational speed of the lb method is much faster than rof model in general cpu time of the lb method for restored images is about one tenth of rof model

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

nonconforming discontinuous galerkin methods for nonlocal variational problems we present a nonconforming discontinuous galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints the nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions the key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin it works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions a convergence theory is established for two types of finite dimensional spaces with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown

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

on tropical eigenvalues of tridiagonal toeplitz matrices abstract in this paper we determine the tropical algebraic eigenvalues of toeplitz matrices it is proved that these matrices have at most two distinct tropical eigenvalues and the explicit formulas for those eigenvalues are given also the asymptotic behavior of eigenvalues of parameterized tridiagonal toeplitz matrices over the classical field of complex numbers is shown

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 mixed formulation of the bingham fluid flow problem analysis and numerical solution in this paper we introduce a mixed formulation of the bingham fluid flow problem we consider both the original and a regularized version of the problem where a parameter e is introduced forcing the entire domain to be formally a fluid region in general common solvers for the regularized problem experience a performance degradation when the parameter e gets smaller the method studied here introduces an auxiliary tensor variable and shows enhanced numerical properties for small values of e a good performance is also observed for the non regularized case the well posedness for the regularized problem and the equivalence at the continuous level between the original primitive variables and the mixed formulation are demonstrated we analyze properties of linearized problems that are relevant for the convergence of numerical solvers a finite element method for the mixed formulation is discussed numerical results confirm the predicted better performances of the mixed formulation when compared to the primitive variables formulation a comparison to a non regularized solver based on the augmented duvaut lions glowinski formulation of the problem is carried out as well

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

characterization of polynomials and higher order sobolev spaces in terms of functionals involving difference quotients abstract the aim of this paper which deals with a class of singular functionals involving difference quotients is twofold deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a k th order sobolev space one of the main theorems is a new characterization of w k p k n and p 1 for arbitrary open sets r n in particular we provide natural generalizations of the results regarding sobolev spaces summarized in brezis overview article brezis 2002 to the higher order case and extend the work borghol 2007 to a more general setting

can the nonlocal characterization of sobolev spaces by bourgain et al be useful for solving variational problems we question whether the recent characterization of sobolev spaces by bourgain brezis and mironescu 2001 could be useful to solve variational problems on w 1 p omega to answer this we introduce a sequence of functionals so that the seminorm is approximated by an integral operator involving a differential quotient and a radial mollifier then for the approximated formulation we prove existence uniqueness and convergence of the solution to the unique solution of the initial formulation we show that these results can also be extended in the bv case interestingly this approximation leads to a unified implementation for sobolev spaces including with high p values and for the bv space finally we show how this theoretical study can indeed lead to a numerically tractable implementation and we give some image diffusion results as an illustration

on formulae decoupling the total variation of bv functions abstract in this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non local functionals one special case shows that rescalings of the fractional laplacian of a function u s b v converge strictly to the singular portion of d u

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

dirichlet s principle and wellposedness of solutions for a nonlocal p laplacian system abstract we prove dirichlet s principle for a nonlocal p laplacian system which arises in the nonlocal setting of peridynamics when p 2 this nonlinear model includes boundary conditions imposed on a nonzero volume collar surrounding the domain our analysis uses nonlocal versions of integration by parts techniques that resemble the classical green and gauss identities the nonlocal energy functional associated with this elliptic type system exhibits a general kernel which could be weakly singular the coercivity of the system is shown by employing a nonlocal poincare s inequality we use the direct method in calculus of variations to show existence and uniqueness of minimizers for the nonlocal energy from which we obtain the wellposedness of this steady state diffusion system

graph laplacian regularization for image denoising analysis in the continuous domain inverse imaging problems are inherently underdetermined and hence it is important to employ appropriate image priors for regularization one recent popular prior the graph laplacian regularizer assumes that the target pixel patch is smooth with respect to an appropriately chosen graph however the mechanisms and implications of imposing the graph laplacian regularizer on the original inverse problem are not well understood to address this problem in this paper we interpret neighborhood graphs of pixel patches as discrete counterparts of riemannian manifolds and perform analysis in the continuous domain providing insights into several fundamental aspects of graph laplacian regularization for image denoising specifically we first show the convergence of the graph laplacian regularizer to a continuous domain functional integrating a norm measured in a locally adaptive metric space focusing on image denoising we derive an optimal metric space assuming non local self similarity of pixel patches leading to an optimal graph laplacian regularizer for denoising in the discrete domain we then interpret graph laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations e g its tendency to promote piecewise smooth signals under certain settings to verify our analysis an iterative image denoising algorithm is developed experimental results show that our algorithm performs competitively with state of the art denoising methods such as bm3d for natural images and outperforms them significantly for piecewise smooth images

an iterative lagrange multiplier method for constrained total variation based image denoising various effective algorithms have been proposed in the past two decades for nonlinear pdes arising from the unconstrained total variation based image denoising problem regularizing the total variation constrained minimization model such algorithms can be used to obtain a satisfactory result as long as a suitable regularization parameter balancing the trade off between a good fit to the data and a regular solution is given however it is generally difficult to obtain a suitable regularization parameter without which restored images can be unsatisfactory if it is too large then the resulting solution is still contaminated by noise while if too small the solution is a poor approximation of the true noise free solution to provide an automatic method for the regularization parameter when the noise level is known a priori one way is to address the coupled karush kuhn tucker kkt systems from the constrained total variation optimization problem so far much less work has been done on this problem this

characterization of function spaces of vector fields and an application in nonlinear peridynamics abstract the paper introduces a class of nonlocal derivative operators defined over vector fields that turn out to satisfy a nonlocal integration by parts formula we demonstrate that in several function spaces these operators have similar continuity property as the classical differential operators a closed formula for the limit of these operators will be obtained when nonlocality vanishes the limit analysis together with the integration by parts formula enables us to link the nonlocal derivative operators and associated function spaces with the conventional local differential operators and sobolev bounded variations and bounded deformation function spaces as an application we present an existence result and asymptotic analysis in the sense of convergence of some nonlinear variational problems that arise in nonlocal continuum mechanics

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

multimodal image registration using a variational approach this paper presents an approach to obtain a deformation which matches two images acquired from different medical imaging modalities this problem arises in the investigation of human brains two distance functionals for the images are proposed with different pros and cons these functionals are to be minimized we add a smoothing term to the minimization problem which retains certain desired elastic features in the solution at each minimization step an approximate solution for the linearized problem is computed with a multigrid method as an inner iteration furthermore we use a multiresolution minimization approach to obtain a suitable initial guess finally we present some experimental results for registration problems of synthetic images and for a real computer tomography ct magnetic resonance imaging mri registration

faces and maximizer subsets of highest weight modules abstract in this paper we study general highest weight modules v over a complex finite dimensional semisimple lie algebra g we present three formulas for the set of weights of a large family of modules v which include but are not restricted to all simple modules and all parabolic verma modules these formulas are direct and do not involve cancellations and were not previously known in the literature our results extend the notion of the weyl polytope to general highest weight g modules v we also show that for all simple modules the convex hull of the weights is a w j invariant polyhedron for some parabolic subgroup w j we compute its vertices faces and symmetries more generally we also do this for all parabolic verma modules and for all modules v with highest weight not on a simple root hyperplane to show our results we extend the notion of convexity to arbitrary additive subgroups a r of coefficients our techniques enable us to completely classify weak a faces of the support sets wt v in the process extending classical results of satake borel tits vinberg and casselman as well as modern variants by chari dolbin ridenour and cellini marietti to general highest weight modules

geometry of total variation regularized l p model in this paper the geometry and scale selection properties of the total variation tv regularized l p model are rigorously analyzed some intrinsic features different from the tv l 1 model are derived and demonstrated numerical algorithms based on recent developed augmented lagrangian methods are implemented and numerical results consistent with the theoretical results are provided

an mathcal o 1 k convergence rate for the variable stepsize bregman operator splitting algorithm an earlier paper proved the convergence of a variable stepsize bregman operator splitting algorithm bosvs for minimizing phi bu h u where h and phi are convex functions and phi is possibly nonsmooth the algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems in this paper the convergence rate of bosvs is analyzed when h u au f 2 where a is a matrix it is shown that for an ergodic approximation bf u k obtained by averaging k bosvs iterates the error in the objective value phi b bf u k h bf u k is cal o 1 k when the optimization problem has a unique solution u we obtain the estimate bf u k u cal o 1 sqrt k the theoretical analysis is compared to observed convergence rates for partially parallel magnetic resonance image reconstruction problems where a is a large dense ill conditioned matrix

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

conditioning analysis of nonlocal integral operators in fractional sobolev spaces we study the conditioning of nonlocal integral operators with singular and integrable kernels in fractional sobolev spaces these operators are used for instance in peridynamics formulation and nonlocal diffusion in one dimension 1d we present sharp quantification of the extremal eigenvalues in all three parameters size of nonlocality mesh size and regularity of the fractional sobolev space we accomplish sharpness both rigorously and numerically for the minimal eigenvalue we obtain sharpness analytically by using a nonlocal characterization of sobolev spaces we verify this estimate by exploiting the toeplitz property of the stiffness matrix however the analytical approach fails to give sharp quantification of the maximal eigenvalue hence in 1d we take an algebraic approach by directly working with the stiffness matrix entries which have complicated expressions due to all three parameters we systematically characterize the nonzero entries and dramatically simplify their expressions by u

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

faces of highest weight modules and the universal weyl polyhedron abstract let v be a highest weight module over a kac moody algebra g and let conv v denote the convex hull of its weights we determine the combinatorial isomorphism type of conv v i e we completely classify the faces and their inclusions in the special case where g is semisimple this brings closure to a question studied by cellini and marietti 2015 7 for the adjoint representation and by khare 2016 2017 17 18 for most modules the determination of faces of finite dimensional modules up to the weyl group action and some of their inclusions also appears in previous works of satake 1960 25 borel and tits 1965 3 vinberg 1990 26 and casselman 1997 6 for any subset of the simple roots we introduce a remarkable convex cone which we call the universal weyl polyhedron which controls the convex hulls of all modules parabolically induced from the corresponding levi factor namely the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules as well as how faces degenerate as the highest weight gets increasingly singular to our knowledge this cone is new in finite and infinite type we further answer a question of michel brion by showing that the localization of conv v along a face is always the convex hull of the weights of a parabolically induced module finally as we determine the inclusion relations between faces representation theoretically from the set of weights without recourse to convexity we answer a similar question for highest weight modules over symmetrizable quantum groups

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

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

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

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

tropical bounds for eigenvalues of matrices abstract let 1 n denote the eigenvalues of a n n matrix ordered by nonincreasing absolute value and let 1 n denote the tropical eigenvalues of an associated n n matrix obtained by replacing every entry of the original matrix by its absolute value we show that for all 1 k n 1 k c n k 1 k where c n k is a combinatorial constant depending only on k and on the pattern of the matrix this generalizes an inequality by friedland 1986 corresponding to the special case k 1

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

wavelet frame based algorithm for 3d reconstruction in electron microscopy in electron microscopy three dimensional 3d reconstruction is one key component in many computerized techniques for solving 3d structures of large protein assemblies using electron microscopy images of particles main challenges in 3d reconstruction include very low signal to noise ratio and very large scale of data sets involved in the computation motivated by the recent advances of sparsity based regularization in the wavelet frame domain for solving various linear inverse problems in imaging science we proposed a wavelet tight frame based 3d reconstruction approach that exploits the sparsity of the 3d density map in a wavelet tight frame system the proposed approach not only runs efficiently in terms of cpu time but also requires a much lower memory footprint than existing framelet based regularization methods the convergence of the proposed iterative scheme and the functional it minimizes is also examined together with the connection to existing wavelet frame based regularizations the numeric

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

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

max plus singular values in this paper we prove a new characterization of the max plus singular values of a max plus matrix as the max plus eigenvalues of an associated max plus matrix pencil this new characterization allows us to compute max plus singu lar values quickly and accurately as well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially pa rameterized we show experimentally that max plus singular values give order of magnitude approximations to the classical singular values of parameter inde pendent classical matrices we also discuss hungarian scaling which is a diagonal scaling strategy for preprocessing classical linear systems we show that hungarian scaling can dramatically reduce the 2 norm condition number and that this action can be explained using our new theory for max plus singular values keywords tropical algebra max plus algebra singular values diagonal scaling condition number optimal assignment problem we also discuss hungarian scaling which is a diagonal scaling strategy for preprocessing classical linear systems we show that hungarian scaling can dramatically reduce the d norm condition number and that this action can be explained using our new theory for max plus singular values

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