variational bayesian image restoration based on a product of t distributions image prior image priors based on products have been recognized to offer many advantages because they allow simultaneous enforcement of multiple constraints however they are inconvenient for bayesian inference because it is hard to find their normalization constant in closed form in this paper a new bayesian algorithm is proposed for the image restoration problem that bypasses this difficulty an image prior is defined by imposing student t densities on the outputs of local convolutional filters a variational methodology with a constrained expectation step is used to infer the restored image numerical experiments are shown that compare this methodology to previous ones and demonstrate its advantages

primal and dual bregman methods with application to optical nanoscopy measurements in nanoscopic imaging suffer from blurring effects modeled with different point spread functions psf some apparatus even have psfs that are locally dependent on phase shifts additionally raw data are affected by poisson noise resulting from laser sampling and photon counts in fluorescence microscopy in these applications standard reconstruction methods em filtered backprojection deliver unsatisfactory and noisy results starting from a statistical modeling in terms of a map likelihood estimation we combine the iterative em algorithm with total variation tv regularization techniques to make an efficient use of a priori information typically tv based methods deliver reconstructed cartoon images suffering from contrast reduction we propose extensions to em tv based on bregman iterations and primal and dual inverse scale space methods in order to obtain improved imaging results by simultaneous contrast enhancement besides further generalizations of the primal and dual scale space methods in terms of general convex variational regularization methods we provide error estimates and convergence rates for exact and noisy data we illustrate the performance of our techniques on synthetic and experimental biological data

common structure of techniques for choosing smoothing parameters in regression problems on tente de relier deux methodes de choix du parametre de lissage l une basee sur la mesure du risque et l autre sur le degre d ajustement de la regression aux donnees

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

normal boundary intersection a new method for generating the pareto surface in nonlinear multicriteria optimization problems this paper proposes an alternate method for finding several pareto optimal points for a general nonlinear multicriteria optimization problem such points collectively capture the trade off among the various conflicting objectives it is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the pareto set given an evenly distributed set of parameters a property which the popular method of minimizing weighted combinations of objective functions lacks further this method can handle more than two objectives while retaining the computational efficiency of continuation type algorithms this is an improvement over continuation techniques for tracing the trade off curve since continuation strategies cannot easily be extended to handle more than two objectives

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

a fast adaptive parameter estimation for total variation image restoration estimation of the regularization parameter which strikes a balance between the data fidelity and regularity is essential for successfully solving ill posed image restoration problems based on the classical total variation tv model and prevalent alternating direction method of multipliers we hammer out a fast algorithm being able to simultaneously estimate the regularization parameter and restore the degraded image by applying variable splitting technique to both the regularization term and data fidelity term we overcome the nondifferentiability of tv and achieve a closed form to update the regularization parameter in each iteration the solution is guaranteed to satisfy morozov s discrepancy principle furthermore we present a convergence proof for the proposed algorithm on the premise of a variable regularization parameter experimental results demonstrate that the proposed algorithm is superior in speed and competitive in accuracy compared with several state of the art methods besides the proposed method can be smoothly extended to the multichannel image restoration

an iterative linear expansion of thresholds for ell_ 1 based image restoration this paper proposes a novel algorithmic framework to solve image restoration problems under sparsity assumptions as usual the reconstructed image is the minimum of an objective functional that consists of a data fidelity term and an l1 regularization however instead of estimating the reconstructed image that minimizes the objective functional directly we focus on the restoration process that maps the degraded measurements to the reconstruction our idea amounts to parameterize the process as a linear combination of few elementary thresholding functions let and to solve the linear weighting coefficients by minimizing the objective functional it is then possible to update the thresholding functions and to iterate this process i let the key advantage of such a linear parametrization is that the problem size reduces dramatically each time we only need to solve an optimization problem over the dimension of the linear coefficients typically less than 10 instead of the whole image dimension with the elementary thresholding functions satisfying certain constraints a global convergence of the iterated let algorithm is guaranteed experiments on several test images over a wide range of noise levels and different types of convolution kernels clearly indicate that the proposed framework usually outperforms state of the art algorithms in terms of both the cpu time and the number of iterations

generalized higher degree total variation hdtv regularization we introduce a family of novel image regularization penalties called generalized higher degree total variation hdtv these penalties further extend our previously introduced hdtv penalties which generalize the popular total variation tv penalty to incorporate higher degree image derivatives we show that many of the proposed second degree extensions of tv are special cases or are closely approximated by a generalized hdtv penalty additionally we propose a novel fast alternating minimization algorithm for solving image recovery problems with hdtv and generalized hdtv regularization the new algorithm enjoys a tenfold speed up compared with the iteratively reweighted majorize minimize algorithm proposed in a previous paper numerical experiments on 3d magnetic resonance images and 3d microscopy images show that hdtv and generalized hdtv improve the image quality significantly compared with tv

total nuclear variation and jacobian extensions of total variation for vector fields we explore a class of vectorial total variation vtv measures formed as the spatial sum of a pixel wise matrix norm of the jacobian of a vector field we give a theoretical treatment that indicates that while color smearing and affine coupling bias often reported as gray scale bias are typically cited as drawbacks for vtv these are actually fundamental to smoothing vector direction i e smoothing hue and saturation in color images additionally we show that encouraging different vector channels to share a common gradient direction is equivalent to minimizing jacobian rank we thus propose total nuclear variation tnv and since nuclear norm is the convex envelope of matrix rank we argue that tnv is the optimal convex regularizer for enforcing shared directions we also propose extended jacobians which use larger neighborhoods than the conventional finite difference operator and we discuss efficient vtv optimization algorithms in simple color image denoising experiments tnv outperformed other common vtv regularizers and was further improved by using extended jacobians tnv was also competitive with the method of non local means often outperforming it by 0 25 to 2 db when using extended jacobians

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

on min norm and min max methods of multi objective optimization this paper examines pareto optimality of solutions to multi objective problems scalarized in the min norm compromise programming generalized goal programming or unrestricted min max formulations issues addressed include among others uniqueness in solution or objective space penalization for over achievement of goals min max reformulation of goal programming inferiority in tchebycheff norm minimization strength and weakness of weighted bound optimization quasi satisficing decision making just attaining or even over passing the goals trading off by modifying weights or goals non convex pareto frontier new general necessary and sufficient conditions for both pareto optimality and weak pareto optimality are presented various formulations are compared in theoretical performance with respect to the goal point location ideas for advanced goal programming and interactive decision making are introduced

finding periodic solutions of ordinary differential equations via homotopy method this paper deals with the problem of finding periodic solutions of ordinary differential equations in our approach by means of the homotopy method the equation considered is linked to a simpler equation by introducing a parameter as a result the problem of finding periodic solutions is transformed into that of finding suitable solutions of the cauchy problem following the path of solutions of the cauchy problems starting from the cauchy problem of a simpler equation one can obtain the desired periodic solutions in general the solutions of the cauchy problem of the simpler equation are easily determined relative to periodic solutions of the equation considered consequently an efficient and global method of finding periodic solutions is established

an adaptive scalarization method in multiobjective optimization this paper presents a new method for the numerical solution of nonlinear multiobjective optimization problems with an arbitrary partial ordering in the objective space induced by a closed pointed convex cone this algorithm is based on the well known scalarization approach by pascoletti and serafini and adaptively controls the scalarization parameters using new sensitivity results the computed image points give a nearly equidistant approximation of the whole pareto surface the effectiveness of this new method is demonstrated with various test problems and an applied problem from medicine

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

coherence enhancing diffusion filtering the completion of interrupted lines or the enhancement of flow like structures is a challenging task in computer vision human vision and image processing we address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the so called interest operator second moment matrix structure tensor an m dimensional formulation of this method is analysed with respect to its well posedness and scale space properties an efficient scheme is presented which uses a stabilization by a semi implicit additive operator splitting aos and the scale space behaviour of this method is illustrated by applying it to both 2 d and 3 d 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

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

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

behavioral analysis of anisotropic diffusion in image processing in this paper we analyze the behavior of the anisotropic diffusion model of perona and malik 1990 the main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem so its behavior can be analyzed based on the shape of the corresponding energy surface we show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem it is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space we give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima the mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator and the other that flows normally to the edges and thus acts as an enhancement operator

nonlocal regularization of inverse problems a unified variational framework we introduce a unifying energy minimization framework for nonlocal regularization of inverse problems in contrast to the weighted sum of square differences between image pixels used by current schemes the proposed functional is an unweighted sum of inter patch distances we use robust distance metrics that promote the averaging of similar patches while discouraging the averaging of dissimilar patches we show that the first iteration of a majorize minimize algorithm to minimize the proposed cost function is similar to current nonlocal methods the reformulation thus provides a theoretical justification for the heuristic approach of iterating nonlocal schemes which re estimate the weights from the current image estimate thanks to the reformulation we now understand that the widely reported alias amplification associated with iterative nonlocal methods are caused by the convergence to local minimum of the nonconvex penalty we introduce an efficient continuation strategy to overcome this problem the similarity of the proposed criterion to widely used nonquadratic penalties e g total variation and lp semi norms opens the door to the adaptation of fast algorithms developed in the context of compressive sensing we introduce several novel algorithms to solve the proposed nonlocal optimization problem thanks to the unifying framework these fast algorithms are readily applicable for a large class of distance metrics

efficient algorithms for hybrid regularizers based image denoising and deblurring to better eliminate the staircase effect and simultaneously preserve edge details this paper investigates a hybrid regularizers model for image denoising and deblurring this technique closely incorporates the advantages of the classical total variation tv filter and the fourth order filter computationally we develop an extremely efficient relaxation scheme and alternating minimization algorithm and give the rigorous convergence analyses there in detail provided experimental results distinctly illustrate the high efficiency of the addressed numerical algorithms also in comparison with the recovered results by the state of the art models simulations manifestly demonstrate the competitive performance of our proposed scheme in reducing the blocky images and sharply maintaining the edge features

a parallel alternating direction method with application to compound l1 regularized imaging inverse problems we derive a parallel alternating direction method of multipliers padmm and apply it to compound l1 regularized imaging inverse problems the proposed method is capable of locating the saddle point of large scale convex minimization problems with the sum of several nonsmooth but proximable terms using an operator splitting strategy the objective is decomposed into subproblems that are conveniently individually and simultaneously solved with the assistance of the moreau decomposition our method excludes auxiliary variables that exist in the admm and possesses a compacter structure thus the proposed method is preferable in distributed computation the convergence proof and convergence rate analysis are presented application to both image restoration and image compressed sensing demonstrates the effectiveness and efficiency of the proposed method

fractional tikhonov regularization with a nonlinear penalty term tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least squares problems with a severely ill conditioned matrix and an error contaminated data vector right hand side this regularization method replaces the given problem by a penalized least squares problem it is well known that tikhonov regularization in standard form may yield approximate solutions that are too smooth i e the computed approximate solution may lack many details that the desired solution of the associated but unavailable error free problem might possess fractional tikhonov regularization methods have been introduced to remedy this shortcoming however the computed solution determined by fractional tikhonov methods in standard form may display undesirable spurious oscillations this paper proposes that fractional tikhonov methods be equipped with a nonlinear penalty term such as a tv norm penalty term to reduce unwanted oscillations numerical examples illustrate the benefits of this approach

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

alternating direction method for balanced image restoration this paper presents an efficient algorithm for solving a balanced regularization problem in the frame based image restoration the balanced regularization is usually formulated as a minimization problem involving an l2 data fidelity term an l1 regularizer on sparsity of frame coefficients and a penalty on distance of sparse frame coefficients to the range of the frame operator in image restoration the balanced regularization approach bridges the synthesis based and analysis based approaches and balances the fidelity sparsity and smoothness of the solution our proposed algorithm for solving the balanced optimal problem is based on a variable splitting strategy and the classical alternating direction method this paper shows that the proposed algorithm is fast and efficient in solving the standard image restoration with balanced regularization more precisely a regularized version of the hessian matrix of the l2 data fidelity term is involved and by exploiting the related fast tight parseval frame and the special structures of the observation matrices the regularized hessian matrix can perform quite efficiently for the frame based standard image restoration applications such as circular deconvolution in image deblurring and missing samples in image inpainting numerical simulations illustrate the efficiency of our proposed algorithm in the frame based image restoration with balanced regularization

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

a multi level non linear multi objective decision making under fuzziness this paper studies a three level non linear multi objective decision making tln modm problem with linear or non linear constraints and in which the objective function at every level are non linear functions which are to be maximized this paper makes an extension work of abo sinna j operat res soc india opsearch 38 5 2001 484 495 which deal with a bi level non linear multi objective decision making problem under fuzziness the three level programming tlp problem whether from the stand point of the three planner stackelberg behavior or from the interactive organizational behavior is a very practical problem and encountered frequently in actual practice this paper proposes a three planner multi objective decision making model and solution method for solving this problem this method uses the concepts of tolerance membership function and multi objective optimization at each level to develop a fuzzy max min decision model for generating pareto optimal satisfactory solution for tln modm problem the first level decision maker fldm specifies his her objective functions and decisions with possible tolerances which are described by membership functions of fuzzy set theory then the second level decision maker sldm specifies his her objective functions and decisions in the view of the fldm with possible tolerances which are described by membership functions of fuzzy set theory finally the third level decision maker tldm uses the preference information for the fldm and sldm to solves his her problem subject to the two upper level decision makers restrictions an illustrative numerical example is given to demonstrate the obtained results

higher degree total variation hdtv regularization for image recovery we introduce novel image regularization penalties to overcome the practical problems associated with the classical total variation tv scheme motivated by novel reinterpretations of the classical tv regularizer we derive two families of functionals involving higher degree partial image derivatives we term these families as isotropic and anisotropic higher degree tv hdtv penalties respectively the isotropic penalty is the mixed norm of the directional image derivatives while the anisotropic penalty is the separable norm of directional derivatives these functionals inherit the desirable properties of standard tv schemes such as invariance to rotations and translations preservation of discontinuities and convexity the use of mixed norms in isotropic penalties encourages the joint sparsity of the directional derivatives at each pixel thus encouraging isotropic smoothing in contrast the fully separable norm in the anisotropic penalty ensures the preservation of discontinuities while continuing to smooth along the line like features this scheme thus enhances the linenlike image characteristics analogous to standard tv we also introduce efficient majorize minimize algorithms to solve the resulting optimization problems the numerical comparison of the proposed scheme with classical tv penalty current second degree methods and wavelet algorithms clearly demonstrate the performance improvement specifically the proposed algorithms minimize the staircase and ringing artifacts that are common with tv and wavelet schemes while better preserving the singularities we also observe that anisotropic hdtv penalty provides consistently improved reconstructions compared with the isotropic hdtv penalty

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

convex non convex image segmentation a convex non convex variational model is proposed for multiphase image segmentation we consider a specially designed non convex regularization term which adapts spatially to the image structures for a better control of the segmentation boundary and an easy handling of the intensity inhomogeneities the nonlinear optimization problem is efficiently solved by an alternating directions methods of multipliers procedure we provide a convergence analysis and perform numerical experiments on several images showing the effectiveness of this procedure

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

an anisotropic fourth order diffusion filter for image noise removal fourth order nonlinear diffusion filters used for image noise removal are mainly isotropic filters it means that the spatially varying diffusivity determined by a diffusion function is applied on the image regardless of the orientation of its local features however the optimal choice of parameters in the numerical solver of these filters for having a minimal distortion of the image features results in forming speckle noise on the denoised image and a very slow convergence rate especially when the noise level is moderately high in this paper a new fourth order nonlinear diffusion filter is introduced which has an anisotropic behavior on the image features in the proposed filter it is shown that a suitable choice for a set of diffusivity functions to unevenly control the strength of the diffusion on the directions of the level set and gradient leads to a good edge preservation capability compared to the other diffusion or regularization filters the comparison of the results obtained by the proposed filter with those of the other second and fourth order filters shows that the proposed method produces a noticeable improvement in the quality of denoised images evaluated subjectively and quantitatively

adaptive smoothing respecting feature directions accurate extraction of the image feature directions is essential to image smoothing and other image processing tasks we show that the gradient based feature direction extraction method can be very erroneous the gradient is too local and it cannot detect oscillations we have developed two new methods the hessian method an approach using higher order differentiations and the gabor 1946 method an approach using space frequency analysis

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

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

improved bounds for subband adaptive iterative shrinkage thresholding algorithms this paper presents new methods for computing the step sizes of the subband adaptive iterative shrinkage thresholding algorithms proposed by bayram selesnick and vonesch unser the method yields tighter wavelet domain bounds of the system matrix thus leading to improved convergence speeds it is directly applicable to non redundant wavelet bases and we also adapt it for cases of redundant frames it turns out that the simplest and most intuitive setting for the step sizes that ignores subband aliasing is often satisfactory in practice we show that our methods can be used to advantage with reweighted least squares penalty functions as well as l1 penalties we emphasize that the algorithms presented here are suitable for performing inverse filtering on very large datasets including 3d data since inversions are applied only to diagonal matrices and fast transforms are used to achieve all matrix vector products

multiobjective bilevel optimization in this work nonlinear non convex multiobjective bilevel optimization problems are discussed using an optimistic approach it is shown that the set of feasible points of the upper level function the so called induced set can be expressed as the set of minimal solutions of a multiobjective optimization problem this artificial problem is solved by using a scalarization approach by pascoletti and serafini combined with an adaptive parameter control based on sensitivity results for this problem the bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated for the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable an algorithm is presented for the first time and applied to two problems a theoretical example and a problem arising in applications

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

a new nonconvex approach for image restoration with gamma noise abstract in this paper a new total generalized variational tgv model for restoring images with multiplicative noise is proposed which contains a nonconvex fidelity term and a tgv term we use a difference of convex functions algorithm dca to deal with the proposed model for multiplicative noise removal there exist many models and algorithms most of which focus on convex approximation so that numerical algorithms with guaranteed convergence can be designed unlike these algorithms we use the dca algorithm to remove multiplicative noise by numerical experiments it is shown that the proposed approach leads to a better solution compared with the gradient projection algorithm for solving the classic multiplicative noise removal models we prove that the sequence generated by the dca algorithm converges to a stationary point which satisfies the first order optimality condition finally we demonstrate the performance of our whole scheme by numerical examples a comparison with other methods is provided as well numerical results demonstrate that the proposed algorithm significantly outperforms some previous methods for multiplicative gamma noise removal

robust scale space filter using second order partial differential equations this paper describes a robust scale space filter that adaptively changes the amount of flux according to the local topology of the neighborhood in a manner similar to modeling heat or temperature flow in physics the robust scale space filter is derived by coupling fick s law with a generalized continuity equation in which the source or sink is modeled via a specific heat capacity the filter plays an essential part in two aspects first an evolution step size is adaptively scaled according to the local structure enabling the proposed filter to be numerically stable second the influence of outliers is reduced by adaptively compensating for the incoming flux we show that classical diffusion methods represent special cases of the proposed filter by analyzing the stability condition of the proposed filter we also verify that its evolution step size in an explicit scheme is larger than that of the diffusion methods the proposed filter also satisfies the maximum principle in the same manner as the diffusion our experimental results show that the proposed filter is less sensitive to the evolution step size as well as more robust to various outliers such as gaussian noise impulsive noise or a combination of the two

stochastic multiobjective problems with complementarity constraints and applications in healthcare management abstract we consider a class of stochastic multiobjective problems with complementarity constraints smopccs in this paper we derive the first order optimality conditions including the clarke mordukhovich strong type stationarity in the pareto sense for the smopcc since these first order optimality systems involve some unknown index sets we reformulate them as nonlinear equations with simple constraints then we introduce an asymptotic method to solve these constrained equations furthermore we apply this methodology results to a patient allocation problem in healthcare management

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

using the l curve for determining optimal regularization parameters the l curve is a plot in ordinary or doubly logarithmic scale of the norm of tikhonov regularized solutions of an ill posed problem versus the norm of the residuals we show that the popular criterion of choosing the parameter corresponding to the point with maximal curvature of the l curve does not yield a convergent regularization strategy to solve the ill posed problem nevertheless the l curve can be used to compute the regularization parameters produced by morozov s discrepancy principle and by an order optimal variant of the discrepancy principle proposed by engl and gfrerer in an alternate way

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

periodic solutions of 2n 1 th order ordinary differential equations abstract the goal of this paper is to prove the existence and uniqueness of periodic solutions for a class of the odd order differential equations here the leray schauder degree theory and the schwarz inequality are used

optimally isotropic laplacian operator laplacian operators used in the literature for digital image processing are not rotationally invariant we examine the anisotropy of 3 spl times 3 laplacian operators for images quantized in square pixels and find the operator which has the minimum overall anisotropy

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

total variation and high order total variation adaptive model for restoring blurred images with cauchy noise abstract in this paper we propose a novel model to restore an image corrupted by blur and cauchy noise the model is composed of a data fidelity term and two regularization terms including total variation and high order total variation total variation provides well preserved edge features but suffers from staircase effects in smooth regions whereas high order total variation can alleviate staircase effects moreover we introduce a strategy for adaptively selecting regularization parameters we develop an efficient alternating minimization algorithm for solving the proposed model numerical examples suggest that the proposed method has the advantages of better preserving edges and reducing staircase effects

hessian schatten norm regularization for linear inverse problems we introduce a novel family of invariant convex and non quadratic functionals that we employ to derive regularized solutions of ill posed linear inverse imaging problems the proposed regularizers involve the schatten norms of the hessian matrix which are computed at every pixel of the image they can be viewed as second order extensions of the popular total variation tv semi norm since they satisfy the same invariance properties meanwhile by taking advantage of second order derivatives they avoid the staircase effect a common artifact of tv based reconstructions and perform well for a wide range of applications to solve the corresponding optimization problems we propose an algorithm that is based on a primal dual formulation a fundamental ingredient of this algorithm is the projection of matrices onto schatten norm balls of arbitrary radius this operation is performed efficiently based on a direct link we provide between vector projections onto norm balls and matrix projections onto schatten norm balls finally we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data

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

a parallel primal dual splitting method for image restoration we develop a parallel primal dual splitting method to solve large scale image restoration problems which involve the sum of several linear operator coupled nonsmooth but proximable terms with the proposed method the objective function is decomposed into pieces that can be processed individually no inverse operator is involved in our method and the highly parallel structure makes it preferable in distributed computation the convergence is proven and the convergence rate is analyzed besides its equivalence to the relaxed parallel linearized alternating direction method of multipliers pladmm is addressed applications to image restoration problems with compound l1 regularizer and comparisons with state of the art methods are detailed to show the superiority of the proposed method

iterative image restoration combining total variation minimization and a second order functional a noise removal technique using partial differential equations pdes is proposed here it combines the total variational tv filter with a fourth order pde filter the combined technique is able to preserve edges and at the same time avoid the staircase effect in smooth regions a weighting function is used in an iterative way to combine the solutions of the tv filter and the fourth order filter numerical experiments confirm that the new method is able to use less restrictive time step than the fourth order filter numerical examples using images with objects consisting of edge flat and intermediate regions illustrate advantages of the proposed model

restoration of poissonian images using alternating direction optimization much research has been devoted to the problem of restoring poissonian images namely for medical and astronomical applications however the restoration of these images using state of the art regularizers such as those based upon multiscale representations or total variation is still an active research area since the associated optimization problems are quite challenging in this paper we propose an approach to deconvolving poissonian images which is based upon an alternating direction optimization method the standard regularization or maximum a posteriori map restoration criterion which combines the poisson log likelihood with a nonsmooth convex regularizer log prior leads to hard optimization problems the log likelihood is nonquadratic and nonseparable the regularizer is nonsmooth and there is a nonnegativity constraint using standard convex analysis tools we present sufficient conditions for existence and uniqueness of solutions of these optimization problems for several types of regularizers total variation frame based analysis and frame based synthesis we attack these problems with an instance of the alternating direction method of multipliers admm which belongs to the family of augmented lagrangian algorithms we study sufficient conditions for convergence and show that these are satisfied either under total variation or frame based analysis and synthesis regularization the resulting algorithms are shown to outperform alternative state of the art methods both in terms of speed and restoration accuracy

an alternating direction algorithm for total variation reconstruction of distributed parameters augmented lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems the alternating direction method of multipliers admm is one of such formulations optimization methods very recently the number of applications of the admm or variants of it to solve inverse problems in image and signal processing has increased at an exponential rate the reason for this interest is that admm decomposes a difficult optimization problem into a sequence of much simpler problems in this paper we use the admm to reconstruct piecewise smooth distributed parameters of elliptical partial differential equations from noisy and linear blurred observations of the underlying field the distributed parameters are estimated by solving an inverse problem with total variation tv regularization the proposed instance of the admm solves in each iteration an and a decoupled optimization problems an operator splitting is used to simplify the treatment of the tv regularizer avoiding its smooth approximation and yielding a simple yet effective admm reconstruction method compared with previously proposed approaches the competitiveness of the proposed method with respect to the state of the art is illustrated in simulated 1 d and 2 d elliptical equation problems which are representative of many real applications

a tv l2 h 1 pde model for effective denoising abstract in this paper we have proposed a higher order non linear pde model containing diffusion terms of t v l 2 and t v h 1 model for effective image denoising fourier spectral method for space and convexity splitting method for time have been used to solve the proposed pde model stability analysis for the time discretized equation has been derived numerical experiment has been done on some test images and the results have been compared with the results of some existing models

a proximal iteration for deconvolving poisson noisy images using sparse representations we propose an image deconvolution algorithm when the data is contaminated by poisson noise the image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms our key contributions are as follows first we handle the poisson noise properly by using the anscombe variance stabilizing transform leading to a nonlinear degradation equation with additive gaussian noise second the deconvolution problem is formulated as the minimization of a convex functional with a data fidelity term reflecting the noise properties and a nonsmooth sparsity promoting penalty over the image representation coefficients e g lscr1 norm an additional term is also included in the functional to ensure positivity of the restored image third a fast iterative forward backward splitting algorithm is proposed to solve the minimization problem we derive existence and uniqueness conditions of the solution and establish convergence of the iterative algorithm finally a gcv based model selection procedure is proposed to objectively select the regularization parameter experimental results are carried out to show the striking benefits gained from taking into account the poisson statistics of the noise these results also suggest that using sparse domain regularization may be tractable in many deconvolution applications with poisson noise such as astronomy and microscopy

a proximal based decomposition method for convex minimization problems this paper presents a decomposition method for solving convex minimization problems at each iteration the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables we derive this algorithm from rockafellar s proximal method of multipliers which involves an augmented lagrangian with an additional quadratic proximal term the algorithm preserves the good features of the proximal method of multipliers with the additional advantage that it leads to a decoupling of the constraints and is thus suitable for parallel implementation we allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem s data the method is globally convergent and at a linear rate the method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets

a fractional order adaptive regularization primal dual algorithm for image denoising this paper aims to develop a fractional order model and a primal dual algorithm for image denoising where a regularization parameter can be adjusted adaptively according to morozov discrepancy principle at each iteration to ensure that the denoised image retains in a specific set in the light of saddle point theory the convergence of our proposed algorithm is guaranteed simulations with comparisons are carried out to demonstrate the effectiveness of our proposed algorithm for image denoising

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

solving constrained total variation image restoration and reconstruction problems via alternating direction methods in this paper we study alternating direction methods for solving constrained total variation image restoration and reconstruction problems alternating direction methods can be implementable variants of the classical augmented lagrangian method for optimization problems with separable structures and linear constraints the proposed framework allows us to solve problems of image restoration impulse noise removal inpainting and image cartoon texture decomposition as the constrained model is employed we need only to input the noise level and the estimation of the regularization parameter is not required in these imaging problems experimental results for such imaging problems are presented to illustrate the effectiveness of the proposed method we show that the alternating direction method is very efficient for solving image restoration and reconstruction problems

augmented lagrangian method for total generalized variation based poissonian image restoration instead of adopting the traditional total variation as a regularizer this article introduces a second order total generalized variation regularization scheme for deconvolving poissonian image numerically an efficient augmented lagrangian method associated with alternating minimization method is described to obtain the optimal solution recursively in addition we provide the rigorous convergence analysis for the resulting algorithm at great length finally compared with the total variation based efficient strategies numerical simulations definitely indicate the competitive performance of our proposed approach to deblurring poissonian image both in terms of restoration accuracy and edge preserving ability

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

an algorithm based on particle swarm optimization for multiobjective bilevel linear problems improved mopso for multiobjective at the upper level bilevel linear problems a hybrid scheme for the selection of the global best particles is proposed an adaptive mutation is also introduced the incorporation of these mechanisms led to significantly better results this paper presents an improved multiple objective particle swarm optimization mopso algorithm to solve bilevel linear programming problems with multiple objective functions at the upper level the algorithm aims to produce a good approximation of the entire pareto front of the problem we have previously designed a mopso algorithm for the same class of problems in which several techniques for the global best selection were tested including a new one the algorithm revealed a good convergence towards the pareto front but the diversity of the solutions was a drawback the algorithm we propose herein uses a hybrid strategy for the global best selection and an adaptive mutation operator the incorporation of these mechanisms led to an improved algorithm which also showed better overall performance than considering alternative options usually employed in mopso algorithms the algorithm and computational results are presented

augmented lagrangian method for tv l1 l2 based colour image restoration abstract in this paper we introduce a total variation l 1 l 2 regularization scheme with adapting the parameter for image restoration involving blurry and noisy colour images numerically an efficient augmented lagrangian method associated with alternating minimization method is described to obtain the optimal solution recursively we provide the convergence analysis for the resulting algorithm experimental results show that our proposed model and algorithm have good signal to noise ratio snr and improvement in signal to noise ratio isnr values for a motion blur with different kinds of noises