convergence of em image reconstruction algorithms with gibbs smoothing p j green has defined an osl one step late algorithm that retains the e step of the em algorithm for image reconstruction in emission tomography but provides an approximate solution to the m step further modifications of the osl algorithm guarantee convergence to the unique maximum of the log posterior function convergence is proved under a specific set of sufficient conditions several of these conditions concern the potential function of the gibb s prior and a number of candidate potential functions are identified generalization of the osl algorithm to transmission tomography is also considered

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

a fast automatic algorithm for image denoising by a regularization method a new fast iterative algorithm is proposed to denoise images affected by gaussian additive noise by solving a constrained optimization problem a family of functionals parameterized by a few hyperparameters already proposed in the literature to solve denoising problems is considered the algorithm estimates jointly the image and the hyperparameters therefore providing an automatic method each member of the family is made up of a coherence with the data term and a term enforcing a roughness penalty and preserving jump discontinuities assuming we know the noise variance an adequacy constraint is also considered the algorithm computes a member of the family and a minimizer of it which satisfies the constraint a convergence proof is provided we then consider a heuristic version of the algorithm which gives restorations of comparable quality whose computational complexity is a linear function of the pixels number experimental results on synthetic and real data are presented moreover numerical comparisons with several fast denoising methods are provided

error bounds for monotone linear complementarity problems we give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered when the point satisfies the linear inequalities of the complementarity problem the residual consists of the complementarity condition plus its square root this latter term is essential and without it the error bound cannot hold we also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

the mean field theory in em procedures for markov random fields in many signal processing and pattern recognition applications the hidden data are modeled as markov processes and the main difficulty of using the maximisation em algorithm for these applications is the calculation of the conditional expectations of the hidden markov processes it is shown how the mean field theory from statistical mechanics can be used to calculate the conditional expectations for these problems efficiently the efficacy of the mean field theory approach is demonstrated on parameter estimation for one dimensional mixture data and two dimensional unsupervised stochastic model based image segmentation experimental results indicate that in the 1 d case the mean field theory approach provides results comparable to those obtained by baum s 1987 algorithm which is known to be optimal in the 2 d case where baum s algorithm can no longer be used the mean field theory provides good parameter estimates and image segmentation for both synthetic and real world images

approximate maximum likelihood hyperparameter estimation for gibbs priors the parameters of the prior the hyperparameters play an important role in bayesian image estimation of particular importance for the case of gibbs priors is the global hyperparameter spl beta which multiplies the hamiltonian here we consider maximum likelihood ml estimation of spl beta from incomplete data i e problems in which the image which is drawn from a gibbs prior is observed indirectly through some degradation or blurring process important applications include image restoration and image reconstruction from projections exact ml estimation of spl beta from incomplete data is intractable for most image processing here we present an approximate ml estimator that is computed simultaneously with a maximum a posteriori map image estimate the algorithm is based on a mean field approximation technique through which multidimensional gibbs distributions are approximated by a separable function equal to a product of one dimensional 1 d densities we show how this approach can be used to simplify the ml estimation problem we also show how the gibbs bogoliubov feynman gbf bound can be used to optimize the approximation for a restricted class of problems we present the results of a monte carlo study that examines the bias and variance of this estimator when applied to image restoration

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

weak sharpness for gap functions in vector variational inequalities abstract in this paper characterizations of the set of solutions for vvi are presented by using scalarization approaches the set of solutions of vvi is shown to be the set of weak sharpness for gap functions of some scalarization of vvi and for gap functions of vvi under semi strong monotonicity some examples are given to illustrate these results

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

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

extension of hoffman s error bound to polynomial systems given any linear system defined by linear inequalities equalities in re n hoffman s error bound says that the distance from any point x in re n to the solution set of the linear system is bounded by a constant independent of x times a certain residual function evaluated at x this paper considers arbitrary possibly nonconvex polynomial systems defined by polynomial inequalities equalities it is shown that the solution sets of such systems are in general holder continuous as the right hand side changes in contrast to the lipschitzian continuity for the linear case also for any convex quadratic inequality system possessing an interior solution the authors show that its solution set behaves in a lipschitzian manner as the right hand side varies this result sharpens the earlier error bound results by mangasarian and robinson for general convex differentiable inequality systems which in addition to the interiority assumption require either an asymptotic constraint qualification con

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

conic positive definiteness and sharp minima of fractional orders in vector optimization problems abstract motivated by the fact that the usual positive definiteness does not work in an infinite space we introduce the concept of s positive definiteness with respect to an ordering cone in a general banach space and show that the s positive definiteness plays the same role as the usual positive definiteness in the finite dimensional case as applications we study sharp and weak sharp minima of fractional orders in vector optimization

piecewise linear multicriteria programs the continuous case and its discontinuous generalization in this paper we study piecewise linear multicriteria programs that is multicriteria programs with either a continuous or discontinuous piecewise linear objective function and a polyhedron set constraint we obtain an algebraic representation of a semi closed polyhedron and apply it to show that the image of a semi closed polyhedron under a continuous linear function is always one semi closed polyhedron we establish that the weak pareto solution point set of a piecewise linear multicriteria program is the union of finitely many semi closed polyhedra we propose an algorithm for finding the pareto point set of a continuous piecewise linear bi criteria program and generalize it to the discontinuous case we apply our algorithm to solve the discontinuous bi criteria portfolio selection problem with an l risk measure and transaction costs and show that this algorithm can be improved by using an ideal point strategy

an empirical study of the simulation of various models used for images markov random fields are typically used as priors in bayesian image restoration methods to represent spatial information in the image commonly used markov random fields are not in fact capable of representing the moderate to large scale clustering present in naturally occurring images and can also be time consuming to simulate requiring iterative algorithms which can take hundreds of thousands of sweeps of the image to converge markov mesh models a causal subclass of markov random fields are however readily simulated we describe an empirical study of simulated realizations from various models used in the literature and we introduce some new mesh type models we conclude however that while large scale clustering may be represented by such models strong directional effects are also present for all but very limited parameterizations it is emphasized that the results do not detract from the use of markov random fields as representers of local spatial properties which is their main purpose in the implementation of bayesian statistical approaches to image analysis brief allusion is made to the issue of parameter estimation

a maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography the expectation maximization method for maximum likelihood image reconstruction in emission tomography based on the poisson distribution of the statistically independent components of the image and measurement vectors is extended to a maximum aposteriori image reconstruction using a multivariate gaussian a priori probability distribution of the image vector the approach is equivalent to a penalized maximum likelihood estimation with a special choice of the penalty function the expectation maximization method is applied to find the a posteriori probability maximizer a simple iterative formula is derived for a penalty function that is a weighted sum of the squared deviations of image vector components from their a priori mean values the method is demonstrated to be superior to pure likelihood maximization in that the penalty function prevents the occurrence of irregular high amplitude patterns in the image with a large number of iterations the so called checkerboard effect or noise artifact

statistical regularization in linearized microwave imaging through mrf based map estimation hyperparameter estimation and image computation the application of a markov random fields mrf based maximum a posteriori map estimation method for microwave imaging is presented in this paper the adopted mrf family is the so called gaussian mrf gmrf whose energy function is quadratic in order to implement the map estimation first the mrf hyperparameters are estimated by means of the expectation maximization em algorithm extended in this case to complex and nonhomogeneous images then it is implemented by minimizing a cost function whose gradient is fully analytically evaluated thanks to the quadratic nature of the energy function of the mrf well posedness and efficiency of the proposed method can be simultaneously guaranteed numerical results also performed on real data show the good performance of the method also when compared with conventional techniques like tikhonov regularization

minimizing and stationary sequences of constrained optimization problems we study some fundamental asymptotic properties associated with the well posedness of constrained optimization problems emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic karush kuhn tucker kkt optimality conditions unlike the subdifferential approach used by auslender cominetti and crouzeix we use a residual function approach that is closely tied to the theory of error bounds this approach handles constraints explicitly and allows the effective treatment of infeasible stationary sequences the asymptotic kkt conditions provide an asymptotic optimality certificate for inequality constrained programs specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with holderian minima are discussed an application of the results to the convergence of a family of newton type iterative descent methods involving singular quasi newton matrices for solving a constrained minimization problem is also presented

weak sharp minima for piecewise linear multiobjective optimization in normed spaces abstract in a general normed space we consider a piecewise linear multiobjective optimization problem we prove that a cone convex piecewise linear multiobjective optimization problem always has a global weak sharp minimum property by a counter example we show that the weak sharp minimum property does not necessarily hold if the cone convexity assumption is dropped moreover under the assumption that the ordering cone is polyhedral we prove that a not necessarily cone convex piecewise linear multiobjective optimization problem always has a bounded weak sharp minimum property

fully bayesian estimation of gibbs hyperparameters for emission computed tomography data in recent years many investigators have proposed gibbs prior models to regularize images reconstructed from emission computed tomography data unfortunately hyperparameters used to specify gibbs priors can greatly influence the degree of regularity imposed by such priors and as a result numerous procedures have been proposed to estimate hyperparameter values from observed image data many of these procedures attempt to maximize the joint posterior distribution on the image scene to implement these methods approximations to the joint posterior densities are required because the dependence of the gibbs partition function on the hyperparameter values is unknown here the authors use recent results in markov chain monte carlo mcmc sampling to estimate the relative values of gibbs partition functions and using these values sample from joint posterior distributions on image scenes this allows for a fully bayesian procedure which does not fix the hyperparameters at some estimated or specified value but enables uncertainty about these values to be propagated through to the estimated intensities the authors utilize realizations from the posterior distribution for determining credible regions for the intensity of the emission source the authors consider two different markov random field mrf models the power model and a line site model as applications they estimate the posterior distribution of source intensities from computer simulated data as well as data collected from a physical single photon emission computed tomography spect phantom

the distance to a polyhedron abstract we give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space e we show that in the case when e r n endowed with the l norm these estimates are in general better than a recent estimate by cook gerards schrijver and tardos 2

completely rank nonincreasing linear maps abstract we give purely algebraic characterizations of the maps that are approximate compressions or skew compressions of a unital representation of a c algebra the key techniques used also relate to closures of joint similarity orbits of matrices and elementary maps on b h

asymptotic constraint qualifications and global error bounds for convex inequalities in this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions i the bounded excess condition ii slater condition together with the asymptotic constraint qualification defined by auslender and crouzeix 1 and iii positivity of normal directional derivatives of the maximum of the constraint functions introduced by lewis and pang 12

simultaneous parameter estimation and segmentation of gibbs random fields using simulated annealing an adaptive segmentation algorithm is developed which simultaneously estimates the parameters of the underlying gibbs random field grf and segments the noisy image corrupted by additive independent gaussian noise the algorithm which aims at obtaining the maximum a posteriori map segmentation is a simulated annealing algorithm that is interrupted at regular intervals for estimating the grf parameters maximum likelihood ml estimates of the parameters based on the current segmentation are used to obtain the next segmentation it is proven that the parameter estimates and the segmentations converge in distribution to the ml estimate of the parameters and the map segmentation with those parameter estimates respectively due to computational difficulties however only an approximate version of the algorithm is implemented the approximate algorithm is applied on several two and four region images with different noise levels and with first order and second order neighborhoods

the lagrange multiplier rule in set valued optimization the known lagrange multiplier rule is extended to set valued constrained optimization problems using the contingent epiderivative as differentiability notion a necessary optimality condition for weak minimizers is derived which is also a sufficient condition under generalized convexity assumptions

global error bounds with fractional exponents using the partial order induced by a proper weakly lower semicontinuous function on a reflexive banach space x we give a sufficient condition for f to have error bounds with fractional exponents application is given to identify the set of such exponents for quadratic functions

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

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

quality and precision of parametric images created from pet sinogram data by direct reconstruction proof of concept we have previously implemented the direct reconstruction of dense kinetic model parameter images parametric images from sinogram data and compared it to conventional image domain kinetic parameter estimation methods although it has been shown that the direct reconstruction algorithm estimates the kinetic model parameters with lower root mean squared error than the conventional image domain techniques some theoretical obstacles remain these obstacles include the difficulty of evaluating the accuracy and precision of the estimated parameters in image domain techniques the reconstructed time activity curve tac and the model predicted tac are compared and the goodness of fit is evaluated as a measure of the accuracy and precision of the estimated parameters this approach cannot be applied to the direct reconstruction technique as there are no reconstructed tacs in this paper we propose ways of evaluating the precision and goodness of fit of the kinetic model parameters estimated by the direct reconstruction algorithm specifically precision of the estimates requires the calculation of variance images for each parameter and goodness of fit is addressed by reconstructing the difference between the measured and the fitted sinograms we demonstrate that backprojecting the difference from sinogram space to image space creates error images that can be examined for goodness of fit and model selection purposes the presence of nonrandom structures in the error images may indicate an inadequacy of the kinetic model that has been incorporated into the direct reconstruction algorithm we introduce three types of goodness of fit images we propose and demonstrate a number of runs image as a means of quantifying the adequacy or deficiency of the model we further propose and demonstrate images of the f statistic and the change in the akaike information criterion as devices for identifying the statistical advantage of one model over another at each voxel as direct reconstruction to parametric images proliferates it will be essential for imagers to adopt methods such as those proposed herein to assess the accuracy and precision of their parametric images

hybrid poisson polynomial objective functions for tomographic image reconstruction from transmission scans this paper describes rapidly converging algorithms for computing attenuation maps from poisson transmission measurements using penalized likelihood objective functions we demonstrate that an under relaxed cyclic coordinate ascent algorithm converges faster than the convex algorithm of lange see ibid vol 4 no 10 p 1430 1438 1995 which in turn converges faster than the expectation maximization em algorithm for transmission tomography to further reduce computation one could replace the log likelihood objective with a quadratic approximation however we show with simulations and analysis that the quadratic objective function leads to biased estimates for low count measurements therefore we introduce hybrid poisson polynomial objective functions that use the exact poisson log likelihood for detector measurements with low counts but use computationally efficient quadratic or cubic approximations for the high count detector measurements we demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias

on generalized second order derivatives and taylor expansions in nonsmooth optimization a representation of cominetti and correa s generalized second order directional derivative siam j control optim 28 1990 pp 789 809 is given and then applied to obtain a taylor theorem type result a conjecture in siam j control optim 28 1990 pp 789 809 concerning functions of the form max 1 leq i leq n g i x is proved under a strengthened assumption but not true otherwise

single image defogging by multiscale depth fusion restoration of fog images is important for the deweathering issue in computer vision the problem is ill posed and can be regularized within a bayesian context using a probabilistic fusion model this paper presents a multiscale depth fusion mdf method for defog from a single image a linear model representing the stochastic residual of nonlinear filtering is first proposed multiscale filtering results are probabilistically blended into a fused depth map based on the model the fusion is formulated as an energy minimization problem that incorporates spatial markov dependence an inhomogeneous laplacian markov random field for the multiscale fusion regularized with smoothing and edge preserving constraints is developed a nonconvex potential adaptive truncated laplacian is devised to account for spatially variant characteristics such as edge and depth discontinuity defog is solved by an alternate optimization algorithm searching for solutions of depth map by minimizing the nonconvex potential in the random field the mdf method is experimentally verified by real world fog images including cluttered depth scene that is challenging for defogging at finer details the fog free images are restored with improving contrast and vivid colors but without over saturation quantitative assessment of image quality is applied to compare various defog methods experimental results demonstrate that the accurate estimation of depth map by the proposed edge preserved multiscale fusion should recover high quality images with sharp details

the mean field theory in em procedures for blind markov random field image restoration a markov random field mrf model based em expectation maximization procedure for simultaneously estimating the degradation model and restoring the image is described the mrf is a coupled one which provides continuity inside regions of smooth gray tones and discontinuity at region boundaries constraints for the restoration problem which is in general ill posed the computational difficulty associated with the em procedure for mrfs is resolved by using the mean field theory from statistical mechanics an orthonormal blur decomposition is used to reduce the chances of undesirable locally optimal estimates experimental results on synthetic and real world images show that this approach provides good blur estimates and restored images the restored images are comparable to those obtained by a wiener filter in mean square error but are most visually pleasing

weak sharp minima in multicriteria linear programming in this short note we study the existence of weak sharp minima in multicriteria linear programming it is shown that weak sharp minimality holds for certain residual functions and gap functions

a local update strategy for iterative reconstruction from projections a method for bayesian reconstruction which relies on updates of single pixel values rather than the entire image at each iteration is presented the technique is similar to gauss seidel gs iteration for the solution of differential equations on finite grids the computational cost per iteration of the gs approach is found to be approximately equal to that of gradient methods for continuously valued images gs is found to have significantly better convergence at modes representing high spatial frequencies in addition gs is well suited to segmentation when the image is constrained to be discretely valued it is shown that bayesian segmentation using gs iteration produces useful estimates at much lower signal to noise ratios than required for continuously valued reconstruction the convergence properties of gradient ascent and gs for reconstruction from integral projections are analyzed and simulations of both maximum likelihood and maximum a posteriori cases are included

error bounds for convex polynomials the purpose of this paper is to investigate error bounds for convex polynomials we prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact as an application we deduce various types of error bounds for unconstrained and polyhedral constrained convex polynomials

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

a cumulant expansion technique for simultaneous markov random field image restoration and hyperparameter estimation we investigate hyperparameter estimation for incomplete data in markov random field image restoration assuming linear dependence of energies with respect to hyperparameters framework we use a cumulant expansion technique widely known in statistical physics and signal processing new insight is given on maximum likelihood estimation of hyperparameters of the prior regularization and contour probability distribution functions pdfs for an explicit joint boundary pixel process aimed to preserve discontinuities in particular the case where the prior regularization potential is an homogeneous function of pixels is fully analyzed a generalized stochastic gradient gsg algorithm with a fast sampling technique is devised aiming to achieve simultaneous hyperparameter estimation and pixel restoration image restoration performances of posterior mean performed during gsg convergence and of simulated annealing performed after gsg convergence are compared experimentally results and perspectives are given

metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization in this paper we consider the metric subregularity property for a piecewise linear multifunction with respect to a piecewise linear constraint as well as the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem of these properties we pay special attention to the global ones we first provide a result on a certain relationship between two nonnegative piecewise linear numerical functions for which the kernel of one of them is contained in the kernel of the other using this result we establish the bounded global metric subregularity results for a piecewise linear multifunction with respect to a piecewise linear set as applications we study the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem

on minimizing and critical sequences in nonsmooth optimization let f be a bounded below lower semicontinuous function from a banach space into r cup infty we study the relationships between minimizing and critical sequences of f where the criticality condition is given in terms of some subdifferential partial here the objective function f is not supposed to be convex or smooth our work extends that of auslender and crouzeix and that of chou ng and pang

error bounds for piecewise convex quadratic programs and applications in this paper we establish a local error estimate for feasible solutions of a piecewise convex quadratic program and a global error estimate for feasible solutions of a convex piecewise quadratic program these error estimates provide a unified approach for deriving many old and new error estimates for linear programs linear complementarity problems convex quadratic programs and affine variational inequality problems the approach reveals the fact that each error estimate is a consequence of some reformulation of the original problem as a piecewise convex quadratic program or a convex piecewise quadratic program in a sense even robinson s result on the upper lipschitz continuity of a polyhedral mapping can be considered as a special case of error estimates for approximate solutions of a piecewise convex quadratic program as an application we derive new global error estimates for iterates of the proximal point algorithm for solving a convex piecewise quadratic program

a generalized em algorithm for 3 d bayesian reconstruction from poisson data using gibbs priors a generalized expectation maximization gem algorithm is developed for bayesian reconstruction based on locally correlated markov random field priors in the form of gibbs functions and on the poisson data model for the m step of the algorithm a form of coordinate gradient ascent is derived the algorithm reduces to the em maximum likelihood algorithm as the markov random field prior tends towards a uniform distribution three different gibbs function priors are examined reconstructions of 3 d images obtained from the poisson model of single photon emission computed tomography are presented

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

global error bounds for convex inequality systems in banach spaces we study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system specifically we obtain an error bound result which unifies many existing results assuming a slater condition we also derive two characterizations for a convex inequality system to possess a global error bound one is in terms of metric regularity and the other is in terms of an associated convex inequality system as a consequence we show that in font bbbabstract msbm10 at 8pt mbox bbbabstract r n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system finally we discuss some applications of these results to convex programs

unsupervised bayesian convex deconvolution based on a field with an explicit partition function this paper proposes a non gaussian markov field with a special feature an explicit partition function to the best of our knowledge this is an original contribution moreover the explicit expression of the partition function enables the development of an unsupervised edge preserving convex deconvolution method the method is fully bayesian and produces an estimate in the sense of the posterior mean numerically calculated by means of a monte carlo markov chain technique the approach is particularly effective and the computational practicability of the method is shown on a simple simulated example

weak sharp minima for set valued vector variational inequalities with an application abstract in this paper the notion of weak sharp minima is employed to the investigation of set valued vector variational inequalities the gap function t for set valued strong vector variational inequalities for short svvi is proved to be less than the gap function t for set valued weak vector variational inequalities for short wvvi under certain conditions which implies that the solution set of svvi is equivalent to the solution set of wvvi moreover it is shown that weak sharp minima for the solution sets of svvi and wvvi hold for min 1 i n p t i and for gap functions t and t under the assumption of strong pseudomonotonicity where p t i is a gap function for i th component of svvi and wvvi as an application the weak pareto solution set of vector optimization problems for short vop is proved to be weak sharp minimum for min 1 i n p g i when each component g i of objective function is strongly convex

computable error bounds for convex inequality systems in reflexive banach spaces in 1952 a j hoffman proved a fundamental result of an error bound on the distance from any point to the solution set of a linear system in hbox bbb r n in siam j control 13 1975 pp 271 273 robinson extended hoffman s theorem to any system of convex inequalities in a normed linear space which satisfies the slater constraint qualification and has a bounded solution set this paper studies any system of convex inequalities in a reflexive banach space which has an unbounded solution set it is shown that hoffman s error bound holds for such a system when a related convex system which defines the recession cone of the solution set for the system satisfies the slater constraint qualification

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

partition function estimation of gibbs random field images using monte carlo simulations a monte carlo simulation technique for estimating the partition function of a general gibbs random field image is proposed by expressing the partition function as an expectation an importance sampling approach for estimating it using monte carlo simulations is developed as expected the resulting estimators are unbiased and consistent computations can be performed iteratively by using simple monte carlo algorithms with remarkable success as demonstrated by simulations the work concentrates on binary second order gibbs random fields defined on a rectangular lattice however the proposed methods can be easily extended to more general gibbs random fields their potential contribution to optimal parameter estimation and hypothesis testing problems for general gibbs random field images using a likelihood approach is anticipated

an iterative gibbsian technique for reconstruction of m ary images abstract the reconstruction of m ary images corrupted by independent noise is treated the original image is modeled by a markov random field mrf whose parameters are unknown likewise the probabilistic structure of the noise is unknown this paper presents an iterative procedure which performs the parameter estimation and image reconstruction tasks at the same time the procedure that we call gibbsian em algorithm is a generalization to the mrf context of a general algorithm known as the em algorithm used to approximate maximum likelihood estimates for incomplete data problems a number of experiments are presented in the case of gaussian noise and binary noise showing that the gibbsian em algorithm is useful and effective for image reconstruction and segmentation

identification and restoration of noisy blurred images using the expectation maximization algorithm a maximum likelihood approach to the blur identification problem is presented the expectation maximization algorithm is proposed to optimize the nonlinear likelihood function in an efficient way in order to improve the performance of the identification algorithm low order parametric image and blur models are incorporated into the identification method the resulting iterative technique simultaneously identifies and restores noisy blurred images

on image reconstruction algorithms for binary electromagnetic geotomography in this paper we discuss the selected image reconstruction methods of binary tomography in the context of their application to geophysical imaging we restrict our considerations to a discrete version of high frequency electromagnetic geotomography which we label as binary electromagnetic geotomography beg basically such an imaging technique may be applied to detect subsurface anomalies air filled voids whose attenuation coefficient is very low nearly zero value and considerably different from that for the background the assumption for a binary representation of the image to be reconstructed substantially relaxes image reconstruction problems related to ill posedness that comes from an intrinsic limitation of an angular range of projections we test two algorithms for binary tomography where the penalty term is based on the markov random field mrf model the mean field reference distribution and mean field annealing are applied to estimate the global maximizer of the gibbs boltzmann distribution associated with the objective function we also apply the projected gradient algorithm that uses a binary steering very efficient implementations of the algorithms are also given the numerical results are presented for noise free noisy and real data

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

error bounds for vector valued functions necessary and sufficient conditions abstract in this paper we attempt to extend the definition and existing local error bound criteria to vector valued functions or more generally to functions taking values in a normed linear space some new derivative like objects slopes and subdifferentials are introduced and a general classification scheme of error bound criteria is presented

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

parallelizable bayesian tomography algorithms with rapid guaranteed convergence bayesian tomographic reconstruction algorithms generally require the efficient optimization of a functional of many variables in this setting as well as in many other optimization tasks functional substitution fs has been widely applied to simplify each step of the iterative process the function to be minimized is replaced locally by an approximation having a more easily manipulated form e g quadratic but which maintains sufficient similarity to descend the true functional while computing only the substitute we provide two new applications of fs methods in iterative coordinate descent for bayesian tomography the first is a modification of our coordinate descent algorithm with one dimensional 1 d newton raphson approximations to an alternative quadratic which allows convergence to be proven easily in simulations we find essentially no difference in convergence speed between the two techniques we also present a new algorithm which exploits the fs method to allow parallel updates of arbitrary sets of pixels using computations similar to iterative coordinate descent the theoretical potential speed up of parallel implementations is nearly linear with the number of processors if communication costs are neglected

second order sufficient conditions for error bounds in banach spaces recently huang and ng presented second order sufficient conditions for error bounds of continuous and ga teaux differentiable functions in banach spaces wu and ye dropped the assumption of huang and ng on ga teaux differentiability but required the space to be a hilbert space we carry on this research in two directions first we extend wu and ye s result to some non hilbert spaces second same as huang and ng we work on banach spaces but provide different second order sufficient conditions that may allow the function to be non ga teaux differentiable

reflexivity algebraic reflexivity and linear interpolation introduction this paper was motivated by a discovery that most finite dimensional subalgebras of b h for h an infinite dimensional hilbert space are reflexive with the only obstructions to reflexivity being finite rank considerations proofs section 2 do not depend on multiplicative structure nor on topology so extend to linear subspaces of transformations in an abstract setting abstract reflexivity has been studied in 3 4 7 primarily for singly generated algebras reflexivity properties can be interpreted as linear interpolation properties and we shall adopt this point of view the results of section 2 extend in section 3 to algebraic reflexivity counterparts for countably generated algebraically linear subspaces of bounded linear transformations acting on a banach space as consequences in section 4 we obtain generalizations of two single operator results one is a multivariate version of the result due to kaplansky 10 theorem 15 that a bounded locally algebraic operator acting on a banach space is algebraic the other is a multivariate version of the result due to douglas and foias 5 for hilbert space and to hadwin 7 for a general banach space extending work of fillmore 6 that a bounded non algebraic operator acting on a banach space is topologically algebraically reflexive in section 5 we give an application kindly suggested by d hadwin concerning joint strong similarity orbits of n tupules of operators we wish to thank e azoff k davidson j erdos d hadwin j kraus c lance and a sourour for conversations and some correspon

a unified analysis of hoffman s bound via fenchel duality in 1952 hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual subsequently this bound has been studied extensively and has found many practical uses in this paper we consider an extension of hoffman s bound to a partially infinite dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form ax a in k where a is a continuous linear operator from some normed linear space into mathbf r m a is an element of the range space of a and k is a nonempty closed convex cone in mathbf r m when specialized to the finite dimensional case we unify and extend many existing results on hoffman s bound our analysis is based on the use of fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of k much of our analysis als

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

perturbation of error bounds our aim in the current article is to extend the developments in kruger et al siam j optim 20 6 3280 3296 2010 doi 10 1137 100782206 and more precisely to characterize in the banach space setting the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations we propose new concepts of arbitrary convex and linear perturbations of the given function defining the system under consideration which turn out to be a useful tool in our analysis the characterizations of error bounds for families of perturbations can be interpreted as estimates of the radius of error bounds the definitions and characterizations are illustrated by examples

likelihood analysis of spatial point patterns summary the likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns approximation methods of the normalizing factor of gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness or hardness of repulsive interactions the approximations are useful up to a considerably high density the validity of our procedure is demonstrated by some computer experi ments some real data are analysed

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

estimating optimal parameters for mrf stereo from a single image pair this paper presents a novel approach for estimating the parameters for mrf based stereo algorithms this approach is based on a new formulation of stereo as a maximum a posterior map problem in which both a disparity map and mrf parameters are estimated from the stereo pair itself we present an iterative algorithm for the map estimation that alternates between estimating the parameters while fixing the disparity map and estimating the disparity map while fixing the parameters the estimated parameters include robust truncation thresholds for both data and neighborhood terms as well as a regularization weight the regularization weight can be either a constant for the whole image or spatially varying depending on local intensity gradients in the latter case the weights for intensity gradients are also estimated our approach works as a wrapper for existing stereo algorithms based on graph cuts or belief propagation automatically tuning their parameters to improve performance without requiring the stereo code to be modified experiments demonstrate that our approach moves a baseline belief propagation stereo algorithm up six slots in the middlebury rankings

hyperparameter estimation for satellite image restoration using a mcmc maximum likelihood method abstract the satellite image deconvolution problem is ill posed and must be regularized herein we use an edge preserving regularization model using a function involving two hyperparameters our goal is to estimate the optimal parameters in order to automatically reconstruct images we propose to use the maximum likelihood estimator mle applied to the observed image we need sampling from prior and posterior distributions since the convolution prevents use of standard samplers we have developed a modified geman yang algorithm using an auxiliary variable and a cosine transform we present a markov chain monte carlo maximum likelihood mcmcml technique which is able to simultaneously achieve the estimation and the reconstruction

global regularity theorems for a system of nondifferentiable convex inequalities and linear equalities the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector applications to stability theory are then obtained

completely rank nonincreasing linear maps on spaces of operators we discuss some results and conjectures concerning rank nonincreasing rank preserving completely rank preserving and completely rank nonincreasing linear maps on spaces of operators we show that the main conjectures are equivalent to a statement about closures of joint similarity orbits of k tuples of matrices

on the convergence of monte carlo maximum likelihood calculations summary monte carlo maximum likelihood for normalized families of distributions can be used for an extremely broad class of models given any family he 0 e 0 of non negative integrable functions maximum likelihood estimates in the family obtained by normalizing the functions to integrate to 1 can be approximated by monte carlo simulation the only regularity conditions being a compactification of the parameter space such that the evaluation maps 0 h0 x remain continuous then with probability 1 the monte carlo approximant to the log likelihood hypoconverges to the exact log likelihood its maximizer converges to the exact maximum likelihood estimate approximations to profile likelihoods hypoconverge to the exact profile and level sets of the approximate likelihood support regions converge to the exact sets in painleve kuratowski set convergence the same results hold when there are missing data if a wald type integrability condition is satisfied asymptotic normality of the monte carlo error and convergence of the monte carlo approximation to the observed fisher information are also shown

a condition number for differentiable convex inequalities for a system of differentiable convex inequalities a new bound is given for the absolute error in an infeasible point in terms of the absolute residual by using this bound a condition number is defined for the system of inequalities which gives a bound for the relative error in an infeasible point in terms of the relative residual

sharp estimates for hoffman s constant for systems of linear inequalities and equalities we extend the formulae given by aze and corvellec belousov and andronov and ng and zheng for the hoffman constant associated to the system of linear inequalities ax leq b and relate them to that established by li

inference for discrete markov fields the simplest nontrivial case abstract markov fields provide a general context for describing the strength and structure of spatial interactions the gibbs markov equivalence theorem preston 1974 parameterizes markov fields via their neighborhood structures yielding exponential families in canonical form likelihood inference is therefore apparently straightforward the requisite normalizing constants however are obstreperous even when asymptotic characterizations can be obtained substantial location errors arise during implementation moreover markov fields can exhibit phase transitions and long range interactions thereby creating identifiability problems these issues are illustrated in the simplest nontrivial case the classical ising model of ferromagnetism

a unified approach to statistical tomography using coordinate descent optimization over the past years there has been considerable interest in statistically optimal reconstruction of cross sectional images from tomographic data in particular a variety of such algorithms have been proposed for maximum a posteriori map reconstruction from emission tomographic data while map estimation requires the solution of an optimization problem most existing reconstruction algorithms take an indirect approach based on the expectation maximization em algorithm we propose a new approach to statistically optimal image reconstruction based on direct optimization of the map criterion the key to this direct optimization approach is greedy pixel wise computations known as iterative coordinate decent icd we propose a novel method for computing the icd updates which we call icd newton raphson we show that icd newton raphson requires approximately the same amount of computation per iteration as em based approaches but the new method converges much more rapidly in our experiments typically five to ten iterations other advantages of the icd newton raphson method are that it is easily applied to map estimation of transmission tomograms and typical convex constraints such as positivity are easily incorporated

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

a note on reflexivity abstract we prove some new results on hadwin s general version of reflexivity that reduce the study of e reflexivity or e hyperreflexivity of a linear subspace to a smaller linear subspace by applying our abstract results we present a simple proof of d hadwin s theorem which states that every c algebra is approximately hyperreflexive we also prove that the image of any c algebra under any bounded unital homomorphism into the operators on a banach space is approximately reflexive we introduce a new version of reflexivity called approximate algebraic reflexivity and study its properties

weak sharp minima revisited part ii application to linear regularity and error bounds the notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems it has been studied extensively by several authors this paper is the second of a series on this subject where the basic results on weak sharp minima in part i are applied to a number of important problems in convex programming in part ii we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming we obtain both new results and reproduce several existing results from a fresh perspective

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

merit functions in vector optimization we study the weak domination property and weakly efficient solutions in vector optimization problems in particular scalarization of these problems is obtained by virtue of some suitable merit functions some natural conditions to ensure the existence of error bounds for merit functions are also given

gradient based iterative image reconstruction scheme for time resolved optical tomography currently available tomographic image reconstruction schemes for optical tomography ot are mostly based on the limiting assumptions of small perturbations and a priori knowledge of the optical properties of a reference medium furthermore these algorithms usually require the inversion of large full ill conditioned jacobian matrixes in this work a gradient based iterative image reconstruction giir method is presented that promises to overcome current limitations the code consists of three major parts 1 a finite difference time resolved diffusion forward model is used to predict detector readings based on the spatial distribution of optical properties 2 an objective function that describes the difference between predicted and measured data 3 an updating method that uses the gradient of the objective function in a line minimization scheme to provide subsequent guesses of the spatial distribution of the optical properties for the forward model the reconstruction of these properties is completed once a minimum of this objective function is found after a presentation of the mathematical background two and three dimensional reconstruction of simple heterogeneous media as well as the clinically relevant example of ventricular bleeding in the brain are discussed numerical studies suggest that intraventricular hemorrhages can be detected using the giir technique even in the presence of a heterogeneous background