lagrangean surrogate relaxation for generalized assignment problems abstract this work presents lagrangean surrogate relaxation to the problem of maximum profit assignment of n tasks to m agents n m such that each task is assigned to only one agent subject to capacity constraints on the agents the lagrangean surrogate relaxation combines usual lagrangean and surrogate relaxations relaxing first a set of constraints in the surrogate way then the lagrangean relaxation of the surrogate constraint is obtained and approximately optimized one dimensional dual the lagrangean surrogate is compared with the usual lagrangean relaxation on a computational study using a large set of instances the dual bounds are the same for both relaxations but the lagrangean surrogate can give improved local bounds at the application of a subgradient method resulting in less computational times three relaxations are derived for the problem the first relaxation considers a vector of multipliers for the capacity constraints the second for the assignment constraints and the other for the lagrangean decomposition constraints relaxation multipliers are used with efficient constructive heuristics to find good feasible solutions the application of a lagrangean surrogate approach seems very promising for large scale problems

generalized bundle methods we study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties this setting covers several algorithms from the literature that have been so far regarded as distinct under a different hypothesis on the stabilizing term and or the function to be minimized we prove finite termination asymptotic convergence and finite convergence to an optimal point with or without limits on the number of serious steps and or requiring the proximal parameter to go to infinity the convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm i e the management of the bundle of subgradients beta strategy and of the proximal parameter t strategy we extensively exploit a dual view of bundle methods which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space where nonlinear subproblems are approximately solved at each step with an inner linearization approach this allows us to precisely characterize the changes in the subproblems during the serious steps since the dual problem is not tied to the local concept of varepsilon subdifferential for some of the proofs a generalization of inf compactness called compactness is required this concept is related to that of asymptotically well behaved functions

an mrf model based approach to simultaneous recovery of depth and restoration from defocused images in this paper we propose a map markov random field mrf based scheme for recovering the depth and the focused image of a scene from two defocused images the space variant blur parameter and the focused image of the scene are both modeled as mrfs and their map estimates are obtained using simulated annealing the scheme is amenable to the incorporation of smoothness constraints on the spatial variations of the blur parameter as well as the scene intensity it also allows for inclusion of line fields to preserve discontinuities the performance of the proposed scheme is tested on synthetic as well as real data and the estimates of the depth are found to be better than that of the existing window based depth from defocus technique the quality of the space variant restored image of the scene is quite good even under severe space varying blurring conditions

the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

ergodic primal convergence in dual subgradient schemes for convex programming lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large structured optimization prob lems the dual subgradient scheme does not automatically produce primal feasible solutions there is an abundance of techniques for computing such solutions via penalty functions tangential approximation schemes or the solution of auxiliary primal programs all of which require a fair amount of computational effort we consider a subgradient optimization scheme applied to a lagrangean dual formulation of a convex program and construct at minor cost an ergodic sequence of subproblem solutions which converges to the primal solution set numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained compared to those generated in the basic subgradient scheme

reciprocal focus profile a focus profile having a steeper peak is more resistant to image noise in the autofocus af process of a digital camera however a focus profile of such shape normally has a flatter out of focus region on either side of the profile resulting in a slow af process due to the lack of clue about where the lens should move when the lens is in such regions to address the problem we provide a statistical analysis of the focus profile and show that a strictly monotonic transformation of the focus profile preserves the accuracy of the af on the basis of this analysis we propose a new focus profile representation that transforms the focus profile to the reciprocal domain in which the reciprocal focus profile is modeled by a polynomial function this transformation makes the af mathematically tractable and boosts the search speed experimental results are shown to demonstrate the advantage of the proposed representation

shape from focus the shape from focus method presented here uses different focus levels to obtain a sequence of object images the sum modified laplacian sml operator is developed to provide local measures of the quality of image focus the operator is applied to the image sequence to determine a set of focus measures at each image point a depth estimation algorithm interpolates a small number of focus measure values to obtain accurate depth estimates a fully automated shape from focus system has been implemented using an optical microscope and tested on a variety of industrial samples experimental results are presented that demonstrate the accuracy and robustness of the proposed method these results suggest shape from focus to be an effective approach for a variety of challenging visual inspection tasks

c 1 stably shadowable chain components are hyperbolic abstract let f be a diffeomorphism on a closed manifold and p be a hyperbolic periodic point of f denote c f p the chain component of f that contains p we say c f p is c 1 stably shadowable if there is a c 1 neighborhood u of f such that for every g u c g p g has the shadowing property where p g is the continuation of p we prove in this paper that if c f p is c 1 stably shadowable then c f p is hyperbolic

convergence of approximate and incremental subgradient methods for convex optimization we present a unified convergence framework for approximate subgradient methods that covers various stepsize rules including both diminishing and nonvanishing stepsizes convergence in objective values and convergence to a neighborhood of the optimal set we discuss ways of ensuring the boundedness of the iterates and give efficiency estimates our results are extended to incremental subgradient methods for minimizing a sum of convex functions which have recently been shown to be promising for various large scale problems including those arising from lagrangian relaxation

shape from focus using multilayer feedforward neural networks the conventional shape from focus sff methods have inaccuracies because of piecewise constant approximation of the focused image surface fis we propose a scheme for sff based on representation of three dimensional 3 d fis in terms of neural network weights the neural networks are trained to learn the shape of the fis that maximizes the focus measure

the volume algorithm revisited relation with bundle methods we revise the volume algorithm va for linear programming and relate it to bundle methods when first introduced va was presented as a subgradient like method for solving the original problem in its dual form in a way similar to the serious null steps philosophy of bundle methods va produces green yellow or red steps in order to give convergence results we introduce in va a precise measure for the improvement needed to declare a green or serious step this addition yields a revised formulation rva that is halfway between va and a specific bundle method that we call bva we analyze the convergence properties of both rva and bva finally we compare the performance of the modified algorithms versus va on a set of rectilinear steiner problems of various sizes and increasing complexity derived from real world vlsi design instances

multifocus image fusion by combining with mixed order structure tensors and multiscale neighborhood a new fusion scheme based on multiscale neighborhood technique is proposed for multifocus image fusion a new focus measure by combining the structure tensors of mixed order differential is developed an effective focused region detection method is presented based on multiscale neighborhood technique an averaging fusion scheme is presented based on the fusion decision maps of different scale for uncertain regions our method outperforms the traditional methods in various experiments in this study we propose a new method for multifocus image fusion by combining with the structure tensors of mixed order differentials and the multiscale neighborhood in this method the structure tensor of an integral differential is utilized to detect the high frequency regions and the structure tensor of the fractional differential is used to detect the low frequency regions to improve the performance of the fusion method we propose a new focus measure based on the multiscale neighborhood technique to generate the initial fusion decision maps by exploiting the advantages of different scales next based on the multiscale neighborhood technique a post processing method is used to update the initial fusion decision maps during the fusion process the pixels located in the focused inner regions are selected to produce the fused image in order to avoid discontinuities in the transition zone between the focused and defocused regions we propose a new averaging scheme based on the fusion decision maps at different scales our experimental results demonstrate that the proposed method outperformed the conventional multifocus image fusion methods in terms of both their subjective and objective quality

a moment preserving approach for depth from defocus abstract for range sensing using depth from defocus methods the distance d of a point object from the lens can be evaluated by the concise depth formula d p q d b where p and q are constants for a given camera setting and db is the diameter of the blur circle for the point object on the image detector plane the amount of defocus db is traditionally estimated from the spatial parameter of a gaussian point spread function using a complex iterative solution in this paper we use a straightforward and computationally fast method to estimate the amount of defocus from a single camera the observed gray level image is initially converted into a gradient image using the sobel edge operator for the edge point of interest the proportion of the blurred edge region pe in a small neighborhood window is then calculated using the moment preserving technique the value of pe increases as the amount of defocus increases and therefore is used as the description of degradation of the point spread function in addition to the use of the geometric depth formula for depth estimation artificial neural networks are also proposed in this study to compensate for the estimation errors from the depth formula experiments have shown promising results that the rms depth errors are within 5 for the depth formula and within 2 for the neural networks

convergence analysis of deflected conditional approximate subgradient methods subgradient methods for nondifferentiable optimization benefit from deflection i e defining the search direction as a combination of the previous direction and the current subgradient in the constrained case they also benefit from projection of the search direction onto the feasible set prior to computing the steplength that is from the use of conditional subgradient techniques however combining the two techniques is not straightforward especially if an inexact oracle is available which can only compute approximate function values and subgradients we present a convergence analysis of several different variants both conceptual and implementable of approximate conditional deflected subgradient methods our analysis extends the available results in the literature by using the main stepsize rules presented so far while allowing deflection in a more flexible way furthermore to allow for diminishing square summable rules where the stepsize is tightly controlled a priori we propose a new class of deflection restricted approaches where it is the deflection parameter rather than the stepsize which is dynamically adjusted using the target value of the optimization sequence for both polyak type and diminishing square summable stepsizes we propose a correction of the standard formula which shows that in the inexact case knowledge about the error computed by the oracle which is available in several practical applications can be exploited in order to strengthen the convergence properties of the method the analysis allows for several variants of the algorithm at least one of them is likely to show numerical performances similar to these of heavy ball subgradient methods popular within backpropagation approaches to train neural networks while possessing stronger convergence properties

primal convergence from dual subgradient methods for convex optimization when solving a convex optimization problem through a lagrangian dual reformulation subgradient optimization methods are favorably utilized since they often find near optimal dual solutions quickly however an optimal primal solution is generally not obtained directly through such a subgradient approach unless the lagrangian dual function is differentiable at an optimal solution we construct a sequence of convex combinations of primal subproblem solutions a so called ergodic sequence which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen we generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions in contrast to previously proposed rules they exploit more information from later subproblem solutions than from earlier ones we evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed

improvements in shape from focus for holographic reconstructions with regard to focus operators neighborhood size and height value interpolation this paper presents a shape from focus method which is improved with regard to the mathematical operator used for contrast measurement the selection of the neighborhood size surface refinement through interpolation and surface postprocessing three dimensional models of living human faces are presented with such a high resolution that single hairs are visible

a novel algorithm for estimation of depth map using image focus for 3d shape recovery in the presence of noise three dimensional shape recovery from one or multiple observations is a challenging problem of computer vision in this paper we present a new focus measure for the estimation of a depth map using image focus this depth map can subsequently be used in techniques and algorithms leading to the recovery of a three dimensional structure of the object a requirement of a number of high level vision applications the proposed focus measure has shown robustness in the presence of noise as compared to the earlier focus measures this new focus measure is based on an optical transfer function implemented in the fourier domain the results of the proposed focus measure have shown drastic improvements in estimation of a depth map with respect to the earlier focus measures in the presence of various types of noise including gaussian shot and speckle noises the results of a range of focus measures are compared using root mean square error and correlation metric measures

non parametric blur map regression for depth of field extension real camera systems have a limited depth of field dof which may cause an image to be degraded due to visible misfocus or too shallow dof in this paper we present a blind deblurring pipeline able to restore such images by slightly extending their dof and recovering sharpness in regions slightly out of focus to address this severely ill posed problem our algorithm relies first on the estimation of the spatially varying defocus blur drawing on local frequency image features a machine learning approach based on the recently introduced regression tree fields is used to train a model able to regress a coherent defocus blur map of the image labeling each pixel by the scale of a defocus point spread function a non blind spatially varying deblurring algorithm is then used to properly extend the dof of the image the good performance of our algorithm is assessed both quantitatively using realistic ground truth data obtained with a novel approach based on a plenoptic camera and qualitatively with real images

on the convergence of conditional e subgradient methods for convex programs and convex concave saddle point problems abstract the paper provides two contributions first we present new convergence results for conditional e subgradient algorithms for general convex programs the results obtained here extend the classical ones by polyak sov math doklady 8 1967 593 ussr comput math math phys 9 1969 14 introduction to optimization optimization software new york 1987 as well as the recent ones in math program 62 1993 261 eur j oper res 88 1996 382 math program 81 1998 23 to a broader framework secondly we establish the application of this technique to solve non strictly convex concave saddle point problems such as primal dual formulations of linear programs contrary to several previous solution algorithms for such problems a saddle point is generated by a very simple scheme in which one component is constructed by means of a conditional e subgradient algorithm while the other is constructed by means of a weighted average of the inexact subproblem solutions generated within the subgradient method the convergence result extends those of minimization methods for non differentiable functions springer verlag berlin 1985 oper res lett 19 1996 105 math program 86 1999 283 for lagrangian saddle point problems in linear and convex programming and of int j numer meth eng 40 1997 1295 for a linear quadratic saddle point problem arising in topology optimization in contact mechanics

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

multiresolution 3 d range segmentation using focus cues this paper describes a novel system for computing a three dimensional 3 d range segmentation of an arbitrary visible scene using focus information the process of range segmentation is divided into three steps an initial range classification a surface merging process and a 3 d multiresolution range segmentation first range classification is performed to obtain quantized range estimates the range classification is performed by analyzing focus cues within a bayesian estimation framework a combined energy functional measures the degree of focus and the gibbs distribution of the class field the range classification provides an initial range segmentation second a statistical merging process is performed to merge the initial surface segments this gives a range segmentation at a coarse resolution third 3 d multiresolution range segmentation 3 d mrs is performed to refine the range segmentation into finer resolutions the proposed range segmentation method does not require initial depth estimates it allows the analysis of scenes containing multiple objects and it provides a rich description of the 3 d structure of a scene

removal of partial occlusion from single images this paper examines large partial occlusions in an image which occur near depth discontinuities when the foreground object is severely out of focus we model these partial occlusions using matting with the alpha value determined by the convolution of the blur kernel with a pinhole projection of the occluder the main contribution is a method for removing the image contribution of the foreground occluder in regions of partial occlusion which improves the visibility of the background scene the method consists of three steps first the region of complete occlusion is estimated using a curve evolution method second the alpha value at each pixel in the partly occluded region is estimated third the intensity contribution of the foreground occluder is removed in regions of partial occlusion experiments demonstrate the method s ability to remove the effects of partial occlusion in single images with minimal user input

variational depth from focus reconstruction this paper deals with the problem of reconstructing a depth map from a sequence of differently focused images also known as depth from focus dff or shape from focus we propose to state the dff problem as a variational problem including a smooth but nonconvex data fidelity term and a convex nonsmooth regularization which makes the method robust to noise and leads to more realistic depth maps in addition we propose to solve the nonconvex minimization problem with a linearized alternating directions method of multipliers allowing to minimize the energy very efficiently a numerical comparison to classical methods on simulated as well as on real data is presented

a library for continuous convex separable quadratic knapsack problems abstract the continuous convex separable quadratic knapsack problem cqknp is an easy but useful model that has very many different applications although the problem can be solved quickly it must typically be solved very many times within approaches to much more difficult models hence an efficient solution approach is required we present and discuss a small open source library for its solution that we have recently developed and distributed

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

tunable halfband pair wavelet filter banks and application to multifocus image fusion traditional maximally flat wavelet filters are highly regular but suffer from poor frequency selectivity because of their wide transition band in this paper an efficient method is proposed for the design of biorthogonal perfect reconstruction wavelet filter banks known as halfband pair filter banks hpfb to be used in several applications in image processing and pattern recognition the formulation is based on representation of a general halfband polynomial in the variable x we first derive filter coefficients in the polynomial domain in the variable x in terms of the coefficients of the corresponding function in z domain using convex optimization techniques and due to the simple structure of a parametric polynomial in general we can impose some free parameters to provide a tuning opportunity to optimize and control the wavelet filter characteristics perfect reconstruction and desired number of vanishing moments nvm are incorporated into the design procedure the method is systematic renders a reasonable optimization problem and it offers wavelet filters ranging from the maximally flat to the sharpest transition band therefore it can provide a useful design tool with a fine tuning option which is required in many applications such as watermarking detection segmentation fusion denoising and feature extraction the application of the wavelet pairs which have sharper transition band and better frequency selectivity is shown in multifocus imaging to obtain a fully focused image from a set of registered input images at varying foci by employing the distance transform and exponentially decaying function on the subbands in the wavelet domain various images are tested and experimental results compare favorably to the results in the literature

using logical surrogate information in lagrangean relaxation an application to symmetric traveling salesman problems abstract the traveling salesman problem tsp is a classical combinatorial optimization problem which has been intensively studied the lagrangean relaxation was first applied to the tsp in 1970 the lagrangean relaxation limit approximates what is known today as held and karp hk bound a very good bound less than 1 from optimal for a large class of symmetric instances it became a reference bound for new heuristics mainly for the very large scale instances where the use of exact methods is prohibitive a known problem for the lagrangean relaxation application is the definition of a convenient step size control in subgradient like methods even preserving theoretical convergence properties a wrong defined control can affect the performance and increase computational times we show in this work how to accelerate a classical subgradient method while conserving good approximations to the hk bounds the surrogate and lagrangean relaxations are combined using the local information of the relaxed constraints it results in a one dimensional search that corrects the possibly wrong step size and is independent of the used step size control comparing with the ordinary subgradient method and beginning with the same initial multiplier the computational times are almost twice as fast for medium instances and greatly improved for some large scale tsplib instances

a storm of feasibility pumps for nonconvex minlp one of the foremost difficulties in solving mixed integer nonlinear programs either with exact or heuristic methods is to find a feasible point we address this issue with a new feasibility pump algorithm tailored for nonconvex mixed integer nonlinear programs feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed integer relaxation of the original problems such approaches currently exist in the literature for mixed integer linear programs and convex mixed integer nonlinear programs both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time in nonconvex mixed integer nonlinear programming such a property does not hold and therefore special care has to be exercised in order to allow feasibility pump algorithms to rely only on local optima of the continuous relaxation based on a new high level view of feasibility pump algorithms as a special case of the well known successive projection method we show that many possible different variants of the approach can be developed depending on how several different orthogonal implementation choices are taken a remarkable twist of feasibility pump algorithms is that unlike most previous successive projection methods from the literature projection is naturally taken in two different norms in the two different subproblems to cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems these actually seem to significantly improve the practical performance of the approach we present extensive computational results on the minlplib showing the effectiveness and efficiency of our algorithm

shape from focus using fast discrete curvelet transform a new method for focus measure computation is proposed to reconstruct 3d shape using image sequence acquired under varying focus plane adaptive histogram equalization is applied to enhance varying contrast across different image regions for better detection of sharp intensity variations fast discrete curvelet transform fdct is employed for enhanced representation of singularities along curves in an input image followed by noise removal using bivariate shrinkage scheme based on locally estimated variance the fdct coefficients with high activity are exploited to detect high frequency variations of pixel intensities in a sequence of images finally focus measure is computed utilizing neighborhood support of these coefficients to reconstruct the shape and a well focused image of the scene being probed

a proximal bundle method with approximate subgradient linearizations we give a proximal bundle method for minimizing a convex function f over a closed convex set it only requires evaluating f and its subgradients with an accuracy epsilon 0 which is fixed but possibly unknown it asymptotically finds points that are epsilon optimal when applied to lagrangian relaxation it allows for epsilon accurate solutions of lagrangian subproblems and finds epsilon optimal solutions of convex programs

rational filters for passive depth from defocus a fundamental problem in depth from defocus is the measurement of relative defocus between images the performance of previously proposed focus operators are inevitably sensitive to the frequency spectra of local scene textures as a result focus operators such as the laplacian of gaussian result in poor depth estimates an alternative is to use large filter banks that densely sample the frequency space though this approach can result in better depth accuracy it sacrifices the computational efficiency that depth from defocus offers over stereo and structure from motion we propose a class of broadband operators that when used together provide invariance to scene texture and produce accurate and dense depth maps since the operators are broadband a small number of them are sufficient for depth estimation of scenes with complex textural properties in addition a depth confidence measure is derived that can be computed from the outputs of the operators this confidence measure permits further refinement of computed depth maps experiments are conducted on both synthetic and real scenes to evaluate the performance of the proposed operators the depth detection gain error is less than 1 irrespective of texture frequency depth accuracy is found to be 0 5 1 2 of the distance of the object from the imaging optics

a novel iterative shape from focus algorithm based on combinatorial optimization shape from focus sff is a technique to estimate the depth and 3d shape of an object from a sequence of images obtained at different focus settings in this paper the sff is presented as a combinatorial optimization problem the proposed algorithm tries to find the combination of pixel frames which produces maximum focus measure computed over pixels lying on those frames to reduce the high computational complexity a local search method is proposed after the estimate of the initial depth map solution of an object the neighborhood is defined and an intermediate image volume is generated from the neighborhood the updated depth map solution is found from the intermediate image volume this update process of the depth map solution continues until the amount of improvement is negligible the results of the proposed sff algorithm have shown significant improvements in both the accuracy of the depth map estimation and the computational complexity with respect to the existing sff methods

optimal depth estimation by combining focus measures using genetic programming three dimensional 3d shape reconstruction is a fundamental problem in machine vision applications shape from focus sff is one of the passive optical methods for 3d shape recovery that uses degree of focus as a cue to estimate 3d shape in this approach usually a single focus measure operator is applied to measure the focus quality of each pixel in the image sequence however the applicability of a single focus measure is limited to estimate accurately the depth map for diverse type of real objects to address this problem we develop optimal composite depth ocd function through genetic programming gp for accurate depth estimation the ocd function is constructed by optimally combining the primary information extracted using one or more focus measures the genetically developed composite function is then used to compute the optimal depth map of objects the performance of the developed nonlinear function is investigated using both the synthetic and the real world image sequences experimental results demonstrate that the proposed estimator is more useful in computing accurate depth maps as compared to the existing sff methods moreover it is found that the heterogeneous function is more effective than homogeneous function

analysis of focus measure operators for shape from focus shape from focus sff has widely been studied in computer vision as a passive depth recovery and 3d reconstruction method one of the main stages in sff is the computation of the focus level for every pixel of an image by means of a focus measure operator in this work a methodology to compare the performance of different focus measure operators for shape from focus is presented and applied the selected operators have been chosen from an extensive review of the state of the art the performance of the different operators has been assessed through experiments carried out under different conditions such as image noise level contrast saturation and window size such performance is discussed in terms of the working principles of the analyzed operators

inexact subgradient methods for quasi convex optimization problems in this paper we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi convex constrained optimization problem the inexactness stems from computation errors and noise which come from practical considerations and applications assuming that the computational errors and noise are deterministic and bounded we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima in both cases using the constant and diminishing stepsize rules we describe convergence results in both objective values and iterates and finite convergence to approximate optimality we also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the cobb douglas production efficiency problem the numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm especially for the large scale problems

generation of all in focus images by noise robust selective fusion of limited depth of field images the limited depth of field of some cameras prevents them from capturing perfectly focused images when the imaged scene covers a large distance range in order to compensate for this problem image fusion has been exploited for combining images captured with different camera settings thus yielding a higher quality all in focus image since most current approaches for image fusion rely on maximizing the spatial frequency of the composed image the fusion process is sensitive to noise in this paper a new algorithm for computing the all in focus image from a sequence of images captured with a low depth of field camera is presented the proposed approach adaptively fuses the different frames of the focus sequence in order to reduce noise while preserving image features the algorithm consists of three stages 1 focus measure 2 selectivity measure 3 and image fusion an extensive set of experimental tests has been carried out in order to compare the proposed algorithm with state of the art all in focus methods using both synthetic and real sequences the obtained results show the advantages of the proposed scheme even for high levels of noise

ergodic primal convergence in dual subgradient schemes for convex programming ii the case of inconsistent primal problems consider the utilization of a lagrangian dual method which is convergent for consistent convex optimization problems when it is used to solve an infeasible optimization problem its inconsistency will then manifest itself through the divergence of the sequence of dual iterates will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program we answer this question in the affirmative for a convex program and an associated subgradient algorithm for its lagrange dual we show that the primal dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle point problem in which in the inconsistent case the primal part amounts to finding a solution in the primal space such that the euclidean norm of the infeasibility in the relaxed constraints is minimized the dual part amounts to identifying a feasible steepest ascent direction for the lagrangian dual function we present convergence results for a conditional varepsilon subgradient optimization algorithm applied to the lagrangian dual problem and the construction of an ergodic sequence of primal subproblem solutions this composite algorithm yields convergence of the primal dual sequence to the set of saddle points of the associated homogeneous lagrangian function for linear programs convergence to the subset in which the primal objective is at minimum is also achieved

an inexact bundle variant suited to column generation we give a bundle method for constrained convex optimization instead of using penalty functions it shifts iterates towards feasibility by way of a slater point assumed to be known besides the method accepts an oracle delivering function and subgradient values with unknown accuracy our approach is motivated by a number of applications in column generation in which constraints are positively homogeneous so that zero is a natural slater point and an exact oracle may be time consuming finally our convergence analysis employs arguments which have been little used so far in the bundle community the method is illustrated on a number of cutting stock problems

primal dual subgradient methods for convex problems in this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure our methods are primal dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem besides other advantages this useful feature provides the methods with a reliable stopping criterion the proposed schemes differ from the classical approaches divergent series methods mirror descent methods by presence of two control sequences the first sequence is responsible for aggregating the support functions in the dual space and the second one establishes a dynamically updated scale between the primal and dual spaces this additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set however we always assume the uniform boundedness of subgradients we present the variants of subgradient schemes for nonsmooth convex minimization minimax problems saddle point problems variational inequalities and stochastic optimization in all situations our methods are proved to be optimal from the view point of worst case black box lower complexity bounds

a focus fusion framework with anisotropic depth map smoothing focus fusion is the task of combining a set of images focused at different depths into a single image that is entirely in focus the crucial point of all focus fusion methods is the decision about the in focus areas to this end we present a general framework for focus fusion that introduces a modern regularisation strategy on these per pixel decisions we assume that neighbouring pixels in the fused image belong to similar depth layers following this assumption we smooth the depth map with a sophisticated anisotropic diffusion process combined with a robust data fidelity term the experiments with synthetic and real world data demonstrate that our new model yields a better quality than several existing focus fusion methods moreover our methodology is general and can be applied to improve many fusion approaches highlightswe show that depth map regularisation is an important tool for focus fusion we incorporate modern concepts such as a coupled anisotropic diffusion term we substantially improve the runtime with a fast gpu implementation we evaluate different in focus measures we compare the overall performance to several methods from the literature

conditional subgradient optimization theory and applications abstract we generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients firstly we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization secondly we consider a particular choice of conditional subgradients obtained by projections which leads to an easily implementable modification of traditional subgradient optimization schemes to evaluate the subgradient projection method we consider its use in three applications uncapacitated facility location two person zero sum matrix games and multicommodity network flows computational experiments show that the subgradient projection method performs better than traditional subgradient optimization in some cases the difference is considerable these results suggest that our simply modification may improve subgradient optimization schemes significantly this finding is important as such schemes are very popular especially in the context of lagrangean relaxation

development of autofocusing algorithm based on fuzzy transforms in this paper we propose a new passive auto focusing algorithm to acquire images with enhanced quality through a digital camera the focus measure operations play a key role in a passive auto focusing method the proposed focus measure operation is based on a localized version of image variance method the localization of the image variance focus measure operation can be implemented with the use of fuzzy partition and fuzzy transform in order to validate our approach experimental comparative studies are conducted on test images with the use of different focus measure operators the selected operators have been chosen from an extensive review of the state of the art approaches a comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods

on the extension of constrained optimization algorithms from differentiable to nondifferentiable problems this paper presents three general schemes for extending differentiable optimization algorithms to nondifferentiable problems it is shown that the armijo gradient method phase i phase ii methods of feasible directions and exact penalty function methods have conceptual analogs for problems with locally lipschitz functions and implementable analogs for problems with semismooth functions the exact penalty method has required the development of a new optimality condition

nonlinear approach for enhancement of image focus volume in shape from focus mostly shape from focus algorithms use local averaging using a fixed rectangle window to enhance the initial focus volume in this linear filtering the window size affects the accuracy of the depth map a small window is unable to suppress the noise properly whereas a large window oversmoothes the object shape moreover the use of any window size smoothes focus values uniformly consequently an erroneous depth map is obtained in this paper we suggest the use of iterative 3 d anisotropic nonlinear diffusion filtering andf to enhance the image focus volume in contrast to linear filtering andf utilizes the local structure of the focus values to suppress the noise while preserving edges the proposed scheme is tested using image sequences of synthetic and real objects and results have demonstrated its effectiveness

the volume algorithm producing primal solutions with a subgradient method we present an extension to the subgradient algorithm to produce primal as well as dual solutions it can be seen as a fast way to carry out an approximation of dantzig wolfe decomposition this gives a fast method for producing approximations for large scale linear programs it is based on a new theorem in linear programming duality we present successful experience with linear programs coming from set partitioning set covering max cut and plant location

new approaches for optimizing over the semimetric polytope the semimetric polytope is an important polyhedral structure lying at the heart of several hard combinatorial problems therefore linear optimization over the semimetric polytope is crucial for a number of relevant applications building on some recent polyhedral and algorithmic results about a related polyhedron the rooted semimetric polytope we develop and test several approaches based over lagrangian relaxation and application of non differentiable optimization algorithms for linear optimization over the semimetric polytope we show that some of these approaches can obtain very accurate primal and dual solutions in a small fraction of the time required for the same task by state of the art general purpose linear programming technology in some cases good estimates of the dual optimal solution but not of the primal solution can be obtained even quicker

real time focus range sensor structures of dynamic scenes can only be recovered using a real time range sensor depth from defocus offers an effective solution to fast and dense range estimation however accurate depth estimation requires theoretical and practical solutions to a variety of problems including recovery of textureless surfaces precise blur estimation and magnification variations caused by defocusing both textured and textureless surfaces are recovered using an illumination pattern that is projected via the same optical path used to acquire images the illumination pattern is optimized to maximize accuracy and spatial resolution in computed depth the relative blurring in two images is computed using a narrow band linear operator that is designed by considering all the optical sensing and computational elements of the depth from defocus system defocus invariant magnification is achieved by the use of an additional aperture in the imaging optics a prototype focus range sensor has been developed that has a workspace of 1 cubic foot and produces up to 512 spl times 480 depth estimates at 30 hz with an average rms error of 0 2 several experimental results are included to demonstrate the performance of the sensor

reconstructing arbitrarily focused images from two differently focused images using linear filters we present a novel filtering method for reconstructing an all in focus image or an arbitrarily focused image from two images that are focused differently the method can arbitrarily manipulate the degree of blur of the objects using linear filters without segmentation the filters are uniquely determined from a linear imaging model in the fourier domain an effective and accurate blur estimation method is developed the simulation results show that the accuracy and computational time of the proposed method are improved compared with the previous iterative method and that the effects of blur estimation error on the quality of the reconstructed image are very small the method performs well for real images acquired without visible artifacts

on the identification of active constraints nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied results of this type have only been established for a few algorithms and then under restrictive hypothesis the main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations this result is obtained with a nondegeneracy assumption which is equivalent in the standard nonlinear programming problem to the assumption that there is a set of strictly complementary lagrange multipliers as an important consequence of the authors results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown

e optimal solutions in nondifferentiable convex programming and some related questions in this paper we present e optimality conditions of the kuhn tucker type for points which are within e of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints linear equality constraints and abstract constraints such e optimality conditions are of interest for theoretical consideration as well as from the computational point of view some illustrative applications are made thus we derive an expression for the e subdifferential of a general convex max function we also show how the e optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints

an active set method for solving linearly constrained nonsmooth optimization problems an algorithm for solving linearly constrained optimization problems is proposed the search direction is computed by a bundle principle and the constraints are treated through an active set strategy difficulties that arise when the objective function is nonsmooth require a clever choice of a constraint to relax a certain nondegeneracy assumption is necessary to obtain convergence

the effect of deterministic noise in subgradient methods in this paper we study the influence of noise on subgradient methods for convex constrained optimization the noise may be due to various sources and is manifested in inexact computation of the subgradients and function values assuming that the noise is deterministic and bounded we discuss the convergence properties for two cases the case where the constraint set is compact and the case where this set need not be compact but the objective function has a sharp set of minima for example the function is polyhedral in both cases using several different stepsize rules we prove convergence to the optimal value within some tolerance that is given explicitly in terms of the errors in the first case the tolerance is nonzero but in the second case the optimal value can be obtained exactly provided the size of the error in the subgradient computation is below some threshold we then extend these results to objective functions that are the sum of a large number of convex functions in which case an incremental subgradient method can be used

partial hyperbolicity and robust transitivity throughout this paper m denotes a three dimensional boundaryless compact manifold and diff m the space of gl diffeomorphisms defined on m endowed with the usual cl topology a invariant set a is transitive if a w x for some xea here w x is the forward limit set of x the accumulation points of the positive orbit of x the maximal invariant set of in an open set u denoted by a u is the set of points whose whole orbit is contained in u i e a u i e z i u the set a u is robustly transitive if ar is transitive for every diffeomorphism r clclose to a diffeomorphism ediff m is transitive if m w x for some xem i e if a m m is transitive analogously is robustly transitive if every r glclose to also is transitive i e if a m m is robustly transitive in this paper we focus our attention on forms of hyperbolicity uniform partial and strong partial of a maximal invariant set a u derived from its robust transitivity observe that u can be equal to m and then is robustly transitive on one hand in dimension one there do not exist robustly transitive diffeomorphisms the diffeomorphisms with finitely many hyperbolic periodic points morse smale are open and dense in diff s1 on the other hand for two dimensional diffeomorphisms every robustly transitive set a u is a basic set i e a u is hyperbolic transitive and the periodic points of are dense in a u in particular every robustly transitive surface diffeomorphism is anosov and the unique surface which supports such diffeomorphisms is the torus t 2 these assertions follow from m3 and m4 in dimension bigger than or equal to three besides anosov hyperbolic diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type as far as we know three types of such diffeomorphisms have been constructed skew products

a stabilized structured dantzig wolfe decomposition method we discuss an algorithmic scheme which we call the stabilized structured dantzig wolfe decomposition method for solving large scale structured linear programs it can be applied when the subproblem of the standard dantzig wolfe approach admits an alternative master model amenable to column generation other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron stabilization is achieved by the same techniques developed for the standard dantzig wolfe approach and it is equally useful to improve the performance as shown by computational results obtained on an application to the multicommodity capacitated network design problem

two well known properties of subgradient optimization the subgradient method is both a heavily employed and widely studied algorithm for non differentiable optimization nevertheless there are some basic properties of subgradient optimization that while well known to specialists seem to be rather poorly known in the larger optimization community this note concerns two such properties both applicable to subgradient optimization using the divergent series steplength rule the first involves convergence of the iterative process and the second deals with the construction of primal estimates when subgradient optimization is applied to maximize the lagrangian dual of a linear program the two topics are related in that convergence of the iterates is required to prove correctness of the primal construction scheme

improving shape from focus using defocus cue the shape from focus sff method uses a sequence of frames to estimate the structure of a 3 d object its accuracy depends on the step size by which the translational table is moved while capturing the images existing sff algorithms use an ad hoc interpolation strategy to account for the error due to the finite step size we propose an improved sff method that uses relative defocus blur derived from actual image data to arrive at the final estimates of the structure of the object a space variant image restoration scheme is also proposed to obtain a focused image of the 3 d object the reconstructed 3 d structure as well as the quality of the restored image are superior for the proposed method in comparison to traditional sff

shape from focus by tensor voting in this correspondence we address the task of recovering shape from focus sff as a perceptual organization problem in 3 d using tensor voting depth hypotheses from different focus operators are validated based on their likelihood to be part of a coherent 3 d surface thereby exploiting scene geometry and focus information to generate reliable depth estimates the proposed method is fast and yields significantly better results compared with existing sff methods

sampling for shape from focus in optical microscopy shape from focus sff which relies on image focus as a cue within sequenced images represents a passive technique in recovering object shapes in scenes although numerous methods have been recently proposed less attention has been paid to particular factors affecting them in regard to sff one such critical factor impacting system application is the total number of images a large data set requires a huge amount of computation power whereas decreasing the number of images causes shape reconstruction to be crude and erroneous the total number of images is inversely proportional to interframe distance or sampling step size in this paper interframe distance or sampling step size criteria for sff systems have been formulated in particular light ray focusing is approximated by the use of a gaussian beam followed by the formulation of a sampling expression using nyquist sampling consequently a fitting function for focus curves is also obtained experiments are performed on simulated and real objects to validate the proposed schemes

a shadowing lemma for quasi hyperbolic strings of flows abstract in this paper we prove a shadowing lemma for pseudo orbits made by quasi hyperbolic strings we allow singularities in question and hence in particular the quasi hyperbolic strings are formulated by the rescaled linear poincare flow instead of the usual linear poincare flow we also introduce the sectional poincare map and rescaled sectional poincare map for lipschitz vector fields on banach spaces in the article

bundle methods for sum functions with easy components applications to multicommodity network design we propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum function where some of the components are easy that is they are lagrangian functions of explicitly known compact convex programs this corresponds to a stabilized partial dantzig wolfe decomposition where suitably modified representations of the easy convex subproblems are inserted in the master problem as an alternative to iteratively inner approximating them by extreme points thus providing the algorithm with exact information about a part of the dual objective function the resulting master problems are potentially larger and less well structured than the standard ones ruling out the available specialized techniques and requiring the use of general purpose solvers for their solution this strongly favors piecewise linear stabilizing terms as opposed to the more usual quadratic ones which in turn may have an adverse effect on the convergence speed of the algorithm so that the overall performance may depend on appropriate tuning of all these aspects yet very good computational results are obtained in at least one relevant application the computation of tight lower bounds for fixed charge multicommodity min cost flow problems

depth from defocus a spatial domain approach a new method named stm is described for determining distance of objects and rapid autofocusing of camera systems stm uses image defocus information and is based on a new spatial domain convolution deconvolution transform the method requires only two images taken with different camera parameters such as lens position focal length and aperture diameter both images can be arbitrarily blurred and neither of them needs to be a focused image therefore stm is very fast in comparison with depth from focus methods which search for the lens position or focal length of best focus the method involves simple local operations and can be easily implemented in parallel to obtain the depth map of a scene stm has been implemented on an actual camera system named sparcs experiments on the performance of stm and their results on real world planar objects are presented the results indicate that the accuracy of stm compares well with depth from focus methods and is useful in practical applications the utility of the method is demonstrated for rapid autofocusing of electronic cameras

stably average shadowable homoclinic classes abstract let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold m and let h f p be the homoclinic class of f containing p in this paper we show that if h f p is locally maximal and every hyperbolic periodic point in h f p is uniformly far away from being nonhyperbolic and h f p has the average shadowing property then h f p is hyperbolic