tree approximations to markov random fields methods for approximately computing the marginal probability mass functions and means of a markov random field mrf by approximating the lattice by a tree are described applied to the a posteriori mrf these methods solve bayesian spatial pattern classification and image restoration problems the methods are described several theoretical results concerning fixed point problems are proven and four numerical examples are presented including comparison with optimal estimators and the iterated conditional mode estimator and including two agricultural optical remote sensing problems

map estimation via agreement on trees message passing and linear programming we develop and analyze methods for computing provably optimal maximum a posteriori probability map configurations for a subclass of markov random fields defined on graphs with cycles by decomposing the original distribution into a convex combination of tree structured distributions we obtain an upper bound on the optimal value of the original problem i e the log probability of the map assignment in terms of the combined optimal values of the tree problems we prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common an important implication is that any such shared configuration must also be a map configuration for the original distribution next we develop two approaches to attempting to obtain tight upper bounds a a tree relaxed linear program lp which is derived from the lagrangian dual of the upper bounds and b a tree reweighted max product message passing algorithm that is related to but distinct from the max product algorithm in this way we establish a connection between a certain lp relaxation of the mode finding problem and a reweighted form of the max product min sum message passing algorithm

stereo without epipolar lines a maximum flow formulation this paper describes a new algorithm for solving the stereo correspondence problem with a global 2 d optimization by transforming it into a maximum flow problem in a graph this transformation effectively removes explicit use of epipolar geometry thus allowing direct use of multiple cameras with arbitrary geometries the maximum flow solved both efficiently and globally yields a minimum cut that corresponds to a disparity surface for the whole image at once this global and efficient approach to stereo analysis allows the reconstruction to proceed in an arbitrary volume of space and provides a more accurate and coherent depth map than the traditional stereo algorithms in particular smoothness is applied uniformly instead of only along epipolar lines while the global optimality of the depth surface is guaranteed results show improved depth estimation as well as better handling of depth discontinuities while the worst case running time is o n 1 5 d 1 5 log nd the observed average running time is o n 1 2 d 1 3 for an image size of n pixels and depth resolution d

the piecewise smooth mumford shah functional on an arbitrary graph the mumford shah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and despite being introduced over two decades ago it is still in widespread use present day optimization of the mumford shah functional is predominated by active contour methods until recently these formulations necessitated optimization of the contour by evolving via gradient descent which is known for its overdependence on initialization and the tendency to produce undesirable local minima in order to reduce these problems we reformulate the corresponding mumford shah functional on an arbitrary graph and apply the techniques of combinatorial optimization to produce a fast low energy solution in contrast to traditional optimization methods use of these combinatorial techniques necessitates consideration of the reconstructed image outside of its usual boundary requiring additionally the inclusion of regularization for generating these values the energy of the solution provided by this graph formulation is compared with the energy of the solution computed via traditional gradient descent based narrow band level set methods this comparison demonstrates that our graph formulation and optimization produces lower energy solutions than the traditional gradient descent based contour evolution methods in significantly less time finally we demonstrate the usefulness of the graph formulation to apply the mumford shah functional to new applications such as point clustering and filtering of nonuniformly sampled images

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

ranking with partial information a method and an application a method is presented for ranking multiattributed alternatives using a weighted additive evaluation function with partial information about the weighting scaling constants the method is applied to evaluate materials for use in nuclear waste containment the paper derives conditions to determine whether a pair of alternatives can be ranked given the partial information about weighting constants and presents an algorithm that partially rank orders the complete set of alternatives based on the pairwise ranking information

the utility efficient set and its interactive reduction we assume a decision situation under risk with incomplete information on preferences modelled as a vector utility function we consider an additive aggregation of its components and partial information on the scaling constants we develop the concept of utility efficiency to identify efficient strategies in discrete problems when the information about the scaling constants of the decision maker is in the form of a polyhedral cone a characterization of the utility efficient set provides a practical way to compute such efficient strategies we then discuss an interactive method based on the assessment of the scaling constants via an interactive paired comparison with its convergence the method is complemented by a procedure to reduce the utility efficient set to aid in the process of reaching a final strategy

on local region models and a statistical interpretation of the piecewise smooth mumford shah functional the mumford shah functional is a general and quite popular variational model for image segmentation in particular it provides the possibility to represent regions by smooth approximations in this paper we derive a statistical interpretation of the full piecewise smooth mumford shah functional by relating it to recent works on local region statistics moreover we show that this statistical interpretation comes along with several implications firstly one can derive extended versions of the mumford shah functional including more general distribution models secondly it leads to faster implementations finally thanks to the analytical expression of the smooth approximation via gaussian convolution the coordinate descent can be replaced by a true gradient descent

structured prediction dual extragradient and bregman projections we present a simple and scalable algorithm for maximum margin estimation of structured output models including an important class of markov networks and combinatorial models we formulate the estimation problem as a convex concave saddle point problem that allows us to use simple projection methods based on the dual extragradient algorithm nesterov 2003 the projection step can be solved using dynamic programming or combinatorial algorithms for min cost convex flow depending on the structure of the problem we show that this approach provides a memory efficient alternative to formulations based on reductions to a quadratic program qp we analyze the convergence of the method and present experiments on two very different structured prediction tasks 3d image segmentation and word alignment illustrating the favorable scaling properties of our algorithm

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

enhancing multiple point geostatistical modeling 1 graph theory and pattern adjustment in recent years higher order geostatistical methods have been used for modeling of a wide variety of large scale porous media such as groundwater aquifers and oil reservoirs their popularity stems from their ability to account for qualitative data and the great flexibility that they offer for conditioning the models to hard quantitative data which endow them with the capability for generating realistic realizations of porous formations with very complex channels as well as features that are mainly a barrier to fluid flow one group of such models consists of pattern based methods that use a set of data points for generating stochastic realizations by which the large scale structure and highly connected features are reproduced accurately the cross correlation based simulation ccsim algorithm proposed previously by the authors is a member of this group that has been shown to be capable of simulating multimillion cell models in a matter of a few cpu seconds the method is however sensitive to pattern s specifications such as boundaries and the number of replicates in this paper the original ccsim algorithm is reconsidered and two significant improvements are proposed for accurately reproducing large scale patterns of heterogeneities in porous media first an effective boundary correction method based on the graph theory is presented by which one identifies the optimal cutting path surface for removing the patchiness and discontinuities in the realization of a porous medium next a new pattern adjustment method is proposed that automatically transfers the features in a pattern to one that seamlessly matches the surrounding patterns the original ccsim algorithm is then combined with the two methods and is tested using various complex two and three dimensional examples it should however be emphasized that the methods that we propose in this paper are applicable to other pattern based geostatistical simulation methods

finding general and explicit solutions 2 1 dimensional broer kaup kupershmidt system nonlinear equation by exp function method abstract in this work we implement a relatively new analytical technique the exp function method for solving nonlinear special form of generalized nonlinear 2 1 dimensional broer kaup kupershmidt equation which may contain high nonlinear terms this method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics some numerical examples are presented to illustrate the efficiency and reliability of exp method it is predicted that exp function method can be found widely applicable in engineering

global minimization for continuous multiphase partitioning problems using a dual approach this paper is devoted to the optimization problem of continuous multi partitioning or multi labeling which is based on a convex relaxation of the continuous potts model in contrast to previous efforts which are tackling the optimal labeling problem in a direct manner we first propose a novel dual model and then build up a corresponding duality based approach by analyzing the dual formulation sufficient conditions are derived which show that the relaxation is often exact i e there exists optimal solutions that are also globally optimal to the original nonconvex potts model in order to deal with the nonsmooth dual problem we develop a smoothing method based on the log sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal dual model and suggests labelings with maximum entropy such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions an expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization numerical experiments also show that our method outperforms several competitive approaches in various aspects such as lower energies and better visual quality

an experimental comparison of different approaches to determining weights in additive utility models several studies this past decade have examined differences between holistic and decomposed approaches to determining weights in additive utility models some have argued that it matters little which procedure is used whereas others strongly favored particular methods in this paper we address this controversy experimentally by comparing five conceptually different approaches in terms of their weights and predictive ability the five methods are 1 multiple linear and non linear regression analyses of ten and fifteen holistic assessments 2 direct decomposed tradeoffs as proposed by keeney and raiffa keeney r l h raiffa 1977 decisions with multiple objectives wiley new york 3 a recent eigen vector technique of saaty saaty t l 1977 a scaling method for priorities in hierarchical structures j math psych 15 3 234 281 involving redundant pairwise comparisons of attributes 4 a straightforward allocation of hundred importance points and 5 unit weighting i e equal weighting after standardizing the attributes the decision task involved college admissions subjects were asked to evaluate hypothetical college applicants on the basis of verbal sat quantitative sat high school grade point average and a measure of extra curricular activity linear as well as nonlinear attribute utility functions were used in constructing the additive models the nonlinear functions were specified graphically by the subjects through selection from five different shapes i e one per attribute to test the predictive ability of the various models each subject made twenty separate pairwise comparisons of alternatives including direction and strength of preference the prediction criteria were percentage correct predictions as well as correlations using these twenty pairs seventy subjects were tested using an order controlled within subject design in comparing the different methods of weight determination monetary incentives were used to enhance motivation in terms of findings the methods generally differed systematically concerning the weights given to the various attributes as well as the variances of the resulting predictions on average however the methods predicted about equally well except for unit weighting which was clearly inferior the findings differ in this regard from the general literature furthermore nonlinear models were found to be inferior to linear ones finally subjects judged the methods to differ significantly in difficulty and trustworthiness which were found to correlate inversely the overall results raise various applied and theoretical issues which are discussed

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

dynamic graph cuts for efficient inference in markov random fields in this paper we present a fast new fully dynamic algorithm for the st mincut max flow problem we show how this algorithm can be used to efficiently compute map solutions for certain dynamically changing mrf models in computer vision such as image segmentation specifically given the solution of the max flow problem on a graph the dynamic algorithm efficiently computes the maximum flow in a modified version of the graph the time taken by it is roughly proportional to the total amount of change in the edge weights of the graph our experiments show that when the number of changes in the graph is small the dynamic algorithm is significantly faster than the best known static graph cut algorithm we test the performance of our algorithm on one particular problem the object background segmentation problem for video it should be noted that the application of our algorithm is not limited to the above problem the algorithm is generic and can be used to yield similar improvements in many other cases that involve dynamic change

multiresolution gauss markov random field models for texture segmentation this paper presents multiresolution models for gauss markov random fields gmrfs with applications to texture segmentation coarser resolution sample fields are obtained by subsampling the sample field at fine resolution although the markov property is lost under such resolution transformation coarse resolution non markov random fields can be effectively approximated by markov fields we present two techniques to estimate the gmrf parameters at coarser resolutions from the fine resolution parameters one by minimizing the kullback leibler distance and another based on local conditional distribution invariance we also allude to the fact that different gmrf parameters at the fine resolution can result in the same probability measure after subsampling and present the results for first and second order cases we apply this multiresolution model to texture segmentation different texture regions in an image are modeled by gmrfs and the associated parameters are assumed to be known parameters at lower resolutions are estimated from the fine resolution parameters the coarsest resolution data is first segmented and the segmentation results are propagated upward to the finer resolution we use the iterated conditional mode icm minimization at all resolutions our experiments with synthetic brodatz texture and real satellite images show that the multiresolution technique results in a better segmentation and requires lesser computation than the single resolution algorithm

a fast optimal spatial prediction method for massive datasets this article considers a class of multiresolution tree structured models that are spatially shifted versions of each other and proposes a new spatial prediction method that averages over the optimal spatial predictors produced from members of this class of models as a consequence the resulting predicted surface is smooth even when the predictors generated separately from individual multiresolution tree structured models are not we call the new predictor the multiresolution spatial murs predictor and develop a computationally efficient algorithm for it the algorithm can handle massive datasets even when some observations are missing moreover the murs predictor can be shown to be the minimum mean squared error predictor for a large class of covariance functions a simulation example for massive datasets shows that the murs method consistently outperforms two commonly used filtering methods total column ozone data remotely sensed from a satellite are analyzed using the new methodology

an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions this paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties the method is an extension of our earlier method published in this journal zionts s wallenius j 1976 an interactive programming method for solving the multiple criteria problem management sci 22 6 652 663 various technical problems present in predecessor versions have been resolved in addition to presenting the supporting theory and algorithm we discuss certain options in implementation and summarize our practical experience with several versions of the method

boundary detection by constrained optimization a statistical framework is used for finding boundaries and for partitioning scenes into homogeneous regions the model is a joint probability distribution for the array of pixel gray levels and an array of labels in boundary finding the labels are binary zero or one representing the absence or presence of boundary elements in partitioning the label values are generic two labels are the same when the corresponding scene locations are considered to belong to the same region the distribution incorporates a measure of disparity between certain spatial features of block pairs of pixel gray levels using the kolmogorov smirnov nonparametric measures of difference between the distributions of these features the number of model parameters is minimized by forbidding label configurations which are assigned probability zero the maximum a posteriori estimator of boundary placements and partitionings is examined the forbidden states introduce constraints into the calculation of these configurations stochastic relaxation methods are extended to accommodate constrained optimization

parametric maximum flow algorithms for fast total variation minimization this report studies the global minimization of anisotropically discretized total variation tv energies with an l p in particular l 1 and l 2 fidelity term using parametric maximum flow algorithms to minimize s t cut representations of these energies the tv l 2 model also known as the rudin osher fatemi rof model is suitable for restoring images contaminated by gaussian noise while the tv l 1 model is able to remove impulsive noise from grayscale images and perform multiscale decompositions of them preliminary numerical results on large scale two dimensional ct and three dimensional brain mr images are presented to illustrate the effectiveness of these approaches

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

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

bayes smoothing algorithms for segmentation of binary images modeled by markov random fields a new image segmentation algorithm is presented based on recursive bayes smoothing of images modeled by markov random fields and corrupted by independent additive noise the bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel given the total noisy image in a recursive way the a posteriori distribution together with a criterion of optimality then determine a bayes estimate of the scene the algorithm presented is an extension of a 1 d bayes smoothing algorithm to 2 d and it gives the optimum bayes estimate for the scene value at each pixel computational concerns in 2 d however necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm in particular the scene noiseless image is modeled as a markov mesh random field a special class of markov random fields and the bayes smoothing algorithm is applied on overlapping strips horizontal vertical of the image consisting of several rows columns it is assumed that the signal scene values vector sequence along the strip is a vector markov chain since signal correlation in one of the dimensions is not fully used along the edges of the strip estimates are generated only along the middle sections of the strips the overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image the bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel discrete or continuous random variables

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

variational iteration method for solving the epidemic model and the prey and predator problem in this article two problems are considered the first one is the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic and the other one is the problem of prey and predator the variational iteration method is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations governing on these problems the results are compared with the results obtained by adomian decomposition and power series methods some plots are presented to show the reliability and simplicity of the method

a linear framework for region based image segmentation and inpainting involving curvature penalization we present the first method to handle curvature regularity in region based image segmentation and inpainting that is independent of initialization to this end we start from a new formulation of length based optimization schemes based on surface continuation constraints and discuss the connections to existing schemes the formulation is based on a cell complex and considers basic regions and boundary elements the corresponding optimization problem is cast as an integer linear program we then show how the method can be extended to include curvature regularity again cast as an integer linear program here we are considering pairs of boundary elements to reflect curvature moreover a constraint set is derived to ensure that the boundary variables indeed reflect the boundary of the regions described by the region variables we show that by solving the linear programming relaxation one gets reasonably close to the global optimum and that curvature regularity is indeed much better suited in the presence of long and thin objects compared to standard length regularity

segmentation given partial grouping constraints we consider data clustering problems where partial grouping is known a priori we formulate such biased grouping problems as a constrained optimization problem where structural properties of the data define the goodness of a grouping and partial grouping cues define the feasibility of a grouping we enforce grouping smoothness and fairness on labeled data points so that sparse partial grouping information can be effectively propagated to the unlabeled data considering the normalized cuts criterion in particular our formulation leads to a constrained eigenvalue problem by generalizing the rayleigh ritz theorem to projected matrices we find the global optimum in the relaxed continuous domain by eigendecomposition from which a near global optimum to the discrete labeling problem can be obtained effectively we apply our method to real image segmentation problems where partial grouping priors can often be derived based on a crude spatial attentional map that binds places with common salient features or focuses on expected object locations we demonstrate not only that it is possible to integrate both image structures and priors in a single grouping process but also that objects can be segregated from the background without specific object knowledge

power watershed a unifying graph based optimization framework in this work we extend a common framework for graph based image segmentation that includes the graph cuts random walker and shortest path optimization algorithms viewing an image as a weighted graph these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework we then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm which we term the power watershed in particular when q 2 the power watershed leads to a multilabel scale and contrast invariant unique global optimum obtained in practice in quasi linear time placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation

bayesian modeling of dynamic scenes for object detection accurate detection of moving objects is an important precursor to stable tracking or recognition in this paper we present an object detection scheme that has three innovations over existing approaches first the model of the intensities of image pixels as independent random variables is challenged and it is asserted that useful correlation exists in intensities of spatially proximal pixels this correlation is exploited to sustain high levels of detection accuracy in the presence of dynamic backgrounds by using a nonparametric density estimation method over a joint domain range representation of image pixels multimodal spatial uncertainties and complex dependencies between the domain location and range color are directly modeled we propose a model of the background as a single probability density second temporal persistence is proposed as a detection criterion unlike previous approaches to object detection which detect objects by building adaptive models of the background the foregrounds modeled to augment the detection of objects without explicit tracking since objects detected in the preceding frame contain substantial evidence for detection in the current frame finally the background and foreground models are used competitively in a map mrf decision framework stressing spatial context as a condition of detecting interesting objects and the posterior function is maximized efficiently by finding the minimum cut of a capacitated graph experimental validation of the proposed method is performed and presented on a diverse set of dynamic scenes

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

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

cluster expansions for the deterministic computation of bayesian estimators based on markov random fields we describe a family of approximations denoted by cluster approximations for the computation of the mean of a markov random field mrf this is a key computation in image processing when applied to the a posteriori mrf the approximation is to account exactly for only spatially local interactions application of the approximation requires the solution of a nonlinear multivariable fixed point equation for which we prove several existence uniqueness and convergence of algorithm results four numerical examples are presented including comparison with monte carlo calculations

statistical mechanical approach to image processing the basic frameworks and techniques of the bayesian approach to image restoration are reviewed from the statistical mechanical point of view first a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem second the basic formulation of the statistical estimation from the observed degraded image by using the bayes formula is demonstrated the relationship between bayesian statistics and statistical mechanics is also listed particularly it is explained that some correlation inequalities on the nishimori line of the random spin model also play an important role in bayesian image restoration third the framework of bayesian image restoration for binary images by means of the ising model is reviewed some practical algorithms for binary image restoration are given by employing the mean field and the bethe approximations finally bayesian image restoration for a grey level image using the gaussian model is reviewed and the gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges the line state is also extended to quantized values

sar image regularization with fast approximate discrete minimization synthetic aperture radar sar images like other coherent imaging modalities suffer from speckle noise the presence of this noise makes the automatic interpretation of images a challenging task and noise reduction is often a prerequisite for successful use of classical image processing algorithms numerous approaches have been proposed to filter speckle noise markov random field mrf modelization provides a convenient way to express both data fidelity constraints and desirable properties of the filtered image in this context total variation minimization has been extensively used to constrain the oscillations in the regularized image while preserving its edges speckle noise follows heavy tailed distributions and the mrf formulation leads to a minimization problem involving nonconvex log likelihood terms such a minimization can be performed efficiently by computing minimum cuts on weighted graphs due to memory constraints exact minimization although theoretically possible is not achievable on large images required by remote sensing applications the computational burden of the state of the art algorithm for approximate minimization namely the alpha expansion is too heavy specially when considering joint regularization of several images we show that a satisfying solution can be reached in few iterations by performing a graph cut based combinatorial exploration of large trial moves this algorithm is applied to joint regularization of the amplitude and interferometric phase in urban area sar images

self validated labeling of markov random fields for image segmentation this paper addresses the problem of self validated labeling of markov random fields mrfs namely to optimize an mrf with unknown number of labels we present graduated graph cuts ggc a new technique that extends the binary s t graph cut for self validated labeling specifically we use the split and merge strategy to decompose the complex problem to a series of tractable subproblems in terms of gibbs energy minimization a suboptimal labeling is gradually obtained based upon a set of cluster level operations by using different optimization structures we propose three practical algorithms tree structured graph cuts tsgc net structured graph cuts nsgc and hierarchical graph cuts hgc in contrast to previous methods the proposed algorithms can automatically determine the number of labels properly balance the labeling accuracy spatial coherence and the labeling cost i e the number of labels and are computationally efficient independent to initialization and able to converge to good local minima of the objective energy function we apply the proposed algorithms to natural image segmentation experimental results show that our algorithms produce generally feasible segmentations for benchmark data sets and outperform alternative methods in terms of robustness to noise speed and preservation of soft boundaries

synthetic aperture imaging using pixel labeling via energy minimization synthetic aperture imaging using an array of cameras which has become popular recently can easily handle the occlusion problem by seeing through occluders unfortunately the resulting image is still blurry because it combines information not only from the region of interest but also from the occluding regions removing the blurriness of synthetic aperture images has become a challenging task for many computer vision applications in this paper we propose a novel method to improve the image quality of synthetic aperture imaging using energy minimization unlike the conventional synthetic aperture imaging method which averages images from all the camera views we reformulate the problem as a labeling problem in particular we use the energy minimization method to label each pixel in each camera view to decide whether or not it belongs to an occluder after that the focusing at the desired depth is by averaging pixels that are not labeled as occluder the experimental results show that the proposed method outperforms the traditional synthetic aperture imaging method as well as its improved versions which are simply dim and blur occluders in the resulting image to the best of our knowledge our proposed method is the first one for improving the results of synthetic aperture image without using a training set from the input sequence as well it is the first method that makes no assumptions on whether or not the objects in the scenes are static

a linear programming approach to max sum problem a review the max sum labeling problem defined as maximizing a sum of binary i e pairwise functions of discrete variables is a general np hard optimization problem with many applications such as computing the map configuration of a markov random field we review a not widely known approach to the problem developed by ukrainian researchers schlesinger et al in 1976 and show how it contributes to recent results most importantly those on the convex combination of trees and tree reweighted max product in particular we review schlesinger et al s upper bound on the max sum criterion its minimization by equivalent transformations its relation to the constraint satisfaction problem the fact that this minimization is dual to a linear programming relaxation of the original problem and the three kinds of consistency necessary for optimality of the upper bound we revisit problems with boolean variables and supermodular problems we describe two algorithms for decreasing the upper bound we present an example application for structural image analysis

continuous global optimization in multiview 3d reconstruction in this article we introduce a new global optimization method to the field of multiview 3d reconstruction while global minimization has been proposed in a discrete formulation in form of the maxflow mincut framework we suggest the use of a continuous convex relaxation scheme specifically we propose to cast the problem of 3d shape reconstruction as one of minimizing a spatially continuous convex functional in qualitative and quantitative evaluation we demonstrate several advantages of the proposed continuous formulation over the discrete graph cut solution firstly geometric properties such as weighted boundary length and surface area are represented in a numerically consistent manner the continuous convex relaxation assures that the algorithm does not suffer from metrication errors in the sense that the reconstruction converges to the continuous solution as the spatial resolution is increased moreover memory requirements are reduced allowing for globally optimal reconstructions at higher resolutions we study three different energy models for multiview reconstruction which are based on a common variational template unifying regional volumetric terms and on surface photoconsistency the three models use data measurements at increasing levels of sophistication while the first two approaches are based on a classical silhouette based volume subdivision the third one relies on stereo information to define regional costs furthermore this scheme is exploited to compute a precise photoconsistency measure as opposed to the classical estimation all three models are compared on standard data sets demonstrating their advantages and shortcomings for the third one which gives the most accurate results a more exhaustive qualitative and quantitative evaluation is presented

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

dinkelbach ncut an efficient framework for solving normalized cuts problems with priors and convex constraints in this paper we propose a novel framework called dinkelbach ncut dncut which efficiently solves the normalized graph cut ncut problem under general convex constraints as well as under given priors on the nodes of the graph current ncut methods use generalized eigen decomposition which poses computational issues especially for large graphs and can only handle linear equality constraints by using an augmented graph and the iterative dinkelbach method for fractional programming fp we formulate the dncut framework to efficiently solve the ncut problem under general convex constraints and given data priors in this framework the initial problem is converted into a sequence of simpler sub problems i e convex quadratic programs qp s subject to convex constraints the complexity of finding a global solution for each sub problem depends on the complexity of the constraints the convexity of the cost function and the chosen initialization however we derive an initialization which guarantees that each sub problem is a convex qp that can be solved by available convex programming techniques we apply this framework to the special case of linear constraints where the solution is obtained by solving a sequence of sparse linear systems using the conjugate gradient method we validate dncut by performing binary segmentation on real images both with and without linear nonlinear constraints as well as multi class segmentation when possible we compare dncut to other ncut methods in terms of segmentation performance and computational efficiency even though the new formulation is applied to the problem of spectral graph based low level image segmentation it can be directly applied to other applications e g clustering

shortest path constraints for 3d multiobject semiautomatic segmentation via clustering and graph cut we derive shortest path constraints from graph models of structure adjacency relations and introduce them in a joint centroidal voronoi image clustering and graph cut multiobject semiautomatic segmentation framework the vicinity prior model thus defined is a piecewise constant model incurring multiple levels of penalization capturing the spatial configuration of structures in multiobject segmentation qualitative and quantitative analyses and comparison with a potts prior based approach and our previous contribution on synthetic simulated and real medical images show that the vicinity prior allows for the correct segmentation of distinct structures having identical intensity profiles and improves the precision of segmentation boundary placement while being fairly robust to clustering resolution the clustering approach we take to simplify images prior to segmentation strikes a good balance between boundary adaptivity and cluster compactness criteria furthermore allowing to control the trade off compared with a direct application of segmentation on voxels the clustering step improves the overall runtime and memory footprint of the segmentation process up to an order of magnitude without compromising the quality of the result

generalized flows for optimal inference in higher order mrf map use of higher order clique potentials in mrf map problems has been limited primarily because of the inefficiencies of the existing algorithmic schemes we propose a new combinatorial algorithm for computing optimal solutions to 2 label mrf map problems with higher order clique potentials the algorithm runs in time o 2 kn 3 in the worst case k is size of clique and n is the number of pixels a special gadget is introduced to model flows in a higher order clique and a technique for building a flow graph is specified based on the primal dual structure of the optimization problem the notions of the capacity of an edge and a cut are generalized to define a flow problem we show that in this flow graph when the clique potentials are submodular the max flow is equal to the min cut which also is the optimal solution to the problem we show experimentally that our algorithm provides significantly better solutions in practice and is hundreds of times faster than solution schemes like dual decomposition 1 trws 2 and reduction 3 4 5 the framework represents a significant advance in handling higher order problems making optimal inference practical for medium sized cliques

fast approximate maximum a posteriori restoration of multicolour images we propose a new algorithm for the approximation of the maximum a posteriori map restoration of noisy images the image restoration problem is considered in a bayesian setting we assume as prior distribution multicolour markov random fields on a graph whose main restriction is the presence of only pairwise site interactions the noise is modelled as a bernoulli field computing the mode of the posterior distribution is np complete i e can very likely be done only in a time exponential in the number of sites of the underlying graph our algorithm runs in polynomial time and is based on the coding of the colours it produces an image with the following property either a pixel is coloured with one of the possible colours or it is left blank in the first case we prove that this is the colour of the site in the exact map restoration the quality of the approximation is then measured by the number of sites being left blank we assess the performance of the new algorithm by numerical experiments on the simple three colour potts model more rigorously we present a probabilistic analysis of the algorithm the results indicate that the approximation is quite often sufficiently good for the interpretation of the image

multiscale image segmentation using wavelet domain hidden markov models we introduce a new image texture segmentation algorithm hmtseg based on wavelets and the hidden markov tree hmt model the hmt is a tree structured probabilistic graph that captures the statistical properties of the coefficients of the wavelet transform since the hmt is particularly well suited to images containing singularities edges and ridges it provides a good classifier for distinguishing between textures utilizing the inherent tree structure of the wavelet hmt and its fast training and likelihood computation algorithms we perform texture classification at a range of different scales we then fuse these multiscale classifications using a bayesian probabilistic graph to obtain reliable final segmentations since hmtseg works on the wavelet transform of the image it can directly segment wavelet compressed images without the need for decompression into the space domain we demonstrate the performance of hmtseg with synthetic aerial photo and document image segmentations

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

exact solution to median surface problem using 3d graph search and application to parameter space exploration in this paper we propose the generalized median surface problem and present an exact solution by applying a 3d graph search algorithm to our knowledge this is a novel class of median computation problem which has not been considered before in the literature in addition to the theoretic interest in median surface computation we also demonstrate its practical value by means of the task of parameter space exploration without ground truth which is an effective means of dealing with the difficult parameter selection problem for image segmentation we present a concrete application for artery boundary detection in sonography ultrasound imaging it will be shown that the median computation can not only avoid the parameter training but also potentially achieve results comparable with trained parameters the median based approach can thus be a good alternate particularly in situations with no ground truth available highlightsthis paper proposes the median surface problem an optimal graph search algorithm is applied to solve the problem efficiently experimental work is conducted to demonstrate its usability it can potentially be used for segmentation parameter exploration

decision making with incomplete information abstract decision situations with incomplete information are characterized by a decision maker without a precisely defined stable preference structure by probability distributions not known completely or by an inexact evaluation of consequences within the paper a general framework for decision making with incomplete information is presented which shows how to solve problems from descriptive as well as prescriptive decision theory based on this framework an overview of existing methods which are particularly suitable for handling incomplete information is given

tracking cortical activity from m eeg using graph cuts with spatiotemporal constraints this work proposes to use magnetoencephalography meg and electroencephalography eeg source imaging to provide cinematic representations of the temporal dynamics of cortical activations cortical activation maps seen as images of the active brain are scalar maps defined at the vertices of a triangulated cortical surface they can be computed from m eeg data using a linear inverse solver every millisecond taking as input these activation maps and exploiting both the graph structure of the cortical mesh and the high sampling rate of m eeg recordings neural activations are tracked over time using an efficient graph cut based algorithm the method estimates the spatiotemporal support of the active brain regions it consists in computing a minimum cut on a particularly designed weighted graph imposing spatiotemporal regularity constraints on the activation patterns each node of the graph is assigned a label active or non active the method works globally on the full time period of interest can cope with spatially extended active regions and allows the active domain to exhibit topology changes over time the algorithm is illustrated and validated on synthetic data results of the method are provided on two meg cognitive experiments in the visual and somatosensory cortices demonstrating the ability of the algorithm to handle various types of data

graph cuts and efficient n d image segmentation combinatorial graph cut algorithms have been successfully applied to a wide range of problems in vision and graphics this paper focusses on possibly the simplest application of graph cuts segmentation of objects in image data despite its simplicity this application epitomizes the best features of combinatorial graph cuts methods in vision global optima practical efficiency numerical robustness ability to fuse a wide range of visual cues and constraints unrestricted topological properties of segments and applicability to n d problems graph cuts based approaches to object extraction have also been shown to have interesting connections with earlier segmentation methods such as snakes geodesic active contours and level sets the segmentation energies optimized by graph cuts combine boundary regularization with region based properties in the same fashion as mumford shah style functionals we present motivation and detailed technical description of the basic combinatorial optimization framework for image segmentation via s t graph cuts after the general concept of using binary graph cut algorithms for object segmentation was first proposed and tested in boykov and jolly 2001 this idea was widely studied in computer vision and graphics communities we provide links to a large number of known extensions based on iterative parameter re estimation and learning multi scale or hierarchical approaches narrow bands and other techniques for demanding photo video and medical applications

parallel and deterministic algorithms from mrfs surface reconstruction deterministic approximations to markov random field mrf models are derived one of the models is shown to give in a natural way the graduated nonconvexity gnc algorithm proposed by a blake and a zisserman 1987 this model can be applied to smooth a field preserving its discontinuities a class of more complex models is then proposed in order to deal with a variety of vision problems all the theoretical results are obtained in the framework of statistical mechanics and mean field techniques a parallel iterative algorithm to solve the deterministic equations of the two models is presented together with some experiments on synthetic and real images

bittracker a bitmap tracker for visual tracking under very general conditions this paper addresses the problem of visual tracking under very general conditions a possibly non rigid target whose appearance may drastically change over time general camera motion a 3d scene and no a priori information except initialization this is in contrast to the vast majority of trackers which rely on some limited model in which for example the target s appearance is known a priori or restricted the scene is planar or a pan tilt zoom camera is used their goal is to achieve speed and robustness but their limited context may cause them to fail in the more general case the proposed tracker works by approximating in each frame a pdf probability distribution function of the target s bitmap and then estimating the maximum a posteriori bitmap the pdf is marginalized over all possible motions per pixel thus avoiding the stage in which optical flow is determined this is an advantage over other general context trackers that do not use the motion cue at all or rely on the error prone calculation of optical flow using a gibbs distribution with respect to the first order neighborhood system yields a bitmap pdf whose maximization may be transformed into that of a quadratic pseudo boolean function the maximum of which is approximated via a reduction to a maximum flow problem many experiments were conducted to demonstrate that the tracker is able to track under the aforementioned general context

tree based reparameterization framework for analysis of sum product and related algorithms we present a tree based reparameterization trp framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles this class includes the belief propagation bp or sum product algorithm as well as variations and extensions of bp algorithms in this class can be formulated as a sequence of reparameterization updates each of which entails refactorizing a portion of the distribution corresponding to an acyclic subgraph i e a tree or more generally a hypertree the ultimate goal is to obtain an alternative but equivalent factorization using functions that represent exact or approximate marginal distributions on cliques of the graph our framework highlights an important property of the sum product algorithm and the larger class of reparameterization algorithms the original distribution on the graph with cycles is not changed the perspective of tree based updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency we develop interpretations of these results in terms of information geometry the invariance of the distribution in conjunction with the fixed point characterization enables us to derive an exact expression for the difference between the true marginals on an arbitrary graph with cycles and the approximations provided by belief propagation more broadly our analysis applies to any algorithm that minimizes the bethe free energy we also develop bounds on the approximation error which illuminate the conditions that govern their accuracy finally we show how the reparameterization perspective extends naturally to generalizations of bp e g kikuchi 1951 approximations and variants via the notion of hypertree reparameterization

message passing for graph structured linear programs proximal methods and rounding schemes the problem of computing a maximum a posteriori map configuration is a central computational challenge associated with markov random fields there has been some focus on tree based linear programming lp relaxations for the map problem this paper develops a family of super linearly convergent algorithms for solving these lps based on proximal minimization schemes using bregman divergences as with standard message passing on graphs the algorithms are distributed and exploit the underlying graphical structure and so scale well to large problems our algorithms have a double loop character with the outer loop corresponding to the proximal sequence and an inner loop of cyclic bregman projections used to compute each proximal update we establish convergence guarantees for our algorithms and illustrate their performance via some simulations we also develop two classes of rounding schemes deterministic and randomized for obtaining integral configurations from the lp solutions our deterministic rounding schemes use a re parameterization property of our algorithms so that when the lp solution is integral the map solution can be obtained even before the lp solver converges to the optimum we also propose graph structured randomized rounding schemes applicable to iterative lp solving algorithms in general we analyze the performance of and report simulations comparing these rounding schemes

multiple criteria decision making multiattribute utility theory the next ten years management science and decision science have grown exponentially since midcentury two closely related fields central to this growth are multiple criteria decision making mcdm and multiattribute utility theory maut this paper comments on the history of mcdm and maut and discusses topics we believe are important in their continued development and usefulness to management science over the next decade our aim is to identify exciting directions and promising areas for future research

approximation algorithms for classification problems with pairwise relationships metric labeling and markov random fields in a traditional classification problem we wish to assign one of k labels or classes to each of n objects in a way that is consistent with some observed data that we have about the problem an active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified this issue is one of the principal motivations for the framework of markov random fields and it arises in areas such as image processing biometry and document analysis in its most basic form this style of analysis seeks to find a classification that optimizes a combinatorial function consisting of assignment costs based on the individual choice of label we make for each object and separation costs based on the pair of choices we make for two related objects we formulate a general classification problem of this type the metric labeling problem we show that it contains as special cases a number of standard classification frameworks including several arising from the theory of markov random fields from the perspective of combinatorial optimization our problem can be viewed as a substantial generalization of the multiway cut problem and equivalent to a type of uncapacitated quadratic assignment problem we provide the first nontrivial polynomial time approximation algorithms for a general family of classification problems of this type our main result is an o log k log log k approximation algorithm for the metric labeling problem with respect to an arbitrary metric on a set of k labels and an arbitrary weighted graph of relationships on a set of objects for the special case in which the labels are endowed with the uniform metric all distances are the same our methods provide a 2 approximation algorithm

pm pm patchmatch with potts model for object segmentation and stereo matching this paper presents a unified variational formulation for joint object segmentation and stereo matching which takes both accuracy and efficiency into account in our approach depth map consists of compact objects each object is represented through three different aspects 1 the perimeter in image space 2 the slanted object depth plane and 3 the planar bias which is to add an additional level of detail on top of each object plane in order to model depth variations within an object compared with traditional high quality solving methods in low level we use a convex formulation of the multilabel potts model with patchmatch stereo techniques to generate depth map at each image in object level and show that accurate multiple view reconstruction can be achieved with our formulation by means of induced homography without discretization or staircasing artifacts our model is formulated as an energy minimization that is optimized via a fast primal dual algorithm which can handle several hundred object depth segments efficiently performance evaluations in the middlebury benchmark data sets show that our method outperforms the traditional integer valued disparity strategy as well as the original patchmatch algorithm and its variants in subpixel accurate disparity estimation the proposed algorithm is also evaluated and shown to produce consistently good results for various real world data sets kitti benchmark data sets and multiview benchmark data sets

interactively multiphase image segmentation based on variational formulation and graph cuts in this paper the multiple piecewise constant mpc active contour model is extended to deal with multiphase case this proposed multiphase model can be effectively optimized by solving the minimum cuts problem of a specially devised multilayer graph based on the proposed energy functional and its graph cuts optimization an interactively multiphase partition method for image segmentation is presented the user places some scribbles with different colors on the image according to the practical application demand and each group of scribbles with the same color corresponds to a potential image region the distribution of each region can be learned from the input scribbles with some particular color then the corresponding multilayer graph can be constructed and its minimum cuts can be computed to determine the segmentation result of the image numerical experiments show that the proposed interactively multiphase segmentation method can accurately segment the image into different regions according to the input scribbles with different color

partial information dominance and potential optimality in multiattribute utility theory when a multiattribute utility function is only partially specified by prior preference statements what can be said about the relative desirability of actual alternatives this question is addressed for the cases of additively separable cardinal utility with unknown scaling constants multiplicatively separable cardinal utility with unknown scaling constants and additively separable ordinal utility we review previous approaches and treat issues of consistency is the prior information consistent dominance does the prior preference information imply that one outcome is preferred to another and potential optimality are there utility functions of the given form consistent with prior preference information under which a particular outcome is preference optimal in the additive cases a key relationship between dominance and potential optimality may be derived the paper concludes by presenting an example application to a well known nuclear siting study

a survey and comparison of discrete and continuous multi label optimization approaches for the potts model we present a survey and a comparison of a variety of algorithms that have been proposed over the years to minimize multi label optimization problems based on the potts model discrete approaches based on markov random fields as well as continuous optimization approaches based on partial differential equations can be applied to the task in contrast to the case of binary labeling the multi label problem is known to be np hard and thus one can only expect near optimal solutions in this paper we carry out a theoretical comparison and an experimental analysis of existing approaches with respect to accuracy optimality and runtime aimed at bringing out the advantages and short comings of the respective algorithms systematic quantitative comparison is done on the graz interactive image segmentation benchmark this paper thereby generalizes a previous experimental comparison klodt et al 2008 from the binary to the multi label case

minimization of monotonically levelable higher order mrf energies via graph cuts a feature of minimizing images of submodular binary markov random field mrf energies is introduced using this novel feature the collection of minimizing images of levels of higher order monotonically levelable multilabel mrf energies is shown to constitute a monotone collection this implies that these minimizing binary images can be combined to give minimizing images of the multilabel mrf energies thanks to the graph cuts framework the mentioned class of binary mrf energies is known to be minimized by maximum flow computations on appropriately constructed graphs with the aid of these developments an exact and efficient algorithm to minimize monotonically levelable multilabel mrf energies of any order which is composed of a series of maximum flow computations is proposed and an application of the proposed algorithm to image denoising is given

new loss functions in bayesian imaging abstract unlike the development of more accurate prior distributions for use in bayesian imaging the design of more sensible estimators through loss functions has been neglected in the literature we discuss the design of loss functions with a local structure that depend only on a binary misclassification vector the proposed approach is similar to modeling with a markov random field the bayes estimate is calculated in a two step algorithm using markov chain monte carlo and simulated annealing algorithms we present simulation experiments with the ising model where the observations are corrupted with gaussian and flip noise