the image foresting transform theory algorithms and applications the image foresting transform ift is a graph based approach to the design of image processing operators based on connectivity it naturally leads to correct and efficient implementations and to a better understanding of how different operators relate to each other we give here a precise definition of the ift and a procedure to compute it a generalization of dijkstra s algorithm with a proof of correctness we also discuss implementation issues and illustrate the use of the ift in a few applications

optimal surface segmentation in volumetric images a graph theoretic approach efficient segmentation of globally optimal surfaces representing object boundaries in volumetric data sets is important and challenging in many medical image analysis applications we have developed an optimal surface detection method capable of simultaneously detecting multiple interacting surfaces in which the optimality is controlled by the cost functions designed for individual surfaces and by several geometric constraints defining the surface smoothness and interrelations the method solves the surface segmentation problem by transforming it into computing a minimum s hbox t cut in a derived arc weighted directed graph the proposed algorithm has a low order polynomial time complexity and is computationally efficient it has been extensively validated on more than 300 computer synthetic volumetric images 72 ct scanned data sets of different sized plexiglas tubes and tens of medical images spanning various imaging modalities in all cases the approach yielded highly accurate results our approach can be readily extended to higher dimensional image segmentation

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

a posteriori error estimation based on numerical realization of the variational multiscale method this paper presents a numerical realization of the variational multiscale method with the objective of providing a reliable and easy to implement local error estimation technique the variational multiscale framework provides a systematic approach for solution scale decomposition into coarse scales captured by the mesh and fine or subgrid scales in the proposed work the coarse scale errors in the finite element solution are neglected in comparison to the fine scale errors the fine scale variational equation is then localized using a general localization function over an element or a patch of elements to develop a local error estimation technique based on the proposed framework a consistent formulation of a new subdomain error estimator for elliptic problems is derived without the necessity of introducing an error locality assumption the new subdomain error estimator is evaluated numerically within a mesh adaptivity algorithm and it is shown to produce very sharp error estimates that outperform the element residual method estimates

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

existence of solutions for generalized equilibrium problem in g convex space in this paper we introduce and study a kind of generalized equilibrium problem in a g convex space by means of the fixed point theorems we obtain some existence theorems of solutions for the generalized equilibrium problems

guaranteed computable bounds for the exact error in the finite element solution part i one dimensional model problem abstract this paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution and the exact error in any bounded linear functional these bounds are constructed by employing approximate solutions of the element residual problems with equilibrated residual loads the one dimensional setting is used for the clarity of the ideas all the arguments employed can be extended to the higher dimensional case which will be discussed in part ii of this paper the main result presented here is that the computed bounds are guaranteed for the exact error and not the error with respect to an enriched finite element solution like the bounds proposed by other investigators and the bounds are guaranteed for any mesh however coarse it may be the quality of the bounds can be controlled by employing an inexpensive iterative scheme

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985

beyond the flow decomposition barrier we introduce a new approach to the maximum flow problem this approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities our approach leads to an o min n 2 3 m 1 2 m log n 2 m log u time bound for a network with n vertices m arcs and integral arc capacities in the range 1 u this is a fundamental improvement over the previous time bounds we also improve bounds for the gomory hu tree problem the parametric flow problem and the approximate s t cut problem

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis abstract in this article we address adaptive methods for isogeometric analysis based on local refinement guided by recovery based a posteriori error estimates isogeometric analysis was introduced a decade ago and an impressive progress has been made related to many aspects of numerical methods and advanced applications however related to adaptive mesh refinement guided by a posteriori error estimators rather few attempts are pursued besides the use of classical residual based error estimators in this article we explore a feature common for isogeometric analysis iga namely the use of structured tensorial meshes that facilitates superconvergence behavior of the gradient in the galerkin discretization by utilizing the concept of structured mesh refinement using lr b splines our aim is to facilitate superconvergence behavior for locally refined meshes as well superconvergence behavior matches well with the use of recovery based a posteriori estimator in the superconvergent patch recovery spr procedure however to our knowledge so far the spr procedure has not been exploited in the iga community we start out by addressing the existence of derivative superconvergent points in the computed finite element solution based on b splines and lr b splines for an elliptic model problem 1d and 2d poisson then we present some recovery procedures for improving the derivatives or gradient of the isogeometric finite element solution where the spr procedure will be the main focus in particular we show that our spr procedure for the improvement of derivatives fulfills the desired consistency criteria at the end we develop a posteriori error estimator where the improved gradient obtained from the proposed recovery procedures is used numerical results are presented to illustrate the efficiency of using spr procedure for the improvement of derivatives or gradient of computed solution in isogeometric analysis then the proposed a posteriori error estimator based adaptive refinement methodology is tested to solve smooth and non smooth elliptic benchmark problems the focus is put on whether optimal convergence rates are obtained in the computed solution or not as well as the effectivity index of the proposed error estimators

equilibrated patch recovery error estimates simple and accurate upper bounds of the error this paper introduces a new recovery type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error the upper bound property requires the recovered solution to be both statically equilibrated and continuous the equilibrium is obtained locally patch by patch and the continuity is enforced by a postprocessing based on the partition of the unity concept this postprocess is expected to preserve the features of the locally equilibrated stress field nevertheless the postprocess phase modifies the equilibrium which is no longer exactly fulfilled a new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium this requires computing the 2 norm of the error or relating it with the energy norm the guaranteed upper bounds are obtained by using a pessimistic bound of the error 2 norm derived from an eigenvalue problem nevertheless these bounds are not sharp an additional strategy based on a more accurate assessment of the error 2 norm is introduced providing sharp estimates which are practical upper bounds as it is demonstrated in the numerical tests copyright 2006 john wiley sons ltd

logismos b layered optimal graph image segmentation of multiple objects and surfaces for the brain automated reconstruction of the cortical surface is one of the most challenging problems in the analysis of human brain magnetic resonance imaging mri a desirable segmentation must be both spatially and topologically accurate as well as robust and computationally efficient we propose a novel algorithm logismos b based on probabilistic tissue classification generalized gradient vector flows and the logismos graph segmentation framework quantitative results on mri datasets from both healthy subjects and multiple sclerosis patients using a total of 16 thinspace 800 manually placed landmarks illustrate the excellent performance of our algorithm with respect to spatial accuracy remarkably the average signed error was only 0 084 mm for the white matter and 0 008 mm for the gray matter even in the presence of multiple sclerosis lesions statistical comparison shows that logismos b produces a significantly more accurate cortical reconstruction than freesurfer the current state of the art approach p ll 0 001 furthermore logismos b enjoys a run time that is less than a third of that of freesurfer which is both substantial considering the latter takes 10 h subject on average and a statistically significant speedup

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

on the asymptotic exactness of bank weiser s estimator in this paper we analyze an error estimator introduced by bank and weiser we prove that this estimator is asymptotically exact in the energy norm for regular solutions and parallel meshes by considering a simple example we show that this is not true for general meshes

best approximation coincidence and fixed point theorems for set valued maps in r trees abstract suppose x is a closed convex and geodesically bounded subset of a complete r tree m and suppose f x m is an almost lower semicontinuous set valued map whose values are nonempty closed convex suppose also g x x is a continuous onto quasiconvex set valued map with compact convex values then there exists x 0 x such that d g x 0 f x 0 inf x x d x f x 0 as applications we give some coincidence and fixed point results for weakly inward set valued maps our results generalize some well known results in literature

efficient graph based image segmentation this paper addresses the problem of segmenting an image into regions we define a predicate for measuring the evidence for a boundary between two regions using a graph based representation of the image we then develop an efficient segmentation algorithm based on this predicate and show that although this algorithm makes greedy decisions it produces segmentations that satisfy global properties we apply the algorithm to image segmentation using two different kinds of local neighborhoods in constructing the graph and illustrate the results with both real and synthetic images the algorithm runs in time nearly linear in the number of graph edges and is also fast in practice an important characteristic of the method is its ability to preserve detail in low variability image regions while ignoring detail in high variability regions

recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact or reference error lower bounds of the error are also needed in goal oriented adaptivity and for bounds on functional outputs this work introduces a simple and cheap strategy to recover a lower bound estimate from standard upper bound estimates this lower bound may also be used to assess the effectivity of the former estimate and to improve it

automated 3 d extraction of inner and outer surfaces of cerebral cortex from mri automatic computer processing of large multidimensional images such as those produced by magnetic resonance imaging mri is greatly aided by deformable models which are used to extract identify and quantify specific neuroanatomic structures a general method of deforming polyhedra is presented here with two novel features first explicit prevention of self intersecting surface geometries is provided unlike conventional deformable models which use regularization constraints to discourage but not necessarily prevent such behavior second deformation of multiple surfaces with intersurface proximity constraints allows each surface to help guide other surfaces into place using model based constraints such as expected thickness of an anatomic surface these two features are used advantageously to identify automatically the total surface of the outer and inner boundaries of cerebral cortical gray matter from normal human mr images accurately locating the depths of the sulci even where noise and partial volume artifacts in the image obscure the visibility of sulci the extracted surfaces are enforced to be simple two dimensional manifolds having the topology of a sphere even though the data may have topological holes this automatic 3 d cortex segmentation technique has been applied to 150 normal subjects simultaneously extracting both the gray white and gray cerebrospinal fluid interface from each individual the collection of surfaces has been used to create a spatial map of the mean and standard deviation for the location and the thickness of cortical gray matter three alternative criteria for defining cortical thickness at each cortical location were developed and compared these results are shown to corroborate published postmortem and in vivo measurements of cortical thickness

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

motion compensated mega voltage cone beam ct using the deformation derived directly from 2d projection images this paper presents a novel method for respiratory motion compensated reconstruction for cone beam computed tomography cbct the reconstruction is based on a time sequence of motion vector fields which is generated by a dynamic geometrical object shape model the dynamic model is extracted from the 2d projection images of the cbct the process of the motion extraction is converted into an optimal 3d multiple interrelated surface detection problem which can be solved by computing a maximum flow in a 4d directed graph the method was tested on 12 mega voltage mv cbct scans from three patients two sets of motion artifact free 3d volumes full exhale fe and full inhale fi phases were reconstructed for each daily scan the reconstruction was compared with three other motion compensated approaches based on quantification accuracy of motion and size contrast to noise ratio cnr was also quantified for image quality the proposed approach has the best overall performance with a relative tumor volume quantification error of 3 39 3 64 and 8 57 8 31 for fe and fi phases respectively the cnr near the tumor area is 3 85 0 42 fe and 3 58 3 33 fi these results show the clinical feasibility to use the proposed method to reconstruct motion artifact free mvcbct volumes

a posteriori finite element bounds for linear functional outputs of elliptic partial differential equations we present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order coercive linear partial differential equations in two space dimensions the method is based upon the construction of an augmented lagrangian in which the objective is a quadratic energy reformulation of the desired output and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements the bounds on the output for a suitably fine truth mesh discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid flux candidate lagrange multipliers generated by a k element coarser working mesh approximation independent of the form of the original partial differential equation the computation on the truth mesh is reduced to k decoupled subdomain local symmetric neumann problems the technique is illustrated for the convection diffusion and linear elasticity equations

guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains this is the pre peer reviewed version of the following article pares n santos h diez p guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains international journal for numerical methods in engineering 23 marc 2009 vol 79 num 10 p 1203 1244 which has been published in final form at http www3 interscience wiley com journal 122269073 abstract

optimal graph search based segmentation of airway tree double surfaces across bifurcations identification of both the luminal and the wall areas of the bronchial tree structure from volumetric x ray computed tomography ct data sets is of critical importance in distinguishing important phenotypes within numerous major lung diseases including chronic obstructive pulmonary diseases copd and asthma however accurate assessment of the inner and outer airway wall surfaces of a complete 3 d tree structure is difficult due to their complex nature particularly around the branch areas in this paper we extend a graph search based technique logismos to simultaneously identify multiple inter related surfaces of branching airway trees we first perform a presegmentation of the input 3 d image to obtain basic information about the tree topology the presegmented image is resampled along judiciously determined paths to produce a set of vectors of voxels called voxel columns the resampling process utilizes medial axes to ensure that voxel columns of appropriate lengths and directions are used to capture the object surfaces without interference a geometric graph is constructed whose edges connect voxels in the resampled voxel columns and enforce validity of the smoothness and separation constraints on the sought surfaces cost functions with directional information are employed to distinguish inner and outer walls the assessment of wall thickness measurement on a ct scanned double wall physical phantom patterned after an in vivo imaged human airway tree achieved highly accurate results on the entire 3 d tree the observed mean signed error of wall thickness ranged from 0 09 0 24 mm to 0 07 0 23 mm in bifurcating nonbifurcating areas the mean unsigned errors were 0 16 0 12 mm to 0 20 0 11 mm when the airway wall surface was partitioned into meaningful subregions the airway wall thickness accuracy was the same in most tested bifurcation nonbifurcation and carina noncarina regions p ns once validated on phantoms our method was applied to human in vivo volumetric ct data to demonstrate relationships of airway wall thickness as a function of luminal dimension and airway tree generation wall thickness differences between the bifurcation nonbifurcation regions were statistically significant p 0 05 for tree generations 6 7 8 and 9 in carina noncarina regions the wall thickness was statistically different in generations 1 4 5 6 7 and 8

new existence results for equilibrium problems we consider equilibrium problems in the framework of the formulation proposed by blum and oettli which includes variational inequalities nash equilibria in noncooperative games and vector optimization problems for instance as particular cases we establish new sufficient and or necessary conditions for existence of solutions of such problems our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems together with extensions to equilibrium problems of gap functions for variational inequalities then we apply our results to some particular instances of equilibrium problems obtaining results which include among others a new lemma of the alternative for convex optimization problems

a posteriori error estimation for standard finite element analysis a new residual type estimator based on projections of the error on subspaces of locally supported functions is presented the estimator is defined by a standard element by element refinement first an approximation of the energy norm of the error is obtained solving local problems with homogeneous dirichlet boundary conditions a later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces this estimate is a lower bound of the measure of the actual error the estimator does not need to approximate local boundary conditions for the error equation therefore computation of flux jumps is not necessary moreover the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward the presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error as well as its usefulness in adaptive procedures

a unified approach to remeshing strategies for finite element h adaptivity in h adaptivity a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution several authors have introduced different remeshing strategies yielding very different results in this work these methods are included in a unified framework emphasizing the role of the underlying hypotheses moreover an objective tool to evaluate the accuracy of the resulting finite element solution is presented thus a new remeshing strategy is introduced to optimize the accuracy of the adapted solutions the different remeshing strategies are compared with well known numerical examples

a recovery explicit error estimator in energy norm for linear elasticity significant research effort has been devoted to produce one sided error estimates for finite element analyses in particular to provide upper bounds of the actual error typically this has been achieved using residual type estimates one of the most popular and simpler in terms of implementation techniques used in commercial codes is the recovery based error estimator this technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm some attempts to remedy this situation provide bounds depending on unknown constants here a new step towards obtaining error bounds from the recovery based estimates is proposed the idea is 1 to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error 2 to add an explicit type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and 3 to efficiently and accurately evaluate the constants appearing in the bounding expressions thus providing asymptotic bounds the numerical tests with h adaptive refinement process show that the bounding property holds even for coarse meshes providing upper bounds in practical applications

fast geodesic active contours we use an unconditionally stable numerical scheme to implement a fast version of the geodesic active contour model the proposed scheme is useful for object segmentation in images like tracking moving objects in a sequence of images the method is based on the weickert romeney viergever additive operator splitting aos scheme it is applied at small regions motivated by the adalsteinsson sethian level set narrow band approach and uses sethian s 1996 fast marching method for re initialization experimental results demonstrate the power of the new method for tracking in color movies

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

active appearance models we describe a new method of matching statistical models of appearance to images a set of model parameters control modes of shape and gray level variation learned from a training set we construct an efficient iterative matching algorithm by learning the relationship between perturbations in the model parameters and the induced image errors

bounds for quantities of interest and adaptivity in the element free galerkin method a novel approach to implicit residual type error estimation in mesh free methods and an adaptive refinement strategy are presented this allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest the proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux equilibration and resulting in a simple implementation that avoids integrals on edges sides of a domain decomposition mesh this is especially interesting for mesh free methods the adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision copyright 2008 john wiley sons ltd

best approximation coincidence and fixed point theorems for quasi lower semicontinuous set valued maps in hyperconvex metric spaces abstract suppose x is a compact admissible subset of a hyperconvex metric spaces m and suppose f x m is a quasi lower semicontinuous set valued map whose values are nonempty admissible suppose also g x x is a continuous onto quasi convex set valued map with compact admissible values then there exists an x 0 x such that d g x 0 f x 0 inf x x d x f x 0 as applications we give some coincidence and fixed point results for weakly inward set valued maps our results generalize some well known results in literature

an optimal graph theoretic approach to data clustering theory and its application to image segmentation a novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated the data to be clustered are represented by an undirected adjacency graph g with arc capacities assigned to reflect the similarity between the linked vertices clustering is achieved by removing arcs of g to form mutually exclusive subgraphs such that the largest inter subgraph maximum flow is minimized for graphs of moderate size approximately 2000 vertices the optimal solution is obtained through partitioning a flow and cut equivalent tree of g which can be efficiently constructed using the gomory hu algorithm 1961 however for larger graphs this approach is impractical new theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size this algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices the new clustering algorithm is applied to the image segmentation problem the segmentation is achieved by effectively searching for closed contours of edge elements equivalent to minimum cuts in g which consist mostly of strong edges while rejecting contours containing isolated strong edges this method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours

a topology preserving level set method for geometric deformable models active contour and surface models also known as deformable models are powerful image segmentation techniques geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior parameterization independence and ease of implementation however a long claimed advantage of geometric deformable models the ability to automatically handle topology changes turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved we present a new class of geometric deformable models designed using a novel topology preserving level set method which achieves topology preservation by applying the simple point concept from digital topology these new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces moreover since the topology preserving constraint is enforced efficiently through local computations the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm

existence and duality of generalized vector equilibrium problems abstract in this paper we propose some dual formulations of generalized vector equilibrium problems by using such dual formulations we prove the existence of a solution to the generalized vector equilibrium problem under generalized pseudomonotonicity conditions the results of this paper extend and generalize the results of i v konnov and s schaible 2000 j optim theory appl 104 395 408

image segmentation with ratio cut this paper proposes a new cost function cut ratio for segmenting images using graph based methods the cut ratio is defined as the ratio of the corresponding sums of two different weights of edges along the cut boundary and models the mean affinity between the segments separated by the boundary per unit boundary length this new cost function allows the image perimeter to be segmented guarantees that the segments produced by bipartitioning are connected and does not introduce a size shape smoothness or boundary length bias the latter allows it to produce segmentations where boundaries are aligned with image edges furthermore the cut ratio cost function allows efficient iterated region based segmentation as well as pixel based segmentation these properties may be useful for some image segmentation applications while the problem of finding a minimum ratio cut in an arbitrary graph is np hard one can find a minimum ratio cut in the connected planar graphs that arise during image segmentation in polynomial time while the cut ratio alone is not sufficient as a baseline method for image segmentation it forms a good basis for an extended method of image segmentation when combined with a small number of standard techniques we present an implemented algorithm for finding a minimum ratio cut prove its correctness discuss its application to image segmentation and present the results of segmenting a number of medical and natural images using our techniques

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

on scalar equilibrium problem in generalized convex spaces in this paper we prove the existence of a solution to the scalar equilibrium problem in generalized convex space as applications we derive results on the best approximations and simultaneous approximations the results of this paper generalize some known results in the literature

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

cortex segmentation a fast variational geometric approach an automatic cortical gray matter segmentation from a three dimensional 3 d brain images magnetic resonance mr or computed tomography is a well known problem in medical image processing in this paper we first formulate it as a geometric variational problem for propagation of two coupled bounding surfaces an efficient numerical scheme is then used to implement the geodesic active surface model experimental results of cortex segmentation on real 3 d mr data are provided

globally optimal regions and boundaries as minimum ratio weight cycles we describe a new form of energy functional for the modeling and identification of regions in images the energy is defined on the space of boundaries in the image domain and can incorporate very general combinations of modeling information both from the boundary intensity gradients etc and from the interior of the region texture homogeneity etc we describe two polynomial time digraph algorithms for finding the global minima of this energy one of the algorithms is completely general minimizing the functional for any choice of modeling information it runs in a few seconds on a 256 spl times 256 image the other algorithm applies to a subclass of functionals but has the advantage of being extremely parallelizable neither algorithm requires initialization

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

the computation of bounds for linear functional outputs of weak solutions to the two dimensional elasticity equations we present a method for the computation of upper and lower bounds for linear functional outputs of the exact solutions to the two dimensional elasticity equations the method can be regarded as a generalization of the well known complementary energy principle the desired output is cast as the supremum of a quadratic linear convex functional over an infinite dimensional domain using duality the computation of an upper bound for the output of interest is reduced to a feasibility problem for the complementary or dual problem in order to make the problem tractable from a computational perspective two additional relaxations that preserve the bounding properties are introduced first the domain is triangulated and a domain decomposition strategy is used to generate a sequence of independent problems to be solved over each triangle the lagrange multipliers enforcing continuity are approximated using piecewise linear functions over the edges of the triangulation second the solution of the adjoint problem is approximated over the triangulation using a standard galerkin finite element approach a lower bound for the output of interest is computed by repeating the process for the negative of the output reversing the sign of the computed upper bound for the negative of the output yields a lower bound for the actual output the method can be easily generalized to three dimensions however a constructive proof for the existence of feasible solutions for the outputs of interest is only given in two dimensions the computed bound gaps are found to converge optimally that is at the same rate as the finite element approximation an attractive feature of the proposed approach is that it allows for a data set to be generated that can be used to certify and document the computed bounds using this data set and a simple algorithm the correctness of the computed bounds can be established without recourse to the original code used to compute them in the present paper only computational domains whose boundary is made up of straight sided segments and polynomially varying loads are considered two examples are given to illustrate the proposed methodology

guaranteed computable bounds for the exact error in the finite element solution part ii bounds for the energy norm of the error in two dimensions this paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution these bounds are constructed in terms of the solutions of local residual problems with equilibrated residual loads and are rather sharp even for coarse meshes he sharpness of the bounds can be further improved by employing few iterations of a relatively inexpensive iterative scheme he main result is that the bounds are guaranteed for the nergy norm of the exact error unlike the bounds which ave been proposed in 13 14 which are guaranteed only for the nergy norm of the error with respect to an enriched truth esh finite element solution copyright 2000 john wiley sons ltd

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

subdomain based flux free a posteriori error estimators a new residual type flux free error estimator is presented it estimates upper and lower bounds of the error in energy norm the proposed approach precludes the main drawbacks of standard residual type estimators circumvents the need of flux equilibration and results in a simple implementation that uses standard resources available in finite element codes this is specially interesting for 3d applications where the implementation of this technique is as simple as in 2d recall that on the contrary the complexity of the flux equilibration techniques increases drastically in the 3d case the bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs representing quantities of engineering interest the presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest

exact bounds for linear outputs of the advection diffusion reaction equation using flux free error estimates the paper introduces a methodology to compute strict upper and lower bounds for linear functional outputs of the exact solutions of the advection reaction diffusion equation the proposed approach is an alternative to the standard residual type estimators hybrid flux circumventing the need of flux equilibration following a flux free error estimation strategy the presented estimator provides sharper estimates than the ones provided by both the standard hybrid flux techniques and other flux free techniques

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

accurate recovery based upper error bounds for the extended finite element framework this paper introduces a recovery type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method xfem the paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a fem framework the upper bound property requires the recovered solution to be equilibrated and continuous the proposed technique consists of using a recovery technique especially adapted to the xfem framework that yields equilibrium at a local level patch by patch then a postprocess based on the partition of unity concept is used to obtain continuity the result is a very accurate but only nearly statically admissible recovered stress field with small equilibrium defaults introduced by the postprocess sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults as demonstrated by the numerical tests

graph based airway tree reconstruction from chest ct scans evaluation of different features on five cohorts we present a graph based framework for airway tree reconstruction from computerized tomography ct scans and evaluate the performance of different feature categories and their combinations on five lung cohorts the approach consists of two main processing steps first potential airway branch and connection candidates are identified and represented by a graph structure with weighted nodes and edges respectively second an optimization algorithm is utilized for generating an airway detection result by selecting a subset of airway branches and connections based on graph weights derived from image features the performance of the algorithm with different feature categories and their combinations was assessed on a set of 50 lung ct scans from five different cohorts including normal and diseased lungs results show trade offs between feature categories combinations in terms of correctly true positive and incorrectly false positive identified airways also the performance of features in dependence of lung cohort was analyzed across all cohorts a good trade off with high true positive rate tpr and low false positive rate fpr was achieved by a combination of gray value local shape and structural features this combination enabled extracting 91 80 of reference airways tpr in combination with a low fpr of 1 00 in addition this variant was evaluated on the public exact 09 test set and a comparison with other airway detection approaches is provided one of the main advantages of the presented method is that it is robust against local disturbances artifacts or other ambiguities that are frequently occurring in lung ct scans

error estimation including pollution assessment for nonlinear finite element analysis a residual type error estimator for nonlinear finite element analysis is introduced this error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure pollution errors are taken into account by a feedback strategy that is an error estimate based on local computations is used as the input of the pollution analysis this estimator is used in the frame of an adaptive procedure numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions moreover one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms

the background of error estimation and adaptivity in finite element computations the paper summarizes the present state of the art concerning error estimation and adaptive remeshing recent work shows that methods based on recovery result on most accurate error estimators

computing upper and lower bounds for the j integral in two dimensional linear elasticity we present an a posteriori method for computing rigorous upper and lower bounds for the j integral in two dimensional linear elasticity the j integral which is typically expressed as a contour integral is recast as a quadratic continuous functional of the displacement involving only area integration by expanding the quadratic output about an approximate finite element solution the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error the quadratic component is bounded by the energy norm of the error scaled by a continuity constant which is determined explicitly the linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields the method is illustrated with two fracture problems in plane strain elasticity an important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations

three dimensional segmentation of fluid associated abnormalities in retinal oct probability constrained graph search graph cut an automated method is reported for segmenting 3 d fluid associated abnormalities in the retina so called symptomatic exudate associated derangements sead from 3 d oct retinal images of subjects suffering from exudative age related macular degeneration in the first stage of a two stage approach retinal layers are segmented candidate sead regions identified and the retinal oct image is flattened using a candidate sead aware approach in the second stage a probability constrained combined graph search graph cut method refines the candidate seads by integrating the candidate volumes into the graph cut cost function as probability constraints the proposed method was evaluated on 15 spectral domain oct images from 15 subjects undergoing intravitreal anti vegf injection treatment leave one out evaluation resulted in a true positive volume fraction tpvf false positive volume fraction fpvf and relative volume difference ratio rvdr of 86 5 1 7 and 12 8 respectively the new graph cut graph search method significantly outperformed both the traditional graph cut and traditional graph search approaches p 0 01 p 0 04 and has the potential to improve clinical management of patients with choroidal neovascularization due to exudative age related macular degeneration