multiregion image segmentation by parametric kernel graph cuts the purpose of this study is to investigate multiregion graph cut image partitioning via kernel mapping of the image data the image data is transformed implicitly by a kernel function so that the piecewise constant model of the graph cut formulation becomes applicable the objective function contains an original data term to evaluate the deviation of the transformed data within each segmentation region from the piecewise constant model and a smoothness boundary preserving regularization term the method affords an effective alternative to complex modeling of the original image data while taking advantage of the computational benefits of graph cuts using a common kernel function energy minimization typically consists of iterating image partitioning by graph cut iterations and evaluations of region parameters via fixed point computation a quantitative and comparative performance assessment is carried out over a large number of experiments using synthetic grey level data as well as natural images from the berkeley database the effectiveness of the method is also demonstrated through a set of experiments with real images of a variety of types such as medical synthetic aperture radar and motion maps

unsupervised variational image segmentation classification using a weibull observation model studies have shown that the weibull distribution can model accurately a wide variety of images its parameters index a family of distributions which includes the exponential and approximations of the gaussian and the raleigh models widely used in image segmentation this study investigates the weibull distribution in unsupervised image segmentation and classification by a variational method the data term of the segmentation functional measures the conformity of the image intensity in each region to a weibull distribution whose parameters are determined jointly with the segmentation minimization of the functional is implemented by active curves via level sets and consists of iterations of two consecutive steps curve evolution via euler lagrange descent equations and evaluation of the weibull distribution parameters experiments with synthetic and real images are described which verify the validity of method and its implementation

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

color level sets a multi phase method for structural topology optimization with multiple materials in this paper we address the problem of structural shape and topology optimization in a multi material domain a level set method is employed as an alternative approach to the popular homogenization based methods of rule of mixtures for multi material modeling a multi phase level set model is adapted for material and topology representation this model eliminates the need for a material interpolation or phase mixing scheme it only requires m level set functions to represent a structure of n 2m different material phases in a principle similar to combining colors from the three primary colors therefore this multi phase model may be referred to as a color level set representation which has its unique benefits it is flexible to handle complex topologies it substantially reduces the number of model functions when n 3 it automatically avoids the problem of overlap between material phases of a conventional partitioning approach we describe numerical techniques for efficient and robust implementation of the method by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain this would separate the issues of accuracy in numerical calculations of the physical equation and in the level set model propagation a gradient projection method is described for incorporating multiple constraints in the problem finally the benefits and the advantages of the developed method are illustrated with several 2d examples of mean compliance minimization of multi material structures

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

sobolev active contours all previous geometric active contour models that have been formulated as gradient flows of various energies use the same l 2 type inner product to define the notion of gradient recent work has shown that this inner product induces a pathological riemannian metric on the space of smooth curves however there are also undesirable features associated with the gradient flows that this inner product induces in this paper we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with sobolev type inner products we call the resulting flows sobolev active contours sobolev metrics induce favorable regularity properties in their gradient flows in addition sobolev active contours favor global translations but are not restricted to such motions they are also less susceptible to certain types of local minima in contrast to traditional active contours these properties are particularly useful in tracking applications we demonstrate the general methodology by reformulating some standard edge based and region based active contour models as sobolev active contours and show the substantial improvements gained in segmentation

a variational model with hybrid images data fitting energies for segmentation of images with intensity inhomogeneity level set functions based variational image segmentation models provide reliable methods to capture boundaries of objects regions in a given image provided that the underlying intensity has homogeneity the case of images with essentially piecewise constant intensities is satisfactorily dealt with in the well known work of chan vese 2001 and its many variants however for images with intensity inhomogeneity or multiphases within the foreground of objects such models become inadequate because the detected edges and even phases do not represent objects and are hence not meaningful to deal with such problems in this paper we have proposed a new variational model with two fitting terms based on regions and edges enhanced quantities respectively from multiplicative and difference images tests and comparisons will show that our new model outperforms two previous models both synthetic and real life images are used to illustrate the reliability and advantages of our new model graphical abstractdisplay omitted highlightswe model image segmentation problem using product and difference images the convex formulation of the model is also presented the proposed model takes care of the minute details and inhomogeneity in an image this model is compared with lcv and fgm of the active contour model the competing energies are qualitatively tested by using jaccord similarity index

a numerical method for computing minimal surfaces in arbitrary dimension in this paper we propose a numerical method for computing minimal surfaces with fixed boundaries the level set method is used to evolve a codimension 1 surface with fixed codimension 2 boundary in r n under mean curvature flow for n 3 the problem has been approached in d l chopp 1993 and l t cheng d l chopp computing minimal surfaces via level set curvature flow j comput phys 106 1 1993 77 91 and l t cheng the level set method applied to geometrically based motion materials science and image processing ucla cam report 00 20 using the level set method but with a more complicated boundary conditions the method we present can be generalized straightforward to arbitrary dimension and the framework in which it is presented is dimension independent examples are shown for n 2 3 4

threshold dynamics for the piecewise constant mumford shah functional we propose an efficient algorithm for minimizing the piecewise constant mumford shah functional of image segmentation it is based on the threshold dynamics of merriman bence and osher for evolving an interface by its mean curvature we show that a very fast minimization can be achieved by alternating the solution of a linear parabolic partial differential equation and simple thresholding

a direct approach toward global minimization for multiphase labeling and segmentation problems this paper intends to extend the minimization algorithm developed by bae yuan and tai ijcv 2011 in several directions first we propose a new primal dual approach for global minimization of the continuous potts model with applications to the piecewise constant mumford shah model for multiphase image segmentation different from the existing methods we work directly with the binary setting without using convex relaxation which is thereby termed as a direct approach second we provide the sufficient and necessary conditions to guarantee a global optimum moreover we provide efficient algorithms based on a reduction in the intermediate unknowns from the augmented lagrangian formulation as a result the underlying algorithms involve significantly fewer parameters and unknowns than the naive use of augmented lagrangian based methods hence they are fast and easy to implement furthermore they can produce global optimums under mild conditions

contour detection and hierarchical image segmentation this paper investigates two fundamental problems in computer vision contour detection and image segmentation we present state of the art algorithms for both of these tasks our contour detector combines multiple local cues into a globalization framework based on spectral clustering our segmentation algorithm consists of generic machinery for transforming the output of any contour detector into a hierarchical region tree in this manner we reduce the problem of image segmentation to that of contour detection extensive experimental evaluation demonstrates that both our contour detection and segmentation methods significantly outperform competing algorithms the automatically generated hierarchical segmentations can be interactively refined by user specified annotations computation at multiple image resolutions provides a means of coupling our system to recognition applications

robust noise region based active contour model via local similarity factor for image segmentation abstract image segmentation using a region based active contour model could present difficulties when its noise distribution is unknown to overcome this problem this paper proposes a novel region based model for the segmentation of objects or structures in images by introducing a local similarity factor which relies on the local spatial distance within a local window and local intensity difference to improve the segmentation results by using this local similarity factor the proposed method can accurately extract the object boundary while guaranteeing certain noise robustness furthermore the proposed algorithm completely avoids the pre processing steps typical of region based contour model segmentation resulting in a higher preservation of image details experiments performed on synthetic images and real word images demonstrate that the proposed algorithm as compared with the state of art algorithms is more efficient and robust to higher noise level manifestations in the images

implementation of an adaptive finite element approximation of the mumford shah functional we present and detail a method for the numerical solving of the mumford shah problem based on a finite element method and on adaptive meshes we start with the formulation introduced in 13 detail its numerical implementation and then propose a variant which is proved to converge to the mumford shah problem a few experiments are illustrated

level set and total variation regularization for elliptic inverse problems with discontinuous coefficients we propose a level set approach for elliptic inverse problems with piecewise constant coefficients the geometry of the discontinuity of the coefficient is represented implicitly by level set functions the inverse problem is solved using a variational augmented lagrangian formulation with total variation regularization of the coefficient the corresponding euler lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients we use a multiple level set representation which allows the coefficient to have multiple constant regions knowledge of the exact number of regions is not required only an upper bound is needed numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data the method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities

image segmentation by variational methods mumford and shah functional and the discrete approximations in this paper we discuss the links between mumford and shah s variational problem for signal and image segmentation based on an energy functional of a continuous grey level function and the numerical algorithms proposed to solve it these numerical approaches are based on a discrete functional we recall that in one dimension this discrete functional is asymptotically equivalent to the continuous functional this can be summarized in a gamma convergence result we show that the same result holds in dimension two provided that the continuous energy is adapted to the anisotropy of the discrete approaches we display a few experimental results in dimensions one and two

a fast multiphase image segmentation model for gray images abstract in this paper we propose a fast multiphase image segmentation model for gray images our proposed model is based on the piecewise constant multiphase vese chan model the globally convex segmentation method and the split bregman method our proposed model has the advantages of the vese chan model and can deal with piecewise constant multiphase images well besides our proposed model is much more efficient than the vese chan model by applying the globally convex segmentation method and the split bregman method we have tested our proposed model with many synthetic and real images experimental results have demonstrated the efficiency and robustness of our proposed model

level set contour extraction method based on support value filter active contour method is a powerful approach for object contour extraction this paper presents a new object contour extraction method which combines the level set evolution with the support value filter it analyzes image under the least squares support vector machine ls svm framework and uses the support values to represent salient features underlying image the gaussian filter used in conventional level set method to compute the edge indicator is replaced by the support value filter deduced from the mapped ls svm the level set evolution method is implemented on the feature image obtained by convolving the support value filter with the original image experiments are undertaken on the synthetic and real images the experimental results demonstrate that the support value filter can provide a good estimate for the optimal step size in a gradient descent algorithm and the proposed method has advantages over the direct level set evolution method in converging speed and contour extraction accuracy

stochastic motion and the level set method in computer vision stochastic active contours based on recent work on stochastic partial differential equations spdes this paper presents a simple and well founded method to implement the stochastic evolution of a curve first we explain why great care should be taken when considering such an evolution in a level set framework to guarantee the well posedness of the evolution and to make it independent of the implicit representation of the initial curve a stratonovich differential has to be introduced to implement this differential a standard ito plus drift approximation is proposed to turn an implicit scheme into an explicit one subsequently we consider shape optimization techniques which are a common framework to address various applications in computer vision like segmentation tracking stereo vision etc the objective of our approach is to improve these methods through the introduction of stochastic motion principles the extension we propose can deal with local minima and with complex cases where the gradient of the objective function with respect to the shape is impossible to derive exactly finally as an application we focus on image segmentation methods leading to what we call stochastic active contours

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

existence and regularity of minimizers of a functional for unsupervised multiphase segmentation abstract we consider a variational model for image segmentation proposed in sandberg et al 2010 12 in such a model the image domain is partitioned into a finite collection of subsets denoted as phases the segmentation is unsupervised i e the model finds automatically an optimal number of phases which are not required to be connected subsets unsupervised segmentation is obtained by minimizing a functional of the mumford shah type mumford and shah 1989 1 but modifying the geometric part of the mumford shah energy with the introduction of a suitable scale term the results of computer experiments discussed in 12 show that the resulting variational model has several properties which are relevant for applications in this paper we investigate the theoretical properties of the model we study the existence of minimizers of the corresponding functional first looking for a weak solution in a class of phases constituted by sets of finite perimeter then we find various regularity properties of such minimizers particularly we study the structure of triple junctions by determining their optimal angles

a level set model for image classification we present a supervised classification model based on a variational approach this model is devoted to find an optimal partition composed of homogeneous classes with regular interfaces the originality of the proposed approach concerns the definition of a partition by the use of level sets each set of regions and boundaries associated to a class is defined by a unique level set function we use as many level sets as different classes and all these level sets are moving together thanks to forces which interact in order to get an optimal partition we show how these forces can be defined through the minimization of a unique fonctional the coupled partial differential equations pde related to the minimization of the functional are considered through a dynamical scheme given an initial interface set zero level set the different terms of the pde s are governing the motion of interfaces such that at convergence we get an optimal partition as defined above each interface is guided by internal forces regularity of the interface and external ones data term no vacuum no regions overlapping several experiments were conducted on both synthetic and real images

a new stochastic variational pde model for soft mumford shah segmentation abstract in this paper a new stochastic variational pde model is developed using instead of hard segmentation soft segmentation in this way each pixel is allowed to belong to each image pattern with some probability our work proposes a functional with variable exponent which provides a more accurate model for image segmentation and denoising the diffusion resulting from the proposed model is a combination between tv based and isotropic smoothing the modeling procedure computational implementation and results are explored in detail and numerical examples of real and synthetic images are presented

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

natural image statistics for natural image segmentation we integrate a model for filter response statistics of natural images into a variational framework for image segmentation incorporated in a sound probabilistic distance measure the model drives level sets toward meaningful segmentations of complex textures and natural scenes despite its enhanced descriptive power our approach preserves the efficiency of level set based segmentation since each region comprises two model parameters only analyzing thousands of natural images we select suitable filter banks validate the statistical basis of our model and demonstrate that it outperforms variational segmentation methods using second order statistics

curve evolution implementation of the mumford shah functional for image segmentation denoising interpolation and magnification we first address the problem of simultaneous image segmentation and smoothing by approaching the mumford shah 1989 paradigm from a curve evolution perspective in particular we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours each gradient step involves solving an optimal estimation problem for the data within each region connecting curve evolution and the mumford shah functional with the theory of boundary value stochastic processes the resulting active contour model offers a tractable implementation of the original mumford shah model i e without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner various implementations of this algorithm are introduced to increase its speed of convergence we also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions next by generalizing the data fidelity term of the original mumford shah functional to incorporate a spatially varying penalty we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing this more general model leads us to a novel pde based approach for simultaneous image magnification segmentation and smoothing thereby extending the traditional applications of the mumford shah functional which only considers simultaneous segmentation and smoothing

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a variational model for segmentation of overlapping objects with additive intensity value we propose a variant of the mumford shah model for the segmentation of a pair of overlapping objects with additive intensity value unlike standard segmentation models it does not only determine distinct objects in the image but also recover the possibly multiple membership of the pixels to accomplish this some a priori knowledge about the smoothness of the object boundary is integrated into the model additivity is imposed through a soft constraint which allows the user to control the degree of additivity and is more robust than the hard constraint we also show analytically that the additivity parameter can be chosen to achieve some stability conditions to solve the optimization problem involving geometric quantities efficiently we apply a multiphase level set method segmentation results on synthetic and real images validate the good performance of our model and demonstrate the model s applicability to images with multiple channels and multiple objects

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

an axiomatic approach to image interpolation we discuss possible algorithms for interpolating data given in a set of curves and or points in the plane we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolute minimal lipschitz extension model amle is singled out and studied in more detail we show experiments suggesting a possible application the restoration of images with poor dynamic range

integrated segmentation and interpolation of sparse data we address the two inherently related problems of segmentation and interpolation of 3d and 4d sparse data and propose a new method to integrate these stages in a level set framework the interpolation process uses segmentation information rather than pixel intensities for increased robustness and accuracy the method supports any spatial configurations of sets of 2d slices having arbitrary positions and orientations we achieve this by introducing a new level set scheme based on the interpolation of the level set function by radial basis functions the proposed method is validated quantitatively and or subjectively on artificial data and mri and ct scans and is compared against the traditional sequential approach which interpolates the images first using a state of the art image interpolation method and then segments the interpolated volume in 3d or 4d in our experiments the proposed framework yielded similar segmentation results to the sequential approach but provided a more robust and accurate interpolation in particular the interpolation was more satisfactory in cases of large gaps due to the method taking into account the global shape of the object and it recovered better topologies at the extremities of the shapes where the objects disappear from the image slices as a result the complete integrated framework provided more satisfactory shape reconstructions than the sequential approach

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

classification using geometric level sets a variational level set method is developed for the supervised classification problem nonlinear classifier decision boundaries are obtained by minimizing an energy functional that is composed of an empirical risk term with a margin based loss and a geometric regularization term new to machine learning the surface area of the decision boundary this geometric level set classifier is analyzed in terms of consistency and complexity through the calculation of its e entropy for multicategory classification an efficient scheme is developed using a logarithmic number of decision functions in the number of classes rather than the typical linear number of decision functions geometric level set classification yields performance results on benchmark data sets that are competitive with well established methods

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a luenberger observer for reaction diffusion models with front position data we propose a luenberger observer for reaction diffusion models with propagating front features and for data associated with the location of the front over time such models are considered in various application fields such as electrophysiology wild land fire propagation and tumor growth modeling drawing our inspiration from image processing methods we start by proposing an observer for the eikonal curvature equation that can be derived from the reaction diffusion model by an asymptotic expansion we then carry over this observer to the underlying reaction diffusion equation by an inverse asymptotic analysis and we show that the associated correction in the dynamics has a stabilizing effect for the linearized estimation error we also discuss the extension to joint state parameter estimation by using the earlier proposed roukf strategy we then illustrate and assess our proposed observer method with test problems pertaining to electrophysiology modeling including with a realistic model of cardiac atria our numerical trials show that state estimation is directly very effective with the proposed luenberger observer while specific strategies are needed to accurately perform parameter estimation as is usual with kalman filtering used in a nonlinear setting and we demonstrate two such successful strategies

active contours driven by local image fitting energy a new region based active contour model that embeds the image local information is proposed in this paper by introducing the local image fitting lif energy to extract the local image information our model is able to segment images with intensity inhomogeneities moreover a novel method based on gaussian filtering for variational level set is proposed to regularize the level set function it can not only ensure the smoothness of the level set function but also eliminate the requirement of re initialization which is very computationally expensive experiments show that the proposed method achieves similar results to the lbf local binary fitting energy model but it is much more computationally efficient in addition our approach maintains the sub pixel accuracy and boundary regularization properties

discrete region competition for unknown numbers of connected regions we present a discrete unsupervised multiregion competition algorithm for image segmentation over different energy functionals the number of regions present in an image does not need to be known a priori nor their photometric properties the algorithm jointly estimates the number of regions their photometries and their contours the required regularization is provided by defining a region as a connected set of pixels the evolving contours in the image are represented by computational particles that move as driven by an energy minimization algorithm we present an efficient discrete algorithm that allows minimizing a range of well known energy functionals under the topological constraint of regions being connected components the presented framework and algorithms are implemented in the open source insight toolkit image processing library

incorporating topological derivatives into shape derivatives based level set methods shape derivatives and topological derivatives have been incorporated into level set methods to investigate shape optimization problems the shape derivative measures the sensitivity of boundary perturbations while the topological derivative measures the sensitivity of creating a small hole in the interior domain the combination of these two derivatives yields an efficient algorithm which has more flexibility in shape changing and may escape from a local optimal examples on finding the optimal shapes for maximal band gaps in photonic crystal and acoustic drum problems are demonstrated

identification of subsurface structures using electromagnetic data and shape priors we consider the inverse problem of identifying large scale subsurface structures using the controlled source electromagnetic method to identify structures in the subsurface where the contrast in electric conductivity can be small regularization is needed to bias the solution towards preserving structural information we propose to combine two approaches for regularization of the inverse problem in the first approach we utilize a model based reduced composite representation of the electric conductivity that is highly flexible even for a moderate number of degrees of freedom with a low number of parameters the inverse problem is efficiently solved using a standard second order gradient based optimization algorithm further regularization is obtained using structural prior information available e g from interpreted seismic data the reduced conductivity representation is suitable for incorporation of structural prior information such prior information cannot however be accurately modeled with a gaussian distribution to alleviate this we incorporate the structural information using shape priors the shape prior technique requires the choice of kernel function which is application dependent we argue for using the conditionally positive definite kernel which is shown to have computational advantages over the commonly applied gaussian kernel for our problem numerical experiments on various test cases show that the methodology is able to identify fairly complex subsurface electric conductivity distributions while preserving structural prior information during the inversion

semi automatic computer aided lesion detection in dental x rays using variational level set a semi automatic lesion detection framework is proposed to detect areas of lesions from periapical dental x rays using level set method in this framework first a new proposed competitive coupled level set method is used to segment the image into three pathologically meaningful regions using two coupled level set functions tailored for the dental clinical setting a two stage clinical segmentation acceleration scheme is used the method uses a trained support vector machine svm classifier to provide an initial contour for two coupled level sets then based on the segmentation results an analysis scheme is applied firstly the scheme builds an uncertainty map from which those areas with radiolucent will be automatically emphasized by a proposed color emphasis scheme those radiolucent in the teeth or jaw usually suggested possible lesions secondly the scheme employs a method based on the average intensity profile to isolate the teeth and locate two types of lesions periapical lesion pl and bifurcation lesion bl experimental results show that our proposed segmentation method is able to segment the image into pathological meaningful regions for further analysis our proposed framework is able to automatically provide direct visual cues for the lesion detection and when given the orientation of the teeth it is able to automatically locate the pl and bl with a seriousness level marked for further dental diagnosis when used in the clinical setting the framework enables dentist to improve interpretation and to focus their attention on critical areas

a binary level set model and some applications to mumford shah image segmentation in this paper we propose a pde based level set method traditionally interfaces are represented by the zero level set of continuous level set functions instead we let the interfaces be represented by discontinuities of piecewise constant level set functions each level set function can at convergence only take two values i e it can only be 1 or 1 thus our method is related to phase field methods some of the properties of standard level set methods are preserved in the proposed method while others are not using this new method for interface problems we need to minimize a smooth convex functional under a quadratic constraint the level set functions are discontinuous at convergence but the minimization functional is smooth we show numerical results using the method for segmentation of digital images

a multiscale algorithm for image segmentation by variational method most segmentation algorithms are composed of several procedures split and merge small region elimination boundary smoothing each depending on several parameters the introduction of an energy to minimize leads to a drastic reduction of these parameters the authors prove that the most simple segmentation tool the region merging algorithm made according to the simplest energy is enough to compute a local energy minimum belonging to a compact class and to achieve the job of most of the tools mentioned above the authors explain why merging in a variational framework leads to a fast multiscale multichannel algorithm with a pyramidal structure the obtained algorithm is o n ln n where n is the number of pixels of the picture this fast algorithm is applied to make grey level and texture segmentation and experimental results are shown

rapid and brief communication gacv geodesic aided c v method a novel algorithm for image segmentation is proposed the proposed method incorporates geodesic curves and c v method to raise active contours performance on image segmentation moreover we extend our method to color images by practical experiments it is verified that our model obtains better results than original methods especially with respect to images within holes complex background weak edges and noise

reinitialization free level set evolution via reaction diffusion this paper presents a novel reaction diffusion rd method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution lse a diffusion term is introduced into lse resulting in an rd lse equation from which a piecewise constant solution can be derived in order to obtain a stable numerical solution from the rd based lse we propose a two step splitting method to iteratively solve the rd lse equation where we first iterate the lse equation then solve the diffusion equation the second step regularizes the level set function obtained in the first step to ensure stability and thus the complex and costly reinitialization procedure is completely eliminated from lse by successfully applying diffusion to lse the rd lse model is stable by means of the simple finite difference method which is very easy to implement the proposed rd method can be generalized to solve the lse for both variational level set method and partial differential equation based level set method the rd lse method shows very good performance on boundary antileakage the extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed rd lse approach

existence theorem for a minimum problem with free discontinuity set we study the variational problem where is an open set in n n 2g l q l 1 q o andh n 1 is the n 1 dimensional hausdorff measure

a variational framework for multiregion pairwise similarity based image segmentation variational cost functions that are based on pairwise similarity between pixels can be minimized within level set framework resulting in a binary image segmentation in this paper we extend such cost functions and address multi region image segmentation problem by employing a multi phase level set framework for multi modal images cost functions become more complicated and relatively difficult to minimize we extend our previous work proposed for background foreground separation to the segmentation of images in more than two regions we also demonstrate an efficient implementation of the curve evolution which reduces the computational time significantly finally we validate the proposed method on the berkeley segmentation data set by comparing its performance with other segmentation techniques

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

effective level set image segmentation with a kernel induced data term this study investigates level set multiphase image segmentation by kernel mapping and piecewise constant modeling of the image data thereof a kernel function maps implicitly the original data into data of a higher dimension so that the piecewise constant model becomes applicable this leads to a flexible and effective alternative to complex modeling of the image data the method uses an active curve objective functional with two terms an original term which evaluates the deviation of the mapped image data within each segmentation region from the piecewise constant model and a classic length regularization term for smooth region boundaries functional minimization is carried out by iterations of two consecutive steps 1 minimization with respect to the segmentation by curve evolution via euler lagrange descent equations and 2 minimization with respect to the regions parameters via fixed point iterations using a common kernel function this step amounts to a mean shift parameter update we verified the effectiveness of the method by a quantitative and comparative performance evaluation over a large number of experiments on synthetic images as well as experiments with a variety of real images such as medical satellite and natural images as well as motion maps

geometrically induced force interaction for three dimensional deformable models in this paper we propose a novel 3 d deformable model that is based upon a geometrically induced external force field which can be conveniently generalized to arbitrary dimensions this external force field is based upon hypothesized interactions between the relative geometries of the deformable model and the object boundary characterized by image gradient the evolution of the deformable model is solved using the level set method so that topological changes are handled automatically the relative geometrical configurations between the deformable model and the object boundaries contribute to a dynamic vector force field that changes accordingly as the deformable model evolves the geometrically induced dynamic interaction force has been shown to greatly improve the deformable model performance in acquiring complex geometries and highly concave boundaries and it gives the deformable model a high invariancy in initialization configurations the voxel interactions across the whole image domain provide a global view of the object boundary representation giving the external force a long attraction range the bidirectionality of the external force field allows the new deformable model to deal with arbitrary cross boundary initializations and facilitates the handling of weak edges and broken boundaries in addition we show that by enhancing the geometrical interaction field with a nonlocal edge preserving algorithm the new deformable model can effectively overcome image noise we provide a comparative study on the segmentation of various geometries with different topologies from both synthetic and real images and show that the proposed method achieves significant improvements against existing image gradient 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

level set evolution with locally linear classification for image segmentation this paper presents a novel local region based level set model for image segmentation in each local region we define a locally weighted least squares energy to fit a linear classifier with level set representation these local energy functions are then integrated over the whole image domain to develop a global segmentation model the objective function in this model is thereafter minimized via level set evolution in this process the parameters related to the locally linear classifier are iteratively estimated by introducing the locally linear functions to separate background and foreground in local regions our model not only achieves accurate segmentation results but also is robust to initialization extensive experiments are reported to demonstrate that our method holds higher segmentation accuracy and more initialization robustness compared with the classical region based and local region based methods

a variational model for object segmentation using boundary information and shape prior driven by the mumford shah functional in this paper we propose a new variational model to segment an object belonging to a given shape space using the active contour method a geometric shape prior and the mumford shah functional the core of our model is an energy functional composed by three complementary terms the first one is based on a shape model which constrains the active contour to get a shape of interest the second term detects object boundaries from image gradients and the third term drives globally the shape prior and the active contour towards a homogeneous intensity region the segmentation of the object of interest is given by the minimum of our energy functional this minimum is computed with the calculus of variations and the gradient descent method that provide a system of evolution equations solved with the well known level set method we also prove the existence of this minimum in the space of functions with bounded variation applications of the proposed model are presented on synthetic and medical images

multiphase image segmentation via modica mortola phase transition we propose a novel multiphase segmentation model built upon the celebrated phase transition model of modica and mortola in material sciences and a properly synchronized fitting term that complements it the proposed sine sinc model outputs a single multiphase distribution from which each individual segment or phase can be easily extracted theoretical analysis is developed for the gamma convergence behavior of the proposed model and the existence of its minimizers since the model is not quadratic nor convex for computation we adopted the convex concave procedure cccp that has been developed in the literatures of both computational nonlinear pdes and neural computation numerical details and experiments on both synthetic and natural images are presented

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

multiple region segmentation without supervision by adaptive global maximum clustering in this paper we propose a new method of segmenting an image into several sets of pixels with similar intensity values called regions a multiple region segmentation problem is unstable because the result considerably depends on the number of regions given a priori therefore one of the most important tasks in solving the problem is automatically finding the number of regions the method we propose is able to find the reasonable number of distinct regions not only for clean images but also for noisy ones our method is made up of two procedures first we develop the adaptive global maximum clustering in this procedure we deal with an image histogram and automatically obtain the number of significant local maxima of the histogram this number indicates the number of different regions in the image second we derive a simple and fast calculation to segment an image composed of distinct multiple regions then we split an image into multiple regions according to the previous procedure finally we show the efficiency of our method by comparing it with other previous methods

multi region active contours with a single level set function segmenting an image into an arbitrary number of coherent regions is at the core of image understanding many formulations of the segmentation problem have been suggested over the past years these formulations include among others axiomatic functionals which are hard to implement and analyze and graph based alternatives which impose a non geometric metric on the problem we propose a novel method for segmenting an image into an arbitrary number of regions using an axiomatic variational approach the proposed method allows to incorporate various generic region appearance models while avoiding metrication errors in the suggested framework the segmentation is performed by level set evolution yet contrarily to most existing methods here multiple regions are represented by a single non negative level set function the level set function evolution is efficiently executed through the voronoi implicit interface method for multi phase interface evolution the proposed approach is shown to obtain accurate segmentation results for various natural 2d and 3d images comparable to state of the art image segmentation algorithms

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

fast nonparametric active contour adapted to quadratic inhomogeneous intensity fluctuations in the context of unsupervised segmentation of noisy images a minimum description length mdl polygonal active contour technique based on nonparametric modeling of the noise probability density function pdf has been proposed in 2011 this approach allows fast and efficient segmentation of an object without a priori knowledge on the intensity fluctuations nevertheless since the object and the background are assumed to be homogeneous degraded segmentation results are obtained when images present inhomogeneous intensity variations it is shown in this paper that this constraint of homogeneity can be removed still with minimizing a mdl criterion without undetermined parameters and adapted to nonparametric modeling of the noise pdf for that purpose the spatial inhomogeneity of the intensity is modeled with 2d quadratic functions moreover low computation times can be achieved approximately 60ms on 256x256 pixel images using a two step optimization strategy the efficiency and the robustness of this approach are then validated on various synthetic and real images acquired from different sensors

automatic ultrasound image segmentation based on local entropy and active contour model abstract it is still a challenging task to segment ultrasound medical images automatically and accurately because of the poor quality of images to address these problems a two stage automatic segmentation scheme based on local entropy and a proposed active contour model for ultrasound images are put forward in this paper first a new region based active contour model in the level set formulation driven by global and local intensity information is established for the segmentation furthermore for automatically segmenting the ultrasound images a coarse segmentation is performed utilizing local entropy information of the ultrasound images then the coarse segmentation result is used as the initial value of the explicit segmentation based on the proposed model several experiments on real ultrasound images demonstrate that the proposed method outperforms other methods on both visual perception and objective evaluation metrics

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

regressing local to global shape properties for online segmentation and tracking we propose a novel regression based framework that uses online learned shape information to reconstruct occluded object contours our key insight is to regress the global coarse properties of shape from its local properties i e its details we do this by representing shapes using their 2d discrete cosine transforms and by regressing low frequency from high frequency harmonics we learn this regression model using locally weighted projection regression which expedites online incremental learning after sufficient observation of a set of unoccluded shapes the learned model can detect occlusion and recover the full shapes from the occluded ones we demonstrate the ideas using a level set based tracking system that provides shape and pose however the framework could be embedded in any segmentation based tracking system our experiments demonstrate the efficacy of the method on a variety of objects using both real data and artificial data

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

level set segmentation with multiple regions the popularity of level sets for segmentation is mainly based on the sound and convenient treatment of regions and their boundaries unfortunately this convenience is so far not known from level set methods when applied to images with more than two regions this communication introduces a comparatively simple way how to extend active contours to multiple regions keeping the familiar quality of the two phase case we further suggest a strategy to determine the optimum number of regions as well as initializations for the contours

shapes from pixels continuous domain visual signals are usually captured as discrete digital images this operation is not invertible in general in the sense that the continuous domain signal cannot be exactly reconstructed based on the discrete image unless it satisfies certain constraints e g bandlimitedness in this paper we study the problem of recovering shape images with smooth boundaries from a set of samples thus the reconstructed image is constrained to regenerate the same samples consistency as well as forming a shape bilevel image we initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes next we relax the non convex shape constraint to transform the problem into minimizing the total variation over consistent non negative valued images we also introduce a requirement called reducibility that guarantees equivalence between the two problems we illustrate that the reducibility property effectively sets a requirement on the minimum sampling density we also evaluate the performance of the relaxed alternative in various numerical experiments

parametric active membrane for segmentation of multiple objects in an image we propose a topology adaptive active membrane that can segment images of multiple objects present in a scene the parametric active membrane evolves in image space and splits into multiple membranes the shape of the membrane can be constrained according to the shape of the objects present in a scene we have shown that this active membrane model is also suitable for segmenting images of touching objects the proposed segmentation technique unifies the membrane evolution and membrane splitting process the methodology is tested for a number of real images from biomedical and machine vision domains that demonstrate the efficacy of the proposed scheme

a fast segmentation method based on constraint optimization and its applications intensity inhomogeneity and texture segmentation we propose a new constraint optimization energy and an iteration scheme for image segmentation which is connected to edge weighted centroidal voronoi tessellation ewcvt we show that the characteristic functions of the edge weighted voronoi regions are the minimizers may not unique of the proposed energy at each iteration we propose a narrow banding algorithm to accelerate the implementation which makes the proposed method very fast we generalize the cvt segmentation to hand intensity inhomogeneous and texture segmentation by incorporating the global and local image information into the energy functional compared with other approaches such as level set method the experimental results in this paper have shown that our approach greatly improves the calculation efficiency without losing segmentation accuracy

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

uniform discreteness and heisenberg screw motions we give aversion of shimizu s lemma for groups of complex hyperbolic isometries one of whose generators is aheisenberg screw motion our main result interpolates between known results for groups with a generator that is avertical translation or aheisenberg rotation we also give an interpretation of our result in terms of the relation between radii of isometric spheres and their distance from the axis of the heisenberg screw motion

a variant of the level set method and applications to image segmentation in this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases in classical level set approaches the sign of n level set functions are utilized to identify up to 2 phases the novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase if 2 phases should be identified the level set function must approach 2 predetermined constants we just need one level set function to represent 2 unique phases and this gains in storage capacity further the reinitializing procedure requested in classical level set methods is superfluous using our approach the minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the delta and heaviside functions numerical examples are given and we also compare our method with related approaches

joint segmentation of collectively moving objects using a bag of words model and level set evolution in scenes with collectively moving objects to disregard the individual objects and take the entire group into consideration for motion characterization is a promising approach with wide application prospects in contrast to studies on the segmentation of independently moving objects our purpose is to construct a segmentation of these objects to characterize their motions at a macroscopic level in general the collectively moving objects in a group have very similar motion behavior with their neighbors and appear as a kind of global collective motion this paper presents a joint segmentation approach for these collectively moving objects in our model we extract these macroscopic movement patterns based on optical flow field sequences specifically a group of collectively moving objects correspond to a region where the optical flow field has high magnitude and high local direction coherence as a result our problem can be addressed by identifying these coherent optical flow field regions the segmentation is performed through the minimization of a variational energy functional derived from the bayes classification rule specifically we use a bag of words model to generate a codebook as a collection of prototypical optical flow patterns and the class conditional probability density functions for different regions are determined based on these patterns finally the minimization of our proposed energy functional results in the gradient descent evolution of segmentation boundaries which are implicitly represented through level sets the application of our proposed approach is to segment and track multiple groups of collectively moving objects in a large variety of real world scenes

segmentation of overlapping elliptical objects in silhouette images segmentation of partially overlapping objects with a known shape is needed in an increasing amount of various machine vision applications this paper presents a method for segmentation of clustered partially overlapping objects with a shape that can be approximated using an ellipse the method utilizes silhouette images which means that it requires only that the foreground objects and background can be distinguished from each other the method starts with seedpoint extraction using bounded erosion and fast radial symmetry transform extracted seedpoints are then utilized to associate edge points to objects in order to create contour evidence finally contours of the objects are estimated by fitting ellipses to the contour evidence the experiments on one synthetic and two different real data sets showed that the proposed method outperforms two current state of art approaches in overlapping objects segmentation

incorporating patch subspace model in mumford shah type active contours in this paper we propose a unified energy minimization model for segmentation of non smooth image structures e g textures based on mumford shah functional and linear patch model we consider that image patches of a non smooth image structure can be modeled by a patch subspace and image patches of different non smooth image structures belong to different patch subspaces which leads to a computational framework for segmentation of non smooth image structures motivated by the mumford shah model we show that this segmentation framework is equivalent to minimizing a piecewise linear patch reconstruction energy we also prove that the error of segmentation is bounded by the error of the linear patch reconstruction meaning that improving the linear patch reconstruction for each region leads to reduction of the segmentation error in addition we derive an algorithm for the linear patch reconstruction with proven global optimality and linear rate of convergence the segmentation in our method is achieved by minimizing a single energy functional without requiring predefined features hence compared with the previous methods that require predefined texture features our method can be more suitable for handling general textures in unsupervised segmentation as a by product our method also produces a dictionary of optimized orthonormal descriptors for each segmented region we mainly evaluate our method on the brodatz textures the experiments validate our theoretical claims and show the clear superior performance of our methods over other related methods for segmentation of the textures