subspace based and direct algorithms for distorted circular contour estimation circular features are commonly sought in digital image processing the subspace based line detection slide method proposed to estimate the center and the radius of a single circle in this paper we introduce a novel method for estimating several radii while extending the circle estimation to retrieve circular like distorted contours particularly we develop and validate a new model for virtual signal generation by simulating a circular antenna the circle center is estimated by the slide method a variable speed propagation scheme toward the circular antenna yields a linear phase signal therefore a high resolution method provides the radius either the gradient method or the more robust combination of dividing rectangles and spline interpolation can extend this method for free form object segmentation the retrieval of multiple non concentric circles and rotated ellipses is also considered to evaluate the performance of the proposed methods we compare them with a least squares method hough transform and gradient vector flow we apply the proposed method to hand made images while considering some real world images

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

anomalous diffusion of random walk on random planar maps we prove that the simple random walk on the uniform infinite planar triangulation uipt typically travels graph distance at most n 1 4 o n 1 in n units of time together with the complementary lower bound proven by gwynne and miller 2017 this shows that the typical graph distance displacement of the walk after n steps is n 1 4 o n 1 as conjectured by benjamini and curien geom funct anal 2 2 501 531 2013 arxiv 1202 5454 more generally we show that the simple random walks on a certain family of random planar maps in the gamma liouville quantum gravity lqg universality class for gamma in 0 2 including spanning tree weighted maps bipolar oriented maps and mated crt maps typically travels graph distance n 1 d gamma o n 1 in n units of time where d gamma is the growth exponent for the volume of a metric ball on the map which was shown to exist and depend only on gamma by ding and gwynne commun math phys 374 1877 1934 2018 arxiv 1807 01072 since d gamma 2 this shows that the simple random walk on each of these maps is subdiffusive our proofs are based on an embedding of the random planar maps under consideration into mathbb c wherein graph distance balls can be compared to euclidean balls modulo subpolynomial errors this embedding arises from a coupling of the given random planar map with a mated crt map together with the relationship of the latter map to sle decorated lqg

a nonconservative flow field for robust variational image segmentation we introduce a robust image segmentation method based on a variational formulation using edge flow vectors we demonstrate the nonconservative nature of this flow field a feature that helps in a better segmentation of objects with concavities a multiscale version of this method is developed and is shown to improve the localization of the object boundaries we compare and contrast the proposed method with well known state of the art methods detailed experimental results are provided on both synthetic and natural images that demonstrate that the proposed approach is quite competitive

smoothed cross validation for bandwidth selection of a kernel density estimator a generalization of the widely studied least squares cross validation method is considered the essential idea is to do a particular type of presmoothing of the data this is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance the rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions the method is more appealing than other selectors with this property because its motivation is not heavily dependent on precise asymptotic analysis and because its form is simple and intuitive theory is also given for choice of the amount of presmoothing and this is used to derive a data based method for this choice

a geometric model for active contours in image processing we propose a new model for active contours based on a geometric partial differential equation our model is intrinsec stable satisfies the maximum principle and permits a rigorous mathematical analysis it enables us to extract smooth shapes we cannot retrieve angles and it can be adapted to find several contours simultaneously moreover as a consequence of the stability we can design robust algorithms which can be engineed with no parameters in applications numerical experiments are presented

automatic active model initialization via poisson inverse gradient active models have been widely used in image processing applications a crucial stage that affects the ultimate active model performance is initialization this paper proposes a novel automatic initialization approach for parametric active models in both 2 d and 3 d the pig initialization method exploits a novel technique that essentially estimates the external energy field from the external force field and determines the most likely initial segmentation examples and comparisons with two state of the art automatic initialization methods are presented to illustrate the advantages of this innovation including the ability to choose the number of active models deployed rapid convergence accommodation of broken edges superior noise robustness and segmentation accuracy

segmentation of external force field for automatic initialization and splitting of snakes active contours or snakes have been extensively utilized in handling image segmentation and classification problems in traditional active contour models snake initialization is performed manually by users and topological changes such as splitting of the snake cannot be automatically handled in this paper we introduce a new method to solve the snake initialization and splitting problem based on an area segmentation approach the external force field is segmented first and then the snake initialization and splitting can be automatically performed by using the segmented external force field such initialization and splitting produces multiple snakes each of which is within the capture range associated to an object and will be evolved to the object boundary the external force used in this paper is a gradient vector flow with an edge preserving property epgvf which can prevent the snakes from passing over weak boundaries to segment the external force field we represent it with a graph and a graph theory approach can be taken to determine the membership of each pixel experimental results establish the effectiveness of the proposed approach

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

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 central limit theorem for generalized quadratic forms random variables of the form w n mathop sum limits 1 leqq i leqq n mathop sum limits text 1 leqq j leqq n w ijn x i x j are considered with x i independent not necessarily identically distributed and w ijn borel functions such that w ijn x i x j is square integrable and has vanishing conditional expectations e w ijn x i x j x i e w ijn x i x j x j 0 text a text s text a central limit theorem is proved under the condition that the normed fourth moment tends to 3 under some restrictions the condition is also necessary finally conditions on the individual tails of w ijn x i x j and an eigenvalue condition are given that ensure asymptotic normality of w n

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

mac magnetostatic active contour model we propose an active contour model using an external force field that is based on magnetostatics and hypothesized magnetic interactions between the active contour and object boundaries the major contribution of the method is that the interaction of its forces can greatly improve the active contour in capturing complex geometries and dealing with difficult initializations weak edges and broken boundaries the proposed method is shown to achieve significant improvements when compared against six well known and state of the art shape recovery methods including the geodesic snake the generalized version of gradient vector flow gvf snake the combined geodesic and gvf snake and the charged particle model

functional limit theorems for u processes un processus u est une collection de statistiques u indicees par une famille de noyaux symetriques on utilise la theorie limite classique pour demontrer pour le processus u des analogues du theoreme central limite pour les processus empiriques

region based strategies for active contour models the variational method has been introduced by kass et al 1987 in the field of object contour modeling as an alternative to the more traditional edge detection edge thinning edge sorting sequence since the method is based on a pre processing of the image to yield an edge map it shares the limitations of the edge detectors it uses in this paper we propose a modified variational scheme for contour modeling which uses no edge detection step but local computations instead only around contour neighborhoods as well as an anticipating strategy that enhances the modeling activity of deformable contour curves many of the concepts used were originally introduced to study the local structure of discontinuity in a theoretical and formal statement by leclerc zucker 1987 but never in a practical situation such as this one the first part of the paper introduces a region based energy criterion for active contours and gives an examination of its implications as compared to the gradient edge map energy of snakes then a simplified optimization scheme is presented accounting for internal and external energy in separate steps this leads to a complete treatment which is described in the last sections of the paper 4 and 5 the optimization technique used here is mostly heuristic and is thus presented without a formal proof but is believed to fill a gap between snakes and other useful image representations such as split and merge regions or mixed line labels image fields

boundary vector field for parametric active contours snakes are widely used in the fields of computer vision and image processing there are a number of shortcomings of the traditional snakes including the capture range problem and the concave object extraction problem gradient vector flow snake enhances the concave object extraction capability however it suffers from high computational requirement in this paper we present a new external force which is generated by a novel interpolation scheme which reduces the computational requirement significantly and at the same time improves the capture range and concave object extraction capability

snakes shapes and gradient vector flow snakes or active contours are used extensively in computer vision and image processing applications particularly to locate object boundaries problems associated with initialization and poor convergence to boundary concavities however have limited their utility this paper presents a new external force for active contours largely solving both problems this external force which we call gradient vector flow gvf is computed as a diffusion of the gradient vectors of a gray level or binary edge map derived from the image it differs fundamentally from traditional snake external forces in that it cannot be written as the negative gradient of a potential function and the corresponding snake is formulated directly from a force balance condition rather than a variational formulation using several two dimensional 2 d examples and one three dimensional 3 d example we show that gvf has a large capture range and is able to move snakes into boundary concavities

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

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

the amount of noise inherent in bandwidth selection for a kernel density estimator abstract any practical method of constructing a bandwidth must depend only on a statistical sample and should produce some sort of estimate of this bandwidth the purpose of this paper is to show that there a well defined limits to the accuracy of all data driven bandwidth estimates put another way there is an unbridgeable gap between the minimum integrated square error attained using a optimal bandwidth and the minimum achievable integrated square error using a data driven bandwidth estimate additional keywords stochastic processes cross validation and random variables

variational segmentation of vector valued images with gradient vector flow in this paper we extend the gradient vector flow field for robust variational segmentation of vector valued images rather than using scalar edge information we define a vectorial edge map derived from a weighted local structure tensor of the image that enables the diffusion of the gradient vectors in accurate directions through the 4d gradient vector flow equation to reduce the contribution of noise in the structure tensor image channels are weighted according to a blind estimator of contrast the method is applied to biological volume delineation in dynamic pet imaging and validated on realistic monte carlo simulations of numerical phantoms as well as on real images

winding number constrained contour detection salient contour detection can benefit from the integration of both contour cues and region cues however this task is difficult due to different nature of region representations and contour representations to solve this problem this paper proposes an energy minimization framework based on winding number constraints in this framework both region cues such as color texture homogeneity and contour cues such as local contrast and continuity are represented in a joint objective function which has both region and contour labels the key problem is how to design constraints that ensure the topological consistency of the two kinds of labels our technique is based on the topological concept of winding number using a fast method for winding number computation a small number of linear constraints are derived to ensure label consistency our method is instantiated by ratio based energy functions by successfully integrating both region and contour cues our method shows advantages over competitive methods our method is extended to incorporate user interaction which leads to further improvements

area and length minimizing flows for shape segmentation a number of active contour models have been proposed that unify the curve evolution framework with classical energy minimization techniques for segmentation such as snakes the essential idea is to evolve a curve in two dimensions or a surface in three dimensions under constraints from image forces so that it clings to features of interest in an intensity image the evolution equation has been derived from first principles as the gradient flow that minimizes a modified length functional tailored to features such as edges however because the flow may be slow to converge in practice a constant hyperbolic term is added to keep the curve surface moving in the desired direction we derive a modification of this term based on the gradient flow derived from a weighted area functional with image dependent weighting factor when combined with the earlier modified length gradient flow we obtain a partial differential equation pde that offers a number of advantages as illustrated by several examples of shape segmentation on medical images in many cases the weighted area flow may be used on its own with significant computational savings

unimodular hyperbolic triangulations circle packing and random walk we show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six if and only if the triangulation is amenable in the sense of aldous and lyons 1 as a part of this we obtain an alternative proof of the benjamini schramm recurrence theorem 19 secondly in the hyperbolic case we prove that the random walk almost surely converges to a point in the unit circle that the law of this limiting point has full support and no atoms and that the unit circle is a realisation of the poisson boundary finally we show that the simple random walk has positive speed in the hyperbolic metric

active contouring based on gradient vector interaction and constrained level set diffusion this paper presents an extension of our recently introduced mac model to deal with the initialization dependency problem that commonly appears in edge based approaches its dynamic force field unique bidirectionality and constrained diffusion based level set evolution provide great freedom in contour initialization and show significant improvements in initialization independency compared to other edge based techniques it can handle more sophisticated topological changes than splitting and merging it provides new potentials for edge based active contour methods particularly when detecting and localizing objects with unknown location geometry and topology

rags region aided geometric snake an enhanced region aided geometric active contour that is more tolerant toward weak edges and noise in images is introduced the proposed method integrates gradient flow forces with region constraints composed of image region vector flow forces obtained through the diffusion of the region segmentation map we refer to this as the region aided geometric snake or rags the diffused region forces can be generated from any reliable region segmentation technique greylevel or color this extra region force gives the snake a global complementary view of the boundary information within the image which along with the local gradient flow helps detect fuzzy boundaries and overcome noisy regions the partial differential equation pde resulting from this integration of image gradient flow and diffused region flow is implemented using a level set approach we present various examples and also evaluate and compare the performance of rags on weak boundaries and noisy images

boundaries of planar graphs via circle packings we provide a geometric representation of the poisson and martin boundaries of a transient bounded degree triangulation of the plane in terms of its circle packing in the unit disc this packing is unique up to mobius transformations more precisely we show that any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the disk and that the space of extremal positive harmonic functions that is the martin boundary is homeomorphic to the unit circle all our results hold more generally for any good embedding of planar graphs that is an embedding in the unit disc with straight lines such that angles are bounded away from 0 and uniformly and lengths of adjacent edges are comparable furthermore we show that in a good embedding of a planar graph the probability that a random walk exits a disc through a sufficiently wide arc is at least a constant and that brownian motion on such graphs takes time of order r2 to exit a disc of radius r these answer a question recently posed by chelkak 2014