tangent bundle elastica and computer vision visual curve completion an early visual process that completes the occluded parts between observed boundary fragments a k a inducers is a major problem in perceptual organization and a critical step toward higher level visual tasks in both biological and machine vision most computational contributions to solving this problem suggest desired perceptual properties that the completed contour should satisfy in the image plane and then seek the mathematical curves that provide them alternatively few studies including by the authors have suggested to frame the problem not in the image plane but rather in the unit tangent bundler 2 s 1 the space that abstracts the primary visual cortex where curve completion allegedly occurs combining both schools here we propose and develop a biologically plausible theory of elastica in the tangent bundle that provides not only perceptually superior completion results but also a rigorous computational prediction that inducer curvatures greatly affects the shape of the completed curve as indeed indicated by human perception

convergence and stability of numerical solutions to sddes with markovian switching a class of linear stochastic delay differential equations with markovian switching is considered the main aim of this paper is to investigate the convergence and stability of the euler method of the equations it is proved that the euler method is convergent with strong order p 1 2 the ms stable properties of the euler scheme are also studied

crossing preserving coherence enhancing diffusion on invertible orientation scores many image processing problems require the enhancement of crossing elongated structures these problems cannot easily be solved by commonly used coherence enhancing diffusion methods therefore we propose a method for coherence enhancing diffusion on the invertible orientation score of a 2d image in an orientation score the local orientation is represented by an additional third dimension ensuring that crossing elongated structures are separated from each other we consider orientation scores as functions on the euclidean motion group and use the group structure to apply left invariant diffusion equations on orientation scores we describe how we can calculate regularized left invariant derivatives and use the hessian to estimate three descriptive local features curvature deviation from horizontality and orientation confidence these local features are used to adapt a nonlinear coherence enhancing crossing preserving diffusion equation on the orientation score we propose two explicit finite difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence enhancing diffusion the use of curvature leads to improved enhancement of curves with high curvature furthermore the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings

global detection of salient convex boundaries as an important geometric property of many structures or structural components convexity plays an important role in computer vision and image understanding in this paper we describe a general approach that can force various edge grouping algorithms to detect only convex structures from a set of boundary fragments the basic idea is to remove some fragments and fragment connections so that on the remaining ones a prototype edge grouping algorithm that detects closed boundaries without the convexity constraint can only produce convex closed boundaries we show that this approach takes polynomial time and preserves the grouping optimality by not excluding any valid convex boundary from the search space choosing the recently developed ratio contour algorithm as the prototype grouping algorithm we develop a new convex grouping algorithm which can detect convex salient boundaries with good continuity and proximity in a globally optimal fashion to facilitate the application of this convex grouping algorithm we develop a new fragment connection method based on four point bezier curves we demonstrate the performance of this convex grouping algorithm by conducting experiments on both synthetic and real images in addition we provide a comparison with some prior edge grouping algorithms finally we show that the proposed convex grouping algorithm can be further extended to detect convex open boundaries derive region based image hierarchies and incorporate some simple human computer interactions

numerical solutions of stochastic differential delay equations under the generalized khasminskii type conditions the classical existence and uniqueness theorem of the solution to a stochastic differential delay equation sdde requires the local lipschitz condition and the linear growth condition see e g 11 12 and 20 the numerical solutions under these conditions have also been discussed intensively see e g 4 10 13 16 17 18 21 22 and 24 recently mao and rassias 14 and 15 established the generalized khasminskii type existence and uniqueness theorems for sddes where the linear growth condition is no longer imposed these generalized khasminskii type theorems cover a wide class of highly nonlinear sddes but these nonlinear sddes do not have explicit solutions whence numerical solutions are required in practice however there is so far little numerical theory on sddes under these generalized khasminskii type conditions the key aim of this paper is to close this gap

validation of a finite element error estimator the smoothed solution against which the finite element stresses are compared is based on a reformulation of the finite difference method that permits irregular meshes and complex boundary conditions to be analyzed

hierarchical image segmentation part i detection of regular curves in a vector graph the problem of edge detection is viewed as a hierarchy of detection problems where the geometric objects to be detected e g edge points curves regions have increasing complexity and spatial extent an early stage of the proposed hierarchy consists in detecting the regular portions of the visible edges the input to this stage is given by a graph whose vertices are tangent vectors representing local and uncertain information about the edges a model relating the input vector graph to the curves to be detected is proposed an algorithm with linear time complexity is described which solves the corresponding detection problem in a worst case scenario the stability of curve reconstruction in the presence of uncertain information and multiple responses to the same edge is analyzed and addressed explicitly by the proposed algorithm

convergence of numerical solutions to neutral stochastic delay differential equations with markovian switching recently numerical solutions of stochastic differential equations have received a great deal of attention it is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet in the paper the euler maruyama method for neutral stochastic delay differential equations is developed the key aim is to show that the numerical solutions will converge to the true solutions under the local lipschitz condition

the perceptual organization of texture flow a contextual inference approach locally parallel dense patterns sometimes called texture flows define a perceptually coherent structure of particular significance to perceptual organization we argue that with applications ranging from image segmentation and edge classification to shading analysis and shape interpretation texture flows deserve attention equal to edge segment grouping and curve completion this paper develops the notion of texture flow from a geometrical point of view to argue that local measurements of such structures must incorporate two curvatures we show how basic theoretical considerations lead to a unique model for the local behavior of the flow and to a notion of texture flow good continuation this in turn translates to a specification of consistency constraints between nearby flow measurements which we use for the computation of globally piecewise coherent structure through the contextual framework of relaxation labeling we demonstrate the results on synthetic and natural images

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

completion energies and scale the detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping we examine the notion of completion energy of curve elements showing and exploiting its intrinsic dependence on length and width scales we introduce a fast method for computing the most likely completion between two elements by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy we then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields this is done through multiscale procedures i e separate processing at different scales with some interscale interactions such procedures allow the summation of all induction fields to be done in a total of only o n log n operations where n is the number of pixels in the image more important such procedures yield a more realistic dependence of the induction field on the length and width scales the field of a long element is very different from the sum of the fields of its composing short segments

probabilistic motion estimation based on temporal coherence we develop a theory for the temporal integration of visual motion motivated by psychophysical experiments the theory proposes that input data are temporally grouped and used to predict and estimate the motion flows in the image sequence this temporal grouping can be considered a generalization of the data association techniques that engineers use to study motion sequences our temporal grouping theory is expressed in terms of the bayesian generalization of standard kalman filtering to implement the theory we derive a parallel network that shares some properties of cortical networks computer simulations of this network demonstrate that our theory qualitatively accounts for psychophysical experiments on motion occlusion and motion outliers in deriving our theory we assumed spatial factorizability of the probability distributions and made the approximation of updating the marginal distributions of velocity at each point this allowed us to perform local computations and simplified our implementation we argue that these approximations are suitable for the stimuli we are considering for which spatial coherence effects are negligible

numerical analysis of stochastic schemes in geophysics we present and study the stability convergence and order of convergence of a numerical scheme used in geophysics namely the stochastic version of a deterministic implicit leapfrog scheme which has been developed for the approximation of the so called barotropic vorticity model two other schemes which might be useful in the context of geophysical applications are also introduced and discussed

segmentation of multiple salient closed contours from real images using a saliency measure based on the global property of contour closure we have developed a segmentation method which identifies smooth closed contours bounding objects of unknown shape in real images the saliency measure incorporates the gestalt principles of proximity and good continuity that previous methods have also exploited unlike previous methods we incorporate contour closure by finding the eigenvector with the largest positive real eigenvalue of a transition matrix for a markov process where edges from the image serve as states element i j of the transition matrix is the conditional probability that a contour which contains edge j will also contain edge i we show how the saliency measure defined for individual edges can be used to derive a saliency relation defined for pairs of edges and further show that strongly connected components of the graph representing the saliency relation correspond to smooth closed contours in the image finally we report for the first time results on large real images for which segmentation takes an average of about 10 seconds per object on a general purpose workstation

polynomial stability of exact solution and a numerical method for stochastic differential equations with time dependent delay abstract we consider a stochastic differential equations with time dependent delay in this paper we first obtain the existence uniqueness and polynomial stability of exact solution to this equation under suitable conditions then for the numerical method of the corresponding sdde we present a so called modified truncated euler maruyama mtem method and consider the almost sure and mean square polynomial stability of this numerical method by using the well known discrete semimartingale convergence theorem sufficient conditions are obtained for both bounded and unbounded delay to ensure the polynomial stability of the corresponding numerical approximation results suggest that the mtem method replicates the polynomial stability of given sdde under suitable conditions examples are presented to illustrate the conclusion

stochastic completion fields a neural model of illusory contour shape and salience we describe an algorithm and representation level theory of illusory contour shape and salience unlike previous theories our model is derived from a single assumption that the prior probability distribution of boundary completion shape can be modeled by a random walk in a lattice whose points are positions and orientations in the image plane i e the space that one can reasonably assume is represented by neurons of the mammalian visual cortex our model does not employ numerical relaxation or other explicit minimization but instead relies on the fact that the probability that a particle following a random walk will pass through a given position and orientation on a path joining two boundary fragments can be computed directly as the product of two vector field convolutions we show that for the random walk we define the maximum likelihood paths are curves of least energy that is on average random walks follow paths commonly assumed to model the shape of illusory contours a computer model is demonstrated on numerous illusory contour stimuli from the literature

euclidean shift twist symmetry in population models of self aligning objects we consider the symmetry properties of a general class of nonlocal population models describing the aggregation and alignment of oriented objects in two dimensions such objects could be at the level of molecules cells or whole organisms we show that the underlying interaction kernel is invariant under the so called shift twist action of the euclidean group acting on the space mathbf r 2 times s 1 this group action was previously studied within the context of a continuum model of primary visual cortex we use perturbation methods to solve the eigenvalue problem arising from linearization about a homogeneous state and then use equivariant bifurcation theory to identify the various types of doubly periodic patterns that are expected to arise when the homogeneous state becomes unstable we thus establish that two distinct forms of spatio angular order can occur corresponding to scalar and pseudoscalar representations of the euclidean group

trace inference curvature consistency and curve detection an approach is described for curve inference that is based on curvature information the inference procedure is divided into two stages a trace inference stage which is the subject of the present work and a curve synthesis stage it is shown that recovery of the trace of a curve requires estimating local models for the curve at the same time and that tangent and curvature information are sufficient these make it possible to specify powerful constraints between estimated tangents to a curve in terms of a neighborhood relationship called cocircularity and between curvature estimates in terms of a curvature consistency relation because all curve information is quantized special care must be taken to obtain accurate estimates of trace points tangents and curvatures this issue is addressed specifically to the introduction of a smoothness constraint and a maximum curvature constraint the procedure is applied to two types of images artificial images designed to evaluate curvature and noise sensitivity and natural images

delay dependent stability of stochastic split step θ methods for stochastic delay differential equations abstract in this paper the delay dependent asymptotic mean square stability of the stochastic split step method for a scalar linear stochastic delay differential equation with real coefficients is studied the full stability region of this method is given by using root locus technique the necessary and sufficient condition with respect to the equation coefficients time stepsize and method parameter is derived it is proved that the stochastic split step backward euler can preserve the asymptotic mean square stability of the underlying system completely in addition the numerical stability regions of the stochastic split step method and the stochastic method are compared with each other at last we validate our conclusions by numerical experiments

recovering relative depth from low level features without explicit t junction detection and interpretation this work presents a novel computational model for relative depth order estimation from a single image based on low level local features that encode perceptual depth cues such as convexity concavity inclusion and t junctions in a quantitative manner considering information at different scales these multi scale features are based on a measure of how likely is a pixel to belong simultaneously to different objects interpreted as connected components of level sets and hence to be occluded in some of them providing a hint on the local depth order relationships they are directly computed on the discrete image data in an efficient manner without requiring the detection and interpretation of edges or junctions its behavior is clarified and illustrated for some simple images then the recovery of the relative depth order on the image is achieved by global integration of these local features applying a non linear diffusion filtering of bilateral type the validity of the proposed features and the integration approach is demonstrated by experiments on real images and comparison with state of the art monocular depth estimation techniques

characterizing the distribution of completion shapes with corners using a mixture of random processes abstract we derive an analytic expression for the distribution of contours x t generated by fluctuations in x t x t t due to random impulses of two limiting types the first type are frequent but weak while the second are infrequent but strong the result has applications in computational theories of figural completion and illusory contours because it can be used to model the prior probability distribution of short smooth completion shapes punctuated by occasional discontinuities in orientation i e corners this work extends our previous work on characterizing the distribution of completion shapes which dealt only with the case of frequently acting weak impulses

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

learning probabilistic models for contour completion in natural images using a large set of human segmented natural images we study the statistics of region boundaries we observe several power law distributions which likely arise from both multi scale structure within individual objects and from arbitrary viewing distance accordingly we develop a scale invariant representation of images from the bottom up using a piecewise linear approximation of contours and constrained delaunay triangulation to complete gaps we model curvilinear grouping on top of this graphical geometric structure using a conditional random field to capture the statistics of continuity and different junction types quantitative evaluations on several large datasets show that our contour grouping algorithm consistently dominates and significantly improves on local edge detection

embedding gestalt laws in markov random fields the goal of this paper is to study a mathematical framework of 2d object shape modeling and learning for middle level vision problems such as image segmentation and perceptual organization for this purpose we pursue generic shape models which characterize the most common features of 2d object shapes in this paper shape models are learned from observed natural shapes based on a minimax entropy learning theory the learned shape models are gibbs distributions defined on markov random fields mrfs the neighborhood structures of these mrfs correspond to gestalt laws colinearity cocircularity proximity parallelism and symmetry thus both contour based and region based features are accounted for stochastic markov chain monte carlo mcmc algorithms are proposed for learning and model verification furthermore this paper provides a quantitative measure for the so called nonaccidental statistics and thus justifies some empirical observations of gestalt psychology by information theory our experiments also demonstrate that global shape properties can arise from interactions of local features

contour inferences for image understanding we present a new approach to the algorithmic study of planar curves with applications to estimations of contours in images we construct spaces of curves satisfying constraints suited to specific problems exploit their geometric structure to quantify properties of contours and solve optimization and inference problems applications include new algorithms for computing planar elasticae with enhanced performance and speed and geometric algorithms for the estimation of contours of partially occluded objects in images

perceptual grouping by selection of a logically minimal model this paper presents a logic based approach to grouping and perceptual organization called minimal model theory and presents efficient methods for computing interpretations in this framework grouping interpretations are first defined as logical structures built out of atomic qualitative scene descriptors regularities that are derived from considerations of non accidentalness these interpretations can then be partially ordered by their degree of regularity or constraint measured numerically by their logical depth the genericity constraint the principle that interpretations should minimize coincidences in the observed configuration dictates that the preferred interpretation will be the minimum in this partial order i e the interpretation with maximum depth this maximum depth interpretation also called the minimal model or minimal interpretation is in a sense the simplest algebraically minimal interpretation available of the image configuration as a side effect the most salient or most structured part of the scene can be identified as the maximum depth subtree of the minimal model an efficient o n2 method for computing the minimal interpretation is presented along with examples computational experiments show that the algorithm performs well under a wide range of parameter settings

mean square stability and almost sure exponential stability of two step maruyama methods of stochastic delay hopfield neural networks abstract in this paper the two step maruyama methods of stochastic delay hopfield neural networks are studied we have found that under what choices of step size the two step maruyama methods of stochastic delay hopfield networks maintain the stability of exact solutions the mean square stability of two step maruyama methods of stochastic delay hopfield neural networks is investigated under suitable conditions also the almost sure exponential stability of two step maruyama methods of stochastic delay hopfield networks is proved using the semi martingale convergence theorem further the comparisons of stability conditions to the previous results in liu and zhu 2015 rathinasamy 2012 and ronghua et al 2010 are given numerical experiments are provided to illustrate our theoretical results

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

a comparison of measures for detecting natural shapes in cluttered backgrounds we propose a new measure of perceptual saliency and quantitatively compare its ability to detect natural shapes in cluttered backgrounds to five previously proposed measures as defined in the new measure the saliency of an edge is the fraction of closed random walks which contain that edge the transition probability matrix defining the random walk between edges is based on a distribution of natural shapes modeled by a stochastic motion each of the saliency measures in our comparison is a function of a set of affinity values assigned to pairs of edges although the authors of each measure define the affinity between a pair of edges somewhat differently all incorporate the gestalt principles of good continuation and proximity in some form in order to make the comparison meaningful we use a single definition of affinity and focus instead on the performance of the different functions for combining affinity values the primary performance criterion is accuracy we compute false positive rates in classifying edges as signal or noise for a large set of test figures in almost every case the new measure significantly outperforms previous measures

equivalent continuum representation of structures composed of repeated elements a procedure for determining the equivalent continuum properties of a structure composed of repeated patterns of discrete elements with both displacement and rotational coordinates is presented these nodal coordinates are transformed to rigid body and strain gradient variables using a polynomial approximation the maximum number of independent variables that may be retained is determined by applying a ranking procedure to the resulting transformation matrix the possibility of introducing errors by requiring the analyst to supply the strain gradient terms directly is reduced by identifying the appropriate variables through the use of the polynomial expansion and the ranking procedure additional constraints may be imposed in this analysis the equivalent continuum parameters result when a further transformation to the appropriate kinematic variables is applied and the strain energy expression is reduced to these variables three dimensional beamand plate like structures are treated the results correspond to findings using other approaches

euler spiral for shape completion in this paper we address the curve completion problem e g the geometric continuation of boundaries of objects which are temporarily interrupted by occlusion also known as the gap completion or shape completion problem this problem is a significant element of perceptual grouping of edge elements and has been approached by using cubic splines or biarcs which minimize total curvature squared elastica as motivated by a physical analogy our approach is motivated by railroad design methods of the early 1900 s which connect two rail segments by transition curves and by the work of knuth on mathematical typography we propose that in using an energy minimizing solution completion curves should not penalize curvature as in elastica but curvature variation the minimization of total curvature variation leads to an euler spiral solution a curve whose curvature varies linearly with arclength the construction of this curve from a pair of points and tangents at these points is reduced to a nonlinear system of equations involving fresnel integrals whose solution relies on optimization from a suitable initial condition constrained to satisfy given boundary conditions since the choice of an appropriate initial curve is critical in this optimization we analytically derive an optimal solution in the class of biarc curves which is then used as the initial curve the resulting interpolations yield intuitive interpolation across gaps and occlusions and are extensible in contrast to scale invariant elastica in addition euler spiral segments can be used in other applications of curve completions e g modeling boundary segments between curvature extrema or skeletal branch geometry

a new approach to boundary modelling for finite difference applications in solid mechanics procedures for extending the useful scope of the finite difference method in solid mechanics applications are presented the improvements centre around the introduction of the physical nature of the deformations into the equations used to formulate the approximate solution this is accomplished by evaluating the coefficients of taylor series expansions for the displacement approximations in terms of rigid body motions strains and derivatives of strains this procedure is demonstrated with plane stress applications the ability to interpret the derivative approximations physically allows the fictitious nodes typical of the finite difference method to be rationally incorporated into the model as a means of enforcing traction boundary conditions an example problem is solved using both regular and irregular meshes the displacements and stresses compare well with finite element solutions

a bayesian multiple hypothesis approach to edge grouping and contour segmentation a contour segmentation algorithm is presented that takes an edge map and extracts continuous curves of arbitrary smoothness correctly handling curve intersections and capable of extrapolating over significant measurement gaps the algorithm incorporates noise models of the edge detection process and limited scene statistics it is based on an explicit contour model and employs a statistical distance measure to quantify the likelihood of each segmentation hypothesis a bayesian multiple hypothesis tree organizes possible segementations making it possible to postpone grouping decisions until a sufficient amount of information is available we have demonstrated its performance on real and synthetic images

steerable wedge filters for local orientation analysis steerable filters have been used to analyze local orientation patterns in imagery such filters are typically based on directional derivatives whose symmetry produces orientation responses that are periodic with period spl pi independent of image structure we present a more general set of steerable filters that alleviate this problem

convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation the paper deals with convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation it is proved that the semi implicit euler method is convergent with strong order p the conditions under which the method is ms stable and gms stable are determined and the numerical experiments are given

numerical methods for stochastic delay differential equations via the wong zakai approximation we use the wong zakai approximation as an intermediate step to derive numerical schemes for stochastic delay differential equations by approximating the brownian motion with its truncated spectral expansion and then using different discretizations in time we present three schemes a predictor corrector scheme a midpoint scheme and a milstein like scheme we prove that the predictor corrector scheme converges with order half in the mean square sense while the milstein like scheme converges with order one numerical tests confirm the theoretical prediction and demonstrate that the midpoint scheme is of half order convergence numerical results also show that the predictor corrector and midpoint schemes can be of first order convergence under commutative noises when there is no delay in the diffusion coefficients

signature detection and matching for document image retrieval as one of the most pervasive methods of individual identification and document authentication signatures present convincing evidence and provide an important form of indexing for effective document image processing and retrieval in a broad range of applications however detection and segmentation of free form objects such as signatures from clustered background is currently an open document analysis problem in this paper we focus on two fundamental problems in signature based document image retrieval first we propose a novel multiscale approach to jointly detecting and segmenting signatures from document images rather than focusing on local features that typically have large variations our approach captures the structural saliency using a signature production model and computes the dynamic curvature of 2d contour fragments over multiple scales this detection framework is general and computationally tractable second we treat the problem of signature retrieval in the unconstrained setting of translation scale and rotation invariant nonrigid shape matching we propose two novel measures of shape dissimilarity based on anisotropic scaling and registration residual error and present a supervised learning framework for combining complementary shape information from different dissimilarity metrics using lda we quantitatively study state of the art shape representations shape matching algorithms measures of dissimilarity and the use of multiple instances as query in document image retrieval we further demonstrate our matching techniques in offline signature verification extensive experiments using large real world collections of english and arabic machine printed and handwritten documents demonstrate the excellent performance of our approaches

strong superconvergence of the eulermaruyama method for linear stochastic volterra integral equations the eulermaruyama method is presented for linear stochastic volterra integral equations then the strong convergence property is analyzed for convolution kernels and general kernels respectively it is well known that for stochastic ordinary differential equations the strong convergence order of the eulermaruyama method is 12 however the strong superconvergence order of 1 is obtained for linear stochastic volterra integral equations with convolution kernels if the kernel k2 of the diffusion term satisfies k2 0 0 and this strong superconvergence property is inherited by linear stochastic volterra integral equations with general kernels if the kernel k2 of the diffusion term satisfies k2 t t 0 the theoretical results are illustrated by extensive numerical examples

cortical spatiotemporal dimensionality reduction for visual grouping the visual systems of many mammals including humans are able to integrate the geometric information of visual stimuli and perform cognitive tasks at the first stages of the cortical processing this is thought to be the result of a combination of mechanisms which include feature extraction at the single cell level and geometric processing by means of cell connectivity we present a geometric model of such connectivities in the space of detected features associated with spatiotemporal visual stimuli and show how they can be used to obtain low level object segmentation the main idea is to define a spectral clustering procedure with anisotropic affinities over data sets consisting of embeddings of the visual stimuli into higher dimensional spaces neural plausibility of the proposed arguments will be discussed

multistep methods for sdes and their application to problems with small noise in this article the numerical approximation of solutions of ito stochastic differential equations is considered in particular for equations with a small parameter epsilon in the noise coefficient we construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean square consistency numerical stability in the mean square sense and mean square convergence for the special case of two step maruyama schemes we derive conditions guaranteeing their mean square consistency further for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter epsilon simulation results using several explicit and implicit stochastic linear k step schemes k 1 2 illustrate the theoretical findings

stochastic differential equations and geometric flows in previous years curve evolution applied to a single contour or to the level sets of an image via partial differential equations has emerged as an important tool in image processing and computer vision curve evolution techniques have been utilized in problems such as image smoothing segmentation and shape analysis we give a local stochastic interpretation of the basic curve smoothing equation the so called geometric heat equation and show that this evolution amounts to a tangential diffusion movement of the particles along the contour moreover assuming that a priori information about the shapes of objects in an image is known we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise we also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features in contrast to the original geometric heat flow which tends to circularize any closed curve

on exponential mean square stability of two step maruyama methods for stochastic delay differential equations we are concerned with the exponential mean square stability of two step maruyama methods for stochastic differential equations with time delay we propose a family of schemes and prove that it can maintain the exponential mean square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation numerical results for linear and nonlinear equations show that this family of two step maruyama methods exhibits better stability than previous two step maruyama methods

potentials valleys and dynamic global coverings we present a new approach to effect the transition between local and global representations it is based on the notion of a covering or a collection of objects whose union is equivalent to the full one the mathematics of computing global coverings are developed in the context of curve detection where an intermediate representation the tangent field provides a reliable local description of curve structure this local information is put together globally in the form of a potential distribution the elements of the covering are then short curves each of which evolves in parallel to seek the valleys of the potential distribution the initial curve positions are also derived from the tangent field and their evolution is governed by variational principles when stationary configurations are achieved the global dynamic covering is defined by the union of the local dynamic curves

path based clustering for grouping of smooth curves and texture segmentation perceptual grouping organizes image parts in clusters based on psychophysically plausible similarity measures we propose a novel grouping method in this paper which stresses connectedness of image elements via mediating elements rather than favoring high mutual similarity this grouping principle yields superior clustering results when objects are distributed on low dimensional extended manifolds in a feature space and not as local point clouds in addition to extracting connected structures objects are singled out as outliers when they are too far away from any cluster structure the objective function for this perceptual organization principle is optimized by a fast agglomerative algorithm we report on perceptual organization experiments where small edge elements are grouped to smooth curves the generality of the method is emphasized by results from grouping textured images with texture gradients in an unsupervised fashion

fast multiscale clustering and manifold identification we present a novel multiscale clustering algorithm inspired by algebraic multigrid techniques our method begins with assembling data points according to local similarities it uses an aggregation process to obtain reliable scale dependent global properties which arise from the local similarities as the aggregation process proceeds these global properties affect the formation of coherent clusters the global features that can be utilized are for example density shape intrinsic dimensionality and orientation the last three features are a part of the manifold identification process which is performed in parallel to the clustering process the algorithm detects clusters that are distinguished by their multiscale nature separates between clusters with different densities and identifies and resolves intersections between clusters the algorithm is tested on synthetic and real data sets its running time complexity is linear in the size of the data set

on mean square stability of two step maruyama methods for nonlinear neutral stochastic delay differential equations the asymptotic mean square stability of two step maruyama methods is considered for nonlinear neutral stochastic differential equations with constant time delay nsddes under the one sided lipschitz condition and the linear growth condition it is proved that a family of implicit two step maruyama methods can preserve the stability of the analytic solution in mean square sense numerical results for both a nonlinear nsdde and a system show that the family of two step maruyama methods have better stability than previous two step maruyama methods

analysis of some low order quadrilateral reissner mindlin plate elements four quadrilateral elements for the reissner mindlin plate model are considered the elements are the stabilized mitc4 element of lyly stenberg and vihinen 1933 the min4 element of tessler and hughes 1983 the q4bl element of zienkiewicz et al 1993 and the fmin4 element of kikuchi and ishii 1999 for all elements except the q4bl element a unifying variational formulation is introduced and optimal h 1 and l 2 error bounds uniform in the plate thickness are proven moreover we propose a modified q4bl element and show that it admits the optimal h 1 and l 2 error bounds uniform in the plate thickness in particular we study the convergence behavior of all elements regarding the mesh distortion

shape representation and classification using the poisson equation we present a novel approach that allows us to reliably compute many useful properties of a silhouette our approach assigns for every internal point of the silhouette a value reflecting the mean time required for a random walk beginning at the point to hit the boundaries this function can be computed by solving poisson s equation with the silhouette contours providing boundary conditions we show how this function can be used to reliably extract various shape properties including part structure and rough skeleton local orientation and aspect ratio of different parts and convex and concave sections of the boundaries in addition to this we discuss properties of the solution and show how to efficiently compute this solution using multigrid algorithms we demonstrate the utility of the extracted properties by using them for shape classification and retrieval

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

a mixed formulation of c0 linear triangular plate shell element the role of edge shear constraints a simple and effective linear c0 triangular element for plates and shells is developed on the basis of the hellinger reissner mixed variational principle with independently assumed stress and displacement fields two main features are emphasized in this development concerning the assumed transverse shear stress field firstly this assumption results in edge type penalty constraints in the thin plate shell regimes secondly the element is used in the form of a four triangle cross diagonal macroelement mesh as a result the element is shown to be free from shear locking in thin plate shell applications and it satisfied all the appropriate patch tests required for kirchhoff plate models in addition the element exhibits good overall convergence properties in a variety of test problems for plates and shells finally from the computational standpoint the element is very efficient since all stiffness component matrices are derived in explicit forms

a graph spectral method for surface height recovery this paper describes a graph spectral method for 3d surface integration the algorithm takes as its input a 2d field of surface normal estimates delivered for instance by a shape from shading or shape from texture procedure we commence by using the surface normals to obtain an affinity weight matrix whose elements are related to the surface curvature the weight matrix is used to compute a row normalized transition probability matrix and we pose the recovery of the integration path as that of finding the steady state random walk for the markov chain defined by this matrix the steady state random walk is given by the leading eigenvector of the original affinity weight matrix by threading the surface normals together along the path specified by the magnitude order of the components of the leading eigenvector we perform surface integration the height increments along the path are simply related to the traversed path length and the slope of the local tangent plane the method is evaluated on needle maps delivered by a shape from shading algorithm applied to real world data and also on synthetic data the method is compared with the local geometric height reconstruction method of bors hancock and wilson and the global methods of horn and brooks and frankot and chellappa

inferring global perceptual contours from local features we address the problem of contour inference from partial data as obtained from state of the art edge detectors we argue that in order to obtain more pereeptually salient contours it is necessary to impose generic constraints such as continuity and co curvilinearity the implementation is in the form of a convolution with a mask which encodes both the orientation and the strength of the possible continuations we first show how the mask called the extension field is derived then how the contributions from different sites are collected to produce a saliency map we show that the scheme can handle a variety of input data from dot patterns to oriented edgels in a unified manner and demonstrate results on a variety of input stimuli we also present a similar approach to the problem of inferring contours formed by end points in both cases the scheme is non linear non iterative and unified in the sense that all types of input tokens are handled in the same manner

hybrid hexahedral element for solids plates shells and beams by selective scaling a robust two field hexahedral element capable of handling plate shell beam and nearly incompressible material analyses without locking are presented starting with the assumed stress element of pian and tong 7 parasitic strain components leading to locking in plate shell and beam analyses are first identified locking can be alleviated by scaling down selectively the parasitic strain components in the leverage matrix unfortunately the element then fails the patch test however patch test correction and reduction in computation can be achieved by the recently proposed admissible matrix formulation the resulting element is lock free and very efficient all matrices involved in constructing the stiffness matrix can be derived explicitly the accuracy of the element is tested by popular bench mark problems

explicit algorithms for the nonlinear dynamics of shells abstract a finite element formulation and algorithm for the nonlinear analysis of the large deflection materially nonlinear response of impulsively loaded shells is presented a unique feature of this algorithm is the use of a bilinear four node quadrilateral element with single point quadrature and a simple hourglass control which is orthogonal to rigid body modes on an element level and does not compromise the consistency of the equations the geometric nonlinearities are treated by using a corotational description wherein a coordinate system that rotates with the material is embedded at the integration point thus the algorithm is directly applicable to anisotropic materials without any corrections for frame invariance of material property tensors this algorithm can treat about 200 element time steps per cpu second on a cyber 170 730 computer in the explicit time integration mode numerous results are presented for both elastic and elastic plastic problems with large strains that show that the method in most cases is comparable in accuracy with an earlier version of this algorithm employing a cubic triangular plate shell element but substantially faster

a probabilistic spectral framework for grouping and segmentation this paper presents an iterative spectral framework for pairwise clustering and perceptual grouping our model is expressed in terms of two sets of parameters firstly there are cluster memberships which represent the affinity of objects to clusters secondly there is a matrix of link weights for pairs of tokens we adopt a model in which these two sets of variables are governed by a bernoulli model we show how the likelihood function resulting from this model may be maximised with respect to both the elements of link weight matrix and the cluster membership variables we establish the link between the maximisation of the log likelihood function and the eigenvectors of the link weight matrix this leads us to an algorithm in which we iteratively update the link weight matrix by repeatedly refining its modal structure each iteration of the algorithm is a three step process first we compute a link weight matrix for each cluster by taking the outer product of the vectors of current cluster membership indicators for that cluster second we extract the leading eigenvector from each modal link weight matrix third we compute a revised link weight matrix by taking the sum of the outer products of the leading eigenvectors of the modal link weight matrices

subjective surfaces a geometric model for boundary completion we present a geometric model and a computational method for segmentation of images with missing boundaries in many situations the human visual system fills in missing gaps in edges and boundaries building and completing information that is not present boundary completion presents a considerable challenge in computer vision since most algorithms attempt to exploit existing data a large body of work concerns completion models which postulate how to construct missing datas these models are often trained and specific to particular images in this paper we take the following alternative perspective we consider a given reference point within the image and then develop an algorithm which tries to build missing information on the basis of the given point of view and the available information as boundary data to the algorithm starting from this point of view a surface is constructed it is then evolved with the mean curvature flow in the metric induced by the image until a piecewise constant solution is reached we test the computational model on modal completion amodal completion and texture segmentation we extend the geometric model and the algorithm to 3d in order to extract shapes from low signal noise ratio ultrasound image volumes results in 3d echocardiography and 3d fetal echography are also presented

contour and texture analysis for image segmentation this paper provides an algorithm for partitioning grayscale images into disjoint regions of coherent brightness and texture natural images contain both textured and untextured regions so the cues of contour and texture differences are exploited simultaneously contours are treated in the intervening contour framework while texture is analyzed using textons each of these cues has a domain of applicability so to facilitate cue combination we introduce a gating operator based on the texturedness of the neighborhood at a pixel having obtained a local measure of how likely two nearby pixels are to belong to the same region we use the spectral graph theoretic framework of normalized cuts to find partitions of the image into regions of coherent texture and brightness experimental results on a wide range of images are shown

general geometric good continuation from taylor to laplace via level sets good continuation is the gestalt observation that parts often group in particular ways to form coherent wholes perceptual integration of edges for example involves orientation good continuation a property which has been exploited computationally very extensively but more general local global relationships such as for shading or color have been elusive while taylor s theorem suggests certain modeling and smoothness criteria the consideration of level set geometry indicates a different approach using such first principles we derive for the first time a generalization of good continuation to all those visual structures that can be abstracted as scalar functions over the image plane based on second order differential constraints that reflect good continuation our analysis leads to a unique class of harmonic models and a cooperative algorithm for structure inference among the different applications of good continuation here we apply these results to the denoising of shading and intensity distributions and demonstrate how our approach eliminates spurious measurements while preserving both singularities and regular structure a property that facilitates higher level processes which depend so critically on both of these classes of visual structures

an improved treatment of transverse shear in the mindlin type four node quadrilateral element abstract an improved displacement methodology based on mindlin theory is developed and applied to a four node twelve degrees of freedom quadrilateral element special attention is directed toward enhancement of the transverse shear energy representation particularly a non uniform order kinematic interpolation scheme is accompanied by what is termed element appropriate shear correction factors the element matrices are consistently derived from hamilton s variational principle using exact numerical integration throughout the element stiffness matrix possesses correct rank and is well conditioned over the entire mindlin theory range of thickness to length ratios exceptionally good element performance is demonstrated by solutions to a set of static and vibrational problems

error estimation procedure for plate bending elements the elemental errors are identified through the use of an element formulation procedure based on physically interpretable strain gradient interpolation functions the use of this notation allows these errors to be eliminated using rational arguments the discretization errors are identified by comparing the finite element solution to a smoothed superconvergent solution the errors thus identified are used to guide an adaptive mesh refinement procedure that produces improved results

optimal image and video closure by superpixel grouping detecting independent objects in images and videos is an important perceptual grouping problem one common perceptual grouping cue that can facilitate this objective is the cue of contour closure reflecting the spatial coherence of objects in the world and their projections as closed boundaries separating figure from background detecting contour closure in images consists of finding a cycle of disconnected contour fragments that separates an object from its background searching the entire space of possible groupings is intractable and previous approaches have adopted powerful perceptual grouping heuristics such as proximity and co curvilinearity to constrain the search we introduce a new formulation of the problem by transforming the problem of finding cycles of contour fragments to finding subsets of superpixels whose collective boundary has strong edge support few gaps in the image our cost function a ratio of a boundary gap measure to area promotes spatially coherent sets of superpixels moreover its properties support a global optimization procedure based on parametric maxflow extending closure detection to videos we introduce the concept of spatiotemporal closure analogous to image closure we formulate our spatiotemporal closure cost over a graph of spatiotemporal superpixels our cost function is a ratio of motion and appearance discontinuity measures on the boundary of the selection to an internal homogeneity measure of the selected spatiotemporal volume the resulting approach automatically recovers coherent components in images and videos corresponding to objects object parts and objects with surrounding context providing a good set of multiscale hypotheses for high level scene analysis we evaluate both our image and video closure frameworks by comparing them to other closure detection approaches and find that they yield improved performance

sledge sequential labeling of image edges for boundary detection our goal is to detect boundaries of objects or surfaces occurring in an arbitrary image we present a new approach that discovers boundaries by sequential labeling of a given set of image edges a visited edge is labeled as on or off a boundary based on the edge s photometric and geometric properties and evidence of its perceptual grouping with already identified boundaries we use both local gestalt cues e g proximity and good continuation and the global helmholtz principle of non accidental grouping a new formulation of the helmholtz principle is specified as the entropy of a layout of image edges for boundary discovery we formulate a new policy iteration algorithm called sledge training of sledge is iterative in each training image sledge labels a sequence of edges which induces loss with respect to the ground truth these sequences are then used as training examples for learning sledge in the next iteration such that the total loss is minimized for extracting image edges that are input to sledge we use our new low level detector it finds salient pixel sequences that separate distinct textures within the image on the benchmark berkeley segmentation datasets 300 and 500 our approach proves robust and effective we outperform the state of the art both in recall and precision for different input sets of image edges

a tangent bundle theory for visual curve completion visual curve completion is a fundamental perceptual mechanism that completes the missing parts e g due to occlusion between observed contour fragments previous research into the shape of completed curves has generally followed an axiomatic approach where desired perceptual geometrical properties are first defined as axioms followed by mathematical investigation into curves that satisfy them however determining psychophysically such desired properties is difficult and researchers still debate what they should be in the first place instead here we exploit the observation that curve completion is an early visual process to formalize the problem in the unit tangent bundle r2 s1 which abstracts the primary visual cortex v1 and facilitates exploration of basic principles from which perceptual properties are later derived rather than imposed exploring here the elementary principle of least action in v1 we show how the problem becomes one of finding minimum length admissible curves in r2 s1 we formalize the problem in variational terms we analyze it theoretically and we formulate practical algorithms for the reconstruction of these completed curves we then explore their induced visual properties vis a vis popular perceptual axioms and show how our theory predicts many perceptual properties reported in the corresponding perceptual literature finally we demonstrate a variety of curve completions and report comparisons to psychophysical data and other completion models

3d stochastic completion fields for mapping connectivity in diffusion mri the 2d stochastic completion field algorithm introduced by williams and jacobs 1 2 uses a directional random walk to model the prior probability of completion curves in the plane this construct has had a powerful impact in computer vision where it has been used to compute the shapes of likely completion curves between edge fragments in visual imagery motivated by these developments we extend the algorithm to 3d using a spherical harmonics basis to achieve a rotation invariant computational solution to the fokker planck equation describing the evolution of the probability density function underlying the model this provides a principled way to compute 3d completion patterns and to derive connectivity measures for orientation data in 3d as arises in 3d tracking motion capture and medical imaging we demonstrate the utility of the approach for the particular case of diffusion magnetic resonance imaging where we derive connectivity maps for synthetic data on a physical phantom and on an in vivo high angular resolution diffusion image of a human brain

salient closed boundary extraction with ratio contour we present ratio contour a novel graph based method for extracting salient closed boundaries from noisy images this method operates on a set of boundary fragments that are produced by edge detection boundary extraction identifies a subset of these fragments and connects them sequentially to form a closed boundary with the largest saliency we encode the gestalt laws of proximity and continuity in a novel boundary saliency measure based on the relative gap length and average curvature when connecting fragments to form a closed boundary this new measure attempts to remove a possible bias toward short boundaries we present a polynomial time algorithm for finding the most salient closed boundary we also present supplementary preprocessing steps that facilitate the application of ratio contour to real images we compare ratio contour to two closely related methods for extracting closed boundaries elder and zucker s method based on the shortest path algorithm and williams and thornber s method based on spectral analysis and a strongly connected components algorithm this comparison involves both theoretic analysis and experimental evaluation on both synthesized data and real images

tangent bundle curve completion with locally connected parallel networks we propose a theory for cortical representation and computation of visually completed curves that are generated by the visual system to fill in missing visual information e g due to occlusions recent computational theories and physiological evidence suggest that although such curves do not correspond to explicit image evidence along their length their construction emerges from corresponding activation patterns of orientation selective cells in the primary visual cortex previous theoretical work modeled these patterns as least energetic 3d curves in the mathematical continuous space which abstracts the mammalian striate cortex here we discuss the biological plausibility of this theory and present a neural architecture that implements it with locally connected parallel networks part of this contribution is also a first attempt to bridge the physiological literature on curve completion with the shape problem and a shape theory we present completion simulations of our model in natural and synthetic scenes and discuss various observations and predictions that emerge from this theory in the context of curve completion

origins of illusory percepts in digital images abstract here we show the relation between illusory percepts and statistical regularities across scales and orientations to this aim the performance of a computational model for the partitioning of statistical regularities is analyzed on several tasks such as long range boundary completion phase induced contour detection as well as shape and size illusions the system for the automatically learned partitioning of statistical regularities in 2d images is based on a sophisticated band pass filtering operation with fixed scale and orientation sensitivity experimental results are provided to illustrate this analysis on several examples i kanizsa type subjective figures ii phase induced subjective contours iii the zollner illusion and iv the muller lyer illusion

sketches with curvature the curve indicator random field and markov processes while complaints about typical edge operators are common proposals articulating a notion of the perfect edge map are comparatively rare hindering the improvement of contour enhancement techniques to address this situation we suggest that one objective of visual contour computation is the estimation of a clean sketch from a corrupted rendition the latter modeling noisy and low contrast edge or line operator responses to an image our formal model of this clean sketch is the curve indicator random field cirf whose role is to provide a basis for defining edge likelihood models by eliminating the parameter along each curve to create an image of curves for curves modeled with stationary markov processes this ideal edge prior is non gaussian and its moment generating functional has a form closely related to the feynman kac formula this sketch model leads to a nonlinear minimum mean squared error contour enhancement filter that requires the solution of two elliptic partial differential equations the framework is also independent of the order of the contour model allowing us to introduce a markov process model for contour curvature we analyze the distribution of such curves and show that its mode is the euler spiral a curve minimizing changes in curvature example computations using the contour enhancement filter with the curvature based contour model are provided highlighting how the filter is curvature selective even when curvature is absent in the input

improving stability and accuracy of reissner mindlin plate finite elements via algebraic subgrid scale stabilization abstract stabilized finite element methods for the solution of reissner mindlin type plate problems are presented the formulations are based on previously described mixed formulations like the assumed natural strain ans or mitc concept or the discrete shear gap dsg method in particular the algebraic subgrid scale asgs formulation is used for the stabilization term the essential idea is to obtain stable elements and improve coarse mesh accuracy at the same time it is shown how this can be achieved by a proper choice of stabilization parameters on the basis of physical insight into the mechanical behavior of shear deformable plates in this context there is a strong relationship to concepts that have been developed long before stabilization techniques appeared in finite element technology particularly the residual bending flexibility or deflection matching technique

submodeling approach to adaptive mesh refinement a submodeling approach for adaptively refining stress concentrations has been developed the stress concentrations are separated from the remainder of the problem one at time by defining internal boundaries in terms of elemental error measures the fact that the effects of the remainder of the problem are satisfactorily included in the subproblem by interpolating the displacements across the internal boundary is indicated by the accuracy of the results the adaptive refinement is terminated when the maximum elemental error reaches a predetermined value this contrasts to some approaches where a global error measure is used to terminate the analysis the use of the elemental error measure to stop the analysis removes the effect of the far field elements from the assessment of the accuracy of the results at the stress concentration the accuracy of the analysis in the region of interest is used to terminate the analysis the adaptive refinement of a submodel containing a stress concentration allows the computational resources to be focused on the regions of the problem where they can best be utilized for instance if a problem has several high stress regions each one can be treated separately and in detail the 5 elemental error criterion used here to define both the internal boundary and to terminate the analysis was chosen arbitrarily previous experience had shown that this value would produce well converged results for the example problems solved here thus a valuable area of future resarch would be to determine a process for identifying the criteria that would produce the desired mix between accuracy and efficiency for general problems

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation