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

parts of visual form computational aspects underlying recognition is an organization of objects and their parts into classes and hierarchies a representation of parts for recognition requires that they be invariant to rigid transformations robust in the presence of occlusions stable with changes in viewing geometry and be arranged in a hierarchy these constraints are captured in a general framework using notions of a part line and a partitioning scheme a proposed general principle of form from function motivates a particular partitioning scheme involving two types of parts neck based and limb based neck based parts arise from narrowings in shape or the local minima in distance between two points on the boundary while limb based parts arise from a pair of negative curvature minima which have co circular tangents in this paper we present computational support for the limb based and neck based parts by showing that they are invariant robust stable and yield a hierarchy of parts examples illustrate that the resulting decompositions are robust in the presence of occlusion and clutter for a range of man made and natural objects and lead to natural and intuitive parts which can be used for recognition

a shape representation with elastic quadratic polynomials preservation of high curvature points under noisy conditions we present a new shape representation technique for a planar shape using overlapping quadratic splines the technique refines the shape by iteratively updating each spline to reduce the cost associated with c 1 discontinuity it consists of a series of affine commutative linear operators producing a smooth bandpass frequency response the primary purpose of the technique is to remove minute high curvature points while preserving salient ones we compare the performance of the technique against those that are based on either linear splines cubic splines or wavelets we consider three criteria sensitivity of detecting salient high curvature points hausdorff distance between the representation and the original shape and computation time the results show that the proposed technique is highly effective in preserving salient high curvature points with a relatively small hausdorff distance and a computational cost

some solutions of the linear and nonlinear klein gordon equations using the optimal homotopy asymptotic method abstract we investigate the effectiveness of the optimal homotopy asymptotic method oham in solving time dependent partial differential equations to this effect we consider the homogeneous non homogeneous linear and nonlinear klein gordon equations with boundary conditions the results reveal that the method is explicit effective and easy to use

inference and learning with hierarchical shape models in this work we introduce a hierarchical representation for object detection we represent an object in terms of parts composed of contours corresponding to object boundaries and symmetry axes these are in turn related to edge and ridge features that are extracted from the image we propose a coarse to fine algorithm for efficient detection which exploits the hierarchical nature of the model this provides a tractable framework to combine bottom up and top down computation we learn our models from training images where only the bounding box of the object is provided we automate the decomposition of an object category into parts and contours and discriminatively learn the cost function that drives the matching of the object to the image using multiple instance learning using shape based information we obtain state of the art localization results on the uiuc and ethz datasets

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

efficient image retrieval through vantage objects we describe a new indexing structure for general image retrieval that relies solely on a distance function giving the similarity between two images for each image object in the database its distance to a set of m predetermined vantage objects is calculated the m vector of these distances specifies a point in the m dimensional vantage space the database objects that are similar in terms of the distance function to a given query object can be determined by means of an efficient nearest neighbor search on these points we demonstrate the viability of our approach through experimental results obtained with two image databases one consisting of about 5200 raster images of stamps the other containing about 72 000 hieroglyphic polylines

probabilistic methods for finding people finding people in pictures presents a particularly difficult object recognition problem we show how to find people by finding candidate body segments and then constructing assemblies of segments that are consistent with the constraints on the appearance of a person that result from kinematic properties since a reasonable model of a person requires at least nine segments it is not possible to inspect every group due to the huge combinatorial complexity we propose two approaches to this problem in one the search can be pruned by using projected versions of a classifier that accepts groups corresponding to people we describe an efficient projection algorithm for one popular classifier and demonstrate that our approach can be used to determine whether images of real scenes contain people the second approach employs a probabilistic framework so that we can draw samples of assemblies with probabilities proportional to their likelihood which allows to draw human like assemblies more often than the non person ones the main performance problem is in segmentation of images but the overall results of both approaches on real images of people are encouraging

fixed point attractor analysis for a class of neurodynamics nearly all models in neural networks start from the assumption that the input output characteristic is a sigmoidal function on parameter space we present a systematic and feasible method for analyzing the whole spectrum of attractors all saturated all but one saturated all but twosaturated and so on of a neurodynamical system with a saturated sigmoidal function as its input output characteristic we present an argument that claims under a mild condition that only all saturated or all but one saturated attractors are observable for the neurodynamics for any given all saturated configuration all but one saturated configuration the article shows how to construct an exact parameter region r such that if and only if the parameters fall within r then is an attractor a fixed point of the dynamics the parameter region for an all saturated fixed point attractor is independent of the specific choice of a saturated sigmoidal function whereas for an all but one saturated fixed point it

contour completion without region segmentation contour completion plays an important role in visual perception where the goal is to group fragmented low level edge elements into perceptually coherent and salient contours most existing methods for contour completion have focused on pixelwise detection accuracy in contrast fewer methods have addressed the global contour closure effect despite psychological evidences for its importance this paper proposes a purely contour based higher order crf model to achieve contour closure through local connectedness approximation this leads to a simplified problem structure where our higher order inference problem can be transformed into an integer linear program and be solved efficiently compared with the methods based on the same bottom up edge detector our method achieves a superior contour grouping ability measured by rand index a comparable precision recall performance and more visually pleasing results our results suggest that contour closure can be effectively achieved in contour domain in contrast to a popular view that segmentation is essential for this purpose

microaggregation and permutation based anonymization of movement data movement data that is trajectories of mobile objects are automatically collected in huge quantities by technologies such as gps gsm or rfid among others publishing and exploiting such data is essential to improve transportation to understand the dynamics of the economy in a region etc however there are obvious threats to the privacy of individuals if their trajectories are published in a way which allows re identification of the individual behind a trajectory we contribute to the literature on privacy preserving publication of trajectories by presenting a distance measure for trajectories which naturally considers both spatial and temporal aspects of trajectories is computable in polynomial time and can cluster trajectories not defined over the same time span our distance measure can be naturally instantiated using other existing similarity measures for trajectories that are appropriate for anonymization purposes then we propose two heuristics for trajectory anonymization which yield anonymized trajectories formed by fully accurate true original locations the first heuristic is based on trajectory microaggregation using the above distance and on location permutation it effectively achieves trajectory k anonymity the second heuristic is based only on location permutation it gives up trajectory k anonymity and aims at location k diversity the strong point of the second heuristic is that it takes into account reachability constraints when computing anonymized trajectories experimental results on a synthetic data set and a real life data set are presented for similar privacy protection levels and most reasonable parameter choices our two methods offer better utility than comparable previous proposals in the literature

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

groups of adjacent contour segments for object detection we present a family of scale invariant local shape features formed by chains of k connected roughly straight contour segments kas and their use for object class detection kas are able to cleanly encode pure fragments of an object boundary without including nearby clutter moreover they offer an attractive compromise between information content and repeatability and encompass a wide variety of local shape structures we also define a translation and scale invariant descriptor encoding the geometric configuration of the segments within a kas making kas easy to reuse in other frameworks for example as a replacement or addition to interest points ips software for detecting and describing kas is released at http lear inrialpes fr software we demonstrate the high performance of kas within a simple but powerful sliding window object detection scheme through extensive evaluations involving eight diverse object classes and more than 1 400 images we 1 study the evolution of performance as the degree of feature complexity k varies and determine the best degree 2 show that kas substantially outperform ips for detecting shape based classes and 3 compare our object detector to the recent state of the art system by dalal and triggs 2005

application of optimal homotopy asymptotic method for the analytic solution of singular lane emden type equation abstract in this study optimal homotopy asymptotic method is applied on singular initial value lane emden type problems to check the effectiveness and performance of the method it is observed that the method is easy to implement quite valuable to handle singular phenomena and yield excellent results at minimum computational cost computational results of some of the test problems are presented to demonstrate the viability and practical usefulness of the method

the number of synaptic inputs and the synchrony of large sparse neuronal networks the prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons we study synchronization in models of large networks of spiking neurons with random sparse connectivity synchrony occurs only when the average number of synapses m that a cell receives is larger than a critical value mc below mc the system is in an asynchronous state in the limit of weak coupling assuming identical neurons we reduce the model to a system of phase oscillators that are coupled via an effective interaction in this framework we develop an approximate theory for sparse networks of identical neurons to estimate mc analytically from the fourier coefficients of our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it we apply this theory to compute mc for a model of inhibitory networks of integrate and fire i f neurons as a function of the intrinsic neuronal properties e g the refractory period tr the synaptic time constants and the strength of the external stimulus iext the number mc is found to be nonmonotonous with the strength of iext for tr 0 we estimate the minimum value of mc over all the parameters of the model to be 363 8 above mc the neurons tend to fire in smeared one cluster states at high firing rates and smeared two or more cluster states at low firing rates refractoriness decreases mc at intermediate and high firing rates these results are compared to numerical simulations we show numerically that systems with different sizes n behave in the same way provided the connectivity m is such that 1 meff 1 m 1 n remains constant when n varies this allows extrapolating the large n behavior of a network from numerical simulations of networks of relatively small sizes n 800 in our case we find that our theory predicts with remarkable accuracy the value of mc and the patterns of synchrony above mc provided the synaptic coupling is not too large we also study the strong coupling regime of inhibitory sparse networks all of our simulations demonstrate that increasing the coupling strength reduces the level of synchrony of the neuronal activity above a critical coupling strength the network activity is asynchronous we point out a fundamental limitation for the mechanisms of synchrony relying on inhibition alone if heterogeneities in the intrinsic properties of the neurons and spatial fluctuations in the external input are also taken into account

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

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

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

on a graded mesh method for a class of weakly singular volterra integral equations in this paper a class of weakly singular volterra integral equations with an infinite set of solutions is investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solution of this class of equations has been a difficult topic to analyze and has received much previous investigation the aim of this paper is to improve the convergence rates by a graded mesh method the convergence rates are proved and a variety of numerical examples are provided to support the theoretical findings

measuring shapes with desired convex polygons in this paper we have developed a family of shape measures all the measures from the family evaluate the degree to which a shape looks like a predefined convex polygon a quite new approach in designing object shape based measures has been applied in most cases such measures were defined by exploiting some shape properties such properties are optimized e g maximized or minimized by certain shapes and based on this the new shape measures were defined an illustrative example might be the shape circularity measure derived by exploiting the well known result that the circle has the largest area among all the shapes with the same perimeter of course there are many more such examples e g ellipticity linearity elongation and squareness measures are some of them there are different approaches as well in the approach applied here no desired property is needed and no optimizing shape has to be found we start from a desired convex polygon and develop the related shape measure the method also allows a tuning parameter thus there is a new 2 fold family of shape measures dependent on a predefined convex polygon and a tuning parameter that controls the measure s behavior the measures obtained range over the interval 0 1 and pick the maximal possible value equal to 1 if and only if the measured shape coincides with the selected convex polygon that was used to develop the particular measure all the measures are invariant with respect to translations rotations and scaling transformations an extension of the method leads to a family of new shape convexity measures

collocation and iterated collocation methods for a class of weakly singular volterra integral equations we discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular volterra integral equation associated with certain heat conduction problems this work completes the previous studies of numerical methods for this type of equations with noncompact kernel in particular a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces some numerical examples illustrate the theoretical results

rigorous convergence analysis of jacobi spectral galerkin methods for volterra integral equations with noncompact kernels abstract this article deals with the implementation of jacobi spectral and pseudospectral galerkin approaches for solving weakly singular volterra integral equations with noncompact kernels the computational domain of the main problem and intervals of the involved integrals are transformed into 1 1 by some appropriate change of variables therefore one can apply the jacobi galerkin method and impose the inner product conditions on the residual of the considered equation also jacobi gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the jacobi weight function in the pseudospectral galerkin method for both cases of spectral and pseudospectral galerkin schemes rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in l and weighted l 2 norms several numerical examples are also given to confirm the theoretical prediction

prior learning and gibbs reaction diffusion this article addresses two important themes in early visual computation it presents a novel theory for learning the universal statistics of natural images and it proposes a general framework of designing reaction diffusion equations for image processing we studied the statistics of natural images including the scale invariant properties then generic prior models were learned to duplicate the observed statistics based on minimax entropy theory the resulting gibbs distributions have potentials of the form u i spl lambda s spl sigma sub spl alpha 1 sup k spl sigma sub x y spl lambda sup spl alpha f sup spl alpha i x y with s f sup 1 f sup 2 f sup k being a set of filters and spl lambda spl lambda sup 1 spl lambda sup 2 spl lambda sup k the potential functions the learned gibbs distributions confirm and improve the form of existing prior models such as line process but in contrast to all previous models inverted potentials were found to be necessary we find that the partial differential equations given by gradient descent on u i spl lambda s are essentially reaction diffusion equations where the usual energy terms produce anisotropic diffusion while the inverted energy terms produce reaction associated with pattern formation enhancing preferred image features we illustrate how these models can be used for texture pattern rendering denoising image enhancement and clutter removal by careful choice of both prior and data models of this type incorporating the appropriate features

a new method for solving a class of singular two point boundary value problems abstract in this paper we present a new method for solving a class of singular two point boundary value problems for certain ordinary differential equations having singular coefficients the exact solution u x is represented in the form of series in the reproducing kernel space in the mean time the n term approximate solution u n x is obtained by solving a linear system of equations and is proved to converge to the exact solution this method can avoid the gram schmidt orthogonalization process so as to improve the precision and decrease consumedly the runtime when the number of knots is the same especially when the number of knots is large it is efficiently applied to solve some model problems with the comparison between the numerical solutions and the exact solutions the high precision of this method is demonstrated by the numerical examples

piecewise approximation of contours through scale space selection of dominant points this paper describes a method of approximating a shape contour with a polygon the polygon vertices are extracted from the curvature extremes through a scale space description of the contour via linear diffusion these vertices are located on the contour points where the sharper changes of the contour directions occur using a proper strategy a set of extremes that result in a given approximation level is chosen by adding new vertices the approximation level can be improved and a scalable representation of the contour is identified this method results in an approximation that discriminates local from global geometric features and provides a good visual representation of the original contour this polygonal approximation method is used for scalable encoding of the shape contours in this regard an encoding technique suitable for scalable polygonal approximation has been developed we show that encoding the approximated polygons result in a good relation between the distortion and the bitrate finally we show that in addition to coding this method can be efficiently used for shape comparison and shape retrieval

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

the elastic ratio introducing curvature into ratio based globally optimal image segmentation in this paper we present the first globally optimal ratio based image segmentation method allowing to impose curvature regularity of the region boundary the proposed method is fully unsupervised and compares favorably to other such approaches to identify the optimal foreground region in the image the algorithm minimizes the ratio of flux over a weighted sum of length and curvature regularity of the region boundary the key concept is to find cycles in a product graph where each node corresponds to a pair of image locations furthermore our results allow to draw conclusions about certain global optima of a reformulated snakes functional which is independent of parameterization the proposed algorithm allows to find parameter sets where the modified snakes functional has a meaningful solution and simultaneously provides the corresponding global solution

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

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

asymptotic behavior and synchronizability characteristics of a class of recurrent neural networks we propose an approach to the analysis of the influence of the topology of a neural network on its synchronizability in the sense of equal output activity rates given by a particular neural network model the model we introduce is a variation of the zhang model we investigate the time asymptotic behavior of the corresponding dynamical system in particular the conditions for the existence of an invariant compact asymptotic set and apply the results of the synchronizability analysis on a class of random scale free networks and to the classical random networks with poisson connectivity distribution

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

the elastic ratio introducing curvature into ratio based image segmentation we present the first ratio based image segmentation method that allows imposing curvature regularity of the region boundary our approach is a generalization of the ratio framework pioneered by jermyn and ishikawa so as to allow penalty functions that take into account the local curvature of the curve the key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment among ratios whose discrete counterparts can be globally minimized with our approach we focus in particular on the elastic ratio 0l c i c s c s ds vl c 0l c c s qds that depends given an image i on the oriented boundary c of the segmented region candidate minimizing this ratio amounts to finding a curve neither small nor too curvy through which the brightness flux is maximal we prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions we also prove that the discrete minimizers provided by our graph based algorithm converge as the resolution increases to continuous minimizers in contrast to most existing segmentation methods with computable and meaningful i e nondegenerate global optima the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations furthermore our results allow drawing conclusions about global optima of a parameterization independent version of the snakes functional the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution

a multigrid homotopy method for nonlinear inverse problems abstract in the present contribution we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems in which the forward problems are modeled by some forms of partial differential equations the method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest and then carries out the inversion on the coarsest grid by the homotopy method the strategy may give a rapidly and globally convergent method as a practical application this method is used to solve the nonlinear inverse problem of a nonlinear convection diffusion equation which is the saturation equation within the two phase porous media flow we demonstrate the effectiveness and merits of the multigrid homotopy method on two actual model problems

the exact solution of a class of volterra integral equation with weakly singular kernel abstract in this paper the weakly singular volterra integral equations with an infinite set of solutions are investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation the aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory applying weighted integral we provide a new definition for reproducing kernel space and obtain reproducing kernel function using the good properties of reproducing kernel function the only smooth solution is exactly expressed in the form of series the n term approximate solution is obtained by truncating the series meanwhile we prove that the derivative of approximation converges to the derivative of exact solution uniformly the final numerical examples compared with other methods show that the method is efficient

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

an efficiently computable metric for comparing polygonal shapes a method for comparing polygons that is a metric invariant under translation rotation and change of scale reasonably easy to compute and intuitive is presented the method is based on the l sub 2 distance between the turning functions of the two polygons it works for both convex and nonconvex polygons and runs in time o mn log mn where m is the number of vertices in one polygon and n is the number of vertices in the other some examples showing that the method produces answers that are intuitively reasonable are presented

fast calculation of multiple line integrals a line integral is defined as the integral of two dimensional data along a one dimensional straight line of given length and orientation line integrals are used in various forms of edge and line detectors in images and in the computation of the radon transform we present a recursive algorithm which enables approximation of discretized line integrals at all lengths orientations and locations to within a prescribed error bound in at most o n log n log log n operations where n is the number of data points furthermore for most applications in particular where even small amounts of noise are present in the data all of these integrals can be computed to the desired accuracy in about 24n log n operations

robust and efficient detection of salient convex groups this paper describes an algorithm that robustly locates salient convex collections of line segments in an image the algorithm is guaranteed to find all convex sets of line segments in which the length of the gaps between segments is smaller than some fixed proportion of the total length of the lines this enables the algorithm to find convex groups whose contours are partially occluded or missing due to noise we give an expected case analysis of the algorithm performance this demonstrates that salient convexity is unlikely to occur at random and hence is a strong clue that grouped line segments reflect underlying structure in the scene we also show that our algorithm run time is o n sup 2 log n nm when we wish to find the m most salient groups in an image with n line segments we support this analysis with experiments on real data and demonstrate the grouping system as part of a complete recognition system

synergy between object recognition and image segmentation using the expectation maximization algorithm in this work we formulate the interaction between image segmentation and object recognition in the framework of the expectation maximization em algorithm we consider segmentation as the assignment of image observations to object hypotheses and phrase it as the e step while the m step amounts to fitting the object models to the observations these two tasks are performed iteratively thereby simultaneously segmenting an image and reconstructing it in terms of objects we model objects using active appearance models aams as they capture both shape and appearance variation during the e step the fidelity of the aam predictions to the image is used to decide about assigning observations to the object for this we propose two top down segmentation algorithms the first starts with an oversegmentation of the image and then softly assigns image segments to objects as in the common setting of em the second uses curve evolution to minimize a criterion derived from the variational interpretation of em and introduces aams as shape priors for the m step we derive aam fitting equations that accommodate segmentation information thereby allowing for the automated treatment of occlusions apart from top down segmentation results we provide systematic experiments on object detection that validate the merits of our joint segmentation and recognition approach

a wavelet multiscale homotopy method for the parameter identification problem of partial differential equations by introducing the wavelet multiscale method and the homotopy method to the inversion process for the parameter identification problem of partial differential equations a joint inversion method called the wavelet multiscale homotopy method is proposed which is globally convergent computationally efficient and has the anti noise ability as a practical problem the parameter identification problem of the saturation equation in the fractional flow formulation of the two phase porous media flow equations is solved numerical results validate the method s merits

convergence analysis of spectral collocation methods for a class of weakly singular volterra integral equations abstract in this paper we discuss the application of spectral jacobi collocation methods to a certain class of weakly singular volterra integral equations first we use some function transformations and variable changes to transform the equation into a volterra integral equation defined on the standard interval 1 1 then the jacobi gauss quadrature formula is used to approximate the integral operator for the spectral jacobi collocation method a rigorous error analysis in both the l and weighted l 2 norms is given under the assumption that both the kernel function and the source function are sufficiently smooth finally some numerical examples are provided to illustrate the theoretical results

determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method this paper deals with the nonlinear oscillations of a particle which moves on a rotating parabola an analytic approximate technique namely optimal homotopy asymptotic method oham is employed to propose an analytic approach to solve nonlinear oscillations the validity of the oham is independent on whether or not there exist small or large parameters in the considered nonlinear equations our procedure provides us with a convenient way to optimally control the convergence of the approximate solutions an example is given and the results reveal that this procedure is very effective simple and accurate this paper demonstrates the general validity and the great potential of the oham

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

optimal homotopy asymptotic method for solving nonlinear fredholm integral equations of second kind abstract the aim of this communication is to present the effectiveness of optimal homotopy asymptotic method oham for solving nonlinear fredholm integral equations of second kind here we consider the first second and third order fredholm integral equations of second kind comparisons are made with adomian decomposition method discrete adomian decomposition method homotopy perturbation method quasi interpolation method triangular functions method trapezoidal rule and simpson rule the results reveal that the oham is very effective simple and has close agreement with exact solutions

partial shape recognition by sub matrix matching for partial matching guided image labeling we propose a new partial shape recognition algorithm by sub matrix matching using a proximity based shape representation given one or more example object templates and a number of candidate object regions in an image points with local maximum curvature along contours of each are chosen as feature points to compute distance matrices for each candidate object region and example template s a sub matrix matching algorithm is then proposed to determine correspondences for evaluation of partial similarity between an example template and a candidate object region the method is translation rotation scale and reflection invariant applications of the proposed partial matching technique include recognition of partially occluded objects in images as well as significant acceleration of recognition matching of full non occluded objects for object based image labeling by learning from examples the speed up in the latter application comes from the fact that we can now search only those combinations of regions in the neighborhood of potential partial matches as soon as they are identified as opposed to all combinations of regions as was done in our prior work xu et al object formation and retrieval using a learning based hierarchical content description proceedings of the icip kobe japan 1999 experimental results are provided to demonstrate both applications

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

lyapunov exponents and transport in the zhang model of self organized criticality we discuss the role played by lyapunov exponents in the dynamics of zhang s model of self organized criticality we show that a large part of the spectrum the slowest modes is associated with energy transport in the lattice in particular we give bounds on the first negative lyapunov exponent in terms of the energy flux dissipated at the boundaries per unit of time we then establish an explicit formula for the transport modes that appear as diffusion modes in a landscape where the metric is given by the density of active sites we use a finite size scaling ansatz for the lyapunov spectrum and relate the scaling exponent to the scaling of quantities such as avalanche size duration density of active sites etc

radial projection an efficient update rule for relaxation labeling relaxation labeling uses contextual information for finding consistent labelings of graphs although relaxation labeling is parallel and iterative the complexity of updating with standard rules is too costly for practical implementation a description is given of a computationally more efficient updating rule that utilizes radial projection instead of normal projection to avoid the complexities incurred by previous update rules when boundaries to the labeling space are encountered this reduction in complexity is achieved by first restricting support vectors to the positive quadrant and then using radial projection onto the constraint instead of normal projection crucial order information is conserved through smooth convergence towards the optimum and a rate of convergence proportional to the magnitudes of the support functions

dynamical properties of the zhang model of self organized criticality critical exponents of the infinitely slowly driven zhang model of self organized criticality are computed for d52 and 3 with particular emphasis devoted to the various roughening exponents besides confirming recent estimates of some exponents new quantities are monitored and their critical exponents computed among other results it is shown that the three dimensional exponents do not coincide with the bak tang wiesenfeld phys rev lett 59 381 1987 phys rev a 38 364 1988 abelian model and that the dynamical exponent as computed from the correlation length and from the roughness of the energy profile do not necessarily coincide as is usually implicitly assumed an explanation for this is provided the possibility of comparing these results with those obtained from renormalization group arguments is also briefly addressed s1063 651x 98 06107 8 pacs number s 64 60 lx

solving the multiple instance problem with axis parallel rectangles the multiple instance problem arises in tasks where the training examples are ambiguous a single example object may have many alternative feature vectors instances that describe it and yet only one of those feature vectors may be responsible for the observed classification of the object this paper describes and compares three kinds of algorithms that learn axis parallel rectangles to solve the multiple instance problem algorithms that ignore the multiple instance problem perform very poorly an algorithm that directly confronts the multiple instance problem by attempting to identify which feature vectors are responsible for the observed classifications performs best giving 89 correct predictions on a musk odor prediction task the paper also illustrates the use of artificial data to debug and compare these algorithms

graphical templates for model registration a new method of model registration is proposed using graphical templates a graph of landmarks is chosen in the template image all possible candidates for these landmarks are found in the data image using local operators a dynamic programming algorithm on decomposable subgraphs of the template graph finds the optimal match to a subset of the candidate points in polynomial time this combination of local operators to describe points of interest landmarks and a graph to describe their geometric orientation in the plane yields fast and precise matches of the model to the data with no initialization required

numerical solutions of weakly singular volterra integral equations using the optimal homotopy asymptotic method the purpose of this paper is to apply a numerical technique namely the optimal homotopy asymptotic method oham for finding the approximate solutions of a class of volterra integral equations with weakly singular kernels this method uses simple computations with quite acceptable approximate solutions which has close agreement with exact solutions illustrative examples are included to demonstrate the validity and applicability of the present method and a comparison has been made with existing results

waveguide a joint wavelet based image representation and description system data representation and content description are two basic components required by the management of any image database a wavelet based system called the waveguide which integrates these two components in a unified framework is proposed in this work in the waveguide system images are compressed with the state of the art wavelet coding technique and indexed with color texture and object shape descriptors generated in the wavelet domain during the encoding process all the content descriptors are extracted by machines automatically with a low computational complexity and stored with a low memory space extensive experiments are performed to demonstrate the performance of the new approach

what can one learn about self organized criticality from dynamical systems theory we develop a dynamical system approach for the zhang model of self organized criticality for which the dynamics can be described either in terms of iterated function systems or as a piecewise hyperbolic dynamical system of skew product type in this setting we describe the soc attractor and discuss its fractal structure we show how the lyapunov exponents the haussdorf dimensions and the system size are related to the probability distribution of the avalanche size via the ledrappier young formula

numerical solutions of volterra integral equation with weakly singular kernel using scw method numerical methods for weakly singular volterra integral equations are rarely considered in the literature the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration and bring some difficulties in numerical computation in this paper we present a numerical solution of weakly singular volterra integral equations including the abels equations by the second chebyshev wavelet scw we give the scw operational matrix of fractional integration and combine with the block pulse functions bpfs to derive the procedure of solving this kind integral equations the proposed method is illustrated with numerical examples the results reveal that the method is accurate and easy to implement

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

synchronization in networks of excitatory and inhibitory neurons with sparse random connectivity in model networks of e cells and i cells excitatory and inhibitory neurons respectively synchronous rhythmic spiking often comes about from the interplay between the two cell groups the e cells synchronize the i cells and vice versa under ideal conditions homogeneity in relevant network parameters and all to all connectivity for instance this mechanism can yield perfect synchronization we find that approximate imperfect synchronization is possible even with very sparse random connectivity the crucial quantity is the expected number of inputs per cell as long as it is large enough more precisely as long as the variance of the total number of synaptic inputs per cell is small enough tight synchronization is possible the desynchronizing effect of random connectivity can be reduced by strengthening the e i synapses more surprising it cannot be reduced by strengthening the i e synapses however the decay time constant of inhibition plays an important role faster decay yields tighter synchrony in particular in models in which the inhibitory synapses are assumed to be instantaneous the effects of sparse random connectivity cannot be seen

local grayvalue invariants for image retrieval this paper addresses the problem of retrieving images from large image databases the method is based on local grayvalue invariants which are computed at automatically detected interest points a voting algorithm and semilocal constraints make retrieval possible indexing allows for efficient retrieval from a database of more than 1 000 images experimental results show correct retrieval in the case of partial visibility similarity transformations extraneous features and small perspective deformations

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

solving a nonlinear system of second order boundary value problems in this paper a method is presented to obtain the analytical and approximate solutions of linear and nonlinear systems of second order boundary value problems the analytical solution is represented in the form of series in the reproducing kernel space in the mean time the approximate solution un x is obtained by the n term intercept of the analytical solution and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

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

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

filters random fields and maximum entropy frame towards a unified theory for texture modeling this article presents a statistical theory for texture modeling this theory combines filtering theory and markov random field modeling through the maximum entropy principle and interprets and clarifies many previous concepts and methods for texture analysis and synthesis from a unified point of view our theory characterizes the ensemble of images i with the same texture appearance by a probability distribution f i on a random field and the objective of texture modeling is to make inference about f i given a set of observed texture examples in our theory texture modeling consists of two steps 1 a set of filters is selected from a general filter bank to capture features of the texture these filters are applied to observed texture images and the histograms of the filtered images are extracted these histograms are estimates of the marginal distributions of f i this step is called feature extraction 2 the maximum entropy principle is employed to derive a distribution p i which is restricted to have the same marginal distributions as those in 1 this p i is considered as an estimate of f i this step is called feature fusion a stepwise algorithm is proposed to choose filters from a general filter bank the resulting model called frame filters random fields and maximum entropy is a markov random field mrf model but with a much enriched vocabulary and hence much stronger descriptive ability than the previous mrf models used for texture modeling gibbs sampler is adopted to synthesize texture images by drawing typical samples from p i thus the model is verified by seeing whether the synthesized texture images have similar visual appearances to the texture images being modeled experiments on a variety of 1d and 2d textures are described to illustrate our theory and to show the performance of our algorithms these experiments demonstrate that many textures which are previously considered as from different categories can be modeled and synthesized in a common framework

local symmetry detection in natural images using a particle filtering approach in this paper we propose an algorithm to detect smooth local symmetries and contours of ribbon like objects in natural images the detection is formulated as a spatial tracking task using a particle filtering approach extracting one part of a structure at a time using an adaptive local geometric model the method can detect straight reflection symmetries in perfectly symmetrical objects as well as smooth local symmetries in curved elongated objects in addition the proposed approach jointly estimates spine and contours making it possible to generate back ribbon objects experiments for local symmetry detection have been conducted on a recent extension of the berkeley segmentation data sets we also show that it is possible to retrieve specific geometrical objects using intuitive prior structural information

accurate analytical solutions to oscillators with discontinuities and fractional power restoring force by means of the optimal homotopy asymptotic method in this paper a new approach combining the features of the homotopy concept with an efficient computational algorithm which provides a simple and rigorous procedure to control the convergence of the solution is proposed to find accurate analytical explicit solutions for some oscillators with discontinuities and a fractional power restoring force which is proportional to sign x a very fast convergence to the exact solution was proved since the second order approximation lead to very accurate results comparisons with numerical results are presented to show the effectiveness of this method four numerical applications prove the accuracy of the method which works very well for the whole range of initial amplitudes the obtained results prove the validity and efficiency of the method which can be easily extended to other strongly nonlinear problems

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

a nystrom interpolant for some weakly singular nonlinear volterra integral equations we consider a second kind weakly singular nonlinear volterra hammerstein integral equation defined by a compact operator and derive a nystrom type interpolant of the solution based on gauss radau nodes we prove the convergence of the interpolant and derive convergence estimates for equations with nonlinearity of algebraic kind we improve the rate of convergence by using a smoothing transformation some numerical examples are given

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

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

shape matching by random sampling in order to determine the similarity between two planar shapes which is an important problem in computer vision and pattern recognition it is necessary to first match the two shapes as well as possible as sets of allowed transformation to match shapes we consider translations rigid motions and similarities we present a generic probabilistic algorithm based on random sampling for matching shapes which are modelled by sets of curves the algorithm is applicable to the three considered classes of transformations we analyze which similarity measure is optimized by the algorithm and give rigorous bounds on the number of samples necessary to get a prespecified approximation to the optimal match within a prespecified probability

similarity measures for convex polyhedra based on minkowski addition abstract in this paper we introduce and investigate similarity measures for convex polyhedra based on minkowski addition and inequalities for the mixed volume and volume related to the brunn minkowski theory all measures considered are invariant under translations furthermore some of them are also invariant under subgroups of the affine transformation group for the case of rotation and scale invariance we prove that to obtain the measures based on mixed volume it is sufficient to compute certain functionals only for a finite number of critical rotations the paper presents a theoretical framework for comparing convex shapes and contains a complexity analysis of the solution numerical implementations of the proposed approach are not discussed

superconvergence of collocation methods for a class of weakly singular volterra integral equations we discuss the application of spline collocation methods to a certain class of weakly singular volterra integral equations it will be shown that by a special choice of the collocation parameters superconvergence properties can be obtained if the exact solution satisfies certain conditions this is in contrast with the theory of collocation methods for abel type equations several numerical examples are given which illustrate the theoretical results