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

diffeomorphisms groups and pattern matching in image analysis in a previous paper it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group we show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images using lie group ideas we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub optimal gradient pattern matching its links with fluid models are established

computable elastic distances between shapes we dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other the energy is formally dened from a left invariant riemannian distance on an innite dimensional group acting on the curves which can be explicitly computed the obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed an analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem which can eciently be implemented as illustrated by experiments 1 1 generalities the problem of matching two objects together is very impor tant in computer vision and shape recognition in many cases recognition is based on shapes outlines with the help of some suitably designed distance a general principle is to associate with any pair o1 o 2 of objects to be compared a measure of discrepancy d o1 o 2 the recognition of an observed object o may be done by nding from a dictionary of templates the previously recorded object otemp for which d o otemp is minimal clearly the denition of the distance is the crucial step of the method and much research has been done in this direction we shall not try here to provide a review of the huge literature existing on the subject see for example 17 but rather focus on methods related to deformable templates with which we are directly concerned instead of basing recognition on a nite collection of points of interest primitives taken from the outline of an object corners inflexion points etc which is a popular way of handling the problem our purpose is to base the comparison on the whole outline considered as a plane curve the distance we shall dene incorporates some deformation energy between the curves the approach as we will see turns out to be intrinsic and robust to usual euclidean transformations the method is related to the wide literature on snakes 14 7 21 etc in the way that our distance corresponds to some continuous process of deformation of one curve into another it is also related to papers on elastic matching such as 8 however we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes taking into account possible invariance to scaling or euclidean transformations in the case they are required from this point of view our results are indebted to the seminal work of grenander on group theory applied to pattern recognition cf 10 and 11 in particular see also 1 2 12 another source of inspiration may come from mathematical physics since we are going to look to the path process of lowest energy which deforms one object into

a continuum mechanical approach to geodesics in shape space in this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes where shapes are implicitly described as boundary contours of objects the proposed shape metric is derived from a continuum mechanical notion of viscous dissipation a geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized the approach can easily be generalized to shapes given as segment contours of multi labeled images and to geodesic paths between partially occluded objects the proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems which is strictly invariant with respect to rigid body motions and ensures a 1 1 correspondence along the induced flow in shape space when decreasing the time step size the proposed model leads to the minimization of the actual geodesic length where the hessian of the pairwise matching energy reflects the chosen riemannian metric on the underlying shape space if the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field the method is implemented via a level set representation of shapes and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations the numerical relaxation of the energy is performed via an efficient multi scale procedure in space and time various examples for 2d and 3d shapes underline the effectiveness and robustness of the proposed approach

exploiting information geometry to improve the convergence of nonparametric active contours this paper presents a fast converging riemannian steepest descent method for nonparametric statistical active contour models with application to image segmentation unlike other fast algorithms the proposed method is general and can be applied to any statistical active contour model from the exponential family which comprises most of the models considered in the literature this is achieved by first identifying the intrinsic statistical manifold associated with this class of active contours and then constructing a steepest descent on that manifold a key contribution of this paper is to derive a general and tractable closed form analytic expression for the manifold s riemannian metric tensor which allows computing discrete gradient flows efficiently the proposed methodology is demonstrated empirically and compared with other state of the art approaches on several standard test images a phantom positron emission tomography scan and a b mode echography of in vivo human dermis

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

coarse to fine segmentation and tracking using sobolev active contours recently proposed sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining gradients associated to these energies sobolev active contours evolve more globally and are less attracted to certain intermediate local minima than traditional active contours and it is based on a well structured riemannian metric which is important for shape analysis and shape priors in this paper we analyze sobolev active contours using scale space analysis in order to understand their evolution across different scales this analysis shows an extremely important and useful behavior of sobolev contours namely that they move successively from coarse to increasingly finer scale motions in a continuous manner this property illustrates that one justification for using the sobolev technique is for applications where coarse scale deformations are preferred over fine scale deformations along with other properties to be discussed the coarse to fine observation reveals that sobolev active contours are in particular ideally suited for tracking algorithms that use active contours we will also justify our assertion that the sobolev metric should be used over the traditional metric for active contours in tracking problems by experimentally showing how a variety of active contour based tracking methods can be significantly improved merely by evolving the active contour according to the sobolev method

gradient flows for optimizing triangular mesh based surfaces applications to 3d reconstruction problems dealing with visibility this article tackles the problem of using variational methods for evolving 3d deformable surfaces we give an overview of gradient descent flows when the shape is represented by a triangular mesh based surface and we detail the gradients of two generic energy functionals which embody a number of energies used in mesh processing and computer vision in particular we show how to rigorously account for visibility in the surface optimization process we present different applications including 3d reconstruction from multiple views for which the visibility is fundamental the gradient correctly takes into account the visibility changes that occur when a surface moves this forces the contours generated by the reconstructed surface to match with the apparent contours in the input images

variational b spline level set a linear filtering approach for fast deformable model evolution in the field of image segmentation most level set based active contour approaches take advantage of a discrete representation of the associated implicit function we present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expressed on a b spline basis starting from the active contour energy functional we show that this formulation allows us to compute the solution as a restriction of the variational problem on the space spanned by the b splines as a consequence the minimization of the functional is directly obtained in terms of the b spline coefficients we also show that each step of this minimization may be expressed through a convolution operation because the b spline functions are separable this convolution may in turn be performed as a sequence of simple 1 d convolutions which yields an efficient algorithm as a further consequence each step of the level set evolution may be interpreted as a filtering operation with a b spline kernel such filtering induces an intrinsic smoothing in the algorithm which can be controlled explicitly via the degree and the scale of the chosen b spline kernel we illustrate the behavior of this approach on simulated as well as experimental images from various fields

a 2 d coupled bem fem simulation of electro elastostatics at large strain the numerical simulation of nonlinear electro elastostatics is considered in this work using the coupled boundary and finite element method the objective of the work is to properly simulate the deformation of an electroelastic body undergoing large deformation subjected to electric stimulations in the case where the surrounding space has significant influence on the electric field inside the body finite elements are used to model the nonlinear electroelastic body in which both mechanical nonlinearity and electrical nonlinearity are taken into account boundary elements are used to model the surrounding space and account for the large deformation of the boundary of the body

stabilization of parametric active contours using a tangential redistribution term depending on implementation active contours have been classified as geometric or parametric active contours parametric contours irrespective of representation are known to suffer from the problem of irregular bunching and spacing out of curve points during the curve evolution in a spline based implementation of active contours this leads to occasional formation of loops locally and subsequently the curve blows up due to instabilities in this paper we analyze the reason for this problem and propose a solution to alleviate the same we propose an ordinary differential equation ode for controlling the curve parametrization during evolution by including a tangential force we show that the solution of the proposed ode is bounded we demonstrate the effectiveness of the proposed method for segmentation and tracking tasks on closed as well as open contours

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

on a new class of electro elastic bodies ii boundary value problems in part i of this two part paper a new theoretical framework was presented to describe the response of electro elastic bodies the constitutive theory that was developed consists of two implicit c

a subexponential time quantum algorithm for the dihedral hidden subgroup problem we present a quantum algorithm for the dihedral hidden subgroup problem dhsp with time and query complexity 2 o sqrt log n in this problem an oracle computes a function f on the dihedral group d n which is invariant under a hidden reflection in d n by contrast the classical query complexity of dhsp is o sqrt n the algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group the algorithm begins as usual with a quantum character transform which in the case of d n is essentially the abelian quantum fourier transform this yields the name of a group representation of d n which is not by itself useful and a state in the representation which is a valuable but indecipherable qubit the algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of d n once the algorithm obtains certain target representations direct measurements reveal the hidden subgroup

deformotion deforming motion shape average and the joint registration and approximation of structures in images what does it mean for a deforming object to be moving how can we separate the overall motion a finite dimensional group action from the more general deformation a diffeomorphism in this paper we propose a definition of motion for a deforming object and introduce a notion of shape average as the entity that separates the motion from the deformation our definition allows us to derive novel and efficient algorithms to register non identical shapes using region based methods and to simultaneously approximate and align structures in greyscale images we also extend the notion of shape average to that of a moving average in order to track moving and deforming objects through time the algorithms we propose extend prior work on landmark based matching to smooth curves and involve the numerical integration of partial differential equations which we address within the framework of level set methods

joint segmentation and shape regularization with a generalized forward backward algorithm this paper presents a method for the simultaneous segmentation and regularization of a series of shapes from a corresponding sequence of images such series arise as time series of 2d images when considering video data or as stacks of 2d images obtained by slicewise tomographic reconstruction we first derive a model where the regularization of the shape signal is achieved by a total variation prior on the shape manifold the method employs a modified kendall shape space to facilitate explicit computations together with the concept of sobolev gradients for the proposed model we derive an efficient and computationally accessible splitting scheme using a generalized forward backward approach our algorithm treats the total variation atoms of the splitting via proximal mappings whereas the data terms are dealt with by gradient descent the potential of the proposed method is demonstrated on various application examples dealing with 3d data we explain how to extend the proposed combined approach to shape fields which for instance arise in the context of 3d t imaging modalities and show an application in this setup as well

efficient shape priors for spline based snakes parametric active contours are an attractive approach for image segmentation thanks to their computational efficiency they are driven by application dependent energies that reflect the prior knowledge on the object to be segmented we propose an energy involving shape priors acting in a regularization like manner thereby the shape of the snake is orthogonally projected onto the space that spans the affine transformations of a given shape prior the formulation of the curves is continuous which provides computational benefits when compared with landmark based discrete methods we show that this approach improves the robustness and quality of spline based segmentation algorithms while its computational overhead is negligible an interactive and ready to use implementation of the proposed algorithm is available and was successfully tested on real data in order to segment drosophila flies and yeast cells in microscopic images

numerical solution of finite geometry boundary value problems in nonlinear magnetoelasticity this paper provides examples of the numerical solution of boundary value problems in nonlinear magnetoelasticity involving finite geometry based on the theoretical framework developed by dorfmann and co workers specifically using a prototype constitutive model for isotropic magnetoelasticity we consider two two dimensional problems for a block with rectangular cross section and of infinite extent in the third direction in the first problem the deformation induced in the block by the application of a uniform magnetic field far from the block and normal to its larger faces without mechanical load is examined while in the second problem the same magnetic field is applied in conjunction with a shearing deformation produced by in plane shear stresses on its larger faces for each problem the distribution of the magnetic field throughout the block and the surrounding space is illustrated graphically along with the corresponding deformation of the block the rapidly in space changing magnitude of the magnetic field in the neighbourhood of the faces of the block is highlighted

a clique based exact method for optimal winner determination in combinatorial auctions winner determination wdp is a key problem in combinatorial auctions with many applications we develop a clique based branch and bound algorithm for the wdp we show extensive computational results on three test suites of popular wdp benchmark instances we compare the performance with the general cplex 12 4 solver we demonstrate both approaches complement each other given a set of items to sell and a set of combinatorial bids the winner determination problem wdp in combinatorial auctions is to determine an allocation of items to bidders such that the auctioneer s revenue is maximized while each item is allocated to at most one bidder wdp is at the core of numerous relevant applications in multi agent systems e commerce and many others we develop a clique based branch and bound approach for wdp which relies on a transformation of wdp into the maximum weight clique problem to ensure the efficiency of the proposed search algorithm we introduce specific bounding and branching strategies using a dedicated vertex coloring procedure and a specific vertex sorting technique we assess the performance of the proposed algorithm on a large collection of benchmark instances in comparison with the cplex 12 4 solver and other approaches computational results show that this clique based method constitutes a valuable and complementary approach for wdp relative to the existing methods

on the empirical scaling of run time for finding optimal solutions to the travelling salesman problem abstract the travelling salesman problem tsp is one of the most prominent np hard combinatorial optimisation problems after over fifty years of intense study the tsp continues to be of broad theoretical and practical interest using a novel approach to empirical scaling analysis which in principle is applicable to solvers for many other problems we demonstrate that some of the most widely studied types of tsp instances tend to be much easier than expected from previous theoretical and empirical results in particular we show that the empirical median run time required for finding optimal solutions to so called random uniform euclidean rue instances one of the most widely studied classes of tsp instances scales substantially better than 2 n with the number n of cities to be visited the concorde solver for which we achieved this result is the best performing exact tsp solver we are aware of and has been applied to a broad range of real world problems furthermore we show that even when applied to a broad range of instances from the prominent tsplib benchmark collection for the tsp concorde exhibits run times that are surprisingly consistent with our empirical model of concorde s scaling behaviour on rue instances this result suggests that the behaviour observed for the simple random structure underlying rue is very similar to that obtained on the structured instances arising in various applications

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

enhanced discrete bacterial memetic evolutionary algorithm an efficacious metaheuristic for the traveling salesman optimization abstract in this paper we present a novel universal metaheuristic discrete bacterial memetic evolutionary algorithm dbmea which is based on the combination of the bacterial evolutionary algorithm and local search techniques used for solving np hard optimization problems the algorithm was tested on a series of symmetric traveling salesman problems tsp and traveling salesman problem with time windows tsptw benchmarks the size of the symmetric tsp benchmarks went up to 5 000 cities in all cases the dbmea algorithm produced optimal or near optimal solutions and the difference from the known best values was within 0 16 while for large size problems it was much faster than the concorde solver it was found to be slower compared to the helsgaun lin kernighan heuristic which is the most efficient tsp solver method with some slight modifications the same algorithm was also tested on tsp with time windows tsptw benchmark instances in most cases the dbmea procedure found the known best solutions and it was again the second fastest method compared with the state of the art techniques for the tsptw dbmea is called efficacious because it is a universal method it can be efficiently applied to various np hard optimization problems and as in all cases it results in the optimal or a very near optimal solutions while its runtime is very predictable in terms of the size of the problem and the topology of the instance does not affect its runtime significantly even though heuristics developed for a particular type of problem might perform better for that restricted class our novel method proposed here is universally applicable and may be deployed successfully for optimizing other discreet np hard graph search and optimization problems as well

analysis of two dimensional non rigid shapes analysis of deformable two dimensional shapes is an important problem encountered in numerous pattern recognition computer vision and computer graphics applications in this paper we address three major problems in the analysis of non rigid shapes similarity partial similarity and correspondence we present an axiomatic construction of similarity criteria for deformation invariant shape comparison based on intrinsic geometric properties of the shapes and show that such criteria are related to the gromov hausdorff distance next we extend the problem of similarity computation to shapes which have similar parts but are dissimilar when considered as a whole and present a construction of set valued distances based on the notion of pareto optimality finally we show that the correspondence between non rigid shapes can be obtained as a byproduct of the non rigid similarity problem as a numerical framework we use the generalized multidimensional scaling gmds method which is the numerical core of the three problems addressed in this paper

the tau method and a new preconditioner in relations to the order of linear ordinary differential equations using a modified form of the chebyshev or legendre and gegenbauer polynomials some particular integral operators are introduced these are used to give a factorization of the operators arising from the application of the chebyshev or legendre tau method the new tau method presented in this article is then compared with the standard tau method and preconditioned method of cabos the new tau method shows a superior performance an analysis of error and a bound for condition number is given numerical examples applying iterative solvers show dramatic reduction in condition number and improved convergence for the tau method with the new preconditioner

generalized gradients priors on minimization flows this paper tackles an important aspect of the variational problem underlying active contours optimization by gradient flows classically the definition of a gradient depends directly on the choice of an inner product structure this consideration is largely absent from the active contours literature most authors explicitely or implicitely assume that the space of admissible deformations is ruled by the canonical l 2 inner product the classical gradient flows reported in the literature are relative to this particular choice here we investigate the relevance of using i other inner products yielding other gradient descents and ii other minimizing flows not deriving from any inner product in particular we show how to induce different degrees of spatial consistency into the minimizing flow in order to decrease the probability of getting trapped into irrelevant local minima we report numerical experiments indicating that the sensitivity of the active contours method to initial conditions which seriously limits its applicability and efficiency is alleviated by our application specific spatially coherent minimizing flows we show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors

variational principles surface evolution pdes level set methods and the stereo problem we present a novel geometric approach for solving the stereo problem for an arbitrary number of images spl ges 2 it is based upon the definition of a variational principle that must be satisfied by the surfaces of the objects in the scene and their images the euler lagrange equations that are deduced from the variational principle provide a set of partial differential equations pdes that are used to deform an initial set of surfaces which then move toward the objects to be detected the level set implementation of these pdes potentially provides an efficient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformations i e to deal with multiple objects results of an implementation of our theory also dealing with occlusion and visibility are presented on synthetic and real images

new possibilities with sobolev active contours recently the sobolev metric was introduced to define gradient flows of various geometric active contour energies it was shown that the sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important some interesting properties of sobolev gradient flows include that they stabilize certain unstable traditional flows and the order of the evolution pdes are reduced when compared with traditional gradient flows of the same energies in this paper we explore new possibilities for active contours made possible by sobolev metrics the sobolev method allows one to implement new energy based active contour models that were not otherwise considered because the traditional minimizing method render them ill posed or numerically infeasible in particular we exploit the stabilizing and the order reducing properties of sobolev gradients to implement the gradient descent of these new energies we give examples of this class of energies which include some simple geometric priors and new edge based energies we also show that these energies can be quite useful for segmentation and tracking we also show that the gradient flows using the traditional metric are either ill posed or numerically difficult to implement and then show that the flows can be implemented in a stable and numerically feasible manner using the sobolev gradient

shape tracking with occlusions via coarse to fine region based sobolev descent we present a method to track the shape of an object from video the method uses a joint shape and appearance model of the object which is propagated to match shape and radiance in subsequent frames determining object shape self occlusions and dis occlusions of the object from camera and object motion pose difficulties to joint shape and appearance models in tracking they are unable to adapt to new shape and appearance information leading to inaccurate shape detection in this work we model self occlusions and dis occlusions in a joint shape and appearance tracking framework self occlusions and the warp to propagate the model are coupled thus we formulate a joint optimization problem we derive a coarse to fine optimization method advantageous in tracking that initially perturbs the model by coarse perturbations before transitioning to finer scale perturbations seamlessly this coarse to fine behavior is automatically induced by gradient descent on a novel infinite dimensional riemannian manifold that we introduce the manifold consists of planar parameterized regions and the metric that we introduce is a novel sobolev metric experiments on video exhibiting occlusions dis occlusions complex radiance and background show that occlusion dis occlusion modeling leads to superior shape accuracy

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

polynomial time approximation schemes for euclidean traveling salesman and other geometric problems we present a polynomial time approximation scheme for euclidean tsp in fixed dimensions for every fixed c 1 and given any n nodes in r 2 a randomized version of the scheme finds a 1 1 c approximation to the optimum traveling salesman tour in o n log n o c time when the nodes are in r d the running time increases to o n log n o d c d 1 for every fixed c d the running time is n poly log n that is nearly linear in n the algorithmm can be derandomized but this increases the running time by a factor o n d the previous best approximation algorithm for the problem due to christofides achieves a 3 2 aproximation in polynomial time we also give similar approximation schemes for some other np hard euclidean problems minimum steiner tree k tsp and k mst the running times of the algorithm for k tsp and k mst involve an additional multiplicative factor k the previous best approximation algorithms for all these problems achieved a constant factor approximation we also give efficient approximation schemes for euclidean min cost matching a problem that can be solved exactly in polynomial time all our algorithms also work with almost no modification when distance is measured using any geometric norm such as e p for p 1 or other minkowski norms they also have simple parallel i e nc implementations