large scale image geo localization using dominant sets this paper presents a new approach for the challenging problem of geo localization using image matching in a structured database of city wide reference images with known gps coordinates we cast the geo localization as a clustering problem of local image features akin to existing approaches to the problem our framework builds on low level features which allow local matching between images for each local feature in the query image we find its approximate nearest neighbors in the reference set next we cluster the features from reference images using dominant set clustering which affords several advantages over existing approaches first it permits variable number of nodes in the cluster which we use to dynamically select the number of nearest neighbors for each query feature based on its discrimination value second this approach is several orders of magnitude faster than existing approaches thus we obtain multiple clusters different local maximizers and obtain a robust final solution to the problem using multiple weak solutions through constrained dominant set clustering on global image features where we enforce the constraint that the query image must be included in the cluster this second level of clustering also bypasses heuristic approaches to voting and selecting the reference image that matches to the query we evaluate the proposed framework on an existing dataset of 102k street view images as well as a new larger dataset of 300k images and show that it outperforms the state of the art by 20 and 7 percent respectively on the two datasets

recognition using region correspondences recognition systems attempt to recover information about the identity of observed objects and their location in the environment a fundamental problem in recognition is pose estimation this is the problem of using a correspondence between some portions of an object model and some portions of an image to determine whether the image contains an instance of the object and in case it does to determine the transformation that relates the model to the image the current approaches to this problem are divided into methods that use global properties of the object e g centroid and moments of inertia and methods that use local properties of the object e g corners and line segments global properties are sensitive to occlusion and specifically to self occlusion local properties are difficult to locate reliably and their matching involves intensive computation we present a novel method for recognition that uses region information in our approach the model and the image are divided into regions given a match between subsets of regions without any explicit correspondence between different pieces of the regions the alignment transformation is computed the method applies to planar objects under similarity affine and projective transformations and to projections of 3 d objects undergoing affine and projective transformations the new approach combines many of the advantages of the previous two approaches while avoiding some of their pitfalls like the global methods our approach makes use of region information that reflects the true shape of the object but like local methods our approach can handle occlusion

on the sum of laplacian eigenvalues of a signed graph abstract for a signed graph let e denote the number of edges and s k denote the sum of the k largest eigenvalues of the laplacian matrix of we conjecture that for any signed graph with n vertices s k e k 1 2 1 holds for k 1 n we prove the conjecture for any signed graph when k 2 and prove that this conjecture is true for unicyclic and bicyclic signed graphs

variational time integrators for finite dimensional thermo elasto dynamics without heat conduction this paper focuses on the formulation of variational integrators vis for finite dimensional thermo elastic systems without heat conduction the dynamics of these systems happen to have a hamiltonian structure after thermal displacements are introduced it is then possible to formulate integrators by taking advantage of standard methods in vi the class of integrators we construct have some remarkable features a they are symplectic b they exactly conserve the entropy of the system or in other words they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes c they nearly exactly conserve the value of the energy for very long times and d they exactly conserve linear and angular momentum we first describe how to adapt any vi for the classical mechanical systems to integrate adiabatic thermo elastic ones and then formulate three new types of integrators the first class based on a generalized trapezoidal rule gives rise to two first order explicit integrators and a second order implicit one by composing then the two first order integrators we construct a second order explicit one finally we formulate a fourth order implicit integrator which is a symplectic partitioned runge kutta method the performance of these new algorithms is showcased through numerical examples copyright 2011 john wiley sons ltd

galerkin lie group variational integrators based on unit quaternion interpolation abstract lie group variational integrators of arbitrary order are developed using the galerkin method based on unit quaternion interpolation to our knowledge quaternions have not been used before for this purpose though they allow a very simple and efficient way to perform the interpolation the resulting integrators are symplectic and structure preserving in the sense that certain symmetries in the lagrangian of the mechanical system are carried over to the discrete setting which leads to the preservation of the corresponding momentum maps the integrators furthermore exhibit a very good long time energy behavior i e energy is neither dissipated nor gained artificially at the same time the lie group structure is preserved by carefully defining the variations the interpolation method and by solving the non linear system of equations directly on the manifold rather than constraining it in a surrounding space using lagrange multipliers as a consequence we are able to show that lie group variational integrators based on the special orthogonal group are equivalent to the variational integrators for constrained systems using the discrete null space method employed e g in dmocc discrete mechanics and optimal control of constrained systems we show new numerical results on the convergence rates which are substantially higher than the known theoretical bounds and on the relation between accuracy and computational cost

explicit momentum conserving integrator for dynamics of rigid bodies approximating the midpoint lie algorithm we reformulate the midpoint lie algorithm which is implicit in the torque calculation to achieve explicitness in the torque evaluation this is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step thus we obtain two related variants both of which are explicit in the torque calculation but only first order in the time step both variants are momentum conserving and both are symplectic consequently drawing on the properties of the composition of maps we introduce another algorithm that combines the two variants in a single time step the resulting algorithm is explicit momentum conserving symplectic and second order its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

the averaged lagrangian method in this paper we present a new class of symplectic methods namely variational integrators employing the same integral averaging found in the averaged vector field method extensions to higher order are obtained using a collocation approach the obtained methods are time symmetric p series methods the proposed methods are compared with existing integrators the tests focus on highly oscillatory problems like the fermi pasta ulam problem

learning a probabilistic topology discovering model for scene categorization a recent advance in scene categorization prefers a topological based modeling to capture the existence and relationships among different scene components to that effect local features are typically used to handle photographing variances such as occlusions and clutters however in many cases the local features alone cannot well capture the scene semantics since they are extracted from tiny regions e g 4 times 4 patches within an image in this paper we mine a discriminative topology and a low redundant topology from the local descriptors under a probabilistic perspective which are further integrated into a boosting framework for scene categorization in particular by decomposing a scene image into basic components a graphlet model is used to describe their spatial interactions accordingly scene categorization is formulated as an intergraphlet matching problem the above procedure is further accelerated by introducing a probabilistic based representative topology selection scheme that makes the pairwise graphlet comparison trackable despite their exponentially increasing volumes the selected graphlets are highly discriminative and independent characterizing the topological characteristics of scene images a weak learner is subsequently trained for each topology which are boosted together to jointly describe the scene image in our experiment the visualized graphlets demonstrate that the mined topological patterns are representative to scene categories and our proposed method beats state of the art models on five popular scene data sets

time adaptive variational integrators a space time geodesic approach abstract the goal of this paper is to show that the space time geodesic approach of classical mechanics can be used to generate a time adaptive variational integration scheme the only assumption we make is that the lagrangian for the system is in a separable form the geometric structure which is preserved in the integration scheme is made explicit and the algorithm is illustrated with simulation for a compact case a non compact case a chaotic system which arises as a perturbation of an integrable system and the figure eight solution for a three body problem

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

variational integrators for the dynamics of thermo elastic solids with finite speed thermal waves this paper formulates variational integrators for finite element discretizations of deformable bodies with heat conduction in the form of finite speed thermal waves the cornerstone of the construction consists in taking advantage of the fact that the green naghdi theory of type ii for thermo elastic solids has a hamiltonian structure thus standard techniques to construct variational integrators can be applied to finite element discretizations of the problem the resulting discrete in time trajectories are then consistent with the laws of thermodynamics for these systems for an isolated system they exactly conserve the total entropy and nearly exactly conserve the total energy over exponentially long periods of time moreover linear and angular momenta are also exactly conserved whenever the exact system does for definiteness we construct an explicit second order accurate algorithm for affine tetrahedral elements in two and three dimensions and demonstrate its performance with numerical examples

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

on the sum of the laplacian eigenvalues of a tree given an n vertex graph g v e the laplacian spectrum of g is the set of eigenvalues of the laplacian matrix l d a where d and a denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively in this paper we study the laplacian spectrum of trees more precisely we find a new upper bound on the sum of the k largest laplacian eigenvalues of every n vertex tree where k 1 n this result is used to establish that the n vertex star has the highest laplacian energy over all n vertex trees which answers affirmatively to a question raised by radenkovic and gutman 10

spectral collocation variational integrators abstract spectral methods are a popular choice for constructing numerical approximations for smooth problems as they can achieve geometric rates of convergence and have a relatively small memory footprint in this paper we introduce a general framework to convert a spectral collocation method into a shooting based variational integrator for hamiltonian systems we also compare the proposed spectral collocation variational integrators to spectral collocation methods and galerkin spectral variational integrators in terms of their ability to reproduce accurate trajectories in configuration and phase space their ability to conserve momentum and energy as well as the relative computational efficiency of these methods when applied to some classical hamiltonian systems in particular we note that spectrally accurate variational integrators such as the galerkin spectral variational integrators and the spectral collocation variational integrators combine the computational efficiency of spectral methods together with the geometric structure preserving and long time structural stability properties of symplectic integrators

an introduction to lie group integrators basics new developments and applications we give a short and elementary introduction to lie group methods a selection of applications of lie group integrators are discussed finally a family of symplectic integrators on cotangent bundles of lie groups is presented and the notion of discrete gradient methods is generalised to lie groups

variable length signature for near duplicate image matching we propose a variable length signature for near duplicate image matching in this paper an image is represented by a signature the length of which varies with respect to the number of patches in the image a new visual descriptor viz probabilistic center symmetric local binary pattern is proposed to characterize the appearance of each image patch beyond each individual patch the spatial relationships among the patches are captured in order to compute the similarity between two images we utilize the earth mover s distance which is good at handling variable length signatures the proposed image signature is evaluated in two different applications i e near duplicate document image retrieval and near duplicate natural image detection the promising experimental results demonstrate the validity and effectiveness of the proposed variable length signature

an eigendecomposition approach to weighted graph matching problems an approximate solution to the weighted graph matching problem is discussed for both undirected and directed graphs the weighted graph matching problem is that of finding the optimum matching between two weighted graphs which are graphs with weights at each arc the proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem using the eigendecompositions of the adjacency matrices in the case of the undirected graph matching problem or hermitian matrices derived from the adjacency matrices in the case of the directed graph matching problem a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other simulation results are given to evaluate the performance of the proposed method

spectral variational integrators in this paper we present a new variational integrator for problems in lagrangian mechanics using techniques from galerkin variational integrators we construct a scheme for numerical integration that converges geometrically and is symplectic and momentum preserving furthermore we prove that under appropriate assumptions variational integrators constructed using galerkin techniques will yield numerical methods that are arbitrarily high order in particular if the quadrature formula used is sufficiently accurate then the resulting galerkin variational integrator has a rate of convergence at the discrete time steps that is bounded below by the approximation order of the finite dimensional function space in addition we show that the continuous approximating curve that arises from the galerkin construction converges on the interior of the time step at half the convergence rate of the solution at the discrete time steps we further prove that certain geometric invariants also converge with high order and that the error associated with these geometric invariants is independent of the number of steps taken we close with several numerical examples that demonstrate the predicted rates of convergence

a low complexity region based video coder using backward morphological motion field segmentation we introduce a novel region based video compression framework based on morphology to efficiently capture motion correspondences between consecutive frames in an image sequence our coder is built on the observation that the motion field associated with typical image sequences can be segmented into component motion subfield clusters associated with distinct objects or regions in the scene and further that these clusters can be efficiently captured using morphological operators in a backward framework that avoids the need to send region shape information region segmentation is performed directly on the motion field by introducing a small core for each cluster that captures the essential features of the cluster and reliably represents its motion behavior cluster matching is used in lieu of the conventional block matching methods of standard video coders to define a cluster motion representation paradigm furthermore a region based pel recursive approach is applied to find the refinement motion field for each cluster and the cluster motion prediction error image is coded by a novel adaptive scalar quantization method experimental results reveal a 10 20 reduction in prediction error energy and 1 3 db gain in the final reconstructed peak signal to noise ratio psnr over the standard mpeg 1 coder at typical bit rates of 500 kb s to 1 mb s on standard test sequences while also requiring lower computational complexity

an image understanding system using attributed symbolic representation and inexact graph matching this paper presents a powerful image understanding system that utilizes a semantic syntactic or attributed synibolic representation scheme in the form of attributed relational graphs arg s for comprehending the global information contents of images nodes in the arg represent the global image features while the relations between those features are represented by attributed branches between their corresponding nodes the extraction of arg representation from images is achieved by a multilayer graph transducer scheme this scheme is basically a rule based system that uses a combination of model driven and data driven concepts in performing a hierarchical symbolic mapping of the image information content from the spatial domain representation into a global representation further analysis and inter pretation of the imagery data is performed on the extracted arg representation a distance measure between images is defined in terms of the distance between their respective arg representations the distance between two arg s and the inexact matching of their respective components are calculated by an efficient dynamic programming technique the system handles noise distortion and ambiguity in real world images by two means namely through modeling and embedding them into the transducer s mapping rules as well as through the appropriate cost of error transformation for the inexact matching of the arg image representation two illustrative experiments are presented to demonstrate some capabilities of the proposed system experiment i deals with locating objects in multiobject scenes while experiment ii is concerned with target detection in sar images

learning shape classes using a mixture of tree unions this paper poses the problem of tree clustering as that of fitting a mixture of tree unions to a set of sample trees the tree unions are structures from which the individual data samples belonging to a cluster can be obtained by edit operations the distribution of observed tree nodes in each cluster sample is assumed to be governed by a bernoulli distribution the clustering method is designed to operate when the correspondences between nodes are unknown and must be inferred as part of the learning process we adopt a minimum description length approach to the problem of fitting the mixture model to data we make maximum likelihood estimates of the bernoulli parameters the tree unions and the mixing proportions are sought so as to minimize the description length criterion this is the sum of the negative logarithm of the bernoulli distribution and a message length criterion that encodes both the complexity of the union trees and the number of mixture components we locate node correspondences by minimizing the edit distance with the current tree unions and show that the edit distance is linked to the description length criterion the method can be applied to both unweighted and weighted trees we illustrate the utility of the resulting algorithm on the problem of classifying 20 shapes using a shock graph representation

long run accuracy of variational integrators in the stochastic context this paper presents a lie trotter splitting for inertial langevin equations geometric langevin algorithm and analyzes its long time statistical properties the splitting is defined as a composition of a variational integrator with an ornstein uhlenbeck flow assuming that the exact solution and the splitting are geometrically ergodic the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial langevin equations to within the accuracy of the variational integrator in representing the hamiltonian in particular if the variational integrator admits no energy error then the method samples the invariant measure of inertial langevin equations without error numerical validation is provided using explicit variational integrators with first second and fourth order accuracy

error analysis of generalized α lie group time integration methods for constrained mechanical systems generalized alpha methods are very popular in structural dynamics they are methods of newmark type and combine favourable stability properties with second order convergence for unconstrained second order systems in linear spaces recently they were extended to constrained systems in flexible multibody dynamics that have a configuration space with lie group structure in the present paper the convergence of these lie group methods is analysed by a coupled one step error recursion for differential and algebraic solution components it is shown that spurious oscillations in the transient phase result from order reduction that may be avoided by a perturbation of starting values or by index reduction numerical tests for a benchmark problem from the literature illustrate the results of the theoretical investigations

multiscale image segmentation by integrated edge and region detection this paper is concerned with the detection of low level structure in images it describes an algorithm for image segmentation at multiple scales the detected regions are homogeneous and surrounded by closed edge contours previous approaches to multiscale segmentation represent an image at different scales using a scale space however structure is only represented implicitly in this representation structures at coarser scales are inherently smoothed and the problem of structure extraction is unaddressed this paper argues that the issues of scale selection and structure detection cannot be treated separately a new concept of scale is presented that represents image structures at different scales and not the image itself this scale is integrated into a nonlinear transform which makes structure explicit in the transformed domain structures that are stable locally invariant to changes in scale are identified as being perceptually relevant the transform can be viewed as collecting spatially distributed evidence for edges and regions and making it available at contour locations thereby facilitating integrated detection of edges and regions without restrictive models of geometry or homogeneity in this sense it performs gestalt analysis all scale parameters of the transform are automatically determined and the structure of any arbitrary geometry can be identified without any smoothing even at coarse scales

approximating the maximum weight clique using replicator dynamics given an undirected graph with weights on the vertices the maximum weight clique problem mwcp is to find a subset of mutually adjacent vertices a clique having the largest total weight this is a generalization of the problem of finding the maximum cardinality clique of an unweighted graph which is the special case of the mwcp when all vertex weights are equal the problem is np hard for arbitrary graphs and so is the problem of approximating it within a constant factor we present a parallel distributed heuristic for approximating the mwcp based on dynamics principles it centers around a continuous characterization of the mwcp a purely combinatorial problem and lets it be formulated in terms of continuous quadratic programming one drawback is the presence of spurious solutions and we present their characterizations to avoid them we introduce a regularized continuous formulation of the mwcp and show how it completely solves the problem the formulation naturally maps onto a parallel distributed computational network whose dynamical behavior is governed by the replicator equations these are dynamical systems introduced in evolutionary game theory and population genetics to model evolutionary processes on a macroscopic scale we present theoretical results which guarantee that the solutions provided by our clique finding replicator network are actually those sought experimental results confirm the effectiveness of the proposed approach

on a conjecture for the signless laplacian eigenvalues abstract let g be a simple graph with n vertices and e g edges and q 1 g q 2 g q n g 0 be the signless laplacian eigenvalues of g let s k g i 1 k q i g where k 1 2 n f ashraf et al conjectured that s k g e g k 1 2 for k 1 2 n in this paper we give various upper bounds for s k g and prove that this conjecture is true for the following cases connected graph with sufficiently large k unicyclic graphs and bicyclic graphs for all k and tricyclic graphs when k 3 finally we discuss whether the upper bound given in this conjecture is tight or not for c cyclic graphs and propose some problems for future research

on the sum of signless laplacian eigenvalues of a graph abstract for a simple graph g let e g denote the number of edges and s k g denote the sum of the k largest eigenvalues of the signless laplacian matrix of g we conjecture that for any graph g with n vertices s k g e g k 1 2 for k 1 n we prove the conjecture for k 2 for any graph and for all k for regular graphs the conjecture is an analogous to a conjecture by a e brouwer with a similar statement but for the eigenvalues of laplacian matrices of graphs

discrete variational hamiltonian mechanics the main contribution of this paper is to present a canonical choice of a hamiltonian theory corresponding to the theory of discrete lagrangian mechanics we make use of lagrange duality and follow a path parallel to that used for construction of the pontryagin principle in optimal control theory we use duality results regarding sensitivity and separability to show the relationship between generating functions and symplectic integrators we also discuss connections to optimal control theory and numerical algorithms

on generalized distance energy of graphs abstract for the distance matrix d g and diagonal matrix of the vertex transmissions t r g of a simple connected graph g the generalized distance matrix d g is the convex combinations of t r g and d g and is defined as d g t r g 1 d g for 0 1 if 1 2 n are the eigenvalues of d g we define the generalized distance energy of the graph g as e d g i 1 n i 2 w g n where w g is the wiener index of g this is analogous to the energies associated with the distance laplacian and distance signless laplacian matrices of g we obtain upper and lower bounds for the generalized distance energy of graphs in terms of various parameters associated with the structure of the graph g we show that for 1 2 1 the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs and for 0 2 n 3 n 2 the star graph has the minimum generalized distance energy among all trees

unsupervised simplification of image hierarchies via evolution analysis in scale sets framework region based hierarchical image representation is crucial in many computer vision applications however in practice an image hierarchy is usually dense and contains many less informative branches it is expected that a hierarchy should be accurate and simplified which is not only desirable for different applications but also saves considerable computational load for the further analysis to achieve this target this paper proposes a novel approach for unsupervised simplification of region based image hierarchies which employs the global and local evolution analyses of a hierarchy first we introduce a global evolution analysis in the scale sets framework which provides clues for eliminating less informative branches moreover a hybrid unsupervised simplification method is designed utilizing the information from global and local evolution functions a number of experiments on various images have shown that the proposed approach is effective and efficient in removing less informative nodes averagely about 90 of the whole nodes while preserving salient image details and retaining the accuracy

generating functionals and lagrangian partial differential equations the main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values we review the concept of type i ii generating functionals defined on the space of boundary data of a lagrangian field theory on the lagrangian side we define an analogue of jacobi s solution to the hamilton jacobi equation for field theories and we show that by taking variational derivatives of this functional we obtain an isotropic submanifold of the space of cauchy data described by the so called multisymplectic form formula as an example of the latter we show that lorentz s reciprocity principle in electromagnetism is a particular instance of the multisymplectic form formula we also define a hamiltonian analogue of jacobi s solution and we show that this functional is a type ii generating functional we finish the paper by defining a similar framework of generating functions for discrete field theories and we show that for the linear wave equation we recover the multisymplectic conservation law of bridges

spectral variational integrators for semi discrete hamiltonian wave equations abstract in this paper we present a highly accurate hamiltonian structure preserving numerical method for simulating hamiltonian wave equations this method is obtained by applying spectral variational integrators svi to the system of hamiltonian odes which are derived from the spatial semi discretization of the hamiltonian pde the spatial variable is discretized by using high order symmetric finite differences an efficient implementation of svi for high dimensional systems of odes is presented

lagrangian mechanics and variational integrators on two spheres euler lagrange equations and variational integrators are developed for lagrangian mechanical systems evolving on a product of two spheres the geometric structure of a product of two spheres is carefully considered in order to obtain global equations of motion both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints we derive global expressions for the euler lagrange equations on two spheres which are more compact than existing equations written in terms of angles since the variational integrators are derived from hamilton s principle they preserve the geometric features of the dynamics such as symplecticity momentum maps or total energy as well as the structure of the configuration manifold computational properties of the variational integrators are illustrated for several mechanical systems in addition lie group variational integrators can be used to integrate lagrangian flows on more general homogeneous spaces this is achieved by lifting the discrete hamilton s principle on homogeneous spaces to a discrete variational principle on the lie group that is constrained by a discrete connection copyright 2009 john wiley sons ltd

polynomial time metrics for attributed trees we address the problem of comparing attributed trees and propose four novel distance measures centered around the notion of a maximal similarity common subtree the proposed measures are general and defined on trees endowed with either symbolic or continuous valued attributes and can be applied to rooted as well as unrooted trees we prove that our measures satisfy the metric constraints and provide a polynomial time algorithm to compute them this is a remarkable and attractive property since the computation of traditional edit distance based metrics is in general np complete at least in the unordered case we experimentally validate the usefulness of our metrics on shape matching tasks and compare them with an approximation of edit distance

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

pattern vectors from algebraic graph theory graph structures have proven computationally cumbersome for pattern analysis the reason for this is that before graphs can be converted to pattern vectors correspondences must be established between the nodes of structures which are potentially of different size to overcome this problem in this paper we turn to the spectral decomposition of the laplacian matrix we show how the elements of the spectral matrix for the laplacian can be used to construct symmetric polynomials that are permutation invariants the coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner we extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a hermitian property matrix that can be viewed as a complex analogue of the laplacian to embed the graphs in a pattern space we explore whether the vectors of invariants can be embedded in a low dimensional space using a number of alternative strategies including principal components analysis pca multidimensional scaling mds and locality preserving projection lpp experimentally we demonstrate that the embeddings result in well defined graph clusters our experiments with the spectral representation involve both synthetic and real world data the experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure the real world experiments show that the method can be used to locate clusters of graphs

lie group variational integrators for the full body problem we develop the equations of motion for full body models that describe the dynamics of rigid bodies acting under their mutual gravity the equations are derived using a variational approach where variations are defined on the lie group of rigid body configurations both continuous equations of motion and variational integrators are developed in lagrangian and hamiltonian forms and the reduction from the inertial frame to a relative frame is also carried out the lie group variational integrators are shown to be symplectic to preserve conserved quantities and to guarantee exact evolution on the configuration space one of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies

higher order discrete variational problems with constraints an interesting family of geometric integrators for lagrangian systems can be defined using discretizations of the hamilton s principle of critical action this family of geometric integrators is called variational integrators in this paper we derive new variational integrators for higher order lagrangian mechanical system subjected to higher order constraints from the discretization of the variational principles we show that our methods are automatically symplectic and in consequence with a very good energy behavior additionally the symmetries of the discrete lagrangian imply that momentum is conserved by the integrator moreover we extend our construction to variational integrators where the lagrangian is explicitly time dependent finally some motivating applications of higher order problems are considered in particular optimal control problems for explicitly time dependent underactuated systems and an interpolation problem on riemannian manifolds

interactive image segmentation by matching attributed relational graphs a model based graph matching approach is proposed for interactive image segmentation it starts from an over segmentation of the input image exploiting color and spatial information among regions to propagate the labels from the regions marked by the user provided seeds to the entire image the region merging procedure is performed by matching two graphs the input graph representing the entire image and the model graph representing only the marked regions the optimization is based on discrete search using deformed graphs to efficiently evaluate the spatial information note that by using a model based approach different interactive segmentation problems can be tackled binary and multi label segmentation of single images as well as of multiple similar images successful results for all these cases are presented in addition to a comparison between our binary segmentation results and those obtained with state of the art approaches an implementation is available at http structuralsegm sourceforge net

region based hierarchical image matching this paper presents an approach to region based hierarchical image matching where given two images the goal is to identify the largest part in image 1 and its match in image 2 having the maximum similarity measure defined in terms of geometric and photometric properties of regions e g area boundary shape and color as well as region topology e g recursive embedding of regions to this end each image is represented by a tree of recursively embedded regions obtained by a multiscale segmentation algorithm this allows us to pose image matching as the tree matching problem to overcome imaging noise one to one many to one and many to many node correspondences are allowed the trees are first augmented with new nodes generated by merging adjacent sibling nodes which produces directed acyclic graphs dags then transitive closures of the dags are constructed and the tree matching problem reformulated as finding a bijection between the two transitive closures on dags while preserving the connectivity and ancestor descendant relationships of the original trees the proposed approach is validated on real images showing similar objects captured under different types of noise including differences in lighting conditions scales or viewpoints amidst limited occlusion and clutter

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

discrete routh reduction this paper develops the theory of abelian routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry the reduction of variational runge kutta discretizations is considered as well as the extent to which symmetry reduction and discretization commute these reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics the methods are demonstrated for the dynamics of an earth orbiting satellite with a non spherical j2 correction as well as the double spherical pendulum the j2 problem is interesting because in the unreduced picture geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish but this issue does not appear in the reduced algorithm where one can directly observe interesting dynamical structures in the reduced phase space the cotangent bundle of shape space in which the geometric phases have been removed the main feature of the double spherical pendulum example is that it has a non trivial magnetic term in its reduced symplectic form our method is still efficient as it can directly handle the essential non canonical nature of the symplectic structure in contrast a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking darboux theorem to transform the symplectic structure into canonical form thereby incurring additional computational cost our method allows one to design reduced symplectic integrators in a natural way despite the non canonical nature of the symplectic structure

on the sum of k largest singular values of graphs and matrices in the recent years the trace norm of graphs has been extensively studied under the name of graph energy the trace norm is just one of the ky fan k norms given by the sum of the k largest singular values which are studied more generally in the present paper several relations to chromatic number spectral radius spread and to other fundamental parameters are outlined some results are extended to more general matrices

matching free trees maximal cliques and monotone game dynamics motivated by our recent work on rooted tree matching in this paper we provide a solution to the problem of matching two free i e unrooted trees by constructing an association graph whose maximal cliques are in one to one correspondence with maximal common subtrees we then solve the problem using simple payoff monotonic dynamics from evolutionary game theory we illustrate the power of the approach by matching articulated and deformed shapes described by shape axis trees experiments on hundreds of larger uniformly random trees are also presented the results are impressive despite the inherent inability of these simple dynamics to escape from local optima they always returned a globally optimal solution

asynchronous variational integrators we describe a class of asynchronous variational integrators avi for nonlinear elastodynamics the avis are characterized by the following distinguishing attributes i the algorithms permit the selection of independent time steps in each element and the local time steps need not bear an integral relation to each other ii the algorithms derive from a spacetime form of a discrete version of hamilton s principle as a consequence of this variational structure the algorithms conserve local energy and momenta exactly subject to solvability of the local time steps numerical tests reveal that even when local energy balance is not enforced exactly the global and local energy behavior of the avis is quite remarkable a property which can probably be traced to the symplectic nature of the algorithm

a transform for multiscale image segmentation by integrated edge and region detection describes a transform to extract image regions at all geometric and photometric scales it is argued that linear approaches have the shortcoming that they require a priori models of region shape the proposed transform avoids this by letting the structure emerge bottom up from interactions among pixels the transform involves global computations on pairs of pixels followed by vector integration of the results an attraction force field is computed over the image in which pixels belonging to the same region are mutually attracted and the region is characterized by a convergent flow it is shown that the transform possesses properties that allow multiscale segmentation or extraction of original unblurred structure at all different geometric and photometric scales present this is in contrast with much previous work wherein multiscale structure is viewed as the smoothed structure in a multiscale signal decimation scale is an integral parameter of the force computation and the number and values of scale parameters associated with the image can be estimated automatically regions are detected at all a priori unknown scales resulting in automatic construction of a segmentation tree in which each pixel is annotated with descriptions of all the regions it belongs to transform properties are presented for piecewise constant images but hold for more general ones thus the proposed method is intended as a solution to the problem of multiscale integrated edge and region detection or low level image segmentation experimental results are given

error analysis of variational integrators of unconstrained lagrangian systems an error analysis of variational integrators is obtained by blowing up the discrete variational principles all of which have a singularity at zero time step divisions by the time step lead to an order that is one less than observed in simulations a deficit that is repaired with the help of a new past future symmetry

unsupervised category modeling recognition and segmentation in images suppose a set of arbitrary unlabeled images contains frequent occurrences of 2d objects from an unknown category this paper is aimed at simultaneously solving the following related problems 1 unsupervised identification of photometric geometric and topological properties of multiscale regions comprising instances of the 2d category 2 learning a region based structural model of the category in terms of these properties and 3 detection recognition and segmentation of objects from the category in new images to this end each image is represented by a tree that captures a multiscale image segmentation the trees are matched to extract the maximally matching subtrees across the set which are taken as instances of the target category the extracted subtrees are then fused into a tree union that represents the canonical category model detection recognition and segmentation of objects from the learned category are achieved simultaneously by finding matches of the category model with the segmentation tree of a new image experimental validation on benchmark data sets demonstrates the robustness and high accuracy of the learned category models when only a few training examples are used for learning without any human supervision

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators