a central limit theorem for random closed geodesics proof of the chas li maskit conjecture abstract we prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface in the special case of a hyperbolic pair of pants this settles a conjecture of chas li maskit

hierarchical shape segmentation and registration via topological features of laplace beltrami eigenfunctions this work introduces a method to hierarchically segment articulated shapes into meaningful parts and to register these parts across populations of near isometric shapes e g head arms legs and fingers of humans in different body postures the method exploits the isometry invariance of eigenfunctions of the laplace beltrami operator and uses topological features level sets at important saddles for the segmentation concepts from persistent homology are employed for a hierarchical representation for the elimination of topological noise and for the comparison of eigenfunctions the obtained parts can be registered via their spectral embedding across a population of near isometric shapes this work also presents the highly accurate computation of eigenfunctions and eigenvalues with cubic finite elements on triangle meshes and discusses the construction of persistence diagrams from the morse smale complex as well as the relation to size functions

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

comparison theorems and orbit counting in hyperbolic geometry in this article we address an interesting problem in hyperbolic geometry this is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold e g word length displacement in the universal cover etc asymptotically our method involves a tnixture of ideas from both thermodynamic ergodic theory and the automaton associated to strongly markov groups 0 introduction in 5 cannon showed that for fundamental groups of many manifolds of negative curvature including for example compact manifolds the generating function for word length is a rational function to explain the implications of this property we recall that the growth of the quantity n n card g fe r igl n was studied by milnor in his fundamental paper 13 where he obtained estimates using comparison with the growth of volume in the universal cover cannon used purely combinatorial methods to show that the generating function is rational in particular this implies that we can find constants ybi and ci and positive integers ki i 1 n such that n n 1 1 cinkitn the key step in cannon s approach was to associate to the grol1p an automaton in this article we shall augment this with the notion of weighting this allows us to draw upon the well known theory of thermodynamic formalism to prove a number of new results which can be viewed as weighted analogues of the above results it has long been understood that this approach must havew close connections with symbolic dynamics markov partitions subshifts of finite type cf 1 20 21 and in particular the work of bourdon 4 and by extension to the whole paraphernalia of the area of dynamical systems collectively called thermodynamic formalism in this article we shall elaborate on this connection as an integral part of our analysis let teln for some n 2 denote n dimensional hyperbolic space i e the unique n dimensional simply connected complete riemannian manifold wi th all sectional received by the editors may 23 1995 1991 mathematics subject classification primary 20f32 22e40 58e40 secondary 11f72 20f10 58f20

a gromov hausdorff framework with diffusion geometry for topologically robust non rigid shape matching in this paper the problem of non rigid shape recognition is studied from the perspective of metric geometry in particular we explore the applicability of diffusion distances within the gromov hausdorff framework while the traditionally used geodesic distance exploits the shortest path between points on the surface the diffusion distance averages all paths connecting the points the diffusion distance constitutes an intrinsic metric which is robust in particular to topological changes such changes in the form of shortcuts holes and missing data may be a result of natural non rigid deformations as well as acquisition and representation noise due to inaccurate surface construction the presentation of the proposed framework is complemented with examples demonstrating that in addition to the relatively low complexity involved in the computation of the diffusion distances between surface points its recognition and matching performances favorably compare to the classical geodesic distances in the presence of topological changes between the non rigid shapes

2 5d face recognition using patch geodesic moments in this paper we propose a novel patch geodesic distance pgd to transform the texture map of an object through its shape data for robust 2 5d object recognition local geodesic paths within patches and global geodesic paths for patches are combined in a coarse to fine hierarchical computation of pgd for each surface point to tackle the missing data problem in 2 5d images shape adjusted texture patches are encoded into local patterns for similarity measurement between two 2 5d images with different viewing angles and or shape deformations an extensive experimental investigation is conducted on 2 5 face images using the publicly available bu 3dfe and bosphorus databases covering face recognition under expression and pose changes the performance of the proposed method is compared with that of three benchmark approaches the experimental results demonstrate that the proposed method provides a very encouraging new solution for 2 5d object recognition

computational surface flattening a voxel based approach a voxel based method for flattening a surface in 3d space into 2d while best preserving distances is presented triangulation or polyhedral approximation of the voxel data are not required the problem is divided into two main parts voxel based calculation of the minimal geodesic distances between points on the surface and finding a configuration of points in 2d that has euclidean distances as close as possible to these distances the method suggested combines an efficient voxel based hybrid distance estimation method that takes the continuity of the underlying surface into account with classical multidimensional scaling mds for finding the 2d point configuration the proposed algorithm is efficient simple and can be applied to surfaces that are not functions experimental results are shown

discrete morse theory for totally non negative flag varieties abstract in a seminal 1994 paper lusztig 1994 26 lusztig extended the theory of total positivity by introducing the totally non negative part g p 0 of an arbitrary generalized partial flag variety g p he referred to this space as a remarkable polyhedral subspace and conjectured a decomposition into cells which was subsequently proven by the first author rietsch 1998 33 in williams 2007 40 the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron by conjecturing it to be a regular cw complex that is homeomorphic to a closed ball in this article we use discrete morse theory to prove this conjecture up to homotopy equivalence explicitly we prove that the boundaries of the cells are homotopic to spheres and the closures of cells are contractible the latter part generalizes a result of lusztig s 1998 28 that g p 0 the closure of the top dimensional cell is contractible concerning our result on the boundaries of cells even the special case that the boundary of the top dimensional cell g p 0 is homotopic to a sphere is new for all g p other than projective space

nasal similarity measure of 3d faces based on curve shape space abstract we propose a novel method for measuring the nasal similarity among 3d faces firstly we construct a representation for the nose shape which is composed of a set of geodesic curves each crosses the bridge of the nose next using these geodesic curves we formulate a similarity measure to compare among noses in the curve shape space under the riemannian framework the shape space is a quotient space for which the scaling translation and rotation are removed since the nose similarity measure is based on the shape comparison the proposed method has the following advantages 1 the similarity measure is robust to facial expressions since the nose is not affected by facial expressions 2 the geometric features of the nose shape match well with the human perception 3 the similarity measure is independent of the mesh grid because the chosen nose curves are not sensitive to the triangular mesh model we construct a nasal hierarchical structure for noses organization which is based on nose similarity measure results in our experiments we evaluate the performance of the proposed method and compare it with competing methods on three public face databases namely frgc2 0 texas3d and bosphorusdb the results show superiority of the proposed method in terms of both the speed and the accuracy when the nasal measurements are processed in the nasal hierarchical structure and the nasal samples with low sampling rate 5 25 of original point cloud

information theoretic shape matching in this paper we describe two related algorithms that provide both rigid and non rigid point set registration with different computational complexity and accuracy the first algorithm utilizes a nonlinear similarity measure known as correntropy the measure combines second and high order moments in its decision statistic showing improvements especially in the presence of impulsive noise the algorithm assumes that the correspondence between the point sets is known which is determined with the surprise metric the second algorithm mitigates the need to establish a correspondence by representing the point sets as probability density functions pdf the registration problem is then treated as a distribution alignment the method utilizes the cauchy schwarz divergence to measure the similarity distance between the point sets and recover the spatial transformation function needed to register them both algorithms utilize information theoretic descriptors however correntropy works at the realizations level whereas cauchy schwarz divergence works at the pdf level this allows correntropy to be less computationally expensive and for correct correspondence more accurate the two algorithms are robust against noise and outliers and perform well under varying levels of distortion they outperform several well known and state of the art methods for point set registration

snapshots a novel local surface descriptor and matching algorithm for robust 3d surface alignment in this paper a novel local surface descriptor is proposed and applied to the problem of aligning partial views of a 3d object the descriptor is based on taking snapshots of the surface over each point using a virtual camera oriented perpendicularly to the surface this representation has the advantage of imposing minimal loss of information be robust to self occlusions and also be very efficient to compute then we describe an efficient search technique to deal with the rotation ambiguity of our representation and experimentally demonstrate the benefits of our approaches which are pronounced especially when we align views with small overlap

efficient computation of isometry invariant distances between surfaces we present an efficient computational framework for isometry invariant comparison of smooth surfaces we formulate the gromov hausdorff distance as a multidimensional scaling like continuous optimization problem in order to construct an efficient optimization scheme we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances between the samples for isometry invariant comparison of surfaces in the case of partially missing data we present the partial embedding distance which is computed using a similar scheme the main idea is finding a minimum distortion mapping from one surface to another while considering only relevant geodesic distances we discuss numerical implementation issues and present experimental results that demonstrate its accuracy and efficiency

the combinatorics of local constraints in model based recognition and localization from sparse data the problem of recognizing what objects are where in the workspace of a robot can be cast as one of searching for a consistent matching between sensory data elements and equivalent model elements in principle this search space is enormous and to control the potential combinatorial explosion constraints between the data and model elements are needed a set of constraints for sparse sensory data that are applicable to a wide variety of sensors are derived and their characteristics are examined known bounds on the complexity of constraint satisfaction problems are used together with explicit estimates of the effectiveness of the constraints derived for the case of sparse noisy three dimensional sensory data to obtain general theoretical bounds on the number of interpretations expected to be consistent with the data it is shown that these bounds are consistent with empirical results reported previously the results are used to demonstrate the graceful degradation of the recognition technique with the presence of noise in the data and to predict the number of data points needed in general to uniquely determine the object being sensed

from 3d point clouds to pose normalised depth maps we consider the problem of generating either pairwise aligned or pose normalised depth maps from noisy 3d point clouds in a relatively unrestricted poses our system is deployed in a 3d face alignment application and consists of the following four stages i data filtering ii nose tip identification and sub vertex localisation iii computation of the relative face orientation iv generation of either a pose aligned or a pose normalised depth map we generate an implicit radial basis function rbf model of the facial surface and this is employed within all four stages of the process for example in stage ii construction of novel invariant features is based on sampling this rbf over a set of concentric spheres to give a spherically sampled rbf ssr shape histogram in stage iii a second novel descriptor called an isoradius contour curvature signal is defined which allows rotational alignment to be determined using a simple process of 1d correlation we test our system on both the university of york uoy 3d face dataset and the face recognition grand challenge frgc 3d data for the more challenging uoy data our ssr descriptors significantly outperform three variants of spin images successfully identifying nose vertices at a rate of 99 6 nose localisation performance on the higher quality frgc data which has only small pose variations is 99 9 our best system successfully normalises the pose of 3d faces at rates of 99 1 uoy data and 99 6 frgc data

enhanced pose normalization and matching of non rigid objects based on support vector machine modelling the estimation of 3d surface correspondence constitutes a fundamental problem in shape matching and analysis applications in the presence of non rigid shape deformations the ambiguity of surface correspondence increases together with the complexity of registration algorithms in this paper we alleviate this problem by means of 3d pose normalization using one class support vector machines ocsvm in detail we show how ocsvm are employed in order to increase the consistency of translation and scale normalization under articulations extrusions or the presence of outliers to estimate the relative translation and scale of an object we use the 3d distribution of points that is modelled by employing ocsvm to estimate the decision surface corresponding to the surface points of the object under a preset tolerance to outliers by discarding the outliers in the estimation of the object s center and size we compute the canonical pose of the core distribution that is less sensitive to intra class shape variations the effectiveness of the proposed method is demonstrated through the increased stability of translation and scale normalization and further justified by improving the precision of content based 3d object retrieval highlightswe improve non rigid surface correspondence by robust computing of the canonical pose the relative translation and scale are computed by discarding shape outliers by svms the improved surface correspondence in turn improves content based retrieval the improved pose normalization can be employed with any 3d shape descriptor

3d shape retrieval using kernels on extended reeb graphs abstract 3d shape retrieval is becoming an acute issue for numerous applications that span from cad to serious games to biomedicine and all contexts where it is fundamental to automatically retrieve geometric information from a collection of 3d models this paper addresses 3d shape retrieval in terms of a graph based description and the definition of a corresponding similarity measure for this purpose 3d models are represented as bags of shortest paths defined over well chosen extended reeb graphs while the similarity between pairs of extended reeb graphs is addressed through kernels adapted to these descriptions results are comparable with the best results of the literature and the modularity and evolutivity of the method ensure its applicability to other problems from partial shape matching to classification

three dimensional face recognition an expression invariant 3d face recognition approach is presented our basic assumption is that facial expressions can be modelled as isometries of the facial surface this allows to construct expression invariant representations of faces using the bending invariant canonical forms approach the result is an efficient and accurate face recognition algorithm robust to facial expressions that can distinguish between identical twins the first two authors we demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods the numerical methods employed by our approach do not require the facial surface explicitly the surface gradients field or the surface metric are sufficient for constructing the expression invariant representation of any given face it allows us to perform the 3d face recognition task while avoiding the surface reconstruction stage

3d face recognition under expressions occlusions and pose variations we propose a novel geometric framework for analyzing 3d faces with the specific goals of comparing matching and averaging their shapes here we represent facial surfaces by radial curves emanating from the nose tips and use elastic shape analysis of these curves to develop a riemannian framework for analyzing shapes of full facial surfaces this representation along with the elastic riemannian metric seems natural for measuring facial deformations and is robust to challenges such as large facial expressions especially those with open mouths large pose variations missing parts and partial occlusions due to glasses hair and so on this framework is shown to be promising from both empirical and theoretical perspectives in terms of the empirical evaluation our results match or improve upon the state of the art methods on three prominent databases frgcv2 gavabdb and bosphorus each posing a different type of challenge from a theoretical perspective this framework allows for formal statistical inferences such as the estimation of missing facial parts using pca on tangent spaces and computing average shapes

optimal estimation and detection in homogeneous spaces this paper presents estimation and detection techniques in homogeneous spaces that are optimal under the squared error loss function the data is collected on a manifold which forms a homogeneous space under the transitive action of a compact lie group signal estimation problems are addressed by formulating wiener hopf equations for homogeneous spaces the coefficient functions of these equations are the signal correlations which are assumed to be known the resulting coupled integral equations on the manifold are converted to wiener hopf convolutional integral equations on the group these are solved using the peter weyl theory of fourier transform on compact lie groups the computational complexity of this algorithm is reduced using the bi invariance of the correlations with respect to a stabilizer subgroup the theory of matched filtering for isotropic signal fields is developed for signal classification where given a set of template signals on the manifold and a noisy test signal the objective is to optimally detect the template buried in the test signal this is accomplished by designing a filter on the manifold that maximizes the signal to noise ratio snr of the filtered output an expression for the snr is obtained as a ratio of quadratic forms expressed as haar integrals over the transformation group these integrals are expressed in the fourier domain as infinite sums over the irreducible representations simplification of these sums is achieved by invariance properties of the signal function and the noise correlation function the wiener filter and matched filter are developed for an abstract homogeneous space and then specialized to the case of spherical signals under the action of the rotation group applications of these algorithms to denoising of 3d surface data visual navigation with omnidirectional camera and detection of compact embedded objects in the stochastic background are discussed with experimental results

minimum distortion isometric shape correspondence using em algorithm we present a purely isometric method that establishes 3d correspondence between two nearly isometric shapes our method evenly samples high curvature vertices from the given mesh representations and then seeks an injective mapping from one vertex set to the other that minimizes the isometric distortion we formulate the problem of shape correspondence as combinatorial optimization over the domain of all possible mappings which then reduces in a probabilistic setting to a log likelihood maximization problem that we solve via the expectation maximization em algorithm the em algorithm is initialized in the spectral domain by transforming the sampled vertices via classical multidimensional scaling mds minimization of the isometric distortion and hence maximization of the log likelihood function is then achieved in the original 3d euclidean space for each iteration of the em algorithm in two steps by first using bipartite perfect matching and then a greedy optimization algorithm the optimal mapping obtained at convergence can be one to one or many to one upon choice we demonstrate the performance of our method on various isometric or nearly isometric pairs of shapes for some of which the ground truth correspondence is available

3d face recognition using isogeodesic stripes in this paper we present a novel approach to 3d face matching that shows high effectiveness in distinguishing facial differences between distinct individuals from differences induced by nonneutral expressions within the same individual the approach takes into account geometrical information of the 3d face and encodes the relevant information into a compact representation in the form of a graph nodes of the graph represent equal width isogeodesic facial stripes arcs between pairs of nodes are labeled with descriptors referred to as 3d weighted walkthroughs 3dwws that capture the mutual relative spatial displacement between all the pairs of points of the corresponding stripes face partitioning into isogeodesic stripes and 3dwws together provide an approximate representation of local morphology of faces that exhibits smooth variations for changes induced by facial expressions the graph based representation permits very efficient matching for face recognition and is also suited to being employed for face identification in very large data sets with the support of appropriate index structures the method obtained the best ranking at the shrec 2008 contest for 3d face recognition we present an extensive comparative evaluation of the performance with the frgc v2 0 data set and the shrec08 data set

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

icp registration using invariant features investigates the use of euclidean invariant features in a generalization of iterative closest point icp registration of range images pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances it is shown that under ideal noise free conditions correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone in addition monotonic convergence to at least a local minimum is shown to hold for this method when noise is present a method that automatically sets the optimal relative contribution of features and positions is described this method trades off the error in feature values due to noise against the error in positions due to misalignment experimental results suggest that using invariant features decreases the probability of being trapped in a local minimum and may be an effective solution for difficult range image registration problems where the scene is very small compared to the model

topology invariant similarity of nonrigid shapes this paper explores the problem of similarity criteria between nonrigid shapes broadly speaking such criteria are divided into intrinsic and extrinsic the first referring to the metric structure of the object and the latter to how it is laid out in the euclidean space both criteria have their advantages and disadvantages extrinsic similarity is sensitive to nonrigid deformations while intrinsic similarity is sensitive to topological noise in this paper we approach the problem from the perspective of metric geometry we show that by unifying the extrinsic and intrinsic similarity criteria it is possible to obtain a stronger topology invariant similarity suitable for comparing deformed shapes with different topology we construct this new joint criterion as a tradeoff between the extrinsic and intrinsic similarity and use it as a set valued distance numerical results demonstrate the efficiency of our approach in cases where using either extrinsic or intrinsic criteria alone would fail

statistical 3d shape analysis by local generative descriptors in this paper we propose a new approach for surface representation generative models are exploited for encoding the variations of local geometric properties of 3d shapes surfaces are locally modeled as a stochastic process which spans a neighborhood area through a set of circular geodesic pathways captured by a modified version of a hidden markov model hmm named multicircular hmm mc hmm the approach proposed consists of two main phases 1 local geometric feature collection and 2 mc hmm parameter estimation the effectiveness of our proposal is demonstrated by several applicative scenarios all using well known benchmark data sets such as multiple view registration matching of deformable shapes and object recognition on cluttered scenes the results achieved are very promising and open up the use of generative models as geometric descriptors in an extensive range of applications

generalized correlation for shape alignment ordered set valued functions are introduced in this study to represent arbitrary shapes in a simple manner the arithmetic of ordered set valued functions is meaningless hence a novel concept namely generalized correlation is proposed to handle shape alignment problems generalized correlation is an extension of correlation from single valued functions to ordered set valued functions it is especially designed for shape alignment but not for a general mathematical definition moreover efficient algorithms are proposed to compute generalized correlation in 2d and 3d cases experiments indicate that the proposed alignment methods converge rapidly and globally furthermore generalized correlation is used handle registration problems to demonstrate its potential in computer vision tasks

a new 3d matching method of nonrigid and partially similar models using curve analysis the 3d shape matching problem plays a crucial role in many applications such as indexing or modeling by example here we present a novel approach to matching 3d objects in the presence of nonrigid transformation and partially similar models in this paper we use the representation of surfaces by 3d curves extracted around feature points indeed surfaces are represented with a collection of closed curves and tools from shape analysis of curves are applied to analyze and to compare curves the belief functions are used to define a global distance between 3d objects the experimental results obtained on the tosca and the shrec07 data sets show that the system performs efficiently in retrieving similar 3d models

partial similarity of objects or how to compare a centaur to a horse similarity is one of the most important abstract concepts in human perception of the world in computer vision numerous applications deal with comparing objects observed in a scene with some a priori known patterns often it happens that while two objects are not similar they have large similar parts that is they are partially similar here we present a novel approach to quantify partial similarity using the notion of pareto optimality we exemplify our approach on the problems of recognizing non rigid geometric objects images and analyzing text sequences

lra local rigid averaging of stretchable non rigid shapes we present a novel algorithm for generating the mean structure of non rigid stretchable shapes following an alignment process which supports local affine deformations we translate the search of the mean shape into a diagonalization problem where the structure is hidden within the kernel of a matrix this is the first step required in many practical applications where one needs to model bendable and stretchable shapes from multiple observations

iterative point matching for registration of free form curves and surfaces a heuristic method has been developed for registering two sets of 3 d curves obtained by using an edge based stereo system or two dense 3 d maps obtained by using a correlation based stereo system geometric matching in general is a difficult unsolved problem in computer vision fortunately in many practical applications some a priori knowledge exists which considerably simplifies the problem in visual navigation for example the motion between successive positions is usually approximately known from this initial estimate our algorithm computes observer motion with very good precision which is required for environment modeling e g building a digital elevation map objects are represented by a set of 3 d points which are considered as the samples of a surface no constraint is imposed on the form of the objects the proposed algorithm is based on iteratively matching points in one set to the closest points in the other a statistical method based on the distance distribution is used to deal with outliers occlusion appearance and disappearance which allows us to do subset subset matching a least squares technique is used to estimate 3 d motion from the point correspondences which reduces the average distance between points in the two sets both synthetic and real data have been used to test the algorithm and the results show that it is efficient and robust and yields an accurate motion estimate

adaptively merging large scale range data with reflectance properties in this paper we tackle the problem of geometric and photometric modeling of large intricately shaped objects typical target objects we consider are cultural heritage objects when constructing models of such objects we are faced with several important issues that have not been addressed in the past issues that mainly arise due to the large amount of data that has to be handled we propose two novel approaches to efficiently handle such large amounts of data a highly adaptive algorithm for merging range images and an adaptive nearest neighbor search to be used with the algorithm we construct an integrated mesh model of the target object in adaptive resolution taking into account the geometric and or photometric attributes associated with the range images we use surface curvature for the geometric attributes and laser reflectance values for the photometric attributes this adaptive merging framework leads to a significant reduction in the necessary amount of computational resources furthermore the resulting adaptive mesh models can be of great use for applications such as texture mapping as we will briefly demonstrate additionally we propose an additional test for the k d tree nearest neighbor search algorithm our approach successfully omits back tracking which is controlled adaptively depending on the distance to the nearest neighbor since the main consumption of computational cost lies in the nearest neighbor search the proposed algorithm leads to a significant speed up of the whole merging process in this paper we present the theories and algorithms of our approaches with pseudo code and apply them to several real objects including large scale cultural assets

surface geodesic pattern for 3d deformable texture matching this paper presents a surface geodesic pattern sgp representation for matching textured 3d deformable surfaces sgp encodes the local variations of the surface texture derivatives to extract local information from distinctive textural relationships contained in a geodesic neighborhood thus sgp derives its strength from the fusion of surface texture and shape information at the data level in a way that is invariant to non rigid deformations we also propose gabor topography wavelet gtw for direct feature extraction from the range data both features are combined using a multi view sparse representation to achieve higher discrimination capability while matching non rigid 3d surfaces the performance of the proposed method is evaluated extensively on the bosphorus face database the frgc v2 face database and the polyu contact free hand database and the results are compared to state of the art methods experimental results show the effectiveness and superiority of the proposed method in recognizing objects under non rigid surface deformations

on bending invariant signatures for surfaces isometric surfaces share the same geometric structure also known as the first fundamental form for example all possible bendings of a given surface that includes all length preserving deformations without tearing or stretching the surface are considered to be isometric we present a method to construct a bending invariant signature for such surfaces this invariant representation is an embedding of the geometric structure of the surface in a small dimensional euclidean space in which geodesic distances are approximated by euclidean ones the bending invariant representation is constructed by first measuring the intergeodesic distances between uniformly distributed points on the surface next a multidimensional scaling technique is applied to extract coordinates in a finite dimensional euclidean space in which geodesic distances are replaced by euclidean ones applying this transform to various surfaces with similar geodesic structures first fundamental form maps them into similar signature surfaces we thereby translate the problem of matching nonrigid objects in various postures into a simpler problem of matching rigid objects as an example we show a simple surface classification method that uses our bending invariant signatures

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

network comparison embeddings and interiors this paper presents methods to compare networks where relationships between pairs of nodes in a given network are defined we define such network distance by searching for the optimal method to embed one network into another network prove that such distance is a valid metric in the space of networks modulo permutation isomorphisms and examine its relationship with other network metrics the network distance defined can be approximated via multidimensional scaling however the lack of structure in networks results in suboptimal approximations to alleviate such a problem we consider methods to define the interiors of networks we show that comparing interiors induced from a pair of networks yields the same result as the actual network distance between the original networks practical implications are explored by showing the ability to discriminate networks generated by different models

recognition of 3d objects based on implicit polynomials closed form expressions for a new set of 3d rotation invariants that are linear quadratic and angular combinations of implicit polynomial ip coefficients are developed based on these invariants we propose a 3d object recognition method that outperforms recognition based on ip fitting after pose estimation and the mpeg 7 ssd technique

euclidean distance based canonical forms for non rigid 3d shape retrieval retrieval of 3d shapes is a challenging problem especially for non rigid shapes one approach giving favourable results uses multidimensional scaling mds to compute a canonical form for each mesh after which rigid shape matching can be applied however a drawback of this method is that it requires geodesic distances to be computed between all pairs of mesh vertices due to the super quadratic computational complexity canonical forms can only be computed for low resolution meshes we suggest a linear time complexity method for computing a canonical form using euclidean distances between pairs of a small subset of vertices this approach has comparable retrieval accuracy but lower time complexity than using global geodesic distances allowing it to be used on higher resolution meshes or for more meshes to be considered within a time budget highlightsour method computes canonical forms of 3d meshes without using geodesic distances our method maximises the euclidean distance between a small set of feature points our method has a lower time complexity than other approaches yet provides comparable retrieval results

isometric deformation invariant 3d shape recognition intra shape deformations complicate 3d shape recognition and therefore need proper modeling thereto an isometric deformation model is used in this paper the method proposed does not need explicit point correspondences for the comparison of 3d shapes the geodesic distance matrix is used as an isometry invariant shape representation two approaches are described to arrive at a sampling order invariant shape descriptor the histogram of geodesic distance matrix values and the set of largest singular values of the geodesic distance matrix shape comparison is performed by comparison of the shape descriptors using the g 2 distance as dissimilarity measure for object recognition the results obtained demonstrate the singular value approach to outperform the histogram based approach as well as the state of the art multidimensional scaling technique the icp baseline algorithm and other isometric deformation modeling methods found in literature using the tosca database a rank 1 recognition rate of 100 is obtained for the identification scenario while the verification experiments are characterized by a 1 58 equal error rate external validation demonstrates that the singular value approach outperforms all other participants for the non rigid object retrieval contests in shrec 2010 as well as shrec 2011 for 3d face recognition the rank 1 recognition rate is 61 9 and the equal error rate is 11 8 on the bu 3dfe database this decreased performance is attributed to the fact that the isometric deformation assumption only holds to a limited extent for facial expressions this is also demonstrated in this paper

least squares fitting of two 3 d point sets two point sets pi and p i i 1 2 n are related by p i rpi t ni where r is a rotation matrix t a translation vector and ni a noise vector given pi and p i we present an algorithm for finding the least squares solution of r and t which is based on the singular value decomposition svd of a 3 3 matrix this new algorithm is compared to two earlier algorithms with respect to computer time requirements

feature preserved 3d canonical form measuring the dissimilarity between non rigid objects is a challenging problem in 3d shape retrieval one potential solution is to construct the models 3d canonical forms i e isometry invariant representations in 3d euclidean space on which any rigid shape matching algorithm can be applied however existing methods which are typically based on embedding procedures result in greatly distorted canonical forms and thus could not provide satisfactory performance to distinguish non rigid models in this paper we present a feature preserved canonical form for non rigid 3d watertight meshes the basic idea is to naturally deform original models against corresponding initial canonical forms calculated by multidimensional scaling mds specifically objects are first segmented into near rigid subparts and then through properly designed rotations and translations original subparts are transformed into poses that correspond well with their positions and directions on mds canonical forms final results are obtained by solving nonlinear minimization problems for optimal alignments and smoothing boundaries between subparts experiments on two non rigid 3d shape benchmarks not only clearly verify the advantages of our algorithm against existing approaches but also demonstrate that with the help of the proposed canonical form we can obtain significantly better retrieval accuracy compared to the state of the art

correlations for pairs of closed geodesics in this article we consider natural counting problems for closed geodesics on negatively curved surfaces we present asymptotic estimates for pairs of closed geodesics the differences of whose lengths lie in a prescribed family of shrinking intervals related pair correlation problems have been studied in both quantum chaos and number theory

the farthest point strategy for progressive image sampling a new method of farthest point strategy fps for progressive image acquisition an acquisition process that enables an approximation of the whole image at each sampling stage is presented its main advantage is in retaining its uniformity with the increased density providing efficient means for sparse image sampling and display in contrast to previously presented stochastic approaches the fps guarantees the uniformity in a deterministic min max sense within this uniformity criterion the sampling points are irregularly spaced exhibiting anti aliasing properties comparable to those characteristic of the best available method poisson disk a straightforward modification of the fps yields an image dependent adaptive sampling scheme an efficient o n log n algorithm for both versions is introduced and several applications of the fps are discussed

affine invariant geometry for non rigid shapes shape recognition deals with the study geometric structures modern surface processing methods can cope with non rigidity by measuring the lack of isometry deal with similarity or scaling by multiplying the euclidean arc length by the gaussian curvature and manage equi affine transformations by resorting to the special affine arc length definition in classical equi affine differential geometry here we propose a computational framework that is invariant to the full affine group of transformations similarity and equi affine thus by construction it can handle non rigid shapes technically we add the similarity invariant property to an equi affine invariant one and establish an affine invariant pseudo metric as an example we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison

expression invariant representations of faces addressed here is the problem of constructing and analyzing expression invariant representations of human faces we demonstrate and justify experimentally a simple geometric model that allows to describe facial expressions as isometric deformations of the facial surface the main step in the construction of expression invariant representation of a face involves embedding of the facial intrinsic geometric structure into some low dimensional space we study the influence of the embedding space geometry and dimensionality choice on the representation accuracy and argue that compared to its euclidean counterpart spherical embedding leads to notably smaller metric distortions we experimentally support our claim showing that a smaller embedding error leads to better recognition

object recognition through topo geometric shape models using error tolerant subgraph isomorphisms we propose a method for 3 d shape recognition based on inexact subgraph isomorphisms by extracting topological and geometric properties of a shape in the form of a shape model referred to as topo geometric shape model tgsm in a nutshell tgsm captures topological information through a rigid transformation invariant skeletal graph that is constructed in a morse theoretic framework with distance function as the morse function geometric information is then retained by analyzing the geometric profile as viewed through the distance function modeling the geometric profile through elastic yields a weighted skeletal representation which leads to a complete shape signature shape recognition is carried out through inexact subgraph isomorphisms by determining a sequence of graph edit operations on model graphs to establish subgraph isomorphisms with a test graph test graph is recognized as a shape that yields the largest subgraph isomorphism with minimal cost of edit operations in this paper we propose various cost assignments for graph edit operations for error correction that takes into account any shape variations arising from noise and measurement errors

spectral multidimensional scaling an important tool in information analysis is dimensionality reduction there are various approaches for large data simplification by scaling its dimensions down that play a significant role in recognition and classification tasks the efficiency of dimension reduction tools is measured in terms of memory and computational complexity which are usually a function of the number of the given data points sparse local operators that involve substantially less than quadratic complexity at one end and faithful multiscale models with quadratic cost at the other end make the design of dimension reduction procedure a delicate balance between modeling accuracy and efficiency here we combine the benefits of both and propose a low dimensional multiscale modeling of the data at a modest computational cost the idea is to project the classical multidimensional scaling problem into the data spectral domain extracted from its laplace beltrami operator there embedding into a small dimensional euclidean space is accomplished while optimizing for a small number of coefficients we provide a theoretical support and demonstrate that working in the natural eigenspace of the data one could reduce the process complexity while maintaining the model fidelity as examples we efficiently canonize nonrigid shapes by embedding their intrinsic metric into a method often used for matching and classifying almost isometric articulated objects finally we demonstrate the method by exposing the style in which handwritten digits appear in a large collection of images we also visualize clustering of digits by treating images as feature points that we map to a plane

squigraphs for fine and compact modeling of 3 d shapes we propose to superpose global topological and local geometric 3 d shape descriptors in order to define one compact and discriminative representation for a 3 d object while a number of available 3 d shape modeling techniques yield satisfactory object classification rates there is still a need for a refined and efficient identification recognition of objects among the same class in this paper we use morse theory in a two phase approach to ensure the invariance of the final representation to isometric transforms we choose the morse function to be a simple and intrinsic global geodesic function defined on the surface of a 3 d object the first phase is a coarse representation through a reduced topological reeb graph we use it for a meaningful decomposition of shapes into primitives during the second phase we add detailed geometric information by tracking the evolution of morse function s level curves along each primitive we then embed the manifold of these curves into 3 and obtain a single curve by combining phase one and two we build new graphs rich in topological and geometric information that we refer to as squigraphs our experiments show that squigraphs are more general than existing techniques they achieve similar classification rates to those achieved by classical shape descriptors their performance however becomes clearly superior when finer classification and identification operations are targeted indeed while other techniques see their performances dropping squigraphs maintain a performance rate of the order of 97

self intersections in combinatorial topology statistical structure oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses by self intersection number one means the minimum number of transversal self intersection points of representatives of the class we prove that if a class is chosen at random from among all classes of m letters then for large m the distribution of the self intersection number approaches the gaussian distribution the theorem was strongly suggested by a computer experiment with four million curves producing a very nearly gaussian distribution

spectral driven isometry invariant matching of 3d shapes this paper presents a matching method for 3d shapes which comprises a new technique for surface sampling and two algorithms for matching 3d shapes based on point based statistical shape descriptors our sampling technique is based on critical points of the eigenfunctions related to the smaller eigenvalues of the laplace beltrami operator these critical points are invariant to isometries and are used as anchor points of a sampling technique which extends the farthest point sampling by using statistical criteria for controlling the density and number of reference points once a set of reference points has been computed for each of them we construct a point based statistical descriptor pssd for short of the input surface this descriptor incorporates an approximation of the geodesic shape distribution and other geometric information describing the surface at that point then the dissimilarity between two surfaces is computed by comparing the corresponding sets of pssds with bipartite graph matching or measuring the l 1 distance between the reordered feature vectors of a proximity graph here the reordering is given by the fiedler vector of a laplacian matrix associated to the proximity graph our tests have shown that both approaches are suitable for online retrieval of deformed objects and our sampling strategy improves the retrieval performances of isometry invariant matching methods finally the approach based on the fiedler vector is faster than using the bipartite graph matching and it has a similar retrieval effectiveness

combinatorial proof of the inversion formula on the kazhdan lusztig r polynomials in this paper we present a combinatorial proof of the inversion formula on the kazhdan lusztig r polynomials this problem was raised by brenti as a consequence we obtain a combinatorial interpretation of the equidistribution property due to verma stating that any nontrivial interval of a coxeter group in the bruhat order has as many elements of even length as elements of odd length the same argument leads to a combinatorial proof of an extension of verma s equidistribution to the parabolic quotients of a coxeter group obtained by deodhar as another application we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element