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

optimality conditions for simultaneous topology and shape optimization new optimality conditions are derived for a class of shape optimization problems the conditions are established on the boundary by an application of the boundary variations technique and in the interior of an optimal domain by exploiting the topological derivative method an example is provided for which the classical second order sufficient optimality conditions are verified for an optimal simply connected domain however the value of the cost can be improved by the topology variations and therefore the optimal solution can be substantially changed by applying the topology optimization

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

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

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

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

charged free fermions vertex operators and the classical theory of conjugate nets we show that the quantum field theoretical formulation of the function theory has a geometrical interpretation within the classical transformation theory of conjugate nets in particular we prove that i the partial charge transformations preserving the neutral sector are laplace transformations ii the basic vertex operators are levy and adjoint levy transformations and iii the diagonal soliton vertex operators generate fundamental transformations we also show that the bilinear identity for the multicomponent kadomtsev petviashvili hierarchy becomes through a generalized miwa map a bilinear identity for the multidimensional quadrilateral lattice equations

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

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

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

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

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

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

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

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

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

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

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