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This article introduces a novel 2D shape data augmentation approach based on intra-class shape space exploration. The proposed method relies on a geodesic interpolation between shapes, leveraging invariant-based morphing techniques. By blending a 2D shape pair belonging to a given class, we are able to generate nonlinear augmentations, hence covering more variations within the shape space. In particular, we formulate data augmentation as an optimization problem that minimizes the deformations between two shapes using the Generalized Finite Fourier Invariant Descriptor. The proposed augmentation technique is evaluated using numerous Convolution Neural Network architectures for 2D shape classification. The results indicate the superiority of the proposed method as compared to state-of-the-art techniques when considering small-scale datasets. [ABSTRACT FROM AUTHOR]

Triple collision orbits of the three-body problem with nonzero initial velocities have been systematically surveyed. For this purpose, we have formulated the sufficient conditions of the velocities of three bodies so that the total angular momentum of the triple system is zero. The velocity conditions are parameterized by two parameters α 2 and β 1 in the initial condition plane. We introduce the characteristic as the curve in the initial condition plane of the non-free-fall triple collision points (TCPs) parameterized by one parameter, say, α 2 / β 1 . The velocity conditions with full parameters suggest that non-free-fall TCPs (nonff-TCPs) occupy two-dimensional areas in the initial condition plane. We plotted five characteristics passing through eight ff-TCPs, one of which forms a closed circuit. Two ff-TCPs on a characteristics are called a twin. This gives us a criterion of classification of TCPs in addition to the one obtained in Tanikawa and Mikkola (Cel Mech Dyn Astron 133:52, 2001) which connects the directions of the initial and final triangles formed by three bodies. We find that the twin ff-TCP TSM-1 and TSM-2 are connected by all the characteristics, which pass through one of them. A neighborhood of ff-TCP TSM-2 has been confirmed to be occupied by nonff-TCPs. We expect that the same is true with a neighborhood of any ff-TCP. Further, we expect that this is true with a neighborhood of any nonff-TCP. We do not continue the characteristics until its end because the continuation seems endless. [ABSTRACT FROM AUTHOR]

We consider motion effected by shape change. Such motions are ubiquitous in nature and the human made environment, ranging from single cells to platform divers and jellyfish. The shapes may be immersed in various media ranging from the very viscous to air and nearly inviscid fluids. In the absence of external forces these settings are characterized by constant momentum. We exploit this in an algorithm which takes a sequence of changing shapes, say, as modeled by an animator, as input and produces corresponding motion in world coordinates. Our method is based on the geometry of shape change and an appropriate variational principle. The corresponding Euler-Lagrange equations are first order ODEs in the unknown rotations and translations and the resulting time stepping algorithm applies to all these settings without modification as we demonstrate with a broad set of examples. [ABSTRACT FROM AUTHOR]

The morphological differences between two geographical maps can be highlighted by a polycentric distance cartogram resulting from a bidimensional regression. Beyond the communicational interest of the transformations thus produced, the method makes it possible to reveal the differences in structure and therefore constitutes a real research tool. However, bidimensional regression can only compare the shape of two maps. Since the 1990s, geometric morphometrics has revolutionized the morphological analysis of natural structures (and others). It has since been applied in many fields of research but not in cartography. This article describes the theoretical and methodological bases of a method combining bidimensional regression with a geometric morphometrics approach to compare the shape of several geographical maps. Geometric morphometrics and bidimensional regression indeed share common approaches of the statistical shape analysis like homologous landmarks and interpolation grids. However, there is no software in geometric morphometrics capable of directly reading geographical data, which would facilitate the work of cartographers accustomed to GIS software. That is why we present MapMorphy, a tool specifically developed for this task. An example on ancient maps illustrates the method. [ABSTRACT FROM AUTHOR]

The germinal center (GC) is a self-organizing structure produced in the lymphoid follicle during the T-dependent immune response and is an important component of the humoral immune system. However, the impact of the special structure of GC on antibody production is not clear. According to the latest biological experiments, we establish a spatiotemporal stochastic model to simulate the whole self-organization process of the GC including the appearance of two specific zones: the dark zone (DZ) and the light zone (LZ), the development of which serves to maintain an effective competition among different cells and promote affinity maturation. A phase transition is discovered in this process, which determines the critical GC volume for a successful growth in both the stochastic and the deterministic model. Further increase of the volume does not make much improvement on the performance. It is found that the critical volume is determined by the distance between the activated B cell receptor (BCR) and the target epitope of the antigen in the shape space. The observation is confirmed in both 2D and 3D simulations and explains partly the variability of the observed GC size.

Abstract We develop a methodology for classifying defects based on their morphology and induced mechanical response. The proposed approach is fairly general and relies on morphological operators (Angulo and Meyer in 9th international symposium on mathematical morphology and its applications to signal and image processing, pp. 226-237, 2009) and spherical harmonic decomposition as a way to characterize the geometry of the pores, and on the Grassman distance evaluated on FFT-based computations (Willot in C. R., Méc. 343(3):232–245, 2015), for the predicted elastic response. We implement and detail our approach on a set of trapped gas pores observed in X-ray tomography of welded joints, that significantly alter the mechanical reliability of these materials (Lacourt et al. in Int. J. Numer. Methods Eng. 121(11):2581–2599, 2020). The space of morphological and mechanical responses is first partitioned into clusters using the “k-medoids” criterion and associated distance functions. Second, we use multiple-layer perceptron neural networks to associate a defect and corresponding morphological representation to its mechanical response. It is found that the method provides accurate mechanical predictions if the training data contains a sufficient number of defects representing each mechanical class. To do so, we supplement the original set of defects by data augmentation techniques. Artificially-generated pore shapes are obtained using the spherical harmonic decomposition and a singular value decomposition performed on the pores signed distance transform. We discuss possible applications of the present method, and how medoids and their associated mechanical response may be used to provide a natural basis for reduced-order models and hyper-reduction techniques, in which the mechanical effects of defects and structures are decorrelated (Ryckelynck et al. in C. R., Méc. 348(10–11):911–935, 2020).

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The phase-field method is based on the energy minimization principle which is a geometric method for modeling diffusive cracks that are popularly implemented with irreversibility based on Griffith 's criterion. This method requires a length-scale parameter that smooths the sharp discontinuity, which influences the diffuse band and results in mesh-dependent fracture propagation results. Recently, a novel approach based on the optimization on Riemann ian shape spaces has been proposed, where the crack path is realized by techniques from shape optimization. This approach requires the shape derivative, which is derived in a continuous sense and used for a gradient-based algorithm to minimize the energy of the system. In this paper, the novel approach based on shape optimization is presented, followed by an assessment of the predicted crack path in both isotropic and anisotropic brittle material using numerical calculations from a phase-field model. • Novel approach in fracture mechanics based on shape optimization is introduced. • Less computationally expensive approach than phase-field method. • Validation is performed by means of the phase-field method for fracture propagation. • Predicting the crack path successful even for anisotropic brittle materials. [ABSTRACT FROM AUTHOR]

This paper addresses the growing need to process non-Euclidean data, by introducing a geometric deep learning (GDL) framework for building universal feedforward-type models compatible with differentiable manifold geometries. We show that our GDL models can approximate any continuous target function uniformly on compact sets of a controlled maximum diameter. We obtain curvature dependant lower-bounds on this maximum diameter and upper-bounds on the depth of our approximating GDL models. Conversely, we find that there is always a continuous function between any two non-degenerate compact manifolds that any "locally-defined" GDL model cannot uniformly approximate. Our last main result identifies data-dependent conditions guaranteeing that the GDL model implementing our approximation breaks "the curse of dimensionality." We find that any "real-world" (i.e. finite) dataset always satisfies our condition and, conversely, any dataset satisfies our requirement if the target function is smooth. As applications, we confirm the universal approximation capabilities of the following GDL models: Ganea et al. (2018)'s hyperbolic feedforward networks, the architecture implementing Krishnan et al. (2015)'s deep Kalman-Filter, and deep softmax classifiers. We build universal extensions/variants of: the SPD-matrix regressor of Meyer et al. (2011b), and Fletcher et al. (2009)'s Procrustean regressor. In the Euclidean setting, our results imply a quantitative version of Kidger and Lyons (2020)'s approximation theorem and a data-dependent version of Yarotsky and Zhevnerchuk (2020)'s uncursed approximation rates. [ABSTRACT FROM AUTHOR]

An important hypothesis in animal cell biology is that an animal's acute exercise regimen affects some subcellular structures, for example mitochondrial morphology, in its muscle tissue. This paper investigates that hypothesis using a nonparametric metric‐based energy test for comparing mitochondrial populations. It explores two shape spaces—the elastic shape space and Kendall shape space—and five corresponding shape metrics on these spaces. The results overwhelmingly point to the statistical significance of the effect of an acute exercise regimen on the shape of SS‐type mitochondria. Although past studies based on specific morphological features derived from mitochondria failed to detect this significance. In this analysis, a potentially significant factor is cell membership and a k‐sample generalization of the energy test—the DISCO test shows that the cell effect is indeed significant. The energy test cannot be applied directly due to the hierarchical structure of the distance matrix. We propose a compression method to remove the significant cell effect while testing for the exercise effect. With this compression, only the elastic scaled metric shows statistical significance of the exercise factor in this more complicated scenario. This result is because the elastic‐scaled metric is more sensitive to subtle changes in mitochondrial shapes caused by acute exercise. [ABSTRACT FROM AUTHOR]

The differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov–Poincaré metric are defined. Moreover, the covariant derivative associated with the first Sobolev metric is deduced in this paper. The explicit expression of the covariant derivative leads to a definition of the Riemannian shape Hessian with respect to the first Sobolev metric. In this paper, we give a brief overview of various optimization techniques based on the gradients and the Hessian. Since the space of smooth shapes limits the application of the optimization techniques, this paper extends the definition of smooth shapes to H 1 / 2 -shapes, which arise naturally in shape optimization problems. We define a diffeological structure on the new space of H 1 / 2 -shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces. [ABSTRACT FROM AUTHOR]

• ORL tree shape space captures tree geometry in a natural way. • Geodesics in shape space provide smooth interpolations between trees. • Point clouds from range scanners require no previous meshing. [Display omitted] We propose a framework for creating a shape space for biological trees from existing point clouds. Our method allows to freely explore the shapes between given input trees by computing arbitrary points on the geodesics induced by our metric. After establishing correspondences between branches and individual input points, our efficient formulation allows to compute the geometric 3D tree model for a given point in shape space in linear time, allowing an interactive exploration. As our metric captures branch attributes such as length, orientation, and radius, in a natural way, it is possible to explore the shape space beyond the convex hull formed by the input trees and their geodesics. Our method works directly on point clouds, which can be acquired using ranged sensing devices, and does not rely on an intermediate mesh representation. [ABSTRACT FROM AUTHOR]

We develop a methodology for classifying defects based on their morphology and induced mechanical response. The proposed approach is fairly general and relies on morphological operators (Angulo and Meyer in 9th international symposium on mathematical morphology and its applications to signal and image processing, pp. 226-237, 2009) and spherical harmonic decomposition as a way to characterize the geometry of the pores, and on the Grassman distance evaluated on FFT-based computations (Willot in C. R., Méc. 343(3):232–245, 2015), for the predicted elastic response. We implement and detail our approach on a set of trapped gas pores observed in X-ray tomography of welded joints, that significantly alter the mechanical reliability of these materials (Lacourt et al. in Int. J. Numer. Methods Eng. 121(11):2581–2599, 2020). The space of morphological and mechanical responses is first partitioned into clusters using the "k-medoids" criterion and associated distance functions. Second, we use multiple-layer perceptron neural networks to associate a defect and corresponding morphological representation to its mechanical response. It is found that the method provides accurate mechanical predictions if the training data contains a sufficient number of defects representing each mechanical class. To do so, we supplement the original set of defects by data augmentation techniques. Artificially-generated pore shapes are obtained using the spherical harmonic decomposition and a singular value decomposition performed on the pores signed distance transform. We discuss possible applications of the present method, and how medoids and their associated mechanical response may be used to provide a natural basis for reduced-order models and hyper-reduction techniques, in which the mechanical effects of defects and structures are decorrelated (Ryckelynck et al. in C. R., Méc. 348(10–11):911–935, 2020). [ABSTRACT FROM AUTHOR]

Abstract Background In 2009, a novel influenza vaccine was distributed worldwide to combat the H1N1 influenza “swine flu” pandemic. However, antibodies induced by the vaccine display differences in their specificity and cross-reactivity dependent on pre-existing immunity. Here, we present a computational model that can capture the effect of pre-existing immunity on influenza vaccine responses. The model predicts the region of the virus hemagglutinin (HA) protein targeted by antibodies after vaccination as well as the level of cross-reactivity induced by the vaccine. We tested our model by simulating a scenario similar to the 2009 pandemic vaccine and compared the results to antibody binding data obtained from human subjects vaccinated with the monovalent 2009 H1N1 influenza vaccine. Results We found that both specificity and cross-reactivity of the antibodies induced by the 2009 H1N1 influenza HA protein were affected by the viral strain the individual was originally exposed. Specifically, the level of antigenic relatedness between the original exposure HA antigen and the 2009 HA protein affected antigenic-site immunodominance. Moreover, antibody cross-reactivity was increased when the individual’s pre-existing immunity was specific to an HA protein antigenically distinct from the 2009 pandemic strain. Comparison of simulation data with antibody binding data from human serum samples demonstrated qualitative and quantitative similarities between the model and real-life immune responses to the 2009 vaccine. Conclusion We provide a novel method to evaluate expected outcomes in antibody specificity and cross-reactivity after influenza vaccination in individuals with different influenza HA antigen exposure histories. The model produced similar outcomes as what has been previously reported in humans after receiving the 2009 influenza pandemic vaccine. Our results suggest that differences in cross-reactivity after influenza vaccination should be expected in individuals with different exposure histories.

Shape analysis is of great importance in many fields, such as computer vision, medical imaging, and computational biology. This analysis can be performed considering shapes as closed planar curves in the shape space. This approach has been used for the first time to obtain the morphological classification of erythrocytes in digital images of sickle cell disease considering the shape space S1, which has the property of being isometric to an infinite-dimensional Grassmann manifold of two-dimensional subspaces (Younes et al., 2008), without taking advantage of all the features offered by the elastic metric related to the possibility of stretching and bending of the curves. In this paper, we study this deformation in the shape space, S2, which is based on the representation of closed planar curves by means of the square-root velocity function (SRVF) (Srivastava et al., 2011), using the elastic metric of this space to obtain more efficient geodesics and geodesic lengths between planar curves. Supervised classification with this approach achieved an accuracy of 94.3%, classification using templates achieved 94.2% and unsupervised clustering in three groups achieved 94.7%, considering three classes of erythrocytes: normal, sickle, and with other deformations. These results are better than those previously achieved in the morphological analysis of erythrocytes and the method can be used in different applications related to the treatment of sickle cell disease, even in cases where it is necessary to study the process of evolution of the deformation, something that can not be done in a natural way in the feature space.

This paper proposes a new algorithm of behavior context awareness and step length estimation based on shape distance and remained features which is called H-features. We assume that the shape feature of the Inertial Measurement Unit (IMU) signal should represent the context, and the contexts be classified according to the shape similarity. In the shape feature extracting procedure, the H-features(translation, scale, and rotation) are excluded by the Helmert transformation. Because the step length is independent of the contexts and the shape features, the step length is related to the H-features. The theoretical assumptions are verified by an experiment. The relation of H-features and step lengths are trained by neural network and tested. The context classification accuracy is over 97%, and the root of mean square error (RMSE) of step length estimation is under 8.3% which is a better result than the conventional algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes a framework that takes a user provided free-hand sketch as input and realistic 3D shape as result. Specifically, the proposed approach retrieves 3D shape based on a sketch, and then a deformation procedure is carried out for a further consistency with the provided sketch. In the retrieval stage, a locality preserving view selection scheme is proposed, which suggests views that are mostly possible to be views when creating a sketch. The proposed criterion predicates the sketch views accurately while significantly reduces the number of views that need to be rendered. In the deformation stage, retrieved shapes are modified according to sketch. However, it is a tremendously difficult job as free-hand sketches always contain various kinds of drawing errors such as stroke jittering and asymmetry. Extracting plausible deformation from sketch while discarding undesirable drawing errors is difficult. To address these issues, we obtain the plausible deformation contained in sketch by exploring a shape space trained by a collection of shapes in the same category. Furthermore, we also develop a user interface with two different intuitive editing modes based on the shape space established. Experimental results show that more consistent 3D shapes with sketches can be obtained by our method. We also illustrate that our proposed method provides easy and smart ways for both a heuristic and a purposeful shape editing. [ABSTRACT FROM AUTHOR]

Background: In 2009, a novel influenza vaccine was distributed worldwide to combat the H1N1 influenza "swine flu" pandemic. However, antibodies induced by the vaccine display differences in their specificity and cross-reactivity dependent on pre-existing immunity. Here, we present a computational model that can capture the effect of pre-existing immunity on influenza vaccine responses. The model predicts the region of the virus hemagglutinin (HA) protein targeted by antibodies after vaccination as well as the level of cross-reactivity induced by the vaccine. We tested our model by simulating a scenario similar to the 2009 pandemic vaccine and compared the results to antibody binding data obtained from human subjects vaccinated with the monovalent 2009 H1N1 influenza vaccine. Results: We found that both specificity and cross-reactivity of the antibodies induced by the 2009 H1N1 influenza HA protein were affected by the viral strain the individual was originally exposed. Specifically, the level of antigenic relatedness between the original exposure HA antigen and the 2009 HA protein affected antigenic-site immunodominance. Moreover, antibody cross-reactivity was increased when the individual's pre-existing immunity was specific to an HA protein antigenically distinct from the 2009 pandemic strain. Comparison of simulation data with antibody binding data from human serum samples demonstrated qualitative and quantitative similarities between the model and real-life immune responses to the 2009 vaccine. Conclusion: We provide a novel method to evaluate expected outcomes in antibody specificity and cross-reactivity after influenza vaccination in individuals with different influenza HA antigen exposure histories. The model produced similar outcomes as what has been previously reported in humans after receiving the 2009 influenza pandemic vaccine. Our results suggest that differences in cross-reactivity after influenza vaccination should be expected in individuals with different exposure histories. [ABSTRACT FROM AUTHOR]

Shape space is an active research topic in computer vision and medical imaging fields. The distance defined in a shape space may provide a simple and refined index to represent a unique shape. This work studies the Wasserstein space and proposes a novel framework to compute the Wasserstein distance between general topological surfaces by integrating hyperbolic Ricci flow, hyperbolic harmonic map, and hyperbolic power Voronoi diagram algorithms. The resulting hyperbolic Wasserstein distance can intrinsically measure the similarity between general topological surfaces. Our proposed algorithms are theoretically rigorous and practically efficient. It has the potential to be a powerful tool for 3D shape indexing research. We tested our algorithm with human face classification and Alzheimer's disease (AD) progression tracking studies. Experimental results demonstrated that our work may provide a succinct and effective shape index. [ABSTRACT FROM AUTHOR]

Summary: We introduce generalized partially linear models with covariates on Riemannian manifolds. These models, like ordinary generalized linear models, are a generalization of partially linear models on Riemannian manifolds that allow for scalar response variables with error distribution models other than a normal distribution. Partially linear models are particularly useful when some of the covariates of the model are elements of a Riemannian manifold, because the curvature of these spaces makes it difficult to define parametric models. The model was developed to address an interesting application: the prediction of children's garment fit based on three‐dimensional scanning of their bodies. For this reason, we focus on logistic and ordinal models and on the important and difficult case where the Riemannian manifold is the three‐dimensional case of Kendall's shape space. An experimental study with a well‐known three‐dimensional database is carried out to check the goodness of the procedure. Finally, it is applied to a three‐dimensional database obtained from an anthropometric survey of the Spanish child population. A comparative study with related techniques is carried out. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for handling the effects of shape outliers in statistical shape analysis. Usually performed by a variant of classical principal component analysis (PCA), variability analysis may be highly affected by erroneous shapes. Principal components may thus imply aberrant modes, while eigenshapes may not accurately describe variability in a given set of shapes. Our robust analysis is performed using an elastic metric associated with the square-root velocity representation of shapes. This elastic shape analysis allows shape variability to be described with natural and intuitive deformations. The proposed method based on shape outlier detection applies the shape restoration procedure to rectify aberrant shapes. The resultant components are thus obtained from a tangent PCA on the restored database. By performing experiments based on MPEG-7 and HAND databases, we demonstrate that the proposed scheme is effective for shape variability analysis in the presence of outlying shapes. Our method is then compared with two existing schemes for robust data variability analysis: minimum covariance determinant-based PCA and projection pursuit-based PCA. [ABSTRACT FROM AUTHOR]

The Square Root Normal Field (SRNF), introduced by Jermyn et al. in [5], provides a way of representing immersed surfaces in R3, and equipping the set of these immersions with a "distance function" (to be precise, a pseudometric) that is easy to compute. Importantly, this distance function is invariant under reparametrizations (i.e., under self-diffeomorphisms of the domain surface) and under rigid motions of R3. Thus, it induces a distance function on the shape space of immersions, i.e., the space of immersions modulo reparametrizations and rigid motions of R3. In this paper, we give examples of the degeneracy of this distance function, i.e., examples of immersed surfaces (some closed and some open) that have the same SRNF, but are not the same up to reparametrization and rigid motions. We also prove that the SRNF does distinguish the shape of a standard sphere from the shape of any other immersed surface, and does distinguish between the shapes of any two embedded strictly convex surfaces. [ABSTRACT FROM AUTHOR]

Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouv\'e, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61-84; A. Trouv\'e and L. Younes, Found. Comput. Math., 5 (2005), pp. 173-198] casts this problem in the frame of Riemannian geometry and geodesic paths between images. The associated metric in the space of images incorporates dissipation caused by a viscous flow transporting image intensities and its variations along motion paths. In many applications, images are maps from the image domain into a manifold (e.g., in diffusion tensor imaging (DTI), the manifold of symmetric positive definite matrices with a suitable Riemannian metric). In this paper, we propose a generalized metamorphosis model for manifold-valued images, where the range space is a finite-dimensional Hadamard manifold. A corresponding time discrete version was presented in [S. Neumayer, J. Persch, and G. Steidl, SIAM J. Imaging Sci., 11 (2018), pp. 1898-1930] based on the general variational time discretization proposed in [B. Berkels, A. Effland, and M. Rumpf, SIAM J. Imaging Sci., 8 (2015), pp. 1457-1488]. Here, we prove the Mosco--convergence of the time discrete metamorphosis functional to the proposed manifold-valued metamorphosis model, which implies the convergence of time discrete geodesic paths to a geodesic path in the (time continuous) metamorphosis model. In particular, the existence of geodesic paths is established. In particular, the existence of geodesic paths is established. In fact, images as maps into Hadamard manifold are not only relevant in applications, but it is also shown that the joint convexity of the distance function--which characterizes Hadamard manifolds--is a crucial ingredient to establish existence of the metamorphosis model. [ABSTRACT FROM AUTHOR]

Shape analysis is of great importance in many fields, such as computer vision, medical imaging, and computational biology. This analysis can be performed considering shapes as closed planar curves in the shape space. This approach has been used for the first time to obtain the morphological classification of erythrocytes in digital images of sickle cell disease considering the shape space S1, which has the property of being isometric to an infinite-dimensional Grassmann manifold of two-dimensional subspaces (Younes et al., 2008), without taking advantage of all the features offered by the elastic metric related to the possibility of stretching and bending of the curves. In this paper, we study this deformation in the shape space, S2, which is based on the representation of closed planar curves by means of the square-root velocity function (SRVF) (Srivastava et al., 2011), using the elastic metric of this space to obtain more efficient geodesics and geodesic lengths between planar curves. Supervised classification with this approach achieved an accuracy of 94.3%, classification using templates achieved 94.2% and unsupervised clustering in three groups achieved 94.7%, considering three classes of erythrocytes: normal, sickle, and with other deformations. These results are better than those previously achieved in the morphological analysis of erythrocytes and the method can be used in different applications related to the treatment of sickle cell disease, even in cases where it is necessary to study the process of evolution of the deformation, something that can not be done in a natural way in the feature space. [ABSTRACT FROM AUTHOR]

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper, we study a general representation of shapes as currents, which are based on linear spaces and are suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the H - s norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element-based discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples. [ABSTRACT FROM AUTHOR]

We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.