Search

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 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.

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 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]

Copyright of Journal of Shanghai University / Shanghai Daxue Xuebao is the property of Journal of Shanghai University (Natural Sciences) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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.