Geometric Representation Research Articles

Simplicits: Mesh-Free, Geometry-Agnostic Elastic Simulation

The proliferation of 3D representations, from explicit meshes to implicit neural fields and more, motivates the need for simulators agnostic to representation. We present a data-, mesh-, and grid-free solution for elastic simulation for any object in any geometric representation undergoing large, nonlinear deformations. We note that every standard geometric representation can be reduced to an occupancy function queried at any point in space, and we define a simulator atop this common interface. For each object, we fit a small implicit neural network encoding spatially varying weights that act as a reduced deformation basis. These weights are trained to learn physically significant motions in the object via random perturbations. Our loss ensures we find a weight-space basis that best minimizes deformation energy by stochastically evaluating elastic energies through Monte Carlo sampling of the deformation volume. At runtime, we simulate in the reduced basis and sample the deformations back to the original domain. Our experiments demonstrate the versatility, accuracy, and speed of this approach on data including signed distance functions, point clouds, neural primitives, tomography scans, radiance fields, Gaussian splats, surface meshes, and volume meshes, as well as showing a variety of material energies, contact models, and time integration schemes.

Il disegno e il rilievo nell’età del metaverso: come prima, più di prima

In the Metaverse, our relationship with space will change. We will be able to immerse our- selves in this space physically, emotionally, and spiritually. The line between real and im- aginary will become blurred, allowing us to experience new sensations. As creators in this space, we must consider how we represent reality. Do we depict what we imagine alongside what already exists, or do we need to learn how to convey our emotions, opinions, and ex- periences? How do we do this? Perhaps the answer is to proceed as before, but even more so Historically, in Drawing and Surveying, there has always been an attempt to bring these aspects to the forefront, beyond architecture, beyond the city. The tools may have been in- adequate, but this “moral obligation” has probably always been drawers and surveyors. An illustrative example is when Monge’s methods were being defined scientifically, coinciding with encounters with vastly different people and cultures that were challenging to describe with words. Denon’s drawings were instrumental in helping Europeans immerse them- selves in Middle Eastern culture, showcasing architecture and the people, customs, and aesthetics. It attempted to blend the geometric representation of space with the complexity of an entire “world”. This comprehensive survey aimed to capture the emotional and cul- tural aspects using the most advanced methods and techniques available at the time, even incorporating emerging photography. This is perhaps the role that Drawing and Surveying are expected to play in the age of the Metaverse: to facilitate the expansion of describing the intangible through new systems driven by the same intentions. Today, just like in the past, we need to embrace this task with an even greater emphasis.

On the isogeometric analysis of vibration and buckling of bioinspired composite plates using inverse hyperbolic shear deformation theory

Inspired by the remarkable energy absorption and damage tolerance capabilities observed in mantis shrimp crustaceans, this study delves into the intricate realm of biological composites with helicoidal structures. However, the effect of the relative orientation of fibers within the helicoidal structure matrix on the buckling and vibration characteristics still warrants comprehensive investigation for the better design of the bioinspired structures. For the first time, a Non-Uniform Rational B-Splines (NURBS)-based isogeometric analysis (IGA) framework termed NURBio is proposed to investigate the vibration and buckling characteristics of bioinspired laminated composite structures. The framework employs NURBS basis functions for exact complex geometric representation and field approximation and offers superior computational accuracy and efficiency as compared to conventional finite element method (FEM). The present analysis is underpinned by an inverse hyperbolic shear deformation theory (IHSDT) capable of accurately capturing the interlaminar stress distribution. This research investigates the performance of various bioinspired layup configurations encompassing recursive, exponential, semicircular, and Fibonacci helicoidal designs and compares them with those of unidirectional, and cross-ply laminates. A series of numerical examples on the buckling and vibration response of bioinspired composite plates are presented to demonstrate the versatility and efficacy of the proposed isogeometric framework. The influence of different layups, loading and boundary conditions, and geometric properties on the response of different bioinspired plate structures are meticulously examined. The study offers valuable insights into the design and optimization of high-performance bioinspired structures.

Geometric epitope and paratope prediction.

Identifying the binding sites of antibodies is essential for developing vaccines and synthetic antibodies. In this article, we investigate the optimal representation for predicting the binding sites in the two molecules and emphasize the importance of geometric information. Specifically, we compare different geometric deep learning methods applied to proteins' inner (I-GEP) and outer (O-GEP) structures. We incorporate 3D coordinates and spectral geometric descriptors as input features to fully leverage the geometric information. Our research suggests that different geometrical representation information is useful for different tasks. Surface-based models are more efficient in predicting the binding of the epitope, while graph models are better in paratope prediction, both achieving significant performance improvements. Moreover, we analyze the impact of structural changes in antibodies and antigens resulting from conformational rearrangements or reconstruction errors. Through this investigation, we showcase the robustness of geometric deep learning methods and spectral geometric descriptors to such perturbations. The python code for the models, together with the data and the processing pipeline, is open-source and available at https://github.com/Marco-Peg/GEP.

Information geometry analysis example for absolute and relative transmission loss in a shallow ocean

The model manifold, an information geometry tool, is a geometric representation of a model that can quantify the expected information content of modeling parameters. For a normal-mode sound propagation model in a shallow ocean environment, transmission loss (TL) is calculated for a vertical line array and model manifolds are constructed for both absolute and relative TL. For the example presented in this paper, relative TL yields more compact model manifolds with seabed environments that are less statistically distinguishable than manifolds of absolute TL. This example illustrates how model manifolds can be used to improve experimental design for inverse problems.

Bouncing behavior of anisotropic cosmological model

We study locally rotationally symmetric (LRS) Bianchi type-I cosmological model in the presence of general theory of relativity. The source for energy–momentum tensor is assumed a perfect fluid. The field equations have been solved by using special form of the average scale factor [Formula: see text], where [Formula: see text] is initial cosmic time and [Formula: see text] is constant, proposed by Cai et al. [J. High Energy Phys. 710, 071 (2007)]. The physical properties and bouncing behavior of the model are discussed through graphical representation. The violation of strong energy condition (SEC) further supports the behavior of accelerated expansion of the Universe. The stability analysis of the model is discussed through geometrical representation.

Joint stereo 3D object detection and implicit surface reconstruction

We present a new learning-based framework S-3D-RCNN that can recover accurate object orientation in SO(3) and simultaneously predict implicit rigid shapes from stereo RGB images. For orientation estimation, in contrast to previous studies that map local appearance to observation angles, we propose a progressive approach by extracting meaningful Intermediate Geometrical Representations (IGRs). This approach features a deep model that transforms perceived intensities from one or two views to object part coordinates to achieve direct egocentric object orientation estimation in the camera coordinate system. To further achieve finer description inside 3D bounding boxes, we investigate the implicit shape estimation problem from stereo images. We model visible object surfaces by designing a point-based representation, augmenting IGRs to explicitly address the unseen surface hallucination problem. Extensive experiments validate the effectiveness of the proposed IGRs, and S-3D-RCNN achieves superior 3D scene understanding performance. We also designed new metrics on the KITTI benchmark for our evaluation of implicit shape estimation.

Using BIM for the development of accessibility

Accessibility in today's reality is becoming an increasingly important issue as it affects every individual. We are observing the progressive ageing of the Polish population. By applying the principles of universal design, all needs related to mobility, vision, hearing, and other issues are met, thereby creating inclusive spaces that eliminate social exclusion and enhance the quality of life. This article aims to demonstrate the usefulness of BIM in building urban accessibility. For this purpose, a point cloud acquired from laser scanning of a single-family building was utilised. Based on this, a digital BIM model of the actual building was created in Revit and subsequently modernised. This modernisation aimed to remove barriers from the building, as stipulated in the relevant regulation. The paper refers to the theory of ageing in place, a response to the steady increase in average age. The BIM model is not merely a geometric representation of the building, but a digital reconstruction of the object using elements equivalent to those used in reality, allowing for efficient data management.

Geometric Representation of Classes of Concave Functions and Duality

Using a natural representation of a 1/s-concave function on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} as a convex set in Rd+1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{d+1},$$\\end{document} we derive a simple formula for the integral of its s-polar. This leads to convexity properties of the integral of the s-polar function with respect to the center of polarity. In particular, we prove that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions for s-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.

Learning Effective Geometry Representation from Videos for Self-Supervised Monocular Depth Estimation

Recent studies on self-supervised monocular depth estimation have achieved promising results, which are mainly based on the joint optimization of depth and pose estimation via high-level photometric loss. However, how to learn the latent and beneficial task-specific geometry representation from videos is still far from being explored. To tackle this issue, we propose two novel schemes to learn more effective representation from monocular videos: (i) an Inter-task Attention Model (IAM) to learn the geometric correlation representation between the depth and pose learning networks to make structure and motion information mutually beneficial; (ii) a Spatial-Temporal Memory Module (STMM) to exploit long-range geometric context representation among consecutive frames both spatially and temporally. Systematic ablation studies are conducted to demonstrate the effectiveness of each component. Evaluations on KITTI show that our method outperforms current state-of-the-art techniques.

Analytical solution of the (2+1)-dimensional Zoomeron equation by rational sine-Gordon Method

The current study is about the solution of the Zoomeron equation, one of the important models of mathematics and physics. In this study, the rational Sine-Gordon expansion method (RSGEM) is used to obtain various analytical solutions of the model. Compared to other methods, this method is quite effective and the desired results were obtained. Although there are many analytical solutions to the model used in the literature, we present rational type solutions for the first time with this method. We obtained rational hyperbolic function solutions, and also classified all soliton solutions (kink-like, kink, singular kink, anti-kink, dark, bright). In addition, geometric representations of the solutions in two-, and three-dimensional space and contour shape are made with the Mathematica software program.

Spheroid models to elaborate the broken symmetry and equivalent volume of molecules in crystalline phase.

Dense packing of particles has provided powerful models to elaborate the important structural features of matter in various systems such as liquid, glassy, and crystalline phases. The simplest sphere packing models can represent and capture salient properties of the building blocks for covalent, metallic, and ionic crystals; it, however, becomes insufficient to reflect the broken symmetry of the commonly anisotropic molecules in molecular crystals. Here, we develop spheroid models with a minimal degree of anisotropy, which serve as a simple geometrical representation for a rich spectrum of molecules-including both isotropic and anisotropic, convex and concave ones-in crystalline phases. Our models are determined via an inverse packing approach: Given a molecular crystal, an optimal spheroid model is constructed using a contact diagram, which depicts the packing relationship between neighboring molecules within the crystal. The spheroid models are capable of accurately capturing the broken symmetry and characterizing the equivalent volume of molecules in the crystalline phases. Moreover, our model retrieves such molecular information from low-quality x-ray diffraction data with poorly resolved structures, and by using soft spheroids, it can also describe the packing behavior in cocrystals.

Detecting local perturbations of networks in a latent hyperbolic embedding space.

This paper introduces two novel scores for detecting local perturbations in networks. For this, we consider a non-Euclidean representation of networks, namely, their embedding onto the Poincaré disk model of hyperbolic geometry. We numerically evaluate the performances of these scores for the detection and localization of perturbations on homogeneous and heterogeneous network models. To illustrate our approach, we study latent geometric representations of real brain networks to identify and quantify the impact of epilepsy surgery on brain regions. Results suggest that our approach can provide a powerful tool for representing and analyzing changes in brain networks following surgical intervention, marking the first application of geometric network embedding in epilepsy research.

Algebraic and toroidal representation of the genetic code

The genetic code is a set of regulatory principles that control the translation of information encoded in messenger RNA (mRNA) into a sequence of amino acids. This study proposes a model starting from the relationships between its physicochemical properties of nucleobase in the codons. We employed a binary metric and the Kronecker product to represent the physicochemical properties of nucleobases in a vector space. This state space has a hierarchical order, with self-similarity and symmetry properties in the hydrogen bonds of triplets. The state space can be mapped under linear transformations to a toroidal geometry with allometric properties. Furthermore, this geometric representation highlights a charge symmetry that exists in amino acids and in the distribution of essential and non-essential amino acids. These results describe a state space between codon-anticodon interactions that can be interpreted as Bell states. This suggests that there is a quantum phenomenon involved in the mechanisms of information storage in DNA. In addition, toroidal geometry can be used to represent the sequences of codons of the mRNA that encode the sequence of amino acids of the proteins to find similarities or homologies between evolutionary related species.

A geometric algebra-based approach for myoelectric pattern recognition control and faster prosthesis recalibration

ObjectiveAlthough many advancements have been made on myoelectric pattern-recognition, the control of poly-articulated upper-limb prostheses remains insufficiently robust. Electrode-shift, sweat or fatigue degrade the performance of classifiers over time, resulting in unfruitful device usage and frequent re-calibration. To tackle this issue, here we introduce two models − µP6 and µP8 − that combine Geometric Algebra with nearest-neighbor classification. We aim at reducing both the necessary training data and training time and, unlike most current state-of-the-art algorithms, we exploit an alternative geometric representation (and visualization) of the EMG signal as different polygons for different types of gestures, facilitating the explanation of the decision-making process to a layman. Moreover, we explore four abstention strategies to reduce the number of misclassifications. MethodsWe perform an offline analysis on two datasets, alongside two other standard models: nonlinear logistic regression (NLR) and linear discriminant analysis (LDA). ResultsEven with few training data, the proposed algorithms achieve high F1-scores (>0.95), significantly higher or non-significantly different from the values obtained with NLR and LDA, while maintaining relatively low abstentions rates and training times (<2 ms). ConclusionThe proposed algorithms allow to reduce the amount of training data and training times without compromising recognition rates. SignificanceThe proposed algorithms may contribute for a faster prosthesis re-calibration procedure while allowing to re-gain high recognition rates. Furthermore, the decision-making process is explainable and interpretable, potentially improving user trust and acceptance.

A novel geometric method based on conformal geometric algebra applied to the resection problem in two and three dimensions

This paper introduces a novel method for solving the resection problem in two and three dimensions based on conformal geometric algebra (CGA). Advantage is taken because of the characteristics of CGA, which enables the representation of points, lines, planes, and volumes in a unified mathematical framework and offers a more intuitive and geometric understanding of the problem, in contrast to existing purely algebraic methods. Several numerical examples are presented to demonstrate the efficacy of the proposed method and to compare its validity with established techniques in the field. Numerical simulations indicate that our vector geometric algebra implementation is faster than the best-known algorithms to date, suggesting that the proposed GA-based methods can provide a more efficient and comprehensible solution to the two- and three-dimensional resection problem, paving the way for further applications and advances in geodesy research. Furthermore, the method’s emphasis on graphical and geometric representation makes it particularly suitable for educational purposes, allowing the reader to grasp the concepts and principles of resection more effectively. The proposed method has potential applications in a wide range of other fields, including surveying, robotics, computer vision, or navigation.

Text-to-Microstructure Generation Using Generative Deep Learning.

Designing novel materials is greatly dependent on understanding the design principles, physical mechanisms, and modeling methods of material microstructures, requiring experienced designers with expertise and several rounds of trial and error. Although recent advances in deep generative networks have enabled the inverse design of material microstructures, most studies involve property-conditional generation and focus on a specific type of structure, resulting in limited generation diversity and poor human-computer interaction. In this study, a pioneering text-to-microstructure deep generative network (Txt2Microstruct-Net) is proposed that enables the generation of 3D material microstructures directly from text prompts without additional optimization procedures. The Txt2Microstruct-Net model is trained on a large microstructure-caption paired dataset that is extensible using the algorithms provided. Moreover, the model is sufficiently flexible to generate different geometric representations, such as voxels and point clouds. The model's performance is also demonstrated in the inverse design of material microstructures and metamaterials. It has promising potential for interactive microstructure design when associated with large language models and could be a user-friendly tool for material design anddiscovery.

Text-to-3D Non-Deformable Animation: A Proof-of-Concept System using Natural Language Descriptions

The ability to generate 3D animations from natural language descriptions has significant potential in various fields, including entertainment, education, and computer-aided design. This research paper presents a novel proof-of-concept system that transforms text prompts into complete and simple 3D animations. The system architecture comprises two main components: extraction and scene creation. In the extraction phase, a transformer model identifies the necessary objects from the text prompt, generates their geometric representations as vertices and edges, and creates a script describing the animations. The scene creation phase involves constructing the 3D objects, adding them to the scene, and rendering the animation based on the generated script. The implementation details, including the technologies and tools used, code snippets, and challenges faced, are thoroughly discussed. Qualitative and quantitative evaluations are conducted to assess the system's performance, highlighting its strengths and limitations compared to existing text-to-animation approaches. Potential applications of the proposed system are explored, ranging from rapid prototyping of 3D animations to interactive storytelling and educational tools. Furthermore, future research directions, such as improving scalability, generalization, and real-world deployment, are outlined. The research presented in this paper contributes to the emerging field of text-to-animation generation and paves the way for further advancements in transforming natural language descriptions into engaging and immersive 3D experiences.

A hierarchical occupancy network with multi‐height attention for vision‐centric 3D occupancy prediction

AbstractThe precise geometric representation and ability to handle long‐tail targets have led to the increasing attention towards vision‐centric 3D occupancy prediction, which models the real world as a voxel‐wise model solely through visual inputs. Despite some notable achievements in this field, many prior or concurrent approaches simply adapt existing spatial cross‐attention (SCA) as their 2D–3D transformation module, which may lead to informative coupling or compromise the global receptive field along the height dimension. To overcome these limitations, we propose a hierarchical occupancy (HierOcc) network featuring our innovative height‐aware cross‐attention (HACA) and hierarchical self‐attention (HSA) as its core modules to achieve enhanced precision and completeness in 3D occupancy prediction. The former module enables 2D–3D transformation, while the latter promotes voxels’ intercommunication. The key insight behind both modules is our multi‐height attention mechanism which ensures each attention head corresponds explicitly to a specific height, thereby decoupling height information while maintaining global attention across the height dimension. Extensive experiments show that our method brings significant improvements compared to baseline and surpasses all concurrent methods, demonstrating its superiority.

Nonperturbative thermodynamic extrinsic curvature of the anyon gas

Thermodynamic extrinsic curvature is a new mathematical tool in thermodynamic geometry. By using the thermodynamic extrinsic curvature, one may obtain a more complete geometric representation of the critical phenomena and thermodynamics. We introduce nonperturbative thermodynamic extrinsic curvature of an ideal two-dimensional gas of anyons. Using extrinsic curvature, we find new fixed points in nonperturbative thermodynamics of the anyon gas that particles behave as semions. Here, we investigate the critical behavior of thermodynamic extrinsic curvature of two-dimensional Kagome Ising model near the critical point [Formula: see text] in a constant magnetic field and show that it behaves as [Formula: see text] with [Formula: see text], where [Formula: see text] denotes the critical exponent of the specific heat. Then, we consider the three-dimensional spherical model and show that the scaling behavior is [Formula: see text], where [Formula: see text]. Finally, using a general argument, we show that extrinsic curvature [Formula: see text] has two different scaling behaviors for positive and negative [Formula: see text]. For [Formula: see text], our results indicate that [Formula: see text]. However, for [Formula: see text], we found a different scaling behavior, where [Formula: see text].