562 research outputs found

    DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling

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    This paper explores a fully unsupervised deep learning approach for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We use the Siamese configuration to train a neural network to solve the problem of least squares multidimensional scaling for generating maps that approximately preserve geodesic distances. By training with only a few landmarks, we show a significantly improved local and nonlocal generalization of the isometric mapping as compared to analogous non-parametric counterparts. Importantly, the combination of a deep-learning framework with a multidimensional scaling objective enables a numerical analysis of network architectures to aid in understanding their representation power. This provides a geometric perspective to the generalizability of deep learning.Comment: 10 pages, 11 Figure

    Project Elements: A computational entity-component-system in a scene-graph pythonic framework, for a neural, geometric computer graphics curriculum

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    We present the Elements project, a computational science and computer graphics (CG) framework, that offers for the first time the advantages of an Entity-Component-System (ECS) along with the rapid prototyping convenience of a Scenegraph-based pythonic framework. This novelty allows advances in the teaching of CG: from heterogeneous directed acyclic graphs and depth-first traversals, to animation, skinning, geometric algebra and shader-based components rendered via unique systems all the way to their representation as graph neural networks for 3D scientific visualization. Taking advantage of the unique ECS in a a Scenegraph underlying system, this project aims to bridge CG curricula and modern game engines, that are based on the same approach but often present these notions in a black-box approach. It is designed to actively utilize software design patterns, under an extensible open-source approach. Although Elements provides a modern, simple to program pythonic approach with Jupyter notebooks and unit-tests, its CG pipeline is not black-box, exposing for teaching for the first time unique challenging scientific, visual and neural computing concepts.Comment: 8 pages, 8 figures, 2 listings, submitted to EuroGraphics 2023 education trac

    Geometric Clifford Algebra Networks

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    We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r)\mathrm{Pin}(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates\textit{geometric templates} that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods

    Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment

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    International audienceWe introduce the quadric conformal geometric algebra (QCGA) inside the algebra of R 9,6. In particular, this paper presents how three-dimensional quadratic surfaces can be defined by the outer product of conformal geometric algebra points in higher dimensions, or alternatively by a linear combination of basis vectors with coefficients straight from the implicit quadratic equation. These multivector expressions code all types of quadratic surfaces in arbitrary scale, location, and orientation. Furthermore, we investigate two types of definitions of axis aligned quadric surfaces, from contact points and dually from linear combinations of R 9,6 basis vectors
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