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2024 | Book

Advanced Computational Applications of Geometric Algebra

ICACGA 2022, Denver, CO, USA, October 2–5

Editors: David William Honorio Araujo Da Silva, Dietmar Hildenbrand, Eckhard Hitzer

Publisher: Springer Nature Switzerland

Book Series : Springer Proceedings in Mathematics & Statistics

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About this book

How Geometric Algebra can naturally serve for constructing solutions for pattern recognition, machine learning, data compression, games, robotics, quantum computing, data encoding, to cite a few. Moreover, there is ample evidence that further research on GA and related areas can significantly expand the number of real-world applications in a wide variety of areas. A mathematical system that is very easy to handle, highly robust and superior performance for engineering applications. Good thematic introduction for engineers and researchers new to the subject. Extensive illustrations and code examples. Thematically well structured with many hands on examples. Learning about GA and how to use it for daily tasks in engineering research and development.

Table of Contents

Frontmatter
Introduction to Geometric Algebra
Abstract
Geometric algebra was initiated by W.K. Clifford over 140 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This introduction explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, the popular conformal model, and projective geometric algebra. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc.
Eckhard Hitzer, Dietmar Hildenbrand
Physical Geometry by Plane-Based Geometric Algebra
Abstract
Plane-based geometric algebra (PGA) offers a way to represent Euclidean motions that is directly built on the primitives of affine geometry, and thus provides a seamless framework for objects and their movement. We show how this universal treatment includes the actual physical motions of objects with mass under forces and torques. PGA unifies the linear and angular aspects compactly, and in a coordinate-free manner; inertia maps become simply additive (without displacement terms). We demonstrate the simple equations and straightforward numerical code that result. We show explicitly how to embed the vector-based concepts of the usual classical Newtonian mechanics into the 3D PGA framework, and why it is advantageous to do so.
Leo Dorst, Steven De Keninck
Inner Product of Two Oriented Points in Conformal Geometric Algebra in Detail
Abstract
We study in full detail the inner product of oriented points in conformal geometric algebra and its geometric meaning. The notion of oriented point is introduced and the inner product of two general oriented points is computed, analyzed (including symmetry) and graphed in terms of point to point distance, and angles between the distance vector and the local orientation planes of the two points. Seven examples illustrate the results obtained. Finally, the results are extended from dimension three to arbitrary dimensions n.
Eckhard Hitzer
Line–Cyclide Intersection and Colinear Point Quadruples in the Double Conformal Model
Abstract
In this paper, we look at using the double conformal model for ray tracing and for tool positioning in Computer Numerically Controlled (CNC) machining. In particular, we explore the intersection of a line with a cyclide in the double conformal model, and how to extract the four points from the resulting colinear point quadruple. Further, we show how to directly construct a colinear point quadruple from four points, and we show how to find the line containing the points of a colinear point quadruple. We also briefly touch on barycentric coordinates and affine combinations in DCGA. Finally, we discuss applications of the double conformal model in ray tracing and in CNC machining.
Huijing Yao, Stephen Mann, Qinchuan Li
A Geometric Algebra Solution to the Absolute Orientation Problem
Abstract
We show here an alternative solution to the fundamental photogrammetric problem of determining Absolute Orientation by reformulating it in Geometric Algebra. Our work is centered on expressing rotors using Characteristic Multivectors. We compare our algorithm under different point cloud geometries and Gaussian Noise levels with the standard least-squares, Singular Value Decomposition solution for fitting 3D point clouds that are related via 1–1 correspondence developed by Arun in 1987. As with Arun’s formulation, the proposed solution is based solely on the point cloud geometries and does not use any extra information that the data may have or any noise filtering mechanisms. The new algorithm effectively follows that of Arun but replaces the rotation matrix calculation step with a rotor-calculation step. The rotation matrix can easily be recovered from the rotor. We show that our algorithm has very similar performance to the SVD method, indicating that it does not suffer from adverse numerical effects. We also indicate how it might be used in the future, where having this closed-form, non-iterative solution might prove beneficial.
Charalampos Matsantonis, Joan Lasenby
Geometric Algebra Models of Proteins for Three-Dimensional Structure Prediction: A Detailed Analysis
Abstract
A protein can be regarded as a chain of amino acids with a unique folding in the three-dimensional (3D) space. Knowing the folding of a protein is highly desirable since the folding controls the protein properties. However, determining it experimentally is expensive and time consuming: estimating the 3D structure of a protein computationally—known as protein structure prediction (PSP)—can overcome these issues. In this paper, we explore the advantage of using Geometric Algebra (GA) to model proteins for PSP applications. In particular, we employ GA to define a metric over the orientations of the amino acids in the chain. We then encode this metric in matrix form and show how patterns in these images mirror folding patterns of proteins. Lastly, we prove that this metric is predictable through a standard deep learning (DL) architecture for the inference of pairwise amino acid distances. We demonstrate that GA is a powerful tool for obtaining a compact representation of the protein geometry with potential to improve the prediction accuracy of standard PSP pipelines.
Alberto Pepe, Joan Lasenby, Pablo Chacon
GAAlign: Robust Sampling-Based Point Cloud Registration Using Geometric Algebra
Abstract
Geometrical 3D data is often represented in form of point clouds. A common problem is the registration of point clouds with shared underlying geometry, for example to align two 3D scans. This work presents GAAlign, a new formulation of a geometric algebra (GA) based algorithm that aims to solve this problem. While the algorithm itself is a gradient-descent-based approach, the implementation takes advantage of GAALOP, which had to be extended with a specific, so far unsupported GA, namely projective GA. The proposed new robust registration algorithm uses a geometric-algebra-based motor estimation algorithm in the context of a mini-batch-gradient-descent-inspired algorithmic structure and achieves state-of-the-art results. When using synthetically disturbed input data the results show, that GAAlign either outperforms other used algorithms (outliers) or is comparable to the best (Gaussian noise) while having a significantly better runtime as soon as the number of correspondences increases. When used in a real world pipeline, GAAlign also performs on the same level or above compared to state-of-the-art algorithms.
Kai A. Neumann, Dietmar Hildenbrand, Florian Stock, Christian Steinmetz, Maximilian Michel
Geometric Algebra: A Possible Foundation for Digital Twin Modeling and Analysis—A Case Study with PIR Scene
Abstract
Digital twin (DT) refers to a virtual representation of an object or system that spans its entire life cycle, constantly updating with real-time data, and utilizes simulation, machine learning, and reasoning to aid decision-making. However, the complexity of the real world and the urgent need for seamless integration between the physical and virtual spaces have posed challenges in the development and implementation of DT. This is primarily due to the separation of expression and analysis in the traditional modeling approaches. Developing the fundamental mathematical space of DT is a important and critical topic. These problems can be addressed by leveraging Geometric Algebra (GA), which enables effective expression and computation in multidimensional space. The paper studies the concept of DT, employs the essential principles of GA as a tool, and proposes the DT’s modeling and analysis methods. To demonstrate the formulation and construction of DT, a typical scenario involving multi-factor coupling of passive infrared sensors (PIR) is presented as an example. The results showcase the effectiveness of the proposed approach in simulating human-sensor interactions, capturing real-space reaction records, and successfully deriving pedestrian trajectories from PIR data. The study presented in this paper offers valuable insights into how to build DT for complex scenarios and also sheds new light on analyzing human behavior using PIR technology.
Wen Luo, Yilei Yin, Binghuang Pan, Chunye Zhou, Zhaoyuan Yu, Linwang Yuan
A Spinor Model for Cascading Two-Port Networks in Conformal Geometric Algebra
Abstract
Building on the work in [1], this paper shows how Conformal Geometric Algebra (CGA) can be used to model an arbitrary two-port network as a rotation in four dimensional Minkowski space, known as a spinor. This spinor representation is analogous to the scattering transfer matrix in conventional microwave network theory, but has a geometric interpretation. Just as the scattering transfer matrix is parameterized by scattering parameters, so is the spinor model constructed herein. (The direct geometric model of the scattering matrix itself is a different problem not discussed in this paper.) Techniques to translate from two-port scattering matrix data in and out of spinor form are given. Once the translation is laid out, geometric interpretations are proposed for the physical properties of reciprocity, loss, and symmetry and some mathematical groups are identified. Methods to decompose a network into various sub-networks, are given. An example application of interpolating a two-port network is provided, demonstrating an advantage of the spinor model. Since rotations in four dimensional Minkowski space are Lorentz transformations, this model opens up the field of network theory to physicists familiar with relativity, and vice versa. The results of this paper have been numerically tested for consistency using the open-source clifford python package [13].
Alex Arsenovic
Clifford Convolutional Neural Networks: Concepts, Implementation, and an Application for Lymphoblast Image Classification
Abstract
This chapter presents the basic concepts of convolutional neural network (CNN) models on Clifford algebras, referred to as Clifford CNNs. The basic building blocks, such as dense and convolutional layers, are briefly addressed. Besides the mathematical formulation, this chapter describes how to implement Clifford CNNs using standard deep-learning libraries. It also presents an application of Clifford CNNs to a medical image classification task, namely the diagnosis of acute lymphoblastic leukemia (ALL). ALL is a type of cancer in the bloodstream characterized by malformed lymphocytes called lymphoblasts. The image classification task aims to discriminate healthy cells from lymphoblasts. Corroborating with previous results reported in the literature, Clifford CNNs outperform real-valued networks of equivalent size in this application. Precisely, the real-valued and a Clifford CNN achieved an average accuracy of 94.60% and 97.02%, respectively. Moreover, we present smaller versions of Clifford CNNs with roughly 75% fewer parameters, yielding a 96.50% average accuracy. The results reported in this work are comparable to high-end models in the literature despite having several orders of magnitude fewer parameters.
Guilherme Vieira, Marcos Eduardo Valle, Wilder Lopes
Geometric Algebra Algorithm Code Optimised by GAALOP Executing on a Simulated Memristor Crossbar Array
Abstract
GAALOP (Geometric Algebra ALgorithms OPtimizer) is software designed to optimize the execution of multiple steps in a Geometric Algebra algorithm by analysing the steps as a whole and precompiling the intermediate code generated. The output is provided in a format that allows it to target popular software languages and systems such as C++, Python, Java, MATLAB, Mathematica and parallel platforms such as OpenCL, CUDA, FPGA and many more.We present a method of executing Geometric Algebra algorithms using simulated memristor crossbar arrays to perform dot product calculations.
Dietmar Hildenbrand, Ed Saribatir, Atilio Morillo Piña, Wilder Bezerra Lopes, Frederic von Wegner, Peter Storey, Zheng Yan, Shiping Wen, Matthew Arnold
Metadata
Title
Advanced Computational Applications of Geometric Algebra
Editors
David William Honorio Araujo Da Silva
Dietmar Hildenbrand
Eckhard Hitzer
Copyright Year
2024
Electronic ISBN
978-3-031-55985-3
Print ISBN
978-3-031-55984-6
DOI
https://doi.org/10.1007/978-3-031-55985-3

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