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Geometric Neural Networks for Object Recognition
Published in Nancy Arana Daniel, Alma Y. Alanis, Carlos Lopez Franco, Neural Networks for Robotics: An Engineering Perspective, 2019
Nancy Arana Daniel, Alma Y. Alanis, Carlos Lopez Franco
Geometric Algebra also known as Clifford Algebra (CA), in fact represents a family of algebras that depend on a chosen vector space and on a special kind of product (the geometric product). The GA is a mathematical framework where we can find embedded concepts from linear algebra, tensor algebra, quaternion algebra, complex numbers and others. We represent a geometric algebra with Gp,q,r $ G_{p,q,r} $ 1.
The geometric description of linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
The case of quadrics is special. They are closely related to quadratic forms and bilinear forms. The algebraic discipline where these objects are studied is known as geometric algebra. We want to follow the basic theory until a point is reached where we can construct interesting codes and caps.
Efficient ray-tracing procedure for radio wave propagation modeling using homogeneous geometric algebra
Published in Electromagnetics, 2020
Ahmad H. Eid, Heba Y. Soliman, Sherif M. Abuelenin
In this paper, we discussed the utilization of homogeneous geometric algebra in the formulation and implementation of a deterministic (i.e. image-theory based) electromagnetic ray-tracing procedure. Geometric algebra is a unifying mathematical language that can easily represent different geometric objects and handle their transformations. We used Geometric algebra to perform the main geometric processing steps in electromagnetic propagation RT, namely, testing for the intersection between rays and surfaces, computing the intersection point, reflecting points on surfaces (to determine the image sources), and testing for surface inclusion in the lit region created from an image point and another surface. The developed procedure utilizes signed-distance computations which lead to compact GA formulations for the geometric computations. Another advantage of using GA is that, due to its unifying nature in representing geometric entities, the same geometric procedure can be used for both 2D and 3D ray-tracing scenarios without losing the efficiency of the derived implementations.
Towards the next-generation GIS: a geometric algebra approach
Published in Annals of GIS, 2019
Linwang Yuan, Zhaoyuan Yu, Wen Luo
Geometric algebra (GA) provides an ideal tool for the representation and computation of multidimensional geometric objects (Dorst, Fontijne, and Mann 2009; Hitzer et al. 2013). Our previous work (Yuan et al. 2011) addressed the uniform representation of GIS primitives with different dimensions and types. GA multivectors were used as the basic data structure to unify all representations of geographic data (Yuan et al. 2011). On the basis of such representations, typical GIS operations such as the computation of topological relationships (Yuan et al. 2014a; Yu et al. 2016), the analysis of field characteristics (Yu et al. 2013), and network analysis (Yuan et al. 2013a) were implemented to demonstrate the feasibility of the new GA framework.