Applications of Clifford’s Geometric Algebra

Hitzer, Eckhard; Nitta, Tohru; Kuroe, Yasuaki

doi:10.1007/s00006-013-0378-4

Cited by 168 publications

(118 citation statements)

References 86 publications

(135 reference statements)

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“…Because dimensional and topological structures in CGA are consistent with the Grassmann dimensional structure, different dimensional elements can be expressed directly via outer products with conformal points. The structure of the Grassmann hierarchy is similar to the topological structures of different dimensional objects [39], which enables the construction of a unified representation for different dimensional geometries in CGA [40]. The unified representation of the geometric construction is purely algebraic, which indicates that geometric objects can be expressed as an algebraic entity [41].…”

Section: Basic Ideamentioning

confidence: 99%

3D Cadastral Data Model Based on Conformal Geometry Algebra

Zhang

Peng-cheng

et al. 2016

IJGI

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Three-dimensional (3D) cadastral data models that are based on Euclidean geometry (EG) are incapable of providing a unified representation of geometry and topological relations for 3D spatial units in a cadastral database. This lack of unification causes problems such as complex expression structure and inefficiency in the updating of 3D cadastral objects. The inability of current cadastral data models to express cadastral objects in a unified manner can be attributed to the different expressions of dimensional objects. Because the hierarchical Grassmann structure corresponds to the hierarchical structure of dimensions in conformal geometric algebra (CGA), geometric objects in different dimensions can be constructed by outer products in a unified expression form, which enables the direct extension of two-dimensional (2D) spatial representations to 3D spatial representations. The multivector structure in CGA can be employed to organize and store different dimensional objects in a multidimensional and unified manner. With the advantages of CGA in multidimensional expressions, a new 3D cadastral data model that is based on CGA is proposed in this paper. The geometries and topological relations of 3D spatial units can be represented in a unified form within the multivector structure. Detailed methods for 3D cadastral data model design based on CGA and data organization in CGA are introduced. The new cadastral data model is tested and analyzed with experimental data. The results indicate that the geometry and topological relations of 3D cadastral objects can be represented in a multidimensional manner with an intuitive topological structure and a unified dimensional expression.

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Section: Basic Ideamentioning

confidence: 99%

3D Cadastral Data Model Based on Conformal Geometry Algebra

Zhang

Peng-cheng

et al. 2016

IJGI

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“…A tutorial introduction to CFTs and Clifford wavelet transforms can be found in [55]. The Clifford algebra application survey [65] contains an up to date section on applications of Clifford algebra intergral transforms, including CFTs, QFTs and wavelet transforms 4 .…”

Section: Clifford Fourier Transformations In Clifford's Geometric Algmentioning

confidence: 99%

Quaternion and Clifford Fourier Transforms and Wavelets

Hitzer¹,

Sangwine²

2013

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Abstract. We survey the historical development of quaternion and Clifford Fourier transforms and wavelets. Keywords. Quaternions, Clifford Algebra, Fourier Transforms, Wavelet Transforms.The development of hypercomplex Fourier transforms and wavelets has taken place in several different threads, reflected in the overview of the subject presented in this chapter. We present in Section 1 an overview of the development of quaternion Fourier transforms, then in Section 2 the development of Clifford Fourier transforms. Finally, since wavelets are a more recent development, and the distinction between their quaternion and Clifford algebra approach has been much less pronounced than in the case of Fourier transforms, Section 3 reviews the history of both quaternion and Clifford wavelets.We recognise that the history we present here may be incomplete, and that work by some authors may have been overlooked, for which we can only offer our humble apologies. [90,91] on Clifford Fourier and Laplace transforms further explained in Section 2.2. Ernst and Delsuc's quaternion transforms were two-dimensional (that is they had two independent variables) and proposed for application to nuclear magnetic resonance (NMR) imaging. Written in terms of two independent time variables 1 t 1 and t 2 , the forward transforms 1 The two independent time variables arise naturally from the formulation of twodimensional NMR spectroscopy. Quaternion Fourier Transforms (QFT)1

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“…Geometric Algebra (also called Clifford Algebra after the British mathematician William Kingdon Clifford) was first developed as a mathematical language to unify all the different algebraic systems trying to express geometric relations/transformations, e.g., rotation and translation [3][4][5]12]. All the following geometric systems are particular cases (subalgebras) of GA: vector and matrix algebras, complex numbers, and quaternions (see Section 3.3).…”

Section: Fundamentals Of Geometric Algebramentioning

confidence: 99%

“…For an indepth discussion about GA theory & history, and its importance to Physics, please refer to [3][4][5][12][13][14]. For applications of GA in engineering and computer science, check [15][16][17][18][19][20].…”

Section: Fundamentals Of Geometric Algebramentioning

confidence: 99%

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Geometric-algebra adaptive filters.

Lopes

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This document introduces a new class of adaptive filters, namely GeometricAlgebra Adaptive Filters (GAAFs). Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex-numbers, quaternions etc. Exploiting those characteristics, among others, a general leastsquares cost function is posed, from which two types of GAAFs are designed. The first one, called standard, provides a generalization of regular adaptive filters for any subalgebra of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing almost perfect agreement for different levels of measurement noise. The second type, called pose estimation, is designed to estimate rigid transformations -rotation and translation -in n-dimensional spaces. The GA-LMS performance is assessed in a 3-dimensional registration problem, in which it is able to estimate the rigid transformation that aligns two point clouds that share common parts.

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Applications of Clifford’s Geometric Algebra

Cited by 168 publications

References 86 publications

3D Cadastral Data Model Based on Conformal Geometry Algebra

3D Cadastral Data Model Based on Conformal Geometry Algebra

Quaternion and Clifford Fourier Transforms and Wavelets

Geometric-algebra adaptive filters.

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