Carrier Method for the General Evaluation and Control of Pose, Molecular Conformation, Tracking, and the Like

Hitzer, Eckhard; Tachibana, Kanta; Buchholz, Sven; Yu, Isseki

doi:10.1007/s00006-009-0160-9

Cited by 34 publications

(41 citation statements)

References 17 publications

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“…A possible application could be a subspace structure self organizing map (SOM) type of neural network, mapping the topology of whole data subspace structures faithfully to lower dimensions. The problem of the relative angles between arbitrary conformal objects (see [7] for definitions) is thus also solved for all dimensions. Let X = S be a sphere or X = F be a flat object.…”

Section: Resultsmentioning

confidence: 99%

Angles Between Subspaces Computed in Clifford Algebra

Hitzer¹

2010

AIP Conference Proceedings

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We first review the definition of the angle between subspaces and how it is computed using matrix algebra. Then we introduce the Grassmann and Clifford algebra description of subspaces. The geometric product of two subspaces yields the full relative angular information in an explicit manner. We explain and interpret the result of the geometric product of subspaces gaining thus full practical access to the relative orientation information.

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Section: Resultsmentioning

confidence: 99%

Angles Between Subspaces Computed in Clifford Algebra

Hitzer¹

2010

AIP Conference Proceedings

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“…As shown in the table, different dimensional geometric objects follow a uniform manner of expression in CGA. The line and circle and the plane and sphere in CGA achieve a unified expression form and geometric meaning, respectively [34,44].…”

Section: Geometrically and Topologically Unified Representation In Cgamentioning

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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“…are called scalar product, left contraction, right contraction, and (associative) outer product, respectively (compare with [11,15,24]). These definitions extend by linearity to the corresponding products of general multivectors.…”

Section: Appendix a Geometric Interpretation Of Clifford Algebramentioning

confidence: 99%

Algebraic foundations of split hypercomplex nonlinear adaptive filtering

Hitzer

2012

Math Methods in App Sciences

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A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed.

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Carrier Method for the General Evaluation and Control of Pose, Molecular Conformation, Tracking, and the Like

Cited by 34 publications

References 17 publications

Angles Between Subspaces Computed in Clifford Algebra

Angles Between Subspaces Computed in Clifford Algebra

3D Cadastral Data Model Based on Conformal Geometry Algebra

Algebraic foundations of split hypercomplex nonlinear adaptive filtering

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