Vortex Detection in Vector Fields Using Geometric Algebra

Pollock, Stuart; Mann, Stephen

doi:10.1007/s00006-013-0434-0

Cited by 3 publications

(4 citation statements)

References 17 publications

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“…Pollock and Mann [199] formulated a method for detecting vortices in sampled vector fields using the geometric algebra framework. Their vortex-detection algorithm recursively searches for vector samples and by applying the 2D version of the Gaussian Theorem the algorithm extracts vortex cores.…”

Section: ) Clifford-quaternion Fourier and Wavelet Transformsmentioning

confidence: 99%

A Survey on Quaternion Algebra and Geometric Algebra Applications in Engineering and Computer Science 1995–2020

Bayro–Corrochano

2021

IEEE Access

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Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020. After a brief introduction of GA, the applications of GA are reviewed across many fields. We discuss the characteristics of the applications of GA to various problems of computer science and engineering. In addition, the challenges and prospects of various applications proposed by many researchers are analyzed. We analyze the developments using GA in image processing, computer vision, neurocomputing, quantum computing, robot modeling, control, and tracking, as well as improvement of computer hardware performance. We believe that up to now GA has proven to be a powerful geometric language for a variety of applications. Furthermore, there is evidence that this is the appropriate geometric language to tackle a variety of existing problems and that consequently, step-by-step GA-based algorithms should continue to be further developed. We also believe that this extensive review will guide and encourage researchers to continue the advancement of geometric computing for intelligent machines.INDEX TERMS Geometric algebra, Clifford algebra, quaternion algebra, screw theory, signal processing, electrical engineering and power systems, geometric and quantum computing, image processing, computer vision, graphic engineering, artificial intelligence, machine learning, neural networks, control engineering, robotics, biomedical engineering, and biotechnology.

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Section: ) Clifford-quaternion Fourier and Wavelet Transformsmentioning

confidence: 99%

A Survey on Quaternion Algebra and Geometric Algebra Applications in Engineering and Computer Science 1995–2020

Bayro–Corrochano

2021

IEEE Access

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“…1 The 3D vectors can be either u × w or (u × w) × u, the projection of w onto plane orthogonal to u. The projection plane could be the supporting plane of the face [2,5] or the plane orthogonal to the averaged u vectors along the face boundary [14,10].…”

Section: Boundary Samplingmentioning

confidence: 99%

“…Our parity test is similar in spirit to several existing methods for detecting zeros of v [2,14,5,10] in that they also monitor the winding of v on the face boundary in some 2D coordinate system. The key difference is that these existing methods use a single plane to establishing the coordinate system.…”

Section: Comparison and Validationmentioning

confidence: 99%

“…In our implementation, we use the same boundary samples of the cell face M as in our method, and we compute the winding number of v at the samples after projecting to either the supporting plane of M [2,5] (which we call the Face Plane test) or the plane orthogonal to the average of samples of u along ∂ M [14,10] (which we call the Average Plane test). The output of a test is the parity of the winding number.…”

Section: Comparison and Validationmentioning

confidence: 99%

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A Robust Parity Test for Extracting Parallel Vectors in 3D

Cheng

Wang

et al. 2014

IEEE Trans. Visual. Comput. Graphics

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Abstract-Parallel vectors (PV), the loci where two vector fields are parallel, are commonly used to represent curvilinear features in 3D for data visualization. Methods for extracting PV usually operate on a 3D grid and start with detecting seed points on a cell face. We propose, to the best of our knowledge, the first provably correct test that determines the parity of the number of PV points on a cell face. The test only needs to sample along the face boundary and works for any choice of the two vector fields. A discretization of the test is described, validated, and compared with existing tests that are also based on boundary sampling. The test can guide PV-extraction algorithms to ensure closed curves wherever the input fields are continuous, which we exemplify in extracting ridges and valleys of scalar functions.

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