2,621 results on '"geometric algebra"'
Search Results
152. Clifford Algebras and Related Algebras
- Author
-
Bayro-Corrochano, Eduardo and Bayro-Corrochano, Eduardo
- Published
- 2019
- Full Text
- View/download PDF
153. Applications in Neurocomputing
- Author
-
Bayro-Corrochano, Eduardo and Bayro-Corrochano, Eduardo
- Published
- 2019
- Full Text
- View/download PDF
154. Geometric Neurocomputing
- Author
-
Bayro-Corrochano, Eduardo and Bayro-Corrochano, Eduardo
- Published
- 2019
- Full Text
- View/download PDF
155. Kinematics of the 2D and 3D Spaces
- Author
-
Bayro-Corrochano, Eduardo and Bayro-Corrochano, Eduardo
- Published
- 2019
- Full Text
- View/download PDF
156. 2D, 3D, and 4D Geometric Algebras
- Author
-
Bayro-Corrochano, Eduardo and Bayro-Corrochano, Eduardo
- Published
- 2019
- Full Text
- View/download PDF
157. Lie Algebras, Lie Groups, and Algebra of Incidence
- Author
-
Bayro-Corrochano, Eduardo and Bayro-Corrochano, Eduardo
- Published
- 2019
- Full Text
- View/download PDF
158. Programming Issues
- Author
-
Bayro-Corrochano, Eduardo and Bayro-Corrochano, Eduardo
- Published
- 2019
- Full Text
- View/download PDF
159. Geometric Algebra for teaching AC Circuit Theory.
- Author
-
Montoya, Francisco G., Baños, Raúl, Alcayde, Alfredo, and Arrabal‐Campos, Francisco M.
- Subjects
- *
VECTOR calculus , *ELECTRICAL engineering education , *VECTORS (Calculus) , *ELECTRICAL engineers , *ALTERNATING currents , *ALGEBRA - Abstract
Summary: This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non‐sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses. The analysis and understanding of how AC circuits operate in steady state are of a paramount importance for all the electrical engineers and practitioners around the world. Typically, lecturers of most undergraduate courses teach circuit theory using complex phasors, vector calculus, or linear algebra. However, these approaches have some important limitations in practice, which requires the development of strategies to improve teaching‐learning process related to AC circuit analysis. By formulating a new mathematical framework, the paper presents and discusses how GA can be of help in this approach. It is also described how the results obtained by using GA are validated using computer‐based simulations. It is highlighted how the proposed teaching methodology based on GA theory can be effective in helping students learn AC circuit analysis since it has several potential benefits derived from its simplicity, compactness, and use of easily identifiable geometrical concepts. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
160. Basis-free Solution to Sylvester Equation in Clifford Algebra of Arbitrary Dimension.
- Author
-
Shirokov, Dmitry
- Abstract
The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case n = 4 , the proofs for the case n = 5 and the case of arbitrary dimension n. The results can be used in symbolic computation. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
161. Blade Products and Angles Between Subspaces.
- Author
-
Mandolesi, André L. G.
- Abstract
Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Plücker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations for this product and its properties, and to formulas relating other blade products (scalar, inner, outer, etc., including those of Grassmann algebra) to angles between subspaces. Contractions are linked to an asymmetric angle, while commutators and anticommutators involve hyperbolic functions of the angle bivector, shedding new light on their properties. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
162. Multi-Focus Image Fusion Using Geometric Algebra Based Discrete Fourier Transform
- Author
-
Yanping Li and Shengming Jiang
- Subjects
Geometric algebra ,Discrete Fourier Transform (DFT) ,multivector ,multi-focus image fusion ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
For color image fusion, existing traditional algorithms regard each color image pixel as a scalar which may lose the relationships among spectral channels. In this paper, a novel multi-focus image fusion algorithm using geometric algebra based Discrete Fourier Transform (GA-DFT) is proposed. First, a novel Discrete Fourier Transform based on geometric algebra (GA) is provided. Second, the proposed algorithm represents the color image as a GA multivector by mapping the RGB channels into the GA space. Third, the variance is selected as a contrast measure to fuse the source images in GA space. The experimental results demonstrate that the proposed algorithm can achieve the comparable performance with its counterparts in terms of subjective quality evaluation, and outperform most of the traditional algorithms in terms of objective quality metrics, indicating good performance in image fusion applications.
- Published
- 2020
- Full Text
- View/download PDF
163. SVDTWDD Method for High Correct Recognition Rate Classifier With Appropriate Rejection Recognition Regions
- Author
-
Guowei Yang, Shaohua Qi, Teng Yu, Minghua Wan, Zhangjing Yang, Tianming Zhan, Fanlong Zhang, and Zhihui Lai
- Subjects
Classifier ,geometric algebra ,pattern recognition ,support vector machine ,support vector domain description ,incremental learning ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
At present, regions of the same class determined by Support Vector Machines (SVM) classifier, Support Vector Domain Description (SVDD) classifier and Deep Learning (DL) classifier may occupy regions of other classes or unknown classes in feature space. There exists a risk that samples of other classes or unknown classes are wrongly classified as a known class. In this paper, the Support Vector Domain Tightly Wrapping Description Design (SVDTWDD) method with appropriate rejection regions and the corresponding incremental learning algorithm are proposed to overcome the above problem. The main work includes: (1) We develop a construction algorithm of the tightly wrapping set for the homogeneous feature set; (2) Based on the homogeneous feature set and tightly wrapping set, a novel algorithm is presented for obtaining the tightly wrapping surface of the homogeneous feature region; (3) The method for constructing all the public regions outside of the tightly wrapping surface and the intersections of wrapping regions in two different tightly wrapping surfaces, as the rejection region of the classifier; (4) An incremental algorithm is also presented based on the SVD-TWDD method. The experimental results with UCI data sets show that the correct recognition rate of our proposed method is nearly100% even if with a low rejection rate.
- Published
- 2020
- Full Text
- View/download PDF
164. Novel Adaptive Filtering Algorithms Based on Higher-Order Statistics and Geometric Algebra
- Author
-
Yinmei He, Rui Wang, Xiangyang Wang, Jian Zhou, and Yi Yan
- Subjects
Adaptive filters ,geometric algebra ,least-mean fourth ,least-mean mixed-norm ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
Adaptive filtering algorithms based on higher-order statistics are proposed for multi-dimensional signal processing in geometric algebra (GA) space. In this paper, the proposed adaptive filtering algorithms utilize the advantage of GA theory in multi-dimensional signal processing to represent a multi-dimensional signal as a GA multivector. In addition, the original least-mean fourth (LMF) and least-mean mixed-norm (LMMN) adaptive filtering algorithms are extended to GA space for multi-dimensional signal processing. Both the proposed GA-based least-mean fourth (GA-LMF) and GA-based least-mean mixed-norm (GA-LMMN) algorithms need to minimize cost functions based on higher-order statistics of the error signal in GA space. The simulation results show that the proposed GA-LMF algorithm performs better in terms of convergence rate and steady-state error under a much smaller step size. The proposed GA-LMMN algorithm makes up for the instability of GA-LMF as the step size increases, and its performance is more stable in mean absolute error and convergence rate.
- Published
- 2020
- Full Text
- View/download PDF
165. Geometric Algebra Applications in Geospatial Artificial Intelligence and Remote Sensing Image Processing
- Author
-
Uzair Aslam Bhatti, Zhaoyuan Yu, Linwang Yuan, Zeeshan Zeeshan, Saqib Ali Nawaz, Mughair Bhatti, Anum Mehmood, Qurat Ul Ain, and Luo Wen
- Subjects
Geometric algebra ,Clifford algebra ,geometric algebra ,computer vision ,artificial intelligence ,quaternions ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
With the increasing demand for multidimensional data processing, Geometric algebra (GA) has attracted more and more attention in the field of geographical information systems. GA unifies and generalizes real numbers and complex, quaternion, and vector algebra, and converts complicated relations and operations into intuitive algebra independent of coordinate systems. It also provides a solution for solving multidimensional information processing with a high correlation among the dimensions and avoids the loss of information. Traditional methods of computer vision and artificial intelligence (AI) provide robust results in multidimensional processing after being combined with GA and give additional feature analysis facility to remote sensing images. In this paper, we provide a detailed review of GA in different fields of AI and computer vision regarding its applications and the current developments in geospatial research. We also discuss the Clifford-Fourier transform (CFT) and quaternions (sub-algebra of GA) because of their necessity in remote sensing image processing. We focus on how GA helps AI and solves classification problems, as well as improving these methods using geometric algebra processing. Finally, we discuss the issues, challenges, and future perspectives of GA with regards to possible research directions.
- Published
- 2020
- Full Text
- View/download PDF
166. Clifford Geometric Algebra-Based Approach for 3D Modeling of Agricultural Images Acquired by UAVs
- Author
-
Prince Waqas Khan, Yung-Cheol Byun, and Muhammad Ahsan Latif
- Subjects
Clifford algebra ,computer vision ,geometric algebra ,image processing ,precision agriculture ,quaternions ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
Three-dimensional image modeling is essential in many scientific disciplines, including computer vision and precision agriculture. So far, various methods of creating three-dimensional (3D) models have been considered. However, the processing of transformation matrices of each input image data is not controlled. Site-specific crop mapping is essential because it helps farmers determine yield, biodiversity, energy, crop coverage, etc. Clifford Geometric Algebraic understanding of signaling and image processing has become increasingly important in recent years. Geometric Algebraic treats multi-dimensional signals in a holistic way to maintain relationship between side sizes and prevent loss of information. This article has used agricultural images acquired by unmanned aerial vehicles (UAVs) to construct three-dimensional models using Clifford geometric algebra. The qualitative and quantitative performance evaluation results show that Clifford geometric algebra can generate a three-dimensional geometric statistical model directly from drones' RGB images. Through peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and visual comparison, the proposed algorithm's performance is compared with latest algorithms. Experimental results show that proposed algorithm is better than other leading 3D modeling algorithms.
- Published
- 2020
- Full Text
- View/download PDF
167. Dr. Bertlmann’s Socks in a Quaternionic World of Ambidextral Reality
- Author
-
Joy Christian
- Subjects
Bell’s theorem ,determinism ,EPR argument ,Friedmann-Robertson-Walker spacetime ,geometric algebra ,local causality ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
Unlike our basic theories of space and time, quantum mechanics is not a locally causal theory. Moreover, it is widely believed that any hopes of restoring local causality within any realistic theory have been undermined by Bell's theorem and the experimental investigations it has inspired. In this pedagogical paper, John S. Bell's amusing example of Dr. Bertlmann's socks to illustrate the results of these experiments is reconsidered, first within a toy model of a two-dimensional one-sided world of a non-orientable Möbius strip, and then within a real world of three-dimensional quaternionic sphere, S3, which results from an addition of a single point to IR3 at infinity. In the latter quaternionic world, which happens to be the spatial part of a solution of Einstein's field equations of general relativity, the singlet correlations between a pair of entangled fermions can be understood as classically as those between Dr. Bertlmann's colorful socks.
- Published
- 2020
- Full Text
- View/download PDF
168. A Normalized Adaptive Filtering Algorithm Based on Geometric Algebra
- Author
-
Rui Wang, Meixiang Liang, Yinmei He, Xiangyang Wang, and Wenming Cao
- Subjects
Geometric algebra ,normalized least mean fourth ,normalized least mean square ,adaptive filters ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
In this paper, we extend the original Normalized Least Mean Fourth (NLMF) and Normalized Least Mean Square (NLMS) adaptive filtering algorithms into Geometric Algebra (GA) space to enable them to process multidimensional signals. We redefine the cost functions and propose the GA based NLMF and NLMS algorithms (GA-NLMF & GA-NLMS). We take full advantage of the ability of GA to represent multidimensional signals in GA space. GA-NLMS minimizes the cost function of the normalized mean square of the error signal, and remain stable as the input signal of the filter increases. GA-NLMS has fast convergence rate but higher steady-state error. The GA-NLMF algorithm minimizes the cost function of the normalized mean fourth of the error signal. Simulation results show that our proposed GA-NLMS adaptive filtering algorithm outperforms original NLMS algorithm in terms of convergence rate and steady-state error, and GA-NLMF outperforms both NLMF and GA-NLMS algorithms. GA-NLMF has faster convergence rate and lower steady state error, which is proved in the experiments.
- Published
- 2020
- Full Text
- View/download PDF
169. On a matrix álgebra without matrices
- Author
-
Edgar Vera Saravia
- Subjects
geometric algebra ,nested of real vector spaces ,nested of algebras ,Applied mathematics. Quantitative methods ,T57-57.97 ,Mathematics ,QA1-939 - Abstract
We enlarge a previus introduction to the two-dimensional geometric algebra published in Volume 05(02) of SELECCIONES MATEMÁTICAS. It is a friendly introduction of a theme of common interest for university proffessors and students of physics and mathematics.
- Published
- 2019
- Full Text
- View/download PDF
170. Geometric-algebra affine projection adaptive filter.
- Author
-
Ren, Yuetao, Zhi, Yongfeng, and Zhang, Jun
- Subjects
ADAPTIVE filters ,LAGRANGE multiplier ,ALGORITHMS ,COST functions ,DATA structures ,CONSTRAINED optimization - Abstract
Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. The stability of the algorithm is analyzed based on the mean-square deviation. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
171. Transient Performance Analysis of Geometric Algebra Least Mean Square Adaptive Filter.
- Author
-
Wang, Wenyuan and Dogancay, Kutluyil
- Abstract
Recently, a new class of multivector-valued adaptive filters, called geometric-algebra adaptive filters (GA-AFs), have been proposed and applied to 3D registration of point clouds, rotation estimation in computer vision, and so on. To offer a complete theoretical foundation for GA-AFs, we present a transient behavior evaluation of the geometric algebra least mean square (GA-LMS) algorithm. Specifically, the transient mean square deviation (MSD) is obtained by using the results of the mean square theoretical behavior of the GA-LMS. Furthermore, the transient excess mean square error (EMSE) is also given by using the results of MSD. The analytical results rely on the independence theory which has been commonly used in the convergence analysis of real-valued adaptive filter. Considering the non-commutative geometric algebra, the conventional method of analysis is not suitable for the GA-LMS algorithm. Therefore, we propose a novel method to analyze the GA-LMS algorithm under white noise assumption by separating the weight-error array. The obtained theoretical results can be used to accurately predict the transient performance of the GA-LMS. Moreover, the stability condition and steady-state performance are also analyzed. Our proposed steady-state model is more accurate than the previous model. Finally, numerical experiments are presented to confirm the accuracy of the theoretical analysis results. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
172. On Inner Automorphisms Preserving Fixed Subspaces of Clifford Algebras.
- Author
-
Shirokov, Dmitry
- Abstract
In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras—subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras—graded central simple algebras or graded central simple algebras with involution. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
173. On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension.
- Author
-
Shirokov, D. S.
- Abstract
In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension n. The formulas involve only the operations of multiplication, summation, and operations of conjugation without explicit use of matrix representation. We use methods of Clifford algebras (including the method of quaternion typification proposed by the author in previous papers and the method of operations of conjugation of special type presented in this paper) and generalizations of numerical methods of matrix theory (the Faddeev–LeVerrier algorithm based on the Cayley–Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials) to the case of Clifford algebras in this paper. We present the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras. We use this construction in the analytical proof of formulas for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in Clifford algebras. The basis-free formulas for the inverse give us basis-free solutions to linear algebraic equations, which are widely used in computer science, image and signal processing, physics, engineering, control theory, etc. The results of this paper can be used in symbolic computation. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
174. Geometric Algebra Based Kinematics Model and Singularity of a Hybrid Surgical Robot
- Author
-
Tanev, Tanio K., Siciliano, Bruno, Series editor, Khatib, Oussama, Series editor, Lenarčič, Jadran, editor, and Merlet, Jean-Pierre, editor
- Published
- 2018
- Full Text
- View/download PDF
175. Mathematical Background
- Author
-
Xambó-Descamps, Sebastià, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, and Xambó-Descamps, Sebastià
- Published
- 2018
- Full Text
- View/download PDF
176. Conclusion
- Author
-
Vince, John and Vince, John
- Published
- 2018
- Full Text
- View/download PDF
177. Tensor of Order Two and Geometric Properties of 2D Metric Space
- Author
-
Tomáš Stejskal, Jozef Svetlík, and Marcela Lascsáková
- Subjects
tensor ,Lagrange’s identity ,geometric algebra ,tensor glyph ,Mathematics ,QA1-939 - Abstract
A 2D metric space has a limited number of properties through which it can be described. This metric space may comprise objects such as a scalar, a vector, and a rank-2 tensor. The paper provides a comprehensive description of relations between objects in 2D space using the matrix product of vectors, geometric product, and dot product of complex numbers. These relations are also an integral part of the Lagrange’s identity. The entire structure of derived theoretical relationships describing properties of 2D space draws on the Lagrange’s identity. The description of how geometric algebra and tensor calculus are interconnected is given here in a comprehensive and essentially clear manner, which is the main contribution of this paper. A new term in this regard is the total geometric and matrix product, which—in a simple manner—predetermines and defines the existence of differential relations such as the gradient, the divergence, and the curl of a vector field. In addition, geometric interpretation of tensors is pointed out, expressed through angular parameters known from the literature as a tensor glyph. This angular interpretation of the tensor has an unequivocal analytical form, and the paper shows how it is linked to the classical tensor denoted by indices.
- Published
- 2022
- Full Text
- View/download PDF
178. Unified Representation of 3D Multivectors with Pauli Algebra in Rectangular, Cylindrical and Spherical Coordinate Systems
- Author
-
Ben Minnaert, Giuseppina Monti, and Mauro Mongiardo
- Subjects
geometric algebra ,Clifford algebra ,Pauli matrices ,coordinate systems ,multivectors ,Mathematics ,QA1-939 - Abstract
In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology.
- Published
- 2022
- Full Text
- View/download PDF
179. Non-Contextual and Local Hidden-Variable Model for the Peres–Mermin and Greenberger–Horne–Zeilinger Systems.
- Author
-
Held, Carsten
- Subjects
- *
QUANTUM theory - Abstract
A hidden-variable model for quantum–mechanical spin, as represented by the Pauli spin operators, is proposed for systems illustrating the well-known no-hidden-variables arguments by Peres (Phys Lett A 151:107–108, 1990) and Mermin (Phys Rev Lett 65:3373–3376, 1990) and by Greenberger et al. (Bell's theorem, quantum theory, and conceptions of the universe, Kluwer, Dordrecht, 1989). Both arguments rely on an assumption of non-contextuality; the latter argument can also be phrased as a non-locality argument, using a locality assumption. The model suggested here is compatible with both assumptions. This is possible because the scalar values of spin observables are replaced by vectors that are components of orientations. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
180. Two-State Quantum Systems Revisited: A Clifford Algebra Approach.
- Author
-
Amao, Pedro and Castillo, Hernan
- Abstract
We revisit the topic of two-state quantum systems using the Clifford Algebra in three dimensions C l 3 . In this description, both the quantum states and Hermitian operators are written as elements of C l 3 . By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in C l 3 . We then use this approach to revisit the problem of a spin-1/2 particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, Clifford algebra reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of C l 3 . [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
181. Geometric Algebra of Singular Ruled Surfaces.
- Author
-
Li, Yanlin, Wang, Zhigang, and Zhao, Tiehong
- Abstract
Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it's difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces transform into the dual singular curves on the dual unit sphere. By using the research method on the singular curves, we give the definition of the dual evolute of the dual front in the dual unit sphere, we further provide the k-th dual evolute of the dual front. Moreover, we consider the ruled surface corresponding to the dual evolute and k-th dual evolute and provide the developable conditions of these ruled surfaces. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
182. Semantic manipulation through the lens of Geometric Algebra.
- Author
-
Evangelista, Raphael dos S., Pereira, Andre Luiz da S., de Moraes, Rogério Ferreira, and Fernandes, Leandro A.F.
- Subjects
ALGEBRA ,IMAGE retrieval ,SEMANTIC computing ,PROOF of concept ,HAIRSTYLES - Abstract
Semantic image manipulation is a complex problem defined as the ability to change high-level features while keeping the final result visually similar to the original. Recent deep generative solutions show that manipulating semantic features in latent space produces compelling results. Even if we consider those advances, the question remains: Can we algebraically operate high-level visual semantic features in images with meaningful operations? In this paper, we demonstrate the feasibility of interpreting and manipulating image pseudovectors (n − 1 -dimensional subspaces) as the union of visual features (k -dimensional subspaces, for 0 < k < n) operated using Geometric Algebra (GA). Depending on how the latent space is organized, any GA operation would be applicable, enabling the solution to handle an open set of problems without retraining generative models for specific tasks. As a proof of concept, in this paper, we demonstrate how GA operations can be applied to manipulate subspaces in the latent space of faces to perform operations like putting on or taking off clothing accessories, transferring age characteristics, changing hairstyles, and performing semantic queries in sets of images. [Display omitted] • Interpret and manipulate latent codes of images as n − 1 -D subspaces using any Geometric Algebra (GA) equations. • Use the contraction concept from GA to remove some features and build up a subspace with the remaining desired ones. • Compute semantic ratios between two images with GA and map the remaining visual energy to another image. • Perform semantic image retrieval with the GA's orthogonal rejection concept. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
183. Objects of consciousness
- Author
-
Hoffman, Donald D and Prakash, Chetan
- Subjects
consciousness ,quantum theory ,Markov chains ,combination problem ,geometric algebra ,Psychology ,Cognitive Sciences - Abstract
Current models of visual perception typically assume that human vision estimates true properties of physical objects, properties that exist even if unperceived. However, recent studies of perceptual evolution, using evolutionary games and genetic algorithms, reveal that natural selection often drives true perceptions to extinction when they compete with perceptions tuned to fitness rather than truth: Perception guides adaptive behavior; it does not estimate a preexisting physical truth. Moreover, shifting from evolutionary biology to quantum physics, there is reason to disbelieve in preexisting physical truths: Certain interpretations of quantum theory deny that dynamical properties of physical objects have definite values when unobserved. In some of these interpretations the observer is fundamental, and wave functions are compendia of subjective probabilities, not preexisting elements of physical reality. These two considerations, from evolutionary biology and quantum physics, suggest that current models of object perception require fundamental reformulation. Here we begin such a reformulation, starting with a formal model of consciousness that we call a "conscious agent." We develop the dynamics of interacting conscious agents, and study how the perception of objects and space-time can emerge from such dynamics. We show that one particular object, the quantum free particle, has a wave function that is identical in form to the harmonic functions that characterize the asymptotic dynamics of conscious agents; particles are vibrations not of strings but of interacting conscious agents. This allows us to reinterpret physical properties such as position, momentum, and energy as properties of interacting conscious agents, rather than as preexisting physical truths. We sketch how this approach might extend to the perception of relativistic quantum objects, and to classical objects of macroscopic scale.
- Published
- 2014
184. A Novel Least-Mean Kurtosis Adaptive Filtering Algorithm Based on Geometric Algebra
- Author
-
Rui Wang, Yinmei He, Chenyang Huang, Xiangyang Wang, and Wenming Cao
- Subjects
Geometric algebra ,least-mean kurtosis ,geometric calculus ,adaptive filters ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
A novel least-mean kurtosis adaptive filtering algorithm based on geometric algebra (GA-LMK) is proposed for multidimensional signal processing. First, taking advantage of geometric algebra (GA) in terms of the representation of multidimensional signal, the GA-LMK algorithm represents a multidimensional signal as a GA multivector. Second, we extend the original least mean kurtosis (LMK) algorithm in GA space for multidimensional signal processing. The proposed GA-LMK algorithm minimizes the cost function of negated kurtosis of the error signal in GA space, and provides a way to make tradeoff problem between convergence rate and steady-state error. Third, we study the steady-state behavior of the GA-LMK algorithm under Gaussian noises to acquire conditions of misadjustment. The simulation results show that our proposed GA-LMK adaptive filtering algorithm can outperform significantly existing the state-of-the-art algorithms in terms of convergence rate and steady-state error.
- Published
- 2019
- Full Text
- View/download PDF
185. UAV’s Agricultural Image Segmentation Predicated by Clifford Geometric Algebra
- Author
-
Prince Waqas Khan, Guangxia Xu, Muhammad Ahsan Latif, Khizar Abbas, and Ammara Yasin
- Subjects
Clifford algebra ,computer vision ,geometric algebra ,image processing ,image segmentation ,precision agriculture ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
Image segmentation is widely used in the field of agriculture to improve the yields and protecting them from pests, herbs, shrubs, and weeds. Precision agriculture is also contributing to the inter and intra crop monitoring. Recently, unmanned aerial vehicles are used widely for acquiring images. In this paper, we purpose Clifford geometric algebra to enhance the segmented images acquired from the UAVs of different agricultural fields. The Clifford geometric algebra is also sometimes used as a collective term for the diverse range of mathematical fields, both classical and modern algebraic mathematics. Previous image segmentation approaches depend upon the intensity of red, green, and blue colors; but the complete perspective could not be obtained from these approaches. Geometric algebra overcomes this limitation and leads to a genuine color space image processing. It is mainly used in the processing of medical images. Subalgebra of the Clifford algebra is Quaternions. We have used this approach in agricultural images. The image segmentation of foreground and background is enhanced using Clifford geometric algebra; hence, the results obtained are fine-tuned segmented images. The anticipated result of our research would have a positive impact on the amelioration of the condition of the farmers and their livelihood.
- Published
- 2019
- Full Text
- View/download PDF
186. Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space
- Author
-
Joy Christian
- Subjects
Bell’s theorem ,determinism ,EPR argument ,Friedmann-Robertson-Walker spacetime ,Geometric Algebra ,local causality ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
In the context of EPR-Bohm type experiments and spin detections confined to spacelike hypersurfaces, a local, deterministic and realistic model within a Friedmann-Robertson-Walker spacetime with a constant spatial curvature (S3) is presented that describes simultaneous measurements of the spins of two fermions emerging in a singlet state from the decay of a spinless boson. Exact agreement with the probabilistic predictions of quantum theory is achieved in the model without data rejection, remote contextuality, superdeterminism or backward causation. A singularity-free Clifford-algebraic representation of S3 with vanishing spatial curvature and non-vanishing torsion is then employed to transform the model in a more elegant form. Several event-by-event numerical simulations of the model are presented, which confirm our analytical results with the accuracy of 4 parts in 104. Possible implications of our results for practical applications such as quantum security protocols and quantum computing are briefly discussed.
- Published
- 2019
- Full Text
- View/download PDF
187. Geometric Algebra Representation and Ensemble Action Classification Method for 3D Skeleton Orientation Data
- Author
-
Wenming Cao, Yitao Lu, and Zhiquan He
- Subjects
Geometric algebra ,motion recognition ,support vector machine ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 - Abstract
In this paper, we propose a novel human body posture representation based on Geometric Algebra to extract the angles and orientations of the most informative body joints to describe human body postures. As a motion usually consists of a number of postures, which are different even in the same type of motion. We treat the postures of a motion independently. For each posture, a new Geometric Algebra based skeleton posture descriptor is used to construct the feature vectors as the input for the Support Vector Machine classifier to decide its motion type. To get the type of the whole motion, we choose the most frequent class from the sequence of predictions of the motion postures using a simple voting scheme. We have tested the method on a public benchmark SYSU-3D-HIO and an in-house dataset of human exercises. The results have demonstrated the effectiveness of our method.
- Published
- 2019
- Full Text
- View/download PDF
188. On the Relationship between Primal/Dual Cell Complexes of the Cell Method and Primal/Dual Vector Spaces: an Application to the Cantilever Elastic Beam with Elastic Inclusion
- Author
-
Ferretti Elena
- Subjects
cellmethod ,topological equations ,geometric algebra ,bialgebra ,coboundary process ,elastic beams ,elastic inclusions ,Mechanics of engineering. Applied mechanics ,TA349-359 - Abstract
The Cell Method (CM) is an algebraic numerical method based on the use of global variables: the configuration, source and energetic global variables. The configuration variables with their topological equations, on the one hand, and the source variables with their topological equations, on the other hand, define two vector spaces that are a bialgebra and its dual algebra. The operators of these topological equations are generated by the outer product of the geometric algebra, for the primal vector space, and by the dual product of the dual algebra, for the dual vector space. The topological equations in the primal cell complex are coboundary processes on even exterior discrete p−forms, whereas the topological equations in the dual cell complex are coboundary processes on odd exterior discrete p−forms. Being expressed by coboundary processes in two different vector spaces, compatibility and equilibrium can be enforced at the same time, with compatibility enforced on the primal cell complex and equilibrium enforced on the dual cell complex. By way of example, in the present paper compatibility and equilibrium are enforced on a cantilever elastic beam with elastic inclusion. In effect, the CM shows its maximum potentialities right in domains made of several materials, as, being an algebraic approach, can treat any kind of discontinuities of the domain easily.
- Published
- 2019
- Full Text
- View/download PDF
189. Singularities of Serial Robots: Identification and Distance Computation Using Geometric Algebra
- Author
-
Isiah Zaplana, Hugo Hadfield, and Joan Lasenby
- Subjects
serial robotic manipulators ,singularity identification ,geometric algebra ,rotor group ,distance to a singularity ,Mathematics ,QA1-939 - Abstract
The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6×n or n×n matrix for an n-degrees-of-freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C, such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely, motion planning, motion control and bilateral teleoperation.
- Published
- 2022
- Full Text
- View/download PDF
190. Geometric Algebra Applied to Multiphase Electrical Circuits in Mixed Time–Frequency Domain by Means of Hypercomplex Hilbert Transform
- Author
-
Francisco G. Montoya, Raúl Baños, Alfredo Alcayde, Francisco M. Arrabal-Campos, and Javier Roldán-Pérez
- Subjects
geometric algebra ,non-sinusoidal power ,Clifford algebra ,power theory ,geometric electricity ,Mathematics ,QA1-939 - Abstract
In this paper, power flows in electrical circuits are modelled in a mixed time-frequency domain by using geometric algebra and the Hilbert transform for the first time. The use of this mathematical framework overcomes some of the limitations of some of the existing methodologies, in which the so-called “active current” may not lead to the lowest Root Mean Square (RMS) current under distorted supply or unbalanced load. Moreover, this current may contain higher levels of harmonic distortion compared to the supply voltage. The proposed method can be used for sinusoidal and non-sinusoidal power supplies, non-linear loads and single- and multi-phase electrical circuits, and it provides meaningful engineering results with a compact formulation. It can also serve as an advanced tool for developing algorithms in the power electronics field. Several examples have been included to verify the validity of the proposed theory.
- Published
- 2022
- Full Text
- View/download PDF
191. Traffic-Data Recovery Using Geometric-Algebra-Based Generative Adversarial Network
- Author
-
Di Zang, Yongjie Ding, Xiaoke Qu, Chenglin Miao, Xihao Chen, Junqi Zhang, and Keshuang Tang
- Subjects
traffic data recovery ,geometric algebra ,deep learning ,intelligent transportation system ,Chemical technology ,TP1-1185 - Abstract
Traffic-data recovery plays an important role in traffic prediction, congestion judgment, road network planning and other fields. Complete and accurate traffic data help to find the laws contained in the data more efficiently and effectively. However, existing methods still have problems to cope with the case when large amounts of traffic data are missed. As a generalization of vector algebra, geometric algebra has more powerful representation and processing capability for high-dimensional data. In this article, we are thus inspired to propose the geometric-algebra-based generative adversarial network to repair the missing traffic data by learning the correlation of multidimensional traffic parameters. The generator of the proposed model consists of a geometric algebra convolution module, an attention module and a deconvolution module. Global and local data mean squared errors are simultaneously applied to form the loss function of the generator. The discriminator is composed of a multichannel convolutional neural network which can continuously optimize the adversarial training process. Real traffic data from two elevated highways are used for experimental verification. Experimental results demonstrate that our method can effectively repair missing traffic data in a robust way and has better performance when compared with the state-of-the-art methods.
- Published
- 2022
- Full Text
- View/download PDF
192. Sobre el Espacio Euclidiano Bidimensional
- Author
-
Edgar Vera Saravia and Henry Zubieta Rojas
- Subjects
Geometric Algebra ,Euclidean Space ,Afin Space ,Applied mathematics. Quantitative methods ,T57-57.97 ,Mathematics ,QA1-939 - Abstract
We offer a geometric introduction to the two-dimensional Euclidean Space E2 using both the geometric approach of the affine R-Space and the two-dimensional Geometric Algebra. These notes, intented for university proffessors of physics and mathematics, try to emulate the program developed by Felix Kline in the first decade of the 20th century.
- Published
- 2018
- Full Text
- View/download PDF
193. Growing Recognition of Bohm’s Causal Interpretation
- Author
-
Talbot, Chris and Talbot, Chris, editor
- Published
- 2017
- Full Text
- View/download PDF
194. Gamified AR/VR Character Rendering and Animation-Enabling Technologies
- Author
-
Papaefthymiou, Margarita, Kateros, Steve, Georgiou, Stylianos, Lydatakis, Nikos, Zikas, Paul, Bachlitzanakis, Vasileios, Papagiannakis, George, Ioannides, Marinos, editor, Magnenat-Thalmann, Nadia, editor, and Papagiannakis, George, editor
- Published
- 2017
- Full Text
- View/download PDF
195. Creating Your First SceneKit Project
- Author
-
Goodwill, James, Matlock, Wesley, Goodwill, James, and Matlock, Wesley
- Published
- 2017
- Full Text
- View/download PDF
196. Retraction article: Eight-dimensional octonion-like but associative normed division algebra.
- Author
-
Christian, Joy
- Subjects
ASSOCIATIVE algebras ,CLIFFORD algebras ,DIVISION algebras ,TOPOLOGY - Abstract
We present an eight-dimensional even sub-algebra of the 2 4 = 16 -dimensional associative Clifford algebra C l 4 , 0 and show that its eight-dimensional elements denoted as X and Y respect the norm relation ‖ XY ‖ = ‖ X ‖ ‖ Y ‖ , thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
197. Geometric algebra as applied to freeform motion design and improvement
- Author
-
Simpson, Leon and Mullineux, Glen
- Subjects
620.00420285 ,geometric algebra ,freeform motion ,slerp ,Bezier ,moion curvature ,inertia tensor - Abstract
Freeform curve design has existed in various forms for at least two millennia, and is important throughout computer-aided design and manufacture. With the increasing importance of animation and robotics, coupled with the increasing power of computers, there is now interest in freeform motion design, which, in part, extends techniques from curve design, as well as introducing some entirely distinct challenges. There are several approaches to freeform motion construction, and the first step in designing freeform motions is to choose a representation. Unlike for curves, there is no "standard" way of representing freeform motions, and the different tools available each have different properties. A motion can be viewed as a continuously-varying pose, where a pose is a position and an orientation. This immediately presents a problem; the dimensions of rotations and translations are different, and it is not clear how the two can be compared, such as to define distance along a motion. One solution is to treat the rotational and translational components of a motion separately, but this is inelegant and clumsy. The philosophy of this thesis is that a motion is not defined purely by rotations and translations, but that the body following a motion is a part of that motion. Specifically, the part of the body that is accounted for is its inertia tensor. The significance of the inertia tensor is that it allows the rotational and translational parts of a motion to be, in some sense, compared in a dimensionally- consistent way. Using the inertia tensor, this thesis finds the form of kinetic energy in <;1'4, and also discusses extensions of the concepts of arc length and curvature to the space of motions, allowing techniques from curve fairing to be applied to motion fairing. Two measures of motion fairness are constructed, and motion fairing is the process of minimizing the measure of a motion by adjusting degrees of freedom present in the motion's construction. This thesis uses the geometric algebra <;1'4 in the generation offreeform motions, and the fairing of such motions. <;1'4 is chosen for its particular elegance in representing rigid-body transforms, coupled with an equivalence relation between elements representing transforms more general than for ordinary homogeneous coordinates. The properties of the algebra germane to freeform motion design and improvement are given, and two distinct frameworks for freeform motion construction and modification are studied in detail.
- Published
- 2012
198. MRI image analysis for abdominal and pelvic endometriosis
- Author
-
Chi, Wenjun, Brady, Michael, Schnabel, Julia, and McVeigh, Enda
- Subjects
618.1 ,Medical Engineering ,Image understanding ,Gynaecology ,Magnetic Resonance Imaging (MRI) ,endometriosis ,multi-view fusion ,image segmentation ,monogenic signals ,Riesz transform ,geometric algebra ,level sets - Abstract
Endometriosis is an oestrogen-dependent gynaecological condition defined as the presence of endometrial tissue outside the uterus cavity. The condition is predominantly found in women in their reproductive years, and associated with significant pelvic and abdominal chronic pain and infertility. The disease is believed to affect approximately 33% of women by a recent study. Currently, surgical intervention, often laparoscopic surgery, is the gold standard for diagnosing the disease and it remains an effective and common treatment method for all stages of endometriosis. Magnetic resonance imaging (MRI) of the patient is performed before surgery in order to locate any endometriosis lesions and to determine whether a multidisciplinary surgical team meeting is required. In this dissertation, our goal is to use image processing techniques to aid surgical planning. Specifically, we aim to improve quality of the existing images, and to automatically detect bladder endometriosis lesion in MR images as a form of bladder wall thickening. One of the main problems posed by abdominal MRI is the sparse anisotropic frequency sampling process. As a consequence, the resulting images consist of thick slices and have gaps between those slices. We have devised a method to fuse multi-view MRI consisting of axial/transverse, sagittal and coronal scans, in an attempt to restore an isotropic densely sampled frequency plane of the fused image. In addition, the proposed fusion method is steerable and is able to fuse component images in any orientation. To achieve this, we apply the Riesz transform for image decomposition and reconstruction in the frequency domain, and we propose an adaptive fusion rule to fuse multiple Riesz-components of images in different orientations. The adaptive fusion is parameterised and switches between combining frequency components via the mean and maximum rule, which is effectively a trade-off between smoothing the intrinsically noisy images while retaining the sharp delineation of features. We first validate the method using simulated images, and compare it with another fusion scheme using the discrete wavelet transform. The results show that the proposed method is better in both accuracy and computational time. Improvements of fused clinical images against unfused raw images are also illustrated. For the segmentation of the bladder wall, we investigate the level set approach. While the traditional gradient based feature detection is prone to intensity non-uniformity, we present a novel way to compute phase congruency as a reliable feature representation. In order to avoid the phase wrapping problem with inverse trigonometric functions, we devise a mathematically elegant and efficient way to combine multi-scale image features via geometric algebra. As opposed to the original phase congruency, the proposed method is more robust against noise and hence more suitable for clinical data. To address the practical issues in segmenting the bladder wall, we suggest two coupled level set frameworks to utilise information in two different MRI sequences of the same patients - the T2- and T1-weighted image. The results demonstrate a dramatic decrease in the number of failed segmentations done using a single kind of image. The resulting automated segmentations are finally validated by comparing to manual segmentations done in 2D.
- Published
- 2012
199. Resolución geométrica de la cinemática inversa de un robot sin muñeca esférica
- Abstract
Una de las aplicaciones más conocidas del álgebra geométrica en la ingenierìa consiste en proporcionar una formulación compacta de la cinemática de los robots manipuladores serie. Sin embargo, el uso de álgebra geométrica en el campo de la robótica está aún en sus inicios, y todavía hay varios problemas abiertos que pueden ser tratados con esta elegante y compacta formulación. En esta línea, el presente artículo introduce una estrategia basada en el álgebra geométrica conforme para resolver el problema de la cinemática inversa para un robot manipulador de 6 grados de libertad (GdL) sin muñeca esférica, para el cual es conocido que el problema de la cinemática inversa no tiene, en general, solución analítica. Para ello, la estrategia propuesta en este artículo se basará en la explotación de las propiedades algebraicas y geométricas del álgebra geométrica conforme como, por ejemplo, que toda isometría se puede representar de manera compacta como un rotor, y que los objetos geométricos no son más que multivectores., Peer Reviewed, Postprint (author's final draft)
- Published
- 2023
200. Resolución geométrica de la cinemática inversa de un robot sin muñeca esférica
- Abstract
[Resumen] Una de las aplicaciones más conocidas del álgebra geométrica en la ingenierìa consiste en proporcionar una formulación compacta de la cinemática de los robots manipuladores serie. Sin embargo, el uso de álgebra geométrica en el campo de la robótica está aún en sus inicios, y todavía hay varios problemas abiertos que pueden ser tratados con esta elegante y compacta formulación. En esta línea, el presente artículo introduce una estrategia basada en el álgebra geométrica conforme para resolver el problema de la cinemática inversa para un robot manipulador de 6 grados de libertad (GdL) sin muñeca esférica, para el cual es conocido que el problema de la cinemática inversa no tiene, en general, solución analítica. Para ello, la estrategia propuesta en este artículo se basará en la explotación de las propiedades algebraicas y geométricas del álgebra geométrica conforme como, por ejemplo, que toda isometría se puede representar de manera compacta como un rotor, y que los objetos geométricos no son más que multivectores., [Abstract] One of the most known applications of geometric algebra in engineering consists of providing a compact formulation of the kinematics of serial robotic manipulators. However, the use of geometric algebra in the field of robotics is still in its early stages, and there are still several open problems that can be addressed with this elegant and compact formulation. In this context, this work introduces a strategy based on conformal geometric algebra to solve the inverse kinematics problem for a 6 degree-of-freedom (DOF) robotic manipulator without a spherical wrist, for which it is known that the inverse kinematics problem generally does not have an analytical solution. To achieve this, the strategy proposed in this work will rely on exploiting the algebraic and geometric properties of conformal geometric algebra such as, for instance, the fact that every isometry can be represented compactly as a rotor, and that geometric objects are nothing more than multivectors.
- Published
- 2023
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.