16 papers related to Clifford algebra

1) Communicated by Nek Valous, National Center for Tumor Diseases (NCT), Heidelberg.
2) In some cases mathematical symbols are missing.

Title: Parameter-free quaternary orthogonal moments for color image retrieval and recognition Journal: J. of Electronic Imaging, 27(1), 011007 (2018)
Authors: Nisrine Dad; Noureddine En-Nahnahi; Said El Alaoui Ouatik
Link: https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging/volume-27/issue-1/011007/Parameter-free-quaternary-orthogonal-moments-for-color-image-retrieval-and/10.1117/1.JEI.27.1.011007.short

In order to describe color images, the use of the algebra of quaternions in combination with existing image orthogonal moments, meant for binary and grayscale images, has been widely investigated. This is because of their advantages in (1) gathering the three-channel color information in a single feature vector while preserving the correlation between them and in (2) eliminating shape information redundancy. However, the computation of these quaternary orthogonal moments depends on a unit pure quaternion parameter. The optimal value of this latter can be fixed only with the help of experiments and it is application-dependent. We propose a parameter-free formulation of the quaternary orthogonal moments. The general formula for the computation of the proposed moments, whose rotation invariance is achieved by retaining the modulus, is provided. Furthermore, experiments are conducted to evaluate the performance of the proposed modulus-based moment invariants for color image retrieval and recognition.

Title: Using Dual Quaternion to Study Translational Surfaces
Mathematics in Computer Science, March 2018, Volume 12, Issue 1, pp 69–75
Haohao Wang, Ron Goldman

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces can also be generated from two rational space curves by dual quaternion multiplication. Using the mathematics of dual quaternions, we provide a necessary and sufficient condition for a rational tensor product surface to be a translational surface. Examples are provided to illustrate our theorems and flesh out our algorithms.

Title: Type synthesis of 1T2R and 2R1T parallel mechanisms employing conformal geometric algebra
Journal: Mechanism and Machine Theory, Volume 121, March 2018, Pages 475-486
Authors: Yimin Song, Pengpeng Han, PanfengWang
Link: https://www.sciencedirect.com/science/article/pii/S0094114X17305694

This paper proposes an analytical method based on conformal geometrical algebra to implement and unify type synthesis of one translational and two rotational (1T2R) parallel mechanisms and two rotational and one translational (2R1T) parallel mechanisms. Firstly, conformal geometrical algebra is introduced in this paper to describe finite motions and carry out type synthesis of parallel mechanisms in an algebraic way. Secondly, the finite motions of 1T2R and 2R1T parallel mechanisms are described and their relations are explored, and this results in unifying the type synthesis of these two types of parallel mechanisms in the same procedure. Then, a parametric and algebraic generation method is proposed to obtain available limbs and axes layout of joints with these limbs. Finally, the assembly conditions among available limbs are given to synthesize systematically 2R1T parallel mechanisms with non-overconstrained or overconstrained property, as well as 1T2R parallel mechanisms with the same properties. This method proposed in this paper is valid in carrying out finite motion representation and type synthesis of parallel mechanisms in an algebraic and analytical way, and then is beneficial to an automatic manner using computer programming languages. Furthermore, the method may solve type synthesis of special parallel mechanisms, such as parallel mechanisms with varied output rotational axes.

Title: The octonions as a twisted group algebra
Journal: Finite Fields and Their Applications, Volume 50, March 2018, Pages 113-121
Authors: Tathagata Basak
Link: https://www.sciencedirect.com/science/article/pii/S1071579717301314

We show that the octonions can be defined as the -algebra with basis  and multiplication given by , where . While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a computation-free proof of their existence.

Title: Spin actions in Euclidean and Hermitian Clifford analysis in superspace
Journal: Journal of Mathematical Analysis and Applications, Volume 457, Issue 1, 1 January 2018, Pages 23-50
Authors: Hennie De Schepper, Alí Guzmán Adán, Frank Sommen
Link: https://www.sciencedirect.com/science/article/pii/S0022247X17307576

In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group  leaving invariant such an inner product turns out to be an extension of  and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of  by means of the representation . In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of and is isomorphic to the subset  of  of elements that commute with the complex structure J. The realization of  within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.

Title: Smooth orientation interpolation using parametric quintic-polynomial-based quaternion spline curve
Journal: Journal of Computational and Applied Mathematics, Volume 329, February 2018, Pages 256-267
Authors: Jieqing Tan, Yan Xing, Wen Fan, Peilin Hong
Link: https://www.sciencedirect.com/science/article/pii/S0377042717303485

In this paper, a continuous quintic-polynomial-based unit quaternion interpolation spline curve with tension parameters is presented to interpolate a given sequence of solid orientations. The curve in unit quaternion space  is an extension of the quintic polynomial interpolation spline curve in Euclidean space. It preserves the interpolatory property and  continuity. Meanwhile, the unit quaternion interpolation spline curve possesses the local shape adjustability due to the presence of tension parameters. The change of one tension parameter will only affect the adjacent two pieces of curves. Compared with the traditional B-spline unit quaternion interpolation curve and -spline unit quaternion interpolation curve, the proposed curve can automatically interpolate the given data points, without solving the nonlinear system of equations over quaternions to obtain the control points, which greatly improves the computational efficiency. Simulation results demonstrate the effectiveness of the proposed scheme.

Title: Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing
Journal: Signal Processing, Volume 148, July 2018, Pages 193-204
Authors: Min Xiang, Shirin Enshaeifar, Alexander E. Stott, Clive Cheong Took, Yili Xia, Sithan Kanna, Danilo P.Mandic
Link: https://www.sciencedirect.com/science/article/pii/S0165168418300719

Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the non-commutative nature of the quaternion product has been prohibitive to the development of quaternion uncorrelating transforms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance and complementary covariance matrices in the quaternion domain, whereby the quaternion version of the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.

Title: Shimizu’s Lemma for Quaternionic Hyperbolic Space
Journal: Computational Methods and Function Theory, March 2018, Volume 18, Issue 1, pp 159–191
Authors: Wensheng Cao, John R. Parker
Link: https://link.springer.com/article/10.1007%2Fs40315-017-0212-4

We give a generalisation of Shimizu’s lemma to complex or quaternionic hyperbolic space in any dimension for groups of isometries containing an arbitrary parabolic map. This completes a project begun by Kamiya (Hiroshima Math J 13:501–506, 1983). It generalises earlier work of Kamiya, Inkang Kim and Parker. The analogous result for real hyperbolic space is due to Waterman (Adv Math 101:87–113, 1993).

Title: Selectivity in definite quaternion algebras
Journal: Journal of Number Theory, Volume 185, April 2018, Pages 379-395
Authors: Patricio Quiroz
Link: https://www.sciencedirect.com/science/article/pii/S0022314X17303505

We study the embedding problem for commutative orders into Eichler orders in definite quaternion algebras over the rationals by using methods from the theory of quadratic forms, specifically, Minkowski–Siegel’s formula for the representation mass. We characterize when the Gaussian or Eisenstein integers embed into some but not all classes in the genus of an Eichler order. We also give an application to the theory of supersingular elliptic curves.

Title: Single-Image Shadow Detection using Quaternion Cues
Journal: The Computer Journal, 27 January 2018, doi 10.1093/comjnl/bxy004
Authors: Hazem Hiary, Rawan Zaghloul, Moh’d Belal Al-Zoubi
Link: https://academic.oup.com/comjnl/advance-article-abstract/doi/10.1093/comjnl/bxy004/4827073

Shadow detection is an important pre-processing step often used in scene interpretation or shadow removal applications. In this paper, we propose a single-image shadow detection method. Many other methods use multiple images; we use a quaternion representation of colour images to extract shadows from only one input image. The generation of the final binary shadow mask is done via automatic threshold selection. Evaluation is carried out qualitatively and quantitatively over three challenging datasets of indoor and outdoor natural images. The results of qualitative assessment were consistent with the statistical results, where the proposed method improves the performance of shadow detection when compared with state-of-the-art methods.

Title: Distributed measurement of polarization mode coupling in fiber ring based on P-OTDR complete polarization state detection
Journal: Optics Express Vol. 26, Issue 4, pp. 4798-4806 (2018)
Authors: Zejia Huang, Chongqing Wu, Zhi Wang, Jian Wang, and Lanlan Liu
Link: https://www.osapublishing.org/oe/abstract.cfm?uri=oe-26-4-4798

Using a quaternion method, the polarization mode-coupling coefficient can be derived from three components of the Stokes vectors at three adjacent points along a fiber. A complete polarization optical time-domain reflectometry scheme for polarization mode coupling distributed measurement in polarization-maintaining fiber ring is proposed based on the above theoretical derivations. By comparing the measurement results of two opposite incident directions and two orthogonal polarization axes of polarization-maintaining fiber rings with different lengths, the feasibility and repeatability of the measurement scheme are verified experimentally with a positioning spatial resolution of 1 meter.

Title: Orders of quaternion algebras with involution
Journal: Journal of Number Theory, Volume 183, February 2018, Pages 249-268
Authors: Arseniy Sheydvasser
Link: https://www.sciencedirect.com/science/article/pii/S0022314X17302834

We introduce the notion of maximal orders over quaternion algebras with orthogonal involution and give a classification over local and global fields. Over local fields, we show that there is a correspondence between maximal and/or modular lattices and orders closed under the involution.

Title: Optimal blind watermarking for color images based on the U matrix of quaternion singular value decomposition
Journal: Multimedia Tools and Applications, 2018, doi: 10.1007/s11042-018-5652-y
Authors: Feng Liu, Long-Hua Ma, Cong Liu, Zhe-Ming Lu
Link: https://link.springer.com/article/10.1007/s11042-018-5652-y

In this paper, a new blind watermarking for copyright protection of color images based on the U matrix through Quaternion Singular Value Decomposition (QSVD) is proposed. The proposed method represents the color image with a quaternion matrix, so that it can deal with the multichannel information in a holistic way. Then the array of pure quaternion is divided into non-overlapping blocks and we perform QSVD on each block to get its U matrix. The watermark is inserted into the optimally selected coefficients of the quaternion elements in the first column of the U matrix. Besides, in the procedures of watermark insertion and extraction, ensuring higher fidelity and robustness to several possible image attacks have been considered. The experimental results show that the proposed method outperforms existing schemes in terms of robustness and invisibility.

Title: New Set of Quaternion Moments for Color Images Representation and Recognition
Journal: Journal of Mathematical Imaging and Vision, 2018, doi: 10.1007/s10851-018-0786-0
Authors: Khalid M. Hosny, Mohamed M. Darwish
Link: https://link.springer.com/article/10.1007/s10851-018-0786-0

In this paper, a new set of quaternion radial-substituted Chebyshev moments (QRSCMs) is proposed for color image representation and recognition. These new moments are circular moments defined over a unit disk by using a new set of orthogonal basis functions called radial-substituted Chebyshev functions. A new hybrid method is proposed for highly accurate computation of QRSCMs in polar coordinates. In this method, the angular kernel is exactly computed by analytical integration of Fourier function over circular pixels. The radial kernel is computed using a recurrence relation which completely eliminates the coefficient matrix associated with the radial-substituted Chebyshev functions. Rotation, scaling, and translation (RST) invariances for QRSCMs are proved. Numerical experiments were conducted where the results of these experiments show better performance of QRSCMs over existing quaternion moments in terms of image reconstruction capabilities, RST invariances, robust to different noises, and CPU elapsed times.

Title: Multistability and multiperiodicity in impulsive hybrid quaternion-valued neural networks with mixed delays
Journal: Neural Networks, Volume 99, March 2018, Pages 1-18
Authors: Călin-Adrian Popa, Eva Kaslik
Link: https://www.sciencedirect.com/science/article/pii/S0893608017302915

The existence of multiple exponentially stable equilibrium states and periodic solutions is investigated for Hopfield-type quaternion-valued neural networks (QVNNs) with impulsive effects and both time-dependent and distributed delays. Employing Brouwer’s and Leray–Schauder’s fixed point theorems, suitable Lyapunov functionals and impulsive control theory, sufficient conditions are given for the existence of  attractors, showing a substantial improvement in storage capacity, compared to real-valued or complex-valued neural networks. The obtained criteria are formulated in terms of many adjustable parameters and are easily verifiable, providing flexibility for the analysis and design of impulsive delayed QVNNs. Numerical examples are also given with the aim of illustrating the theoretical results.

Title: Measuring the closeness to singularities of a planar parallel manipulator using geometric algebra
Journal: Applied Mathematical Modelling, Volume 57, May 2018, Pages 192-205
Authors: Yao Huijing, Li Qinchuan, Chen Qiaohong, Chai Xinxue
Link: https://www.sciencedirect.com/science/article/pii/S0307904X18300180

A new index for measuring the closeness to the singularities of parallel manipulators using geometric algebra is proposed in this paper. Constraint wrenches acting on the moving platform of a parallel manipulator are derived using the outer product and dual operations. Removing the redundant constraint wrenches, a singularity polynomial is obtained when the coefficient of the outer product of all the non-redundant constraint wrenches equals zero. A singularity surface can be drawn using the singularity polynomial. Similarly, an approximate singularity polynomial and approximate singularity surface can be obtained by imposing a threshold to the singular polynomial. Then the singularity volume is calculated as the space between singularity surface and approximate singularity surface. The new index is derived by calculating the ratio of the non-singularity workspace volume (the workspace volume minus the singularity volume) to the workspace volume. The proposed index is coordinate-free and has a clear geometrical and physical interpretation. This index can be a basis for selecting structural parameters, path planning and mechanism design.

Source: Email from N. Velous, nek.valous_AT_nct-heidelberg.de, 03 Mar. 2018.



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5 responses to “16 papers related to Clifford algebra

  1. piuasu

    are with Clifford Algebra geometry and THE THREE BODIES problem

  2. sadedgu

    in so are approached with Clifford algebra

  3. pagenu

    these are very important and can be applied in differential geometry and other aspects of algebra as sylvester algebra as observed prof dr mircea orasanu ,and also in cases of Laplace operator or Lagrange operator with more applications

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