6 papers related to Clifford algebra


Note: Communicated by Nek Valous, National Center for Tumor Diseases (NCT), Heidelberg.

Title: The correct formulation of Gleason’s theorem in quaternionic Hilbert spaces
Journal: arXiv:1803.06882 [math-ph]
Authors: Valter Moretti, Marco Oppio
Link: https://arxiv.org/abs/1803.06882
Abstract: From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\’er’s theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are σ-additive probability measures on that non-Boolean lattice. Gleason’s theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan’s statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan’s theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason’s theorem.


Title: Color image watermarking scheme based on quaternion Hadamard transform and Schur decomposition
Journal:
Multimedia Tools and Applications, February 2018, Volume 77, Issue 4, pp 4545–4561
Authors:
Jianzhong Li, Chuying Yu, B. B. Gupta, Xuechang Ren
Link:
https://link.springer.com/article/10.1007/s11042-017-4452-0
Abstract:
Based on quaternion Hadamard transform (QHT) and Schur decomposition, a novel color image watermarking scheme is presented. To consider the correlation between different color channels and the significant color information, a new color image processing tool termed as the quaternion Hadamard transform is proposed. Then an efficient method is designed to calculate the QHT of a color image which is represented by quaternion algebra, and the QHT is analyzed for color image watermarking subsequently. With QHT, the host color image is processed in a holistic manner. By use of Schur decomposition, the watermark is embedded into the host color image by modifying the Q matrix. To make the watermarking scheme resistant to geometric attacks, a geometric distortion detection method based upon quaternion Zernike moment is introduced. Thus, all the watermark embedding, the watermark extraction and the geometric distortion parameter estimation employ the color image holistically in the proposed watermarking scheme. By using the detection method, the watermark can be extracted from the geometric distorted color images. Experimental results show that the proposed color image watermarking is not only invisible but also robust against a wide variety of attacks, especially for color attacks and geometric distortions.


Title: Four-image encryption scheme based on quaternion Fresnel transform, chaos and computer generated hologram
Journal: Multimedia Tools and Applications, February 2018, Volume 77, Issue 4, pp 4585–4608
Authors: Chuying Yu, Jianzhong Li, Xuan Li, Xuechang Ren, B. B. Gupta
Link: https://link.springer.com/article/10.1007/s11042-017-4637-6
Abstract: A novel four-image encryption scheme based on the quaternion Fresnel transforms (QFST), computer generated hologram and the two-dimensional (2D) Logistic-adjusted-Sine map (LASM) is presented. To treat the four images in a holistic manner, two types of the quaternion Fresnel transform (QFST) are defined and the corresponding calculation method for a quaternion matrix is derived. In the proposed method, the four original images, which are represented by quaternion algebra, are processed holistically in a vector manner by using QFST first. Then the input complex amplitude, which is constructed by the components of the QFST-transformed plaintext images, is encoded by Fresnel transform with two virtual independent random phase masks (RPM). In order to avoid sending entire RPMs to the receiver side for decryption, the RPMs are generated by utilizing 2D–LASM, which results that the amount of the key data is reduced dramatically. Subsequently, by using Burch’s method and the phase-shifting interferometry, the encrypted computer generated hologram is fabricated. To improve the security and weaken the correlation, the encrypted hologram is scrambled base on 2D–LASM. Experiments demonstrate the validity of the proposed image encryption technique.


Title: Geometric techniques for robotics and HMI: Interpolation and haptics in conformal geometric algebra and control using quaternion spike neural networks
Journal: Robotics and Autonomous Systems, Volume 104, June 2018, Pages 72–84
Authors: Eduardo Bayro-Corrochano, Luis Lechuga-Gutiérrez, Marcela Garza-Burgos
Link: https://www.sciencedirect.com/science/article/pii/S0921889017303317
Abstract: In this work, by reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the interpolation, virtual reality, graphics engineering, haptics. We derive intuitive geometric equations to handle surface operations like in kidney surgery. The interpolation can handle the interpolation and dilation in 3D of points, lines, planes, circles and spheres. With this procedure, we interpolate trajectories of surgical instrument. Using quaternions, we formulate the quaternion spike neural network for control. This new neural network structure is based on Spike Neural Networks and developed using the quaternion algebra. The real valued training algorithm was extended so that it could make adjustments of the weights according to the properties and product of the quaternion algebra. In this spike neural network, we are taking into account two relevant ideas the use of Spike neural network which is the best model for oculo-motor control and the role of geometric computing. As illustration. the quaternion spike neural network is applied for control of robot manipulator. The experimental analysis shows promising possibilities for the use of this powerful geometric language to handle multiple tasks in human–machine interaction and robotics.

Title: On quaternionic complexes over unimodular quaternionic manifolds
Journal: Differential Geometry and its Applications, Volume 58, June 2018, Pages 227-253
Authors: Wei Wang
Link: https://www.sciencedirect.com/science/article/pii/S0926224518300597
Abstract: Penrose’s two-spinor notation for 4-dimensional Lorentzian manifolds is extended to two-component notation for quaternionic manifolds, which is a useful tool for calculation. We can construct a family of quaternionic complexes over unimodular quaternionic manifolds only by elementary calculation. On complex quaternionic manifolds as complexification of quaternionic Kähler manifolds, the existence of these complexes was established by Baston by using twistor transformations and spectral sequences. Unimodular quaternionic manifolds constitute a large nice class of quaternionic manifolds: there exists a very special curvature decomposition; the conformal change of a unimodular quaternionic structure is still unimodular quaternionic; the complexes over such manifolds are conformally invariant. This class of manifolds is the real version of torsion-free QCFs introduced by Bailey and Eastwood. These complexes are elliptic. We also obtain a Weitzenböck formula to establish vanishing of the cohomology groups of these complexes for quaternionic Kähler manifolds with negative scalar curvatures.

Title: Quaternion polar harmonic Fourier moments for color images
Journal: Information Sciences, Available online 16 March 2018, In Press, Accepted Manuscript
Authors: Wang Chunpeng, Wang Xingyuan, Li Yongwei, Xia Zhiqiu, Zhang Chuan
Link: https://www.sciencedirect.com/science/article/pii/S002002551631667X
Abstract: This paper proposes quaternion polar harmonic Fourier moments (QPHFM) for color image processing and analyzes the properties of QPHFM. After extending Chebyshev-Fourier moments (CHFM) to quaternion Chebyshev-Fourier moments (QCHFM), comparison experiments, including image reconstruction and color image object recognition, on the performance of QPHFM and quaternion Zernike moments (QZM), quaternion pseudo-Zernike moments (QPZM), quaternion orthogonal Fourier-Mellin moments (QOFMM), QCHFM, and quaternion radial harmonic Fourier moments (QRHFM) are carried out. Experimental results show QPHFM can achieve an ideal performance in image reconstruction and invariant object recognition in noise-free and noisy conditions. In addition, this paper discusses the importance of phase information of quaternion orthogonal moments in image reconstruction.

Source: Email from N. Velous, nek.valous_AT_nct-heidelberg.de, 26 Mar. 2018.

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