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Abstract: Bijectivity of digitized linear transformations is crucial when transforming 2D/3D objects in computer graphics and computer vision. Although characterisation of bijective digitized rotations in 2D is well known, the extension to 3D is still an open problem. A certification algorithm exists that allows to verify that a digitized 3D rotation defined by a quaternion is bijective. In this paper, we use geometric algebra to represent a bijective digitized rotation as a pair of bijective digitized reflections. Visualization of bijective digitized reflections in 3D using geometric algebra leads to a conjectured characterization of 3D bijective digitized reflections and, thus, rotations. So far, any known quaternion that defines a bijective digitized rotation verifies the conjecture. An approximation method of any digitized reflection by a conjectured bijective one is also proposed.

Abstract: Digital twin (DT) has been applied to increasingly complex systems, including environments, energy, and digital cities, due to advancement of data collecting, high-speed networks, big data, artificial intelligence, and other technologies. Because of the complexity of the actual world and newly suggested criteria for the construction of the linkage between the real and virtual spatial, developing and using DT has been significantly hampered. The classic modeling approaches of separating expression from analysis have grown to be a significant barrier. These problems can be resolved due to the benefits of geometric algebra (GA) expression and computation in multidimensional space. The paper studies the concept of DT, employs the essential principles of GA as a tool, and proposes the DT’s modeling and analysis methods. To investigate how DT is formulated and built, a typical multi-factor coupling scenario of passive infrared sensor (PIR) was presented as an example. The results demonstrate the effectiveness of the approach presented in this paper in simulating human-sensor interactions, producing reaction records in real space, and successfully deriving the pedestrian trajectory from PIR recordings. The study presented in this article offers fresh perspectives on how to build DT in complicated scenarios and may also shed new light on how to analyze human behavior using PIR.

Abstract: Oriented elements are part of geometry, and they come in two complementary types: intrinsic and extrinsic. Those different orientation types manifest themselves by behaving differently under reflection. Dualization in geometric algebras can be used to encode them; or vice versa, orientation types inform the interpretation of dualization. We employ the Hodge dual, to include important algebras with null elements like PGA. Oriented elements can be combined using the meet operation, and the dual join (which is here introduced for that purpose). Software written to process one orientation type can be employed to process the complementary type consistently.

Abstract: The octonion Fourier transform (OFT) is a hypercomplex Fourier transform that extends the quaternion Fourier transform. This paper deals with the generalization of Beurling’s uncertainty principle for octonion-valued signals and on $$\mathbb {R}^3$$ , and therefore extends three uncertainty principles (UP), namely Hardy’s UP, Gelfand–Shilov’s UP, and Cowling–Price’s UP, to the OFT domain.