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Communication Dans Un Congrès Année : 2023

Conjecture on Characterisation of Bijective 3D Digitized Reflections and Rotations

Stéphane Breuils
Eric Andres
Akihiro Sugimoto
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Résumé

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.
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Dates et versions

hal-03764839 , version 1 (30-08-2022)

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Stéphane Breuils, Yukiko Kenmochi, Eric Andres, Akihiro Sugimoto. Conjecture on Characterisation of Bijective 3D Digitized Reflections and Rotations. Computer Graphics International CGI 2022, Sep 2022, Genève, Switzerland. pp.41-53, ⟨10.1007/978-3-031-30923-6_4⟩. ⟨hal-03764839⟩
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