Conjecture on Characterisation of Bijective 3D Digitized Reflections and Rotations.

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.