Bijective Digitized 3D Rotation Based on Beam Shears

Each 3D rotation can be decomposed into three 2D rotations parametrized by three Euler angles. Each of the three 2D rotations can be expressed as a sequence of three 2D shears along coordinate axes, leading to a decomposition of a 3D rotation into nine (beam) shears in total. We define a 3D digitized rotation using nine digitized beam shears, i.e., we round the result of each shear before applying the next one. As digitized shears are bijective, our 3D digitized rotation inherits the same property. Experiments show that the average error of our digitized rotation compared to the continuous one is kept under 1 (around 0.8).