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Full-Angle Quaternions for Robustly Matching Vectors of 3D Rotations

“…Numerous distances for 3D rotations have been proposed in the literature [7,10,11,22]. Most of these are Euclidean distances (and variants) under different representations of 3D rotations.…”

“…The Euler angles distance is the Euclidean distance between Euler angles [11]. The 2 -norm under the half-angle quaternion (HAQ) representation leads to the vectorial/extrinsic quaternion distance [7,10,11] and the inverse cosine quaternion distance [11]. Analysis of geodesics on SO(3) leads to intrinsic distances [10,11,22], which are the 2 -norm of rotation vectors (RV).…”

“…Analysis of geodesics on SO(3) leads to intrinsic distances [10,11,22], which are the 2 -norm of rotation vectors (RV). The 2 -norm in the embedding space R 9 of SO(3) induces the chordal/extrinsic distance [7,11,22] between rotation matrices (RM).…”

“…In computer vision arguably the most widely used 3D rotation distances are the extrinsic distances based on HAQ or RM, due to their closed forms [7,22], allowing to efficiently compute rotation means. However, the HAQ representation is not unique, leading to non-global means.…”

“…In contrast to HAQ, which returns non-global means [7], the mean of 3D rotations under FAQ is global and easy to compute as the normalized sum of FAQs.…”