Full-Angle Quaternions for Robustly Matching Vectors of 3D Rotations Stephan Liwicki 1,* Minh-Tri Pham 2,* Stefanos Zafeiriou 1 Maja Pantic 1,3 Bj¨ orn Stenger 2 1 Computer Science, Imperial College London, United Kingdom 2 Cambridge Research Laboratory, Toshiba Research Europe Ltd, Cambridge, United Kingdom 3 Electrical Engineering, Mathematics and Computer Science, University of Twente, The Netherlands Abstract In this paper we introduce a new distance for robustly matching vectors of 3D rotations. A special representation of 3D rotations, which we coin full-angle quaternion (FAQ), allows us to express this distance as Euclidean. We apply the distance to the problems of 3D shape recognition from point clouds and 2D object tracking in color video. For the former, we introduce a hashing scheme for scale and translation which outperforms the previous state-of-the-art approach on a public dataset. For the latter, we incorporate online subspace learning with the proposed FAQ represen- tation to highlight the benefits of the new representation. 1. Introduction Outliers and noisy data are common problems when matching feature vectors in applications such as image reg- istration [29], image matching [6], shape matching [8], face recognition [20], object tracking [18], and feature learn- ing [23]. Standard distances (e.g. the Euclidean distance be- tween Euclidean points) can be disadvantageous since cor- ruptions may bias the results, e.g.[4]. Because identify- ing outliers may be computationally costly and sometimes difficult, robust distances between vectors that suppress the influence of outliers while preserving the inliers’ geometry have been developed [6, 8, 18, 20]. Most commonly, feature vectors are scalar valued. In this case existing methods tackle the problem of outliers mainly by adopting different distances. Early approaches use variants of the Manhattan distance [5, 13], leading to an increase in robustness, but at the cost of reduced efficiency. Recent works [4, 18, 23] achieve both robustness and effi- ciency by mapping points non-linearly to a space where the distance involving outliers is nearly uniformly distributed, thereby allowing for robust subspace learning. * Main contributors: sl609@imperial.ac.uk, mtpham@crl.toshiba.co.uk Non-scalar features have received less attention in the literature. For example, in [8], the matching of unordered sets of 2D points is considered. In [29], a robust compar- ison of 2D rotations adopting the sum of cosines of angle differences is investigated. The work in [20] extends this approach to match vectors of 3D surface normals, by pro- jecting the 3D normals to 2D rotations. In this paper we address the problem of matching fea- ture vectors of 3D rotations, introducing a robust and ef- ficient distance function. We apply our approach to 3D shape recognition by converting the problem of evaluating shape poses into the problem of robustly matching vectors of direct similarities (i.e. transformations with a uniform scale, rotation, and translation [3]). In particular, we in- troduce concurrent hashing of scale and translation to pro- duce vectors of rotations, which we then evaluate using our distance. We also apply the new FAQ representation to 2D object tracking, where we formulate the problem of robustly matching color patches as an online subspace learning task for vectors of rotations. Our contributions are as follows: 1. We propose a closed-form distance between 3D rota- tions, which allows for robust matching and subspace learning with vectors of 3D rotations. 2. We formulate a new 3D rotation representation, called full-angle quaternion, making our distance Euclidean. 3. We introduce a map such that uniformly distributed di- latations (i.e. transformations composed of a uniform scaling and a translation [3]) correspond to uniformly distributed coordinates in Euclidean space, facilitating efficient hashing. 4. We evaluate our framework on 3D shape recognition and 2D tracking, showing superior performance in com- parison to existing methods. 2. Existing Rotation Distances We first briefly review existing distances for 2D and 3D rotations in literature. 1