Defining and reliably finding a canonical orientation for 3D surfaces is key to many Computer Vision and Robotics applications. This task is commonly addressed by handcrafted algorithms exploiting geometric cues deemed as distinctive and robust by the designer. Yet, one might conjecture that humans learn the notion of the inherent orientation of 3D objects from experience and that machines may do so alike. In this work, we show the feasibility of learning a robust canonical orientation for surfaces represented as point clouds. Based on the observation that the quintessential property of a canonical orientation is equivariance to 3D rotations, we propose to employ Spherical CNNs, a recently introduced machinery that can learn equivariant representations defined on the Special Orthogonal group SO(3). Specifically, spherical correlations compute feature maps whose elements define 3D rotations. Our method learns such feature maps from raw data by a self-supervised training procedure and robustly selects a rotation to transform the input point cloud into a learned canonical orientation. Thereby, we realize the first end-to-end learning approach to define and extract the canonical orientation of 3D shapes, which we aptly dub Compass. Experiments on several public datasets prove its effectiveness at orienting local surface patches as well as whole objects.

R1: We agree that defining a canonical orientation for local patches is mainly aimed at descriptor matching. Moreover, as recently shown in Bai et al. in "D3Feat: Joint Learning of Dense Detection We will add this information to the revised version. In the evaluation in Table 2, we follow the standard protocol used in [39] to perform a fair comparison. R2: We used the more general term "pose" to refer to canonical orientation as achieving translation invariance is usually As suggested, we will use only the term orientation. We will modify it in the final version of the paper.

Exp. Psychology, Oxford ELSC, HebrewU Department of Computing Brain Mind Institute, EPFL Gatsby Unit, UCL Imperial College London Andrew M. Saxe Christopher Summerfield Ali Hummos

The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming h is Lipschitz and M has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence.