Correspondences between 3D keypoints generated by matching local descriptors are a key step in 3D computer vision and graphic applications. Learned descriptors are rapidly evolving and outperforming the classical handcrafted approaches in the field. Yet, to learn effective representations they require supervision through labeled data, which are cumbersome and time-consuming to obtain. Unsupervised alternatives exist, but they lag in performance. Moreover, invariance to viewpoint changes is attained either by relying on data augmentation, which is prone to degrading upon generalization on unseen datasets, or by learning from handcrafted representations of the input which are already rotation invariant but whose effectiveness at training time may significantly affect the learned descriptor. We show how learning an equivariant 3D local descriptor instead of an invariant one can overcome both issues. LEAD (Local EquivAriant Descriptor) combines Spherical CNNs to learn an equivariant representation together with plane-folding decoders to learn without supervision. Through extensive experiments on standard surface registration datasets, we show how our proposal outperforms existing unsupervised methods by a large margin and achieves competitive results against the supervised approaches, especially in the practically very relevant scenario of transfer learning.

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 Ortoghonal 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.