Rigidity Preserving Image Transformations and Equivariance in Perspective

Abstract

We characterize the class of image plane transformations which realize rigid camera motions and call these transformations ‘rigidity preserving’. It turns out that the only rigidity preserving image transformations are homographies corresponding to rotating the camera. In particular, 2D translations of pinhole images are not rigidity preserving. Hence, when using CNNs for 3D inference tasks, it can be beneficial to modify the inductive bias from equivariance w.r.t. translations to equivariance w.r.t. rotational homographies. We investigate how equivariance with respect to rotational homographies can be approximated in CNNs, and test our ideas on 6D object pose estimation. Experimentally, we improve on a competitive baseline.