On Symmetry and Multiple-View Geometry: Structure, Pose, and Calibration from a Single Image

In this paper, we provide a principled explanation of how knowledge in global 3-D structural invariants, typically captured by a group action on a symmetric structure, can dramatically facilitate the task of reconstructing a 3-D scene from one or more images. More importantly, since every symmetric structure admits a “canonical” coordinate frame with respect to which the group action can be naturally represented, the canonical pose between the viewer and this canonical frame can be recovered too, which explains why symmetric objects (e.g., buildings) provide us overwhelming clues to their orientation and position. We give the necessary and sufficient conditions in terms of the symmetry (group) admitted by a structure under which this pose can be uniquely determined. We also characterize, when such conditions are not satisfied, to what extent this pose can be recovered. We show how algorithms from conventional multiple-view geometry, after properly modified and extended, can be directly applied to perform such recovery, from all “hidden images” of one image of the symmetric structure. We also apply our results to a wide range of applications in computer vision and image processing such as camera self-calibration, image segmentation and global orientation, large baseline feature matching, image rendering and photo editing, as well as visual illusions (caused by symmetry if incorrectly assumed).