Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation

• Main Authors: Zygmunt Pizlo, Yunfeng Li, Tadamasa Sawada
• Format: Article
• Language: English
• Published: MDPI AG 2011-06-01
• Series: Symmetry
• Subjects: 3D symmetry; 3D recovery; 3D shape; degenerate views; human perception
• Online Access: http://www.mdpi.com/2073-8994/3/2/365/

Symmetry has been shown to be a very effective a priori constraint in solving a 3D shape recovery problem. Symmetry is useful in 3D recovery because it is a form of redundancy. There are, however, some fundamental limits to the effectiveness of symmetry. Specifically, given two arbitrary curves in a single 2D image, one can always find a 3D mirror-symmetric interpretation of these curves under quite general assumptions. The symmetric interpretation is unique under a perspective projection and there is a one parameter family of symmetric interpretations under an orthographic projection. We formally state and prove this observation for the case of one-to-one and many-to-many point correspondences. We conclude by discussing the role of degenerate views, higher-order features in determining the point correspondences, as well as the role of the planarity constraint. When the correspondence of features is known and/or curves can be assumed to be planar, 3D symmetry becomes non-accidental in the sense that a 2D image of a 3D asymmetric shape obtained from a random viewing direction will not allow for 3D symmetric interpretations.