Symmetry 2011, 3(2), 365-388; doi:10.3390/sym3020365
Article

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

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Received: 10 February 2011; in revised form: 27 May 2011 / Accepted: 30 May 2011 / Published: 10 June 2011
(This article belongs to the Special Issue Symmetry Processing in Perception and Art)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: 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.
Keywords: 3D symmetry; 3D recovery; 3D shape; degenerate views; human perception
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MDPI and ACS Style

Sawada, T.; Li, Y.; Pizlo, Z. Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation. Symmetry 2011, 3, 365-388.

AMA Style

Sawada T, Li Y, Pizlo Z. Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation. Symmetry. 2011; 3(2):365-388.

Chicago/Turabian Style

Sawada, Tadamasa; Li, Yunfeng; Pizlo, Zygmunt. 2011. "Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation." Symmetry 3, no. 2: 365-388.

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