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

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

Department of Psychological Sciences, Purdue University, West Lafayette, IN 47907, USA
* Author to whom correspondence should be addressed.
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)
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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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