{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T05:35:18Z","timestamp":1771479318492,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2011,6,10]],"date-time":"2011-06-10T00:00:00Z","timestamp":1307664000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym3020365","type":"journal-article","created":{"date-parts":[[2011,6,10]],"date-time":"2011-06-10T12:35:40Z","timestamp":1307709340000},"page":"365-388","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation"],"prefix":"10.3390","volume":"3","author":[{"given":"Tadamasa","family":"Sawada","sequence":"first","affiliation":[{"name":"Department of Psychological Sciences, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunfeng","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Psychological Sciences, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zygmunt","family":"Pizlo","sequence":"additional","affiliation":[{"name":"Department of Psychological Sciences, Purdue University, West Lafayette, IN 47907, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2011,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Pizlo, Z. 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