On degeneracy of linear reconstruction from three views: Linear line complex and applications

Gideon P. Stein, Amnon Shashua

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


This paper investigates the linear degeneracies of projective structure estimation from line features across three views. We show that the rank of the linear system of equations for recovering the trilinear tensor of three views reduces to 23 (instead of 26) when the scene is a Linear Line Complex (a set of lines in space intersecting at a common line). The LLC situation is only linearly degenerate, and one can obtain a unique solution when the admissibility constraints of the tensor are accounted for. The line configuration described by an LLC, rather than being some obscure case, is in fact quite typical. It includes, as a particular example, the case of a camera moving down a hallway in an office environment or down an urban street. Furthermore, an LLC situation may occur as an artifact such as in direct estimation from spatio-temporal derivatives of image brightness. Therefore, an investigation into degeneracies and their remedy is important also in practice.

Original languageAmerican English
Pages (from-to)244-251
Number of pages8
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number3
StatePublished - Mar 1999
Externally publishedYes

Bibliographical note

Funding Information:
A.S. wishes to thank Steve Maybank, Shai Avidan, and Nassir Na-vab for helpful comments and acknowledges the financial support from US-IS Binational Science Foundation 94-00120/2, the European ACTS project AC074 “Vanguard,” and DARPA through ARL Contract DAAL01-97-0101. G.S. would like acknowledge the financial support from ONR contract N00014-94-1-0128 and DARPA contracts N00014-94-01-0994 and N00014-97-0363.


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