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Algebraic Relations among Matching Constraints of Multiple Images

Long Quan 1
1 MOVI - Modeling, localization, recognition and interpretation in computer vision
GRAVIR - IMAG - Laboratoire d'informatique GRAphique, VIsion et Robotique de Grenoble, Inria Grenoble - Rhône-Alpes, CNRS - Centre National de la Recherche Scientifique : FR71
Abstract : Given a set of $n\ge2$ uncalibrated views, for any corresponding point across $n$ views, there exist three types of matching constraints: bilinear constraints(for $n\ge2$), trilinear constraints (for $n\ge3$, \cite{shashua94c- }) and quadrilinear constraints (for $n \ge 4$, \cite{hartley94d,triggs95a,fau- geras95a}). The exact algebraic relations among these multi-linear constraints have not been elucidated by previous authors. This paper examines the relations between these matching constraints by singling out the degenerate view and point configurations. The key result that will be established is that for generic view configurations and generic points, all multi-linear constraints may algebraically be reduced to the algebraically independent bilinear constraints- . In other words, all matching constraints are contained in the ideal generated only by the bilinear constraints for generic views and points. As a consequence, $2n-3$ algebraically independent bilinearities from pairs of views completely describe the algebraic/geometric structure of $n$ uncalibrated views for generic views and points. For degenerate points of generic views, each type of constraint reduces differently. The exact reduced form of the matching constraints are also made explicit by computer algebra.
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Submitted on : Wednesday, May 24, 2006 - 12:35:16 PM
Last modification on : Friday, February 4, 2022 - 3:24:57 AM
Long-term archiving on: : Sunday, April 4, 2010 - 11:43:11 PM


  • HAL Id : inria-00073344, version 1



Long Quan. Algebraic Relations among Matching Constraints of Multiple Images. RR-3345, INRIA. 1998. ⟨inria-00073344⟩



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