Checking oriented matroid isomorphism by means of canonical labeling

Authors:
JüRgen Bokowski;Ernesto Staffetti
Affiliations:
Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstrasse 7 D-64289 Darmstadt, Germany;Department of Statistics and Operations Research, Universidad Rey Juan Carlos, Calle Tulipán s/n E-28933 Móstoles, Madrid, Spain
Venue:
Discrete Applied Mathematics
Year:
2013

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Abstract

In this paper a method for establishing the structural equivalence of sets of planar geometric features composed by points and lines is presented. It is based on oriented matroid theory, a setting in which the combinatorial structural properties of these geometric features, such as incidence, order, partitioning, separation, and convexity, can be represented and analyzed in a coordinate-free manner. Projective transformations in computer vision keep in general the convexity property which implies an invariant oriented matroid representation of the planar geometric features under this class of transformations. As long as points and lines are in general position, the oriented matroid representation is also insensitive to small changes in the geometric image features. However the oriented matroid representation depends on the labeling of its elements. Checking the structural equivalence of the above mentioned geometric features represented by means of oriented matroids implies establishing whether two oriented matroid representations are equivalent up to relabeling of their elements. This is the oriented matroid isomorphism problem which is solved in this paper by means of a canonical labeling of the elements.