Oriented Matroids

Pattern Recognition on Oriented Matroids

“…The oriented matroid (E, −A L) on E, given by its set of covectors −A L ⊆ {−, 0, +} E is called the reorientation −A M; see [5,Lemma 4.1.8]. …”

“…The circuits, vectors, and maximal vectors of an oriented matroid M are the cocircuits, covectors, and topes, respectively, of the oriented matroid M * , the dual (or orthogonal) of M. The loops of M are the coloops of M * ; M is acyclic iff M * is totally cyclic, see [5,Proposition 3.4.8].…”

“…The set C * of cocircuits of an extension M is described in the following way [31], [5 = (E, L). Then for every cocircuit Y ∈ C * there is a unique way to extend Y to a cocircuit of M: there is a unique function σ :…”

Robot Vision for Autonomous Object Learning and Tracking

“…The result is a set of cocircuits [10] of sign combinations that relate segmented regions with respect to the convex hull of two selected regions of the scene. The details of this process are explained in [11,12].…”

Realization Spaces of Arrangements of Convex Bodies

“…Consider the orientations of a triple a r u , a s v , a t w ∈ P, and the corresponding curves S r u ,S s v , S t w ∈ S. If u , v, w are distinct, then these points have the same orientation as a 1 u , a 1 v , a 1 w , since each a x i among these points is between a 1 i and a n i in the local sequence of a n i −1 among φ i (Q i ), which implies a x i is in convex position together with A 0 between a 1 i and a n i in counter-clockwise order. Furthermore, the curves have the same orientation as C u ,C v ,C w , which is the same as that of a 1 u , That is, for any realization P i of χ i , there is a non-empty convex region where P i can be augmented by a point q n i to obtain a realization of ω i , and the fibers of the deletion map δ : R 1 (ω i ) → R 1 (χ i ) defined by deleting the point q n i are given by this convex region, which implies R 1 (ω i ) and R 1 (χ i ) are homotopic.…”