We explore the maximum total number of edge crossings and the maximum geometric thickness of the Euclidean minimum-weight (k, ℓ)-tight graph on a planar point set P. In this paper, we show that (10/7－ε)|P| and (11/6－ε)|P| are lower bounds for the maximum total number of edge crossings for any ε ＞ 0 in cases (k,ℓ)=(2,3) and (2,2), respectively. We also show that the lower bound for the maximum geometric thickness is 3 for both cases. In the proofs, we apply the method of arranging isomorphic units regularly. While the method is developed for the proof in case (k,ℓ)=(2,3), it also works for different ℓ.

Recent research on graph drawing focuses on Right-Angle-Crossing (RAC) drawings of 1-plane graphs, where each edge is drawn as a straight line and two crossing edges only intersect at right angles. We give a transformation from a restricted case of the RAC drawing problem to a problem of finding a straight-line drawing of a maximal plane graph where some angles are required to be acute. For a restricted version of the latter problem, we show necessary and sufficient conditions for such a drawing to exist, and design an O(n2)-time algorithm that given an n-vertex plane graph produces a desired drawing of the graph or reports that none exists.

Since for recognition tasks it is known that planar invariants are more easily obtained than others, decomposing a scene in terms of planar parts becomes very interresting. This paper presents a new approach to find the projections of planar surfaces in a pair of images. For this task we introduce the facet concept defined by linked edges (chains) and corners. We use collineations as projective information to match and verify their planarity. Our contribution consists in obtaining from an uncalibrated stereo pair of images a match of "planar" chains based on matched corners. Collineations are constrained by the fundamental matrix information and a Kalman filter approach is used to refine its computation.

Introducing a mathematical model of noise in stereo images, we propose a new criterion for intelligent statistical inference about the scene we are viewing by using the geometric information criterion (geometric AIC). Using synthetic and real-image experiments, we demonstrate that a robot can test whether or not the object is located very far away or the object is a planar surface without using any knowledge about the noise magnitude or any empirically adjustable thresholds.

We can understand and recover a scene even from a picture or a line drawing. A number of methods have been developed for solving this problem. They have scarcely aimed to deal with scenes of multiple objects although they have ability to recognize three-dimensional shapes of every object. In this paper, challenging to solve this problem, we describe a method for deciding configurations of multiple objects. This method employs the assumption of coplanarity and the constraint of occlusion. The assumption of coplanarity generates the candidates of configurations of multiple objects and the constraint of occlusion prunes impossible configurations. By combining this method with a method of shape recovery for individual objects, we have implemented a system acquirig a three-dimensional information of scene including multiple objects from a monocular image.