1,318 research outputs found
On powers of interval graphs and their orders
It was proved by Raychaudhuri in 1987 that if a graph power is an
interval graph, then so is the next power . This result was extended to
-trapezoid graphs by Flotow in 1995. We extend the statement for interval
graphs by showing that any interval representation of can be extended
to an interval representation of that induces the same left endpoint and
right endpoint orders. The same holds for unit interval graphs. We also show
that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the
main result of this note, follows from earlier results of [G. Agnarsson, P.
Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and
dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003].
This version is updated accordingl
Disparity map generation based on trapezoidal camera architecture for multiview video
Visual content acquisition is a strategic functional block of any visual system. Despite its wide possibilities,
the arrangement of cameras for the acquisition of good quality visual content for use in multi-view video
remains a huge challenge. This paper presents the mathematical description of trapezoidal camera
architecture and relationships which facilitate the determination of camera position for visual content
acquisition in multi-view video, and depth map generation. The strong point of Trapezoidal Camera
Architecture is that it allows for adaptive camera topology by which points within the scene, especially the
occluded ones can be optically and geometrically viewed from several different viewpoints either on the
edge of the trapezoid or inside it. The concept of maximum independent set, trapezoid characteristics, and
the fact that the positions of cameras (with the exception of few) differ in their vertical coordinate
description could very well be used to address the issue of occlusion which continues to be a major
problem in computer vision with regards to the generation of depth map
Outer Billiards, Arithmetic Graphs, and the Octagon
Outer Billiards is a geometrically inspired dynamical system based on a
convex shape in the plane.
When the shape is a polygon, the system has a combinatorial flavor. In the
polygonal case, there is a natural acceleration of the map, a first return map
to a certain strip in the plane. The arithmetic graph is a geometric encoding
of the symbolic dynamics of this first return map.
In the case of the regular octagon, the case we study, the arithmetic graphs
associated to periodic orbits are polygonal paths in R^8. We are interested in
the asymptotic shapes of these polygonal paths, as the period tends to
infinity. We show that the rescaled limit of essentially any sequence of these
graphs converges to a fractal curve that simultaneously projects one way onto a
variant of the Koch snowflake and another way onto a variant of the Sierpinski
carpet. In a sense, this gives a complete description of the asymptotic
behavior of the symbolic dynamics of the first return map.
What makes all our proofs work is an efficient (and basically well known)
renormalization scheme for the dynamics.Comment: 86 pages, mildly computer-aided proof. My java program
http://www.math.brown.edu/~res/Java/OctoMap2/Main.html illustrates
essentially all the ideas in the paper in an interactive and well-documented
way. This is the second version. The only difference from the first version
is that I simplified the proof of Main Theorem, Statement 2, at the end of
Ch.
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