Discrete combinatorial geometry

摘要

In computer vision, one of the ultimate aims is the determination of geomettric properties of 3-dimensional objects in our real world from measured data. As an expression intermediate between measured raw data and geometric properties, we need a method of representing objects in computers. For the object representations, geometry which uses only finite-precision numbers is necessary because in computers we can only manipulate finite-precision numbers. In this paper, we develop a new geometry, which we call discrete combinatorial geometry due to the discreteness of the space of finite-precision numbers, applying fundamental definitions of classical combinatorial geometry. Using discrete combinatorial geometry, we introduce a new method for representing curves, surfaces and objects in computers. We also show that our new representation is based on the fact that the boundary of a surface consists of curves and the boundary of an object consists of surfaces.