On properties of analytical approximation for discretizing 2D curves and 3D surfaces

Abstract

Abstract:The morphological discretization is most commonly used for curve and surface discretization, which has been well studied and known to have some important properties, such as preservation of topological properties (e.g., connectivity) of an original curve or surface. To reduce its high computational cost, on the other hand, an approximation of the morphological discretization, called the analytical approximation, was introduced. In this paper, we study the properties of the analytical approximation focusing on discretization of 2D curves and 3D surfaces in the form of y = f (x) (x, y ∈ R) and z = f (x, y) (x, y, z ∈ R). We employ as a structuring element for the morphological discretization, the adjacency norm ball and use only its vertices for the analytical approximation. We show that the discretization of any curve/surface by the analytical approximation can be seen as the morphological discretization of a piecewise linear approximation of the curve/surface. The analytical approximation therefore inherits the properties of the morphological discretization even when it is not equal to the morphological discretization.