On automatic threshold selection for polygonal approximations of digital curves

Polygonal approximation is a very common representation of digital curves. A polygonal approximation depends on a parameter ε, which is the error value. In this paper we present a method for an automatic selection of the error value, ε. Let Γ^(ε) be a polygonal approximation of the original curve Γ, with an error value ε. We define a set of function, {N_S(ε)}_s∈S, such that for a given value of s, N_s(ε) is the number of edges that contain at least s vertices in Γ^(epsi;). The time complexity for computing the set of functions {N_s(ε)}_s∈S is almost linear in n, the number of vertices in Γ. In this paper we analyse the N_s(ε) graph, and show that for adequate values of s a wide plateau is expected to appear at the top of the graph. This plateau corresponds to a stable state in the multi-scale representation of {Γ^(ε)}_ε∈E. We show that the functions {N_s(ε)}_s∈S are a statistical representation of some kind of scale-space image.