Polynomial volume estimation and its applications

Abstract

Given a compact set S ⊂ R d we consider the problem of estimating, from a random sample of points, the Lebesgue measure of S, µ(S), and its boundary measure, L(S) (as defined by the Minkowski content of ∂S). This topic has received some attention, especially in the two-dimensional case d = 2, motivated by applications in image analysis. A new method to simultaneously estimate µ(S) and L(S) from a sample of points inside S is proposed. The basic idea is to assume that S has a polynomial volume, that is, that V (r) := µ{x : d(x, S) ≤ r} is a polynomial in r of degree d, for all r in some interval [0, R). We develop a minimum distance approach to estimate the coefficients of V (r) and, in particular µ(S) and L(S), which correspond, respectively, to the independent term and the first degree coefficient of V (r). The strong consistency of the proposed estimators is proved. Some numerical illustrations are given