Classification/comparison of curves by an infinite family of shape invariants

Abstract

In this paper we start with a family of boundary based shape measures I N(γ) = ∫ γ(x(s) 2 + y(s) 2) N ds, N = 1, 2, ..., defined for every curve γ given in an arc-length parametrisation x = x(s), y = y(s), s ε [0, 1] and placed such that the centroid of γ and the origin coincide. We prove I N(γ) ≤ 4 -N, for all N = 1, 2, ... which implies that the sequence I N(γ) converges quickly to 0 and, therefore the first few measures I N(γ) are most useful to compare shapes and to be applied in tasks like object classification, recognition or identification.