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

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 ∈ [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. In order to overcome such a problem, we modify the family I_N(γ) and also introduce a parameter p to define a new family I_N,p(γ), N = 1, 2, ... of shape measures. The new family I_N,p(γ) includes an infinite number of measures which range over intervals wide enough to provide a discrimination capacity enough to distinguish among the shapes. The role of the parameter p is to provide tuning possibilities for the modified family and to expand the number of applications where the measures can be used efficiently. A set of experimental results are provided in order to justify the theoretical considerations.