Short Communication: Curvature Approximation in Images

Abstract

We consider discrete planar curves as they appear in segmented images. In the literature, the curvature of such curves is often estimated via B-spline approximations or by interpolation schemes, while to the best of our knowledge current methods lack of a proof of convergence, see [2,3]. We will not only proof the convergence of our method in the uniform norm for smooth curves, we will also show that our method is able to detect critical points (C 2 -singularities) of our given discrete data, i.e., points where the curvature is undefined. The main idea is to approximate the curve such that the shape of the curve is preserved. Here, we use the Schoenberg splines because of the freedom to choose the knots arbitrarily and because of their variation-diminishing property that leads to an approximation which preserves positivity, monotonicity and convexity. Shape preserving approximation with Schoenberg splinesLet n, k > 0 be integers, let the knot sequence ∆ n = {t j } n+k j=−k such that 0 = t −k = · · · = t 0 < t 1 < · · · < t n = · · · = t n+k = 1 and consider for j = −k, . . . , n − 1 the knot averages ξ j,k := k l=1 t j+l /k and the B-splines N j,k (t) which are normalized such that they form a partition of unity. Then the Schoenberg operator of degree k is defined forNote that the operator can be similarly defined for vector valued functions on [0, 1], as will be needed in Section 2. It has been shown by Marsden [4] that this operator preserves the positivity, the monotonicity, and the convexity of continuous functions. Furthermore, Marsden has shown that not only the approximation converges uniformly on [0, 1] but also the first two derivatives, while in the latter case the convergence holds only on compact subsets of (0, 1). For more details and properties of the Schoenberg operator we refer to [4,6], shape preserving approximations are discussed in [1]. Curvature approximationIn the following we will show that the curvature derived by the spline approximation (1) converges uniformly towards the real curvature on all compact subsets of (0, 1). To the best of our knowledge existing methods do not provide a proof of convergence, while a comparative study based on numerical experiments can be found in [2]. For a more comprehensive overview of existing methods we refer to [3] and the references therein.Let us consider the planar curve α : [0, 1] → R 2 , α(t) = (x(t), y(t)) T , such that α is twice continuously differentiable. Then the signed curvature is defined byWe approximate the signed curvature of α by the curvature of its spline approximation S ∆n,k α κ ∆n,k (t) = det(DS ∆n,k α(t), D 2 S ∆n,k α(t)) DS ∆n,k α(t) 3 .In the following we will show that |κ(t) − κ ∆n,k (t)| → 0 uniformly on all compact subsets of (0, 1) for n → ∞. To this end, let us denote by δ min := min(t j+1 − t j ) and by δ max := max(t j+1 − t j ), where j = 0, . . . , n − 1, the minimal and the maximal distance of two consecutive knots of the sequence ∆ n .Lemma 2.1 Suppose f ∈ C 2 ([0, 1]) and k > 2. Then the derivatives of the spline appr...