Premium
Short Communication: Curvature Approximation in Images
Author(s) -
Nagler Johannes
Publication year - 2014
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201410455
Subject(s) - curvature , convexity , mathematics , monotonic function , norm (philosophy) , gravitational singularity , spline (mechanical) , mathematical analysis , interpolation (computer graphics) , convergence (economics) , planar , total curvature , geometry , mean curvature , image (mathematics) , computer science , mean curvature flow , artificial intelligence , economic growth , computer graphics (images) , structural engineering , political science , financial economics , law , economics , engineering
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. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)