A unified curvature definition for regular, polygonal, and digital planar curves

In this paper, we propose a new definition of curvature, called visual curvature. It is based on statistics of the extreme points of the height functions computed over all directions. By gradually ignoring relatively small heights, a multi-scale curvature is obtained. The theoretical properties and the experiments presented demonstrate that multi-scale visual curvature is stable, even in the presence of significant noise. To our best knowledge, the proposed definition of visual curvature is the first ever that applies to regular curves as defined in differential geometry as well as to turn angles of polygonal curves. Moreover, it yields stable curvature estimates of curves in digital images even under sever distortions. We also show a relation between multi-scale visual curvature and convexity of simple closed curves.