A Unified Curvature Definition for Regular, Polygonal, and Digital Planar Curves

Abstract

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.

On the Farey sequence and its augmentation for applications to image analysis

“…There exists a multitude of algorithms for polygonal (or poly-chain) approximation of a digital curve; see, for example, the works of Bhowmick and Bhattacharya (2007), Wall and Danielsson (1984), Chung et al (2008), Koutroumbas (2012), Liu et al (2008), Nguyen and Debled-Rennesson (2011), Parvez and Mahmoud (2010), Prasad et al (2012), Rosin (1997), Teh and Chin (1989) or Yin (2004) and the references therein. These are based on various techniques like normalization of area deviation (Wall and Danielsson, 1984), curvature estimation (Liu et al, 2008;Teh and Chin, 1989), dominant point detection (Prasad et al, 2012), particle swarm optimization (Yin, 2004), approximate digital straightness (Bhowmick and Bhattacharya, 2007), etc.…”

“…These are based on various techniques like normalization of area deviation (Wall and Danielsson, 1984), curvature estimation (Liu et al, 2008;Teh and Chin, 1989), dominant point detection (Prasad et al, 2012), particle swarm optimization (Yin, 2004), approximate digital straightness (Bhowmick and Bhattacharya, 2007), etc. Working principles of these algorithms and their comparative study are given in Table 2.…”

Visual Curvature for Space Curves

“…In this section we suggests a concept to perform this work. (This is a kind of 3-dimensional version for the works in [9].) Let C be a space curve which can be smooth, polygonal or digital.…”

“…Many data are given as digital forms and one of the main problems is how to reduce the noise. Visual curvature introduced in [9], defined by using height functions for each direction, is a good candidate for the curvature estimation which can deal with the noises of discrete curves such as polygons and digital curves. In [9], the visual curvature is defined only for the plane curves.…”

“…The concept of visual curvature for space curves is essentially same with the one for plane curves, but technically a little bit different and needs a new characterization of curvature. Similar to [9], we consider the height functions on a curve C for each direction α ∈ S 2 where S 2 is the unit sphere in the 3-dimensional Euclidean space E 3 . In section 2, we show that the turning angle of a space curve in a neighborhood can be expressed by a spherical area and the curvature can be given as a rate of the turning angle.…”