Estimation of the curvature of an interface from a digital 2D image

Abstract

In this paper a method for the estimation of the curvature along a condensed phase interface is presented. In a previous paper in this journal [1] a mathematical relationship was established between this curvature and a template disk located at a given point along the interface. The portion of the computed area of the template disk covering one of the phases was shown to be asymptotically linear in the mean curvature. Instead of utilizing this relationship, an empirical approach was proposed in [1] in order to compensate for discrete uncertainties. In this paper, we show that this linear relationship can be used directly along the interface avoiding the empirical approach proposed earlier. Modifications of the algorithm are however needed, and with good data smoothing techniques, our method provides good quantitative curvature estimates.

Grain boundary energy and curvature in Monte Carlo and cellular automata simulations of grain boundary motion

“…Second, the derivation corrects for the practice of centering the kernel on a pixel adjacent to the boundary, rather than on the boundary itself. While this is known to be a significant source of error [16], there does not appear to be any other formula in the literature that makes the same correction.…”

“…Notice that k has units of m, r has units of m, a v has units of m 3 , and H has units of 1/m. The introduction of this section claimed that using the difference of kernel weights inside and outside the surface cancels a constant term that usually appears [15,16]. Nonetheless, Eqs.…”

“…First, the derivation considers the difference of the kernel weights inside and outside the surface, canceling a constant term that usually appears [15,16] and simplifying the formula. Second, the derivation corrects for the practice of centering the kernel on a pixel adjacent to the boundary, rather than on the boundary itself.…”

“…CA models of curvature-driven grain boundary motion generally calculate the mean curvature explicitly, by some variation on the template method [15,16]. The kernel is centered on a pixel adjacent and interior to the surface being considered, and the curvature is a linear function of the weight of the kernel outside the surface.…”

“…The pressure on a grain boundary may be expressed as a function of the mean curvature by means of the Young-Laplace equation [14]. The mean curvature of a discretized interface is difficult to measure though, particularly when the kernel radius is small [15,16]. 3.…”