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Sources of Error

When performing the curvature computation in the digital mode, several sources of error are encountered in addition to those described in the previous section. One source of error lies in the digitized approximation of the template. As Table 1 shows, the area of a "circular" template depends significantly on the number of pixels used to construct it. Assigning length unit to each pixel edge, the area within the template is not equal to the true area of a circle of the same diameter. But the error in template area becomes quite small when a sufficient number of pixels are used, and Table 1 shows that a 15− or greater diameter gives an excellent approximation to the true circle area (Table 2 shows similar information for the volume of spherical templates). Table 1 also shows that using more pixels to refine the circle shape, by a factor of 2 or 3 beyond a 15− diameter, does not significantly improve the area approximation, while Table 2 seems to indicate that increasing a sphere's diameter over this same range does improve the approximation to a true sphere's volume (although the error must approach zero in the limit of infinite diameter). Employing larger templates also increases the portion of surface included in the curvature computation, and for this reason generally causes additional error due to possible non-uniformity of curvature over larger areas. Furthermore, the CPU time required for each curvature computation will increase roughly as the square of the template diameter in 2D, and as the diameter cubed in 3D. Relatively small templates are therefore particularly advantageous in terms of computational accuracy and speed, as long as the area/volume accuracy of the template is adequate.

Table 2. Deviation in volume of a sphere composed of discrete cubic pixels from the volume of a true sphere of the same diameter.

Another source of error when using digitized templates is the finite resolution of the computation. If the template is composed of NN different values of the curvature are resolvable by this method. Increasing the resolution by increasing N will reduce this problem, but at the same time will tend to magnify the errors described in the previous paragraph. Therefore, for a given application the user must decide the relative importances of accuracy, speed, and resolution when choosing the size of the template.

Further sources of error arise from the discrete approximation of the surface shape as step-like shifts in position, which can alter both the accuracy and precision of the curvature computation. Although a linear relation between curvature and the pixel count is still expected, the slopes and intercepts predicted from Eqs. 16 and 25 are not likely to be correct due to the discrete approximation of the surface and template. When applying the method to a digital image, it is necessary to make an "experimental" determination of the correct values of the slope and intercept in the following way. For a given template diameter, the true mean curvature of a circle (or sphere in 3D) can be plotted against the average value of the pixel counts resulting from each site along the surface. We use averaging because, as will be shown shortly, the pixel-to-pixel variation in computed curvature over a surface with nominally constant non-zero curvature can be substantial, although the average value over the entire surface is quite accurate (see Fig. 4). By repeating for circles (spheres) of varying radii, one can use linear regression to determine the equation of the line that best fits the collection of plotted points. Such a plot is shown for 2D in Fig. 3, using a circular template with diameter 2 b = 15. The predicted continuum equation (now in terms of the physical quantities κ, A, and b),

where < C > is the average pixel count. We center the template on the center of a surface pixel to achieve this result. Slightly different values for the numerators would result if the template were centered on, for example, a pixel corner. In 3D, using a template with diameter 2 b = 9 , the predicted continuum equation,

becomes

The diameter of the template sphere modestly affects the values appearing in the numerators of these fitted relations because digitized templates of different diameter have slightly different shapes. Using a template sphere diameter of 15 , linear regression gives instead of Eq. 32,

Figure 3: Pixel count at surface sites along a circle of radius κ⁻¹, plotted against the true curvature of the circle in the continuum limit. Template circle diameter = 15 . Each point is the average of the pixel count computed for all surface sites, and error bars represent plus or minus 1 standard deviation.

The curvature value obtained for an individual surface pixel, using a relation like Eq. 30 or 32, will generally suffer from precision error. This is because, even along surfaces of constant curvature, the discrete approximation of the surface introduces variation in the computed curvature (indicated by the error bars in Fig. 3). For example, Fig. 4 shows the variation in computed curvature with position along the surface of a digitized circle with a 61 -diameter (κ = 0.033 ⁻¹) (solid line). For this figure, the template diameter is 15 , and Eq. 30 was used to compute the curvature. The arithmetic mean of the 23 computed values is 0.0332 ⁻¹, and therefore the mean has a combined standard uncertainty (u_c) [22], or accuracy, of approximately 0.0001 ⁻¹ (1%). But u_c of the individual computations (precision), measured by one standard deviation, is 0.0297 ⁻¹ (± 89% of the true mean). The combined standard uncertainty of individual curvature computations is similar for constant-curvature surfaces in 3D: using a 9 -diameter template sphere, u_c for individual computations is approximately 0.04 ⁻¹ (± 92% and ± 124% of the mean curvature value for a 41 -diameter and a 61 -diameter sphere, respectively). Precision error this large can potentially affect the sign of the difference in curvature between two surface sites and, consequently, the sign of the driving force for mass transport between them during sintering processes.

Figure 4: Computed curvature along one-eighth of the surface of a circle (diameter = 61 ), using a template circle (diameter = 15 ), showing the effect of averaging the computation over all surface pixels within a 3 x 3 box centered on a given surface pixel P. The parameter θ, plotted along the x-axis, is tan⁻¹(y/x), where x and y are the x− and y-position of P relative to the circle center.

Precision can be increased substantially by using a finer pixel grid. For example, by halving the pixel edge length (from 1 to 0.5 ), u_c of individual curvature computations on a 41 −diameter sphere, using a 9 −diameter template, decreases from 0.04 ⁻¹ to 0.03 ⁻¹ (or from 92% to 68% of the true curvature). In fact, all the errors described in this section can be reduced by refining the pixel grid. However, such a remedy requires rather large increases in memory allocation. If the pixel edge length is halved, then eight times as many pixels are required to represent the same portion of a 3D microstructure and, furthermore, a scalar machine would spend about eight times as much CPU time to perform the computations over that portion.

One simple way to increase precision consists of, for each pixel P, computing an arithmetic mean of the curvatures computed for P and all the neighboring surface sites within a prescribed radius about P. Figure 4 shows that using an arithmetic mean over progressively larger neighborhoods steadily increases the precision of the curvature computation. The value of u_c for the curvature computations is 0.029 ⁻¹ (89% of the true curvature) without averaging, and is reduced to 0.003 ⁻¹ (11% of the true curvature) by averaging over all surface pixels within a radius of 11 . Only slight increases in CPU time are required for averaging. But in complex microstructures, large areas of constant curvature may not exist, so that averaging over large areas may not be feasible without employing a finer pixel grid.

³We center the template on the center of a surface pixel to achieve this result. Slightly different values for the numerators would result if the template were centered on, for example, a pixel corner.