Next: Model Description
If the shape of a surface is available analytically, then it is simple to compute the curvature at any twice-differentiable point of the surface. In the model described in this paper, such a description is not available, since material grains are represented as collections of pixels, in digital image form . A different method becomes necessary.
One method would be to fit a polynomial function to the center of the surface pixel of interest, along with a specified number of surface neighbors, and then evaluate the curvature using the fitted interpolated function . This is a fairly slow process, and seemed to be too complicated for the large number of curvature evaluations necessary for our model.
An alternative method, which at first sight would seem to be only qualitatively related to the curvature, is to center a pixel box on the surface pixel of interest, and count the number of air pixels contained in the box . Fig. 1 illustrates this method using a b x b box, with b = 3 ( b is odd, in general, so that the box may be centered on a pixel ). Gray pixels are solid and white pixels are air. On the left of Fig. 1, the "curvature" at the site indicated by the small black circle is 6, as 6 air pixels are contained in the box, along with 3 material pixels. The right-hand side of Fig. 1 shows an air surface site with curvature equal to 3. The pixel of interest is never included in the counting. For a given b x b box, there are in general b2−1 possible values of the curvature, so to be able to distinguish between small differences in curvature requires boxes with larger values of b. We assume that the surface enters the counting box only once. If another surface region enters the box, then the pixels associated with the second solid should be counted as air pixels. In other words, the probe should only test the local surface environment of the surface pixel selected.
Figure 1: Illustration of the box-counting method for computing the curvature of a surface composed of discrete pixels, for a solid surface (left) and an air surface (right).
We have been able to show, numerically and analytically, that the box-counting method gives averaged numbers that are linearly related to the actual curvature. To show this, we have taken digital circles, of a given radius R, averaged the box-counting curvature over the surface, and compared this result with the true curvature of a smooth circle, which is K = 1/R. The results for various box sizes b are shown in Fig. 2. The straight lines in the figure are fitted to the computed points. The fitted intercept for all values of b is very nearly b(b−1)/2, which is the exact value obtained for a pixel in a flat surface when b is an odd number. The slope of each straight line is proportional to b3, which agrees well with an analytical calculation of the same problem, but with a continuum sphere and counting box, given in the Appendix. The one standard deviation vertical error bars in Fig. 2 indicate the variance of the result as the counting-box traverses the perimeter of the digitized circle. Part of this variance is due to the pixel-scale roughness of the surface of the digital circles. However, as the counting box moves around the circle, it maintains a constant orientation, so that the area inside the box but outside the circle is a function of the position on the circle's surface. The calculation in the Appendix shows that for the infinite resolution limit (many pixels per unit length or the continuum limit) of both circle and counting box, the variance should be proportional to cos−3(θ), where θ is the polar angle modulo(π/4), measured from the x axis to the radius vector connecting the center of the circle and the point of interest on the circle's surface. Details of the analytic calculation and the discussion of the Wullf shape for a digitized surface are given in the Appendix. This angular factor of course also contributes to the error bars in Fig. 2, but the pixel roughness of the surface seems to dominate the error bars, as will be discussed more below.
Figure 2: Showing the air pixel count averaged over the surface of digital circles of radius R versus 1/R, the true curvature of the circle, for various values of the box-size b.
Fig. 3 shows the equivalent computation for digital spheres in three dimensions, where the exact curvature is now K = 2/R. The box is now a cube with edge length b, centered at the pixel of interest. The intercepts are very nearly at b2(b−1)/2, the exact value for a flat digital surface in 3−d, for b odd. The slopes of the lines increase with b as b4, as discussed in the Appendix. In general, for a d dimensional counting cube, of volume bd, applied to the surface of a d dimensional sphere, we expect the slope of pixel count vs. true curvature will be proportional to bd+1.
Figure 3: Showing the air pixel count averaged over the surface of digital spheres of radius R versus 2/R, the true curvature of the sphere, for various values of the box-size b.
For simulations of isotropic surfaces, counting variance along a surface of supposedly constant curvature may lead to spurious results. This is because some anisotropy, like the factor of cos−3(θ) discussed above, is built into the counting procedure when a counting square is used. Schemes can be developed to modify the anisotropy. One such scheme would be to multiply the counting result by a factor which depends on a computed surface normal (i.e., cos−3 (θ), in the limit of large counting boxes). This surface-normal scheme could also be modified to generate particular anisotropies. Another scheme would involve tabulating a numerical correction factor as a function of surface normal for a particular counting box (in the same manner as in Fig. 2). This correction factor, when multiplied by the count, would yield the correct curvature and thus eliminate any systematic cos−3 (θ) contribution to the error bars in Fig. 2. However, in two dimensions, probably the best such scheme, for a general pore-solid structure, would employ a digitized counting circle instead of a square counting box. The counting circle would eliminate anisotropy associated with the counting and therefore would have isotropic results in the limit of large diameters. However, it would still exhibit anisotropy for small counting circles due to the underlying square pixel lattice. This method will be used in future simulations .
In the results presented below, anisotropy was never of sufficient magnitude to give results which were qualitatively inconsistent with the assumption of isotropy, so none of the above measures were utilized. This is probably due to the digital-image "roughness" of the surfaces in the model. Variance due to this roughness probably dominated any angular factors that were derived assuming perfectly smooth surfaces.
We therefore conclude that the box-counting method for curvature computation is actually measuring a quantity that is a linear function of the true curvature at the given point. This means that a model of mass rearrangement based on curvature differences obtained through a box- counting method should be able to simulate real situations, since a true measure of curvature is being used.