Some results on real aggregates

Real aggregate shapes were obtained via the burning algorithm from Fig. 6. Visual comparisons will be made first between numerical image data obtained directly from the x-ray tomograph and images of particles that have been reconstructed from the spherical harmonic expansion.

Figures 12 and 13 compare views of real particles (mottled dark gray) taken directly from the x-ray tomograph, and the corresponding reconstructed particle (light gray). By "reconstruction" is meant that an image rendering program was used to triangulate the particle surface as defined by the spherical harmonic expansion. The image pairs can be seen to match each other closely in both figures. How the spherical harmonic expansion effectively interpolates the surface can be qualitatively seen in these image pairs. This simple visual comparison is reassuring, but not conclusive, as to the accuracy of the spherical harmonic expansion on random shapes.

Figure 12: Comparing the digital image taken directly from the original x-ray tomograph (mottled gray) (digital volume = 9175 voxels), with the spherical harmonic expansion reconstruction (shiny gray). See VRML movie of spherical harmonic aggregate A slight surface texture has been added to the tomographic image in order to see the digital detail more clearly. Use your browser's "Refresh" button to make the animation play again.

Figure 13: Comparing the digital image taken directly from the original x-ray tomograph (mottled gray) (digital volume = 9061 voxels), with the spherical harmonic expansion reconstruction (shiny gray). See VRML movie of spherical harmonic aggregate A slight surface texture has been added to the tomographic image in order to see the digital detail more clearly. Use your browser's "Refresh" button to make the animation play again.

A more quantitative study can be done of the random particles shown in Figs. 12 and  13. But the most important question in this analysis, which must be addressed first, is the following: what is the "right" answer against which the spherical harmonic analysis is to be compared? There are no analytically correct quantities with which to compare as was the case for ellipsoids, except for the result of eq. (25), which is true for all shapes considered herein.

Only the digital volume of the original image exists against which to compare the spherical harmonic-derived volume. The moment of inertia tensor can also be computed using the centers of the voxels to numerically evaluate equation (26). In general, terms involving volume integrals are easily computed for the digital shapes by performing the appropriate sums over the voxels.

However, terms that are calculated by surface integrals are difficult to compute for digital images. The surfaces of digital images are composed of square tiles oriented along the coordinate axes (digitally rough), and overcount the surface area by some amount up to a factor of 3/2, as was discussed above. It is really not possible to integrate over a digital surface without using some kind of interpolation scheme. The spherical harmonic expansion essentially interpolates the digital surface. If there were to be a physical measurement of aggregate surface area, the number computed using the spherical harmonic expansion would be better to compare with rather than the digital measure of surface area, as has been seen previously. Of course, the value of k must be unity for any random or non-random shape aggregate like those considered here. Because of these facts, it is only meaningful, in the following, to compare spherical harmonic results for volume and I_ij, since they are computed via volume integrals, and k, since it is independent of shape, to their digital equivalents.

With these limitations in mind, one can proceed to do similar calculations with the real particles as with the ellipsoids, looking at the effect of the number of spherical harmonic terms used and at the number of Gaussian quadrature points used to originally compute the surface analysis (surface interpolation) and spherical harmonic coefficients, and then used to compute the various properties of the particle considered. These two sets of Gaussian quadratures are kept equal to each other, and are varied together below.

First, consider the particle shown in Fig. 12. One can compute the components of the moment of inertia tensor using the spherical harmonic expansion, as a function of the number of terms used in the expansion. Figure 14 shows these components, as well as the "exact" numbers computed directly from the digital image. By the time N gets past 12 or so, there is quite good agreement ( 1 %) between all five pairs of terms. The term I₂₂ is not shown, as it was similar in value to I₃₃. Even the negative term, I₁₃, shows good agreement between digital calculation and spherical harmonic expansion.

Figure 14: The components of the moment of inertia tensor, for the particle shown in Fig. 12, as a function of the number of spherical harmonic terms, N, used in the expansion of the particle shape. The solid lines are the components as calculated directly from the x-ray tomographic digital image, and serve as the "exact" check on the spherical harmonic result.

Figure 15 shows the results for the volume and k for the particle shown in Fig. 13, as a function of the number of terms taken in the spherical harmonic expansion. The volume has been normalized by the "exact" digital volume. Both a 120 point and a 240 point Gaussian quadrature were used to compute the volume and k integrals. The value of k deviates significantly from unity by N = 10, for the 120 point Gaussian quadrature, but is close to unity all the way to N = 25 when the 240 point quadrature was used. Surprisingly, the volume was relatively insensitive to the differing number of quadrature points, so only the 120 point case is shown. This example shows how the integrated value of the Gaussian curvature can be used for "quality control" of the mathematical analysis.

Figure 15: The volume, normalized by the particle volume of 9061voxels for the particle shown in Fig. 13, and the value of k, as a function of the number of terms, N, used in the spherical harmonic expansion. The two k curves are Gaussian quadratures of 120 and 240 points used for all aspects (surface determination, calculation of spherical harmonic coefficients, calculation of particle properties) of the spherical harmonic expansion calculation.