(3)
As a first test of the SH analysis generated from the CT data, the volumes of the 12 test rocks were computed using the SH approximation for the r(q,f) function. The general integral for volume, V, using spherical polar coordinates, is
(3)
The r-integral can be performed, resulting in:
(4)
The rock volumes were also measured using Archimedes’ method (weight in air vs. weight in water) using an electronic balance accurate to 0.1 mg. The two sets of data are compared in Table 1, along with the voxel volume of each rock obtained from the original tomograph. The error in estimation of the volume for each rock, between the measured volume using Archimedes’ method and spherical harmonic functions, is given in the last column. This error is less than 2 % in all cases and less than 1 % in many. So the spherical harmonic reconstruction captures the experimentally measured volume very accurately. One should note that simply summing up the volume of each voxel in the raw voxel data gives similar results to the spherical harmonic calculation (agreement to within a few tenths of a percent), which then becomes a measure of accuracy for the spherical harmonic reconstruction of the rocks. The best estimate of the uncertainty in the CT-SH calculated results is the comparison to experimental data.
Images based on the spherical harmonic reconstruction qualitatively show that the shape of each rock is closely captured as well, as can be seen by comparing Figs. 2a and 2b to Figs. 1a and 1b, respectively. This comparison shows that the x-ray CT data, which is the foundation for the spherical harmonic surface reconstruction, accurately captures the physical size and shape of each rock.
Table 1. Volume of individual rocks comparing physical measurements to voxel counting and spherical harmonic-based calculations.
Figure 2a: VRML images of the six 0.5 rocks studied (compare with Fig. 1a).
Figure 2b: VRML images of the six 0.75 rocks studied (compare with Fig. 1b).
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