4. Reference rocks

For direct comparison of mathematical analysis and experimental measurement, we used three arbitrary reference rocks that we could image using the x-ray CT, and then remove from the sample to be able to quantitatively and qualitatively compare with the VRML image. For easy manipulation, they were embedded in dry, packed cement powder in cylindrical samples. After x-ray CT imaging, they were taken out of the samples, cleaned, and were available for other measurements and for comparison with the VRML images. To precisely identify the resolution of the non-cubic voxels in the X-ray CT images, the volumes of the three rocks were precisely determined using Archimedes' method of weighing in air and in water (Table 1).

The z-resolution (along the vertical axis of a cylindrical specimen) of the x-ray CT scanner used at the Turner Fairbanks Highway Research Center can be controlled, and is set before each scan. It is determined by the step size of the vertical sample positioning mechanism. The machine used overlapped each vertical slice by 20 %. The x and y resolutions are identical, and are determined by the number of x-ray detectors in the linear array, 512 in this case. If the specimen's maximum horizontal dimension fills the field of view of the detector, the horizontal length per voxel is simply this maximum horizontal dimension divided by 512. This was done for the cylindrical samples to be described in the next section. However, for the individual rocks, only an image of the actual rock was saved, which did not fill the image. The x and y resolutions were then determined by multiplying the actual number of voxels, N, in the image times the volume of each voxel, which was ZXY, where Z is the known z-resolution and X=Y stand for the unknown x and y resolutions, which are always equal. Since the physical volume, V, of the sample is known to a high precision, one simply equates V = N Z X² and solves for X. Table 1 shows the actual resolutions used for each of the standard rocks.

Figure 3 shows visual comparisons, for rocks 2 and 3, of digital camera images of the actual rocks (left) with 3-D VRML images (right) reconstructed from the corresponding spherical harmonic coefficients. The VRML images were manipulated qualitatively using an ordinary browser (with a VRML plug-in) so as to approximately match the orientation of the real rock. The VRML image for rock 2 was created using up to N = 30, while rock 3 used up to N = 20. These limits were determined using the Gaussian curvature integral criterion as discussed in the previous section.

Figure 4, which is similar to Fig.1, focuses on reference rock number 1. Each 2-D image, taken from a 3-D VRML image, shows what the spherical harmonic reconstruction looks like, in approximately the same orientation, for a different number of spherical harmonic coefficients. The simplest shape, shown in the top left hand corner of Fig. 4, is simply a sphere and uses only the a₀₀ coefficient (N = 0). In the bottom right hand corner, just before a digital camera image of the real rock, coefficients up to N = 40 were used to create the computational image (see eq. 1). The shape complexity of the images increases as a larger number of coefficients are used. Figure 4 shows that using spherical harmonics up to about N = 15 captures the basic shape of the rock, while using values of N larger than 15 brings out smaller details of the shape and texture.

Another, more quantitative comparison can be done for these three reference rocks by considering their geometrical dimensions. ASTM D4791 [14] defines the length (L) of an aggregate as the maximum distance between two surface points. The width (W) is defined as the longest surface-surface distance that is perpendicular to the length. The thickness (T) is the largest surface-surface distance that is perpendicular to both the length and the width. These values were computed from the spherical harmonic series with an uncertainty of 1.0 mm, since there is no exact equation for these numbers, which have to be numerically searched for. The values of L, W, and T were measured directly on the reference rocks using digital calipers. The calipers measure with an uncertainty of 0.1 mm, but the actual uncertainty in measuring L, W, and T is larger, also about 1.0 mm, since it is dominated by being able to estimate the perpendicular angle requirements between L, W, and T. The theoretical values and the and T is larger, also about 1.0 mm, since it is dominated by being able to estimate the perpendicular angle requirements between L, W, and T. The theoretical values and the experimental values of L, W, and T (Table 2) generally agree within experimental uncertainty. A comparison between spherical harmonic prediction and experimental measurement where the voxel size was known independently and precisely is given in Ref. [11], which further validates the X-ray CT and spherical harmonic techniques.

Figure 4: This figure shows a series of 2-D images taken from the VRML particle reconstruction of reference rock 1. From top left to bottom right, the highest order of spherical harmonic coefficients used (value of N in eq. (1)) was 0, 2, 4, 6, 9, 12,1 5, 18, and 40. The final image on the bottom right is a digital camera image of the real rock, in the same orientation, about 80 mm in length in the vertical direction.