(1) X-RAY CT and SPHERICAL HARMONICS
In X-ray computed tomography (CT), X-rays interrogate a 3-D sample at many different angles and the absorption is measured. A computer-based reconstruction technique then makes gray level images, where each image is a slice of the sample and the contrast in gray levels is caused by the different x-ray absorption properties of the materials in the sample. These differences usually track density differences (Kak and Slaney, 2001). If these images are stacked together, a 3-D view of the sample interior is obtained. The current limit of voxel size on standard machines, usually available at universities, is about 0.005 mm or 5 mm. One should note that the voxel length out of the plane of the images does not have to be same as that in the plane of the images. X-ray microtomography, which makes use of synchrotron radiation (available at national laboratories, e.g. Brookhaven and Argonne), can reduce the voxel size to the order of one micrometer per voxel. Usually the voxel size is the same in all three directions on these machines. Both techniques were used in this study. CT may be considered as the current ultimate surveying device for small objects.
For the 12 test rocks, three samples were made by suspending the rocks in a wax matrix. Two cylindrical samples were used for the six 0.75 rocks (three rocks in each cylinder), and the third cylindrical sample contained all six 0.5 rocks. The microfine aggregates were suspended in a thin (2 mm diameter) epoxy cylinder. For details of how such samples should be made, and the important variables, see (Erdogan et al., 2004). For the 12 test rocks, scanning was carried out with a voxel size of 0.065 mm in the horizontal direction and 0.0821 mm in the direction of the axis of the cylinder. This resolution gave 140+ voxels in the minimum length dimension of the smallest rocks, so was judged to be more than adequate to capture the details of the rock shape. The microfine aggregates were scanned at 3.968 mm per cubic voxel side, which gave less voxels per unit length (10 – 20), but was still adequate to see rocks above about 40 mm in size.
The resulting 3-D image, made by stacking the many 2-D images of the sample, is a gray-sale image that needs to be thresholded to produce the final image containing the aggregates. In this final image, details below the voxel size have been lost, and it is possible that the surface of the image could be a little smaller or a little larger than reality, due to the thresholding process. However, this CT data does give an accurate value of particle volume, via counting voxels, as will be seen below. But almost no information about the particle surface can be obtained since the object is simply a collection of rectangular parallelepipeds. The CT data can be used, however, to generate spherical harmonic functions (Garboczi, 2002), which can then be used to create a smooth approximation to the function r(q,f), which is the distance from the center of volume to the surface in the direction given by the spherical polar angles (q,f). In Cartesian coordinates, the unit vector in this direction is given by
(1)
Using this function, one can compute any geometric quantity of the rock like volume, surface area, or moment of inertia (Garboczi, 2002).
The spherical harmonic mathematical analysis relies on eq. (2), which states that any sufficiently smooth function r(q,f), where q and f are the azimuthal and polar angles of 3-D spherical coordinates, can be written as a series of spherical harmonic functions, where the Ynm are the complex spherical harmonic (SH) functions and the anm are complex coefficients (Arfken, 1970):
Strictly speaking, the series in eq. (2) becomes exact only as N ое. However, like 2-D Fourier series, a finite value of N is usually found to give an adequate approximation of a given function, within some specified uncertainty limit.
The function r(q,f) gives the distance from the centroid or center of volume, which is equal to the center of mass for a uniform density object, to a given point on the surface of the particle, in a direction specified by the two angles. Using a numerically determined r(q,f) function from a 3-D image, in this case derived from the 3-D x-ray CT images, one can accurately determine the first N = 20 or so coefficients, which are usually enough to satisfactorily represent the particle. In the cases studied in this paper, the values of accurate N range from N = 16 to N = 30, with most cases having a value of N of about 20. However, the spherical harmonic analyses for each of the 12 test rocks all have N = 30 accurate coefficients. Once these coefficients are obtained, one can create approximations for the r(q,f) function, which determines the shape of the particle. Storing up to n = N coefficients requires (N+1)2 complex numbers, which requires far less computer storage space than that required for storing the voxels from the numerical images.
An alternative approach to generate the numerical values of r(q,f) needed for the SH analysis is to perform a surface survey of the body, directly producing the coordinates of a finite number of points on the surface of the body. A suitable technique is LADAR (laser direction and range finding) (Cheok, 2005). The advantage of this approach is that only the surface is sampled, reducing the number of points needed to characterize the particle. If interior information is desired, then the complete CT scan can give it. If only the overall shape is desired, then these interior points are wasted information. At present, the lowest voxel size available in LADAR units is tens of micrometers, so is still an order of magnitude larger than the best CT microtomography results. A spherical harmonic expansion can be created from the LADAR points as easily as from the x-ray CT points (Cheok, 2005).
Both of these approaches offer the advantage of permitting the construction of mathematical models of a body, thus presenting researchers with powerful new approaches to many problems. The accuracy and precision of the models will be a direct function of the number of survey points used and the appropriateness of the form of the equations employed. Hence, one of the tasks facing such researchers is to select the level of approximation appropriate to the task at hand. With suitable data and proven algorithms, these techniques have the potential to generate fundamental advances in the study of particulate materials of all types.
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