Given that a 3-D multi-aggregate image has been obtained, one can proceed to extract individual particles. Ideally, the image should be taken of a system with a fairly low volume percent of aggregate, say 20 %, so that on the average, most particles are not near each other. The image in Fig. 1, however, was of a real concrete at a practical aggregate volume percentage (around 60 %). Because of this fact, when the 3-D image was made, many particles appeared to be in contact. This is also due partly to the fairly coarse resolution of the image, as described above, which would make many close but not touching contacts to appear as real contacts. Also, the very nature of x-ray CT causes the exterior of particles to be a little bit "fuzzy", so a large aggregate volume will result in some artificial "touches." This situation can be handled with a simple erosion and dilation algorithm [6], which breaks apart the tenuously connected aggregates, without significantly changing their size.
We next describe the "burning" algorithm that was used to identify single particles. It is quite analogous to that algorithm used in percolation studies, both in digital, pixel-based models [7-9], and in continuum model studies [10]. Imagine a 3-D cube of pixels, where each pixel is labeled either matrix (1) or particle (2). Assume for now that no particle is touching any other particle. We will also stay away from the boundaries, so that the proper boundary conditions are not a consideration. Scan through the image until a pixel is found that has label "2." Now find all nearest neighbors (back-front, left-right, up-down) of this pixel that also have label "2." Save the locations of these pixels, and then again find all neighbors of these pixels that also have the same label. Iterate this process until no more pixels of label "2" can be found. The collection of pixels found constitutes a single particle. Figure 2 illustrates this process in two dimensions.
Figure 2: Showing a model particle in two dimensions (gray) embedded in a matrix (white). The black pixel shows the first particle pixel that is found by the burning algorithm. The pixels found at each iteration are labeled a-i. Nine iterations were necessary to "find" the complete particle.
The center of mass of this collection of pixels is computed, simply by finding the average of the (i,j,k) labels defining the pixels. Taking this point as the origin for the particle, the pixels making up the particle are then stored in terms of their (i,j,k) label relative to the center of mass of the particle.
A solid particle can be described by the function r (
,
), where r is the distance
from the center of mass point to the particle surface along the direction
specified by the two angles ,
from spherical polar coordinates. The unit
vector along this direction is sin
cos
i + sin
sin j + cos k, where i, j, and k are the usual Cartesian unit vectors. The function r is numerically determined at about 10,000 choices of the angle pairs, using the pixel collection taken from the tomograph.
The function r can be analyzed using spherical harmonics, a mathematical
method often encountered in quantum mechanics [11]
and in shape analysis of molecular orbitals [12,
13]. For any function
f( , ),
defined on the surface of a sphere (0 <
< 2
and 0 <
< ), the spherical harmonics form a complete set [14]:
where Y mn , ) is a spherical harmonic function of order (n,m) and a(n,m) is a numerical coefficient [14]. Typically n=20 to n=30 is a high enough order to go to in the series above to capture the shape of most particles. Work is underway to determine the optimum range of n.
A simple way of seeing how well the spherical harmonic series captures the shape of a real particle is displayed in Fig. 3. This figure shows three sets of real particles, as taken from the interior of Fig. 1, in yellow, alongside the shape as derived from the spherical harmonic expansion, in purple. It is clear, even from these 2-D images, that the spherical harmonic
Figure 3: Three different sets of particles, comparing the real particle (yellow) as taken from the tomograph of Fig. 1 and the particle as re-created from its spherical harmonic expansion (purple). The long dimension of each particle is about 10 mm to 20 mm.
expansion does indeed capture the shape of the real particle well. Seeing the particles in a 3-D imaging package confirms this belief. The actual size of the long dimension of the particles was about 10 mm to 20 mm.
There are two main results of this particle analysis process. First, many functions of the particle shape and size can now be analytically calculated, including volume, surface area, moment of inertia, and others. Second, these particles now have a fairly simple mathematical form, like that of a sphere or ellipsoid, so that they can be incorporated into a model like the hard core soft shell model for concrete microstructure [15, 16]. This will allow real aggregate shapes to play a role in models that before used only simple shapes like spheres and ellipsoids [17]. An additional benefit will be the ability to construct databases of 3-D aggregate shapes corresponding to various aggregate sources.