# Three Dimensional Digital Sieving of Asphal Mixture Based on X-ray Computed Tomography
## 4 Three-Dimensional Digital Sieving  
In the laboratory, a gradation test is performed by mechanical size analysis. A representative sample of aggregates is poured into a column of sieves, which is typically placed in a mechanical sieve shaker. During the shaking process, the mechanical device would move the sieves to cause the particles to bounce, tumble, or otherwise turn so as to present different orientations to the sieving surface [16]. As a result, aggregate larger than the sieve opening remains on the sieve, while the smaller ones pass through. Due to numerous particles in the mixture, high-frequency interaction between the sieve mesh and particles, and the very large computation consumption, it is impractical to simulate the laboratory sieve analysis based on the mechanical model. However, if the sieving process is possible to be simplified as a mathematical problem, the digital gradation test could be solved by numerical calculation. Figure 1: Industrial CT. In order to simplify the problem, an assumption related to passage judgment of aggregate is made first. When an aggregate is right above a sieve opening, whether it can pass that sieve opening is assumed to be equivalent to whether its orthographic projection on the sieve plate can be surrounded by four sides of the square opening. Apparently, this could be determined by the length of the minimum bounding square (MBS) of the projection area. If the length of the MBS of the projection is smaller than the sieve size, the aggregate passes through. It should be noted that the MBS is obtained when the length of circumscribed square achieves the minimum value. Though the problem of particle passage judgment is solved, the displacement of aggregate during shaking is not considered. In the actual mechanical sieving process, the aggregate will experience rotation. Thus, orthographic projection in every direction might be the critical projection that determines the smallest sieve opening that aggregate passes through. In this paper, digital sieve size (DSS) is defined as the minimum length of the minimum bounding squares of all possible orthographic projections. Ideally, each aggregate has only one DSS because it corresponds to the smallest MBS. It could be thought as a virtual sieve opening that aggregate is just able to fit in. Hereinafter, whether an aggregate is able to pass through certain sieve opening is totally depend on the value of DSS of the aggregate. The numerical method used to obtain the DSS of an aggregate is described as follows:  
1. Assume a local rectangular coordinate system \(x^{\prime}y^{\prime}z^{\prime}\), as shown in Figure 2, where \(xyz\) is the global rectangular coordinate system of the voxel data of aggregate particle, (\(\delta\), \(\theta\)) is used to describe the direction of \(z^{\prime}\) axis, \(\delta\) is the angle between \(x\) axis and the projection of \(z^{\prime}\) axis on \(xy\) plane, and \(\theta\) is the angle between the \(z^{\prime}\) axis and the \(xy\) plane. 2. Calculate Euler angles \(\alpha\), \(\beta\), \(\gamma\) based on (\(\delta\), \(\theta\)). Euler angles are used to describe the rotation of a coordinate system [17]. 3. Calculate the rotation matrix \(R\) through Euler angles \(\alpha\), \(\beta\), \(\gamma\), as shown in Equation (1). Perform coordinate conversion according to Equation (2). Hence, voxel data of the particle can be expressed in the local coordinate system \(x^{\prime}y^{\prime}z^{\prime}\) (see Figure 3). \[R\ =\ \left[ \begin{array}{ccc}\cos\beta\cos\gamma&-\cos\beta\sin \gamma&\sin\beta\\ \sin\alpha\sin\beta\cos\gamma\ +\ \cos\alpha\sin\gamma&-\sin\alpha\sin\beta\sin \gamma\ +\ \cos\alpha\cos\gamma&-\sin\alpha\cos\beta\\ -\cos\alpha\sin\beta\cos\gamma\ +\ \sin\alpha\sin\gamma&\cos\alpha\sin\beta\sin \gamma\ +\ \sin\alpha\cos\gamma&\cos\alpha\cos\beta\end{array} \right],\] (1) \[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array} \right]\ =\ R\left[ \begin{array}{c}x\\ y\\ z\end{array} \right].\] (2)
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