## 2 Materials and Methods
### Point Cloud Registration
#### 2.3.2 Conjugate Point Identification  
The relative position and orientation between the \(V_{x}\)-\(V_{y}\)-\(V_{z}\) and \(U_{x}\)-\(U_{y}\)-\(U_{z}\) coordination systems should be known before pairing up three pairs of conjugate points for the registration. Knowledge of which facets were scanned is critical for estimating the relation position and orientation between the coordinate systems. The origin of the \(V_{x}\)-\(V_{y}\)-\(V_{z}\) coordinate system can coincide with one of the four upper corners of the dice, so we can designate \(V_{x}\)-\(V_{y}\)-\(V_{z}\) as the four cardinal directions for illustration (Figure 8). As can be seen, there are 24 possibilities for the \(V_{x}\)-\(V_{y}\)-\(V_{z}\) coordinate system to be constructed, depending on how the die is placed for the scans. From Figure 8, taking the 4-pips face up and the \(V_{x}\)-\(V_{y}\)-\(V_{z}\) as the west coordinate system as example, there are three transformations to generate another coordinate systems  
Figure 8: Twenty-four constructed coordinate systems in the die space, and the transformation. (either north, east, or south) for the \(U_{x}\)-\(U_{y}\)-\(U_{z}\) to register. To transform from the west to the east coordinate system, the point coordinate (\(X^{\prime}\), \(Y^{\prime}\), \(Z^{\prime}\)) take:  
\[\left[ \begin{array}{c}X^{\prime\prime}\\ Y^{\prime\prime}\\ Z^{\prime\prime}\end{array} \right]=\mathbf{R}_{3}(\Omega)\left[\left[ \begin{array} []{c}X^{\prime}\\ Y^{\prime}\\ Z^{\prime}\end{array} \right]+\left[ \begin{array}{c}l_{x}\\ l_{y}\\ l_{z}\end{array} \right]\right] \tag{4}\]  
where \(\mathbf{R}_{3}\) is the rotation matrix about the Z-axis; \(\Omega\) = 180\({}^{\circ}\) for west-to-east; \(l_{x}\), \(l_{y}\), and \(l_{z}\) are the dimensions of the die along the X, Y, and \(Z\) directions, respectively. In this case (transforming from the west to the east), \(l_{z}\) theoretically equals zero. After the relative position and orientation between the \(V_{x}\)-\(V_{y}\)-\(V_{z}\) and \(U_{x}\)-\(U_{y}\)-\(U_{z}\) coordination systems are determined, one point along each axis in \(V_{x}\)-\(V_{y}\)-\(V_{z}\) with length of unity should be identified. As a result, we have:  
\[\left\{ \begin{array}{c}\left(v_{x_{1}}/\left|\overset{\rightarrow}{V}_{x} \right|,v_{x_{2}}/\left|\overset{\rightarrow}{V}_{x}\right|,v_{x_{3}}/\left| \overset{\rightarrow}{V}_{x}\right|\right)\\ \left(v_{y_{1}}/\left|\overset{\rightarrow}{V}_{y}\right|,v_{y_{2}}/\left| \overset{\rightarrow}{V}_{y}\right|,v_{y_{3}}/\left|\overset{\rightarrow}{V} _{y}\right|\right)\\ \left(v_{z_{1}}/\left|\overset{\rightarrow}{V}_{z}\right|,v_{z_{2}}/\left| \overset{\rightarrow}{V}_{z}\right|,v_{z_{3}}/\left|\overset{\rightarrow}{V} _{z}\right|\right)\end{array} \right. \tag{5}\]  
where \(v_{xi}\) = 1, 2, 3, \(v_{yi}\) = 1, 2, 3 and \(v_{zi}\) = 1, 2, 3 are the elements of \(V_{x}\), \(V_{y}\) and \(V_{z}\), respectively. Their conjugate points in the scanner space are simply:  
\[\left\{ \begin{array}{c}\left(1,0,0\right)\\ \left(0,1,0\right)\\ \left(0,0,1\right)\end{array} \right. \tag{6}\]  
Applying Equation (4) to transform the above coordinates, we have:  
\[\left\{ \begin{array}{c}\left(l_{x}-1,l_{y},0\right)\\ \left(l_{x},l_{y}-1,0\right)\\ \left(l_{x},l_{y},1\right)\end{array} \right. \tag{7}\]  
which are the conjugate points for the following points identified analogously from the \(U_{x}\)-\(U_{y}\)-\(U_{z}\) coordinate system:  
\[\left\{ \begin{array}{c}\left(u_{x_{1}}/\left|\overset{\rightarrow}{U}_{x} \right|,u_{x_{2}}/\left|\overset{\rightarrow}{U}_{x}\right|,u_{x_{3}}/\left| \overset{\rightarrow}{U}_{x}\right|\right)\\ \left(u_{y_{1}}/\left|\overset{\rightarrow}{U}_{y}\right|,u_{y_{2}}/\left| \overset{\rightarrow}{U}_{y}\right|,u_{y_{3}}/\left|\overset{\rightarrow}{U} _{y}\right|\right)\\ \left(u_{z_{1}}/\left|\overset{\rightarrow}{U}_{x}\right|,u_{z_{2}}/\left| \overset{\rightarrow}{U}_{x}\right|,u_{z_{3}}/\left|\overset{\rightarrow}{U} _{x}\right|\right)\end{array} \right.