## 2 Proposed Segmentation Methodology
### Attribute Computation Using the Neighboring Points
The magnitudes of the normal position vectors from the planes to the second origin will be different from one another (_i.e._, \(||n_{21}||\
eq||n_{22}||\
eq||n_{23}||\)). Hence, the points belonging to different planes will have different pairs of attributes after two different origins are introduced. These attributes are utilized in the clustering procedure through the voting scheme. It should be noted that the number of origins determines the dimensions of the accumulator array. If there is one origin for planar patch attribute computation, only a one-dimensional accumulator array is necessary for the voting scheme. Since, in the present research, two origins were introduced to reduce the possibility of the above-mentioned ambiguity, a two-dimensional accumulator array was utilized. Figure 4b is a diagram of the accumulator array for the points belonging to the planes in Figure 4a. All of these points are recorded in the accumulator array according to their attributes. As shown in Figure 4b, these points are recorded at the same location along the origin 1-axis. However, they are recorded at different locations along the origin 2-axis, according to the planes to which they belong. The points belonging to the different planes are recorded at the different locations in the two-dimensional accumulator array while avoiding ambiguity. The main advantage of the introduction of two origins while utilizing the magnitudes of the normal position vectors of the points is that the dimensions of the utilized accumulator array are reduced; rather than using 3-dimensional attribute space. The locations of the two origins are selected according to the way in which these origins are well distributed in the utilized dataset. Once the minimum and maximum values of the laser point cloud in question are recognized, the locations of the two origins are set by Equation (6) (also, refer to Figure 5). \[origin\ 1=\left( \begin{array}{c}min\ X\\ min\ Y\\ min\ Z\end{array} \right)+\frac{1}{3}\left( \begin{array}{c}max\ X-min\ X\\ max\ Y-min\ Y\\ max\ Z-min\ Z\end{array} \right),\ origin\ 2=\left( \begin{array}{c}min\ X\\ min\ Y\\ min\ Z\end{array} \right)+\frac{2}{3}\left( \begin{array}{c}max\ X-min\ X\\ max\ Y-min\ Y\\ max\ Z-min\ Z\end{array} \right) \tag{6}\]  
By positioning the two origins in this manner, the possibility of segmentation ambiguity is significantly reduced. However, there might still be the chance of ambiguity that causes the attribute vectors of different planes to be located at the same position in the accumulator array; see Figure 6 as  
Figure 4: (**a**) Situation in which one origin is located at an equal distance from the planes and the other origin is not; and (**b**) the recorded votes in a two-dimensional accumulator array. an example. The position of plane 1 in the attribute space is (n1, n2); n1 and n2 are the magnitudes of the normal position vectors to plane 1 from the two origins. However, plane 2 shares the same locations in the attribute space. Although the likelihood of this ambiguity occurring in reality is extremely low, the possibility cannot be ignored. In this regard, further processing has to be undertaken to remove the ambiguity thoroughly. This additional processing is designed to detect and resolve the segmentation ambiguity, and is implemented in the clustering procedure. The methodology by which coplanar points are derived using these attributes follows.