## 2 Proposed Segmentation Methodology
### Attribute Computation Using the Neighboring Points  
In this research, point attributes were computed based on the neighboring points identified using the defined neighborhood. After introducing an origin, a normal vector from the origin to the plane for a certain point in question was defined as seen in Figure 3. Since this normal vector extends from the reference point (_i.e._, the origin introduced) to the specific position at which the plane and the vector orthogonally intersect with each other, the vector can be also called a position vector. To differentiate the vector seen in Figure 3 from the conventional definitions of a normal vector and a position vector, it is referred to as _a normal position vector_ in this research. Most techniques based on attribute clustering use a voting scheme with an accumulator array that is constructed in the attribute space. The dimensions of the accumulator array depend on the number of attributes utilized by the technique in question. The three components of the normal position vector can be used as attributes in the voting scheme. It should be noted that one normal position vector defines one plane without any mathematical ambiguity. In this regard, the segmentation technique that uses the three components of the normal position vector partitions points into distinct regions without any segmentation ambiguity, excepting that arising from co-planarity (_i.e._, planes sharing the same normal position vector but disconnected in the object space). However, use of these attributes requires the construction of an accumulator array that has three dimensions, one for each vector component. Unfortunately, a voting scheme that uses a three-dimensional accumulator array is computationally expensive. To reduce the three-dimensional accumulator array to two dimensions, the slopes of the normal vector in the x and y directions were used as attributes for the planar patch segmentation, rather than using all three normal vector components to handle airborne laser scanning data [22; 23; 31]. Even though this method reduces the dimensions of the accumulator array, segmentation ambiguity remains a problem. If this method is used, parallel planes that have the same normal vector slopes in the x and y directions but different offsets in the z direction are segmented as one group. This problem was resolved, after completion of segmentation in the two-dimensional attribute space, through an additional process; that is to say, by either spatial analysis of the data or the introduction of a one-dimensional accumulator array for the offsets in the z direction. In this research, the magnitude of the normal position vector, \(\parallel\overrightarrow{n}\parallel\), was utilized as an attribute. The magnitude is also same as the shortest Euclidean distance between the origin and the defined plane. One origin can be used for planar patch attribute computation. However, different planes with the same magnitudes of normal position vectors can exist by chance. Planes tangential to a sphere centered on the origin are examples of this case. The points belonging to these planes will have the  
Figure 3: Vector defined from the origin to the computed plane for a given point, using its neighboring points. same attribute values. Therefore, two origins were introduced in the present research; significantly, this reduces the possibility of the occurrence of the ambiguity whereby points belonging to planes with different slopes and aspects will have the same attribute values. Figure 4a illustrates the situation in which one origin is located at an equal distance from the planes and the other origin is not. If one origin (origin 1) is located at an equal distance from the planes (_i.e._, \(||n_{111}||=||n_{121}||=||n_{13}||\)), the points belonging to these planes will have the same attribute values. Therefore, it would be impossible to separate these points into different groups in the attribute space. This problem can be resolved by introducing another origin (origin 2).