## 2 Proposed Segmentation Methodology
### Clustering of Points
Only one cluster is generated from the points located on the physically different roof patches, due to the intentionally determined locations of the origins. The plane fitting quality measure, which is the RMS of the distances between the points (at the highest peak) and the plane determined, is much worse than the vertical accuracy of the airborne laser scanning data. The plane fitting quality measure for the peak is 0.816 m, which is more than five times the vertical accuracy of the laser data (_i.e._, 0.15 m). This fact indicates that the peak was constructed by mixed points from two different roof patches. Figure 8: Aggregation of points with similar attributes: (**a**) the digital image; (**b**) the Digital Surface Models over the area of interest; (**c**) the accumulator array; and (**d**) the segmented patches. The locations of the two new origins were determined by Equation (7). Afterwards, all of the points registered at the peak cell were taken, and their attributes were recomputed using the two new origins. The new accumulator array was constructed based on the recomputed attributes. Figure 10 shows the scatter plot re-drawn using the recomputed attributes of the points registered at the peak cell. There are two distinctively different clusters in the figure. Moreover, the plane fitting quality measure of the highest peak is 0.043 m, which is much better than the vertical accuracy of the airborne laser scanning data. The scatter plot and plane fitting quality measure indicate that the points located on the physically different roof patches produced different clusters. After the homogeneous points registered at the highest peak cell are identified, the remaining clustering steps can be carried out in the original accumulator array based on those points. Then, the clustering process moves to the other points that did not contribute to the highest peak cell but have similar attributes (_i.e._, the points in the cells neighboring the highest peak). This step is necessary in  
Figure 10: Scatter plot after resolution of ambiguity problem. Figure 9: Scatter plot prior to resolution of ambiguity problem. order to consider the inaccuracy of the computed attributes of certain points, which are generally either located around break lines or affected by noise. More specifically, an initial cluster is first defined by including the points contributing to the highest peak cell. Then, plane parameters are determined by fitting a plane using the points belonging to the initial cluster. Afterwards, the points registered at the cells neighboring the highest peak are identified in the accumulator array. Searching of neighboring cells is carried out by starting from the cells closest to the highest peak and moving thereafter to the more distant ones. As seen in Figure 11a, eight cells neighboring the highest peak cell are searched in the first iteration. Among the points in these cells, only those that satisfy two conditions, (1) the proximity condition and (2) the co-planarity condition, are included in the initial cluster (refer to Figure 7). More specifically, the points registered at the eight cells are first checked to determine if they are spatially close (_i.e._, within approximately twice of the average point spacing of the laser data) to the points belonging to the initial cluster (_i.e._, the proximity condition). Only the points that satisfy the proximity condition are considered as candidates. In other words, the proximity condition is the means of simultaneously checking for the closeness of candidate points in the object domain as well as the similarity of the associated attributes in the parameter domain. Afterwards, the shortest Euclidean distances between the candidate points and the plane defined by the points in the initial cluster are computed. Then, the points that have distances less than the pre-determined threshold are accepted into the cluster (_i.e._, the co-planarity condition).