Generalized line calculation

When there are no more buffers left to be created, the final equations for the lines are calculated in order to create the new generalized lines. The equation for the generalized line is only depending on the parallel lines described by the buffer, which are stored within LP. During the earlier process of joining the parallel lines the values for A and B are already updated, which leaves only the determination of the C value. The determination can be performed by averaging given all parallel lines weighted by their length, but it can also be set equal to the line with the highest priority.

Approach 1: Weighted average lines

The approach as described by Kada & Luo [2006] is based on averaging the buffered lines both their orientation and location. A similar approach is described inKada[2010]. However this is, as described before, for the generalization of 3D buildings using 3D planes. This approach uses the area of the planes as weight for the orientation and location of the generalized plane. For this research this can be translated into the length of the lines being used as weight. Since the orientation of the buffer is already calculated and explained in the previous, only the location is left to solve. In Equation (3.10) the adjusted version from the original weighted average equation byKada [2010] can be found.

∀p ∈ LP : Cnew = n P i=0 length(LPi)(Apx+ Bpy) n P i=0 length(LPi) (3.10)

An example using this weighted average approach is shown in Figure3.6. The green lines are joined since they are parallel while the orange line completely falls within the buffer created by joining the green lines and is thus included. The generalized line resulting from the recalculation of the orientation and location is shown in red.

Figure 3.6: Building footprint with generalized line (red). The green lines are parallel thus joined where the orange line is included with these, resulting in the red line which location is the weighted average of the green lines.

30 Generalization and decomposition using line equations

Using the previously described generalization combined with the average line calculation a generalization is performed on an input data set. More information on the data sets used can be found inChapter 6. Some examples of the resulting generalization are shown in Figure 3.7. Within these figures the original polygons are described by the red line with the points in black, the resulting generalization is shown with the black outline and filled cyan.

Figure 3.7: Original building outlines (red line with black points) overlaid on generalized footprints (1m) using weighted average lines.

Approach 2: Best fitting lines

Based on quality evaluations a new approach improving that ofKada & Luo [2006] is de- veloped byPeter et al.[2008] for preserving facade and original footprint lines. Especially when a block of connecting buildings is processed they should not overlap or introduce gaps after generalization, further reading on this in Section3.1.4. This approach is imple- mented in such a way it can be applied after the generalization as an subsequent step. The lines of the generalized footprint are adjusted to coincide with the main building lines of the original footprint. Selection of the main building lines is similar to the merging using the line buffers as described before. Selection of the best fitting main line is performed using the weight of the original line and the distance between the generalized and original one.

Given these observations the adjustments of the lines can be performed directly when calculating the new location of the generalized line, no subsequent step is needed. The best fitting location of the line is that of the line with the highest weight in the buffer. Since the calculation as described by Peter et al. [2008] is not useful in this case a new one has been formulated. The calculation of the generalized line location is shown in Equation (3.11).

3.1 Generalization using line equations 31

The result of the best fitting approach is that generalized lines are at the same location of one of the original lines. In Figure3.8the location of the generalized line, the red line, is based on the best fitting location thus the line with the highest weight. The selection of the best fitting line for the generalization retains most of the original structure and facade lines of a building, compared to using the weighted average.

Figure 3.8: Building footprint with generalized line (red). The green lines are parallel thus joined where the orange line is included with these, resulting in the red line which location is equal to the green line with the highest priority.

Using the best fitting line method, the results resemble more of the original shape of buildings. In Figure3.9 one can see the results of the generalization using this method. In this figure the red line with the black points resemble the original footprint where the black outline filled with cyan resembles the generalized shape. These examples show that small features are removed correctly and also the extra nodes are gone.

Figure 3.9: Original building outlines (red line with black points) overlaid on generalized footprints (1m) using best fitting lines.

32 Generalization and decomposition using line equations