Automatic Estimation of Excavation Volume from Laser Mobile Mapping Data for Mountain Road Widening III

Artículo patrocinado por Extraco, Misturas, Lógica, Enmacosa e Ingeniería InSitu, dentro del proyecto SITEGI, cofinanciado por el CDTI. (2012). 

Article sponsored by Extraco, Misturas, Lógica, Enmacosa and Ingeniería Insitu inside the SITEGI project, cofinanced by the CDTI. (2012)

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4. Quality Discussion and Validation

There is no data available from other sensors that can be used to verify the computed results.

Instead the quality of the results is analyzed by considering the quality of the input data in combination with an analysis of how this quality propagates into the final volume computations. In addition, the excavation volumes were determined from a second LMMS dataset, acquired in a second run by the same system on the same day. Moreover, a possible measurement plan for further validation of the results is sketched.

4.1. Discussion on the Quality of the Results

Since the total volume is computed by summing up slices, the squared total error equals the squared sum of the errors in the determination of each sliced volume. The random error in the computation of a sliced volume consists of a variance component caused by random measurement errors in the original point cloud. This  component is denoted as s_PTS. Another variance component corresponds to the surface roughness and is denoted s_R. Using the law of error propagation, the relationship between these errors is given by Equation (9).

where s_Total is the total error of the volume computation, s_i is the random error in the estimation of the volume of the i-th slice, while  s_i,PTS and s_i,r denote the point cloud measuring error and roughness of slice i respectively, k is the number of the slices volumes, here equal to 132. Thus, the error of the volume of a single slide is studied first. According to the specifications of the Lynx LMMS and previous error studies [34,35], the range precision and range accuracy is 8 mm and ±10 mm, respectively.

As can be seen in Figure 15, such single slide is divided into 1-m by 0.5-m blocks in the road parallel and the road perpendicular direction, respectively. In each block, the mean and standard deviation of the points in that block are computed. A slice from the north side slope of the road was randomly selected to compute mean and standard deviation of points in each block of the slice. A side view of both the original and the down sampled point cloud is shown in Figure 16.

Figure 15. Single slice volume computation error analysis.

Figure 16. Side view of randomly selected road side slice.
(a) Road side points from original data;
b) Road side slope points from down sampled data.

The resulting standard deviation (st.dev) values for the eight blocks that together form the slice depicted in Figure 8 are given in Table 1. The average number of points per block is reduced from 488 to 36. This table also clearly demonstrates the purpose of the downsampling strategy: close to the road, point density is very high and therefore the reduction in the number of points is high as well. Further from the road, the point density drops and a much larger fraction of the original point is maintained. On top of that, the geometry of the terrain with regard to the lasers on the car has a strong influence on the point density.

Table 1. Mean and standard deviation of points per block in meters.
First three rows: original point cloud; Last three rows: downsampled point cloud.

To summarize the results from Table 1, we determine the differences between the means per block from the original data and the reduced data. The mean of the absolute differences equals 0.18 m. Further validation is needed to verify which means are actually better: the means from the original data are computed based on more points, but some parts of the surface may also be overrepresented in the original point cloud due to local variations in scanning geometry induced by local relief variations. For both the original and the reduced blocks, the st.dev values are comparable, between 0.5 and 0.6 m. 

These st.dev values are larger than the absolute differences between full and reduced data, and also much larger than the quality of the individual points. Therefore, it is concluded that these values are dominated by surface relief which is also clear from Figure 16. 

Assuming a st.dev value per block of 0.55 m, the st.dev per slice equals 1.56 m. Assuming 132 slices, this results in a st.dev for the total volume on one road side of 17.9 m. This st.dev value corresponds to an error below 4%, when compared to a value of 500 m3 of total excavation volume. As the current error is dominated by surface relief, a reduction in the error could be obtained by decreasing the block size.

4.2. Validation Using Data from a Second Run

For validating the results shown in Section 3, in this paragraph the same method will be applied to a second dataset obtained using the same LMMS on the same day. The differences in outcome will be compared to as discussed in Section 4.1.

4.2.1. Description of the Point Cloud Obtained in the Second Run

For the second run, the same system was used but the position of the car on the road was different, as will be shown below. As for the first dataset, the data of the second run consists of a georeferenced point cloud and of a dataset giving the trajectory of the LMMS car during data acquisition. The point cloud of the second run cropped to the same piece of road consists of 6,374,830 points, and has a point density of ~2,000 points per square meter. A side view of the second run point cloud is shown in Figure 17.

Figure 17. Side view of point cloud data from the second run.

4.2.2. Computation Results

Following the same methodology as described in Section 2, the data of the second run was processed, and the excavation volumes for both road sides were computed. The results are shown in Figure 18.

Figure 18. Cumulative volume of a roadside extension determined from point cloud data of the second run.

A comparison of the results from both datasets is given in Table 2. The results show that the difference in excavation volumes for both sides of the road are within the error budget as derived in Section 4.1, which was determined as 4% of the total excavation volume.

Table 2. Comparison of excavation volumes determined from original data and data from second run.

4.2.3. Comparison Analysis

As can be seen from Table 2, there are some differences in the excavation volumes as computed from the original point cloud data and the data from the second run. Recall that volumes are determined from 1 m slices that are further divided in eight blocks, comparing Figure 15. To obtain insight in the differences between the outcomes from the first and the second run, Figure 19 shows differences in height per block in meter of 1.0 m in road parallel direction and 0.5 m in road perpendicular direction. The dotted red line is the abstracted center line of the studied road. The purple and chocolate line in Figure 19 depict the trajectories of the LMMS while collecting the original and the second run point cloud data, respectively.

As shown in Figure 19, most of the blocks have approximately the same height, which demonstrates that the two datasets are consistent. Only the purple circles indicate locations where local height differences in the order of 1–2 m occur. Examining the two point clouds in detail indicates that at those locations hardly any points were sampled in one of the two runs. This local under sampling is probably caused by limited visibility of the roadside from the location of the LMMS acquisition.

Figure 19. Height difference per block between original and second run point cloud data (m).

This effect is illustrated in Figure 20, which shows a schematized cross section roadside geometry. For the trajectory 1, the purple area is invisible from the LMMS and is therefore not sampled. However, the area can be scanned from trajectory 2. A good solution would be to combine data from both runs such that the two point clouds data can supplement each other in such situations.

Figure 20. Geometry relation between LMMS and steep roadside terrain.

Figure 20 shows a cross section of mountainous road geometry. For the trajectory 1, the purple area
has no laser reflection, and thus, has no measured points. However, for trajectory 2, the area can be
scanned and have point cloud data, and vice versa. The point cloud data could supplement with each
other in those similar locations.

4.3. Proposal for Further Validation

There are several options to further validate the results of the methodology proposed in this paper in a field experiment. A general idea is to locally use other, preferably superior measuring methods to sample the geometry of a piece of the road and road side considered, and repeat the computations with these superior data. A traditional method would be to use a total station or RTK-GPS to measure some profiles of 3D road surface points in a local georeferenced datum, and import the obtained data into modeling software such as AutoCAD or 3ds Max, to construct a local road model and compute the volume. This method should give accurate results, but is labor intensive. A total different approach would be to actually perform measurements directly before a planned road extension. In this way, the real volume of the material that is excavated can be measured and compared to the results of the analysis of the corresponding LMMS data.

5. Conclusions

In this paper, a method is proposed for the estimation of the excavation volume of a planned road widening from a Laser Mobile Mapping point cloud. Starting with a LMMS point cloud data sampling a mountainous road, we used a uniform-size voxel to downsample the point cloud data and remove outliers. Then, local normals and 2D slopes were estimated at each resulting grid point to separate road from off-road points. Finally, the volume needed to excavate the road by 4 m on both sides was computed. It was shown on LMMS data representing a mountain road in Spain that the volume to be excavated on the left side differs by 8% to that on the right side. A more detailed analysis of one slice of data indicates that the error in the estimated excavation volume is below 4%. The results were partly validated by a comparison to results from analyzing a second point cloud obtained by the same system on the same day, but from a different trajectory. The resulting excavation volumes as estimated from both datasets differed by 2.5%–3.5%.

A further step would be to use the proposed method for determining the widening of the road of, e.g., 4 m by x meters on the right and (4 − x) meters on the left, with 0 m ≤ x ≤ 4 m, that minimizes the moved volume over a stretch of, say, 100 m of road. Further research is also needed to determine an optimal block size: In this paper, blocks of size 0.5 m by 1 m are used; reducing the block size will decrease the effect of surface relief on the error, but will increase the effect of measurement noise and varying point densities.

Acknowledgments

The authors would like to thank the three anonymous reviewers for their comments in improving
the manuscript. Also, the authors gratefully acknowledge financial support from the China Scholarship
Council. The authors would also like to thank the support from project p10-TIC-6114 JUNTA
ANDALUCIA. This paper is partly supported by IQmulus (FP7-ICT-2011-318787), a project aiming
at a High-volume Fusion and Analysis Platform for Geospatial Point Clouds, Coverages and
Volumetric Data Sets.

Conflicts of Interest

The authors declare no conflict of interest.

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