^{1}

^{2}

^{3}

^{4}

^{1}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

An efficient method for the continuous extraction of subway tunnel cross sections using terrestrial point clouds is proposed. First, the continuous central axis of the tunnel is extracted using a 2D projection of the point cloud and curve fitting using the RANSAC (RANdom SAmple Consensus) algorithm, and the axis is optimized using a global extraction strategy based on segment-wise fitting. The cross-sectional planes, which are orthogonal to the central axis, are then determined for every interval. The cross-sectional points are extracted by intersecting straight lines that rotate orthogonally around the central axis within the cross-sectional plane with the tunnel point cloud. An interpolation algorithm based on quadric parametric surface fitting, using the BaySAC (Bayesian SAmpling Consensus) algorithm, is proposed to compute the cross-sectional point when it cannot be acquired directly from the tunnel points along the extraction direction of interest. Because the standard shape of the tunnel cross section is a circle, circle fitting is implemented using RANSAC to reduce the noise. The proposed approach is tested on terrestrial point clouds that cover a 150-m-long segment of a Shanghai subway tunnel, which were acquired using a LMS VZ-400 laser scanner. The results indicate that the proposed quadric parametric surface fitting using the optimized BaySAC achieves a higher overall fitting accuracy (0.9 mm) than the accuracy (1.6 mm) obtained by the plain RANSAC. The results also show that the proposed cross section extraction algorithm can achieve high accuracy (millimeter level, which was assessed by comparing the fitted radii with the designed radius of the cross section and comparing corresponding chord lengths in different cross sections) and high efficiency (less than 3 s/section on average).

Because underground structures such as tunnels require routine inspections and maintenance for their optimal use, efficient and accurate tunnel inspections are necessary. The shape, width and area of the cross sections of constructed or natural tunnels can be used to determine their structural stability. These estimations are generally conducted based on sparsely sampled points that are surveyed using a total station. This approach requires a considerable amount of time and effort, although non-prism total stations are now available [

The application of laser technology is rapidly expanding and offers decreased costs and increased accuracy. Therefore, 3D laser scanners make it possible to obtain high-accuracy data and high location precision, even under disadvantageous conditions. Tsakiri

A tunnel's internal cross section can be obtained from large quantities of laser scanning data of the tunnel, and precise outbreak and underbreak quantities can be calculated. Yoon

Two methods are commonly used to extract the cross sections of a tunnel from TLS data [

First, the central axis of the subway tunnel is determined using a 2D projection of the tunnel point cloud and multiple model (e.g., straight line, transition curve and curve) fitting using a so-called statistical testing algorithm and the RANSAC (RANdom SAmple Consensus) algorithm. Because the extraction of the central axis by multiple model fitting may suffer from noise in the tunnel point data and bending of the tunnel, we propose a global extraction algorithm based on segment-wise fitting. Oude Elberink

We begin by describing the algorithm used to extract the central axis of a tunnel in Section 2. The next section introduces the algorithm for the continuous extraction of cross sections based on the extracted central axis from the terrestrial point clouds. Section 4 discusses the test results, after which we offer conclusions and suggestions for further research in Section 5.

To extract so-called continuous (

The tunnel point clouds are projected onto the XOY plane, from which we extract the boundary points of both sides of the tunnel. An algorithm for boundary point extraction is proposed using a moving window. _{1}). We then calculate the differences between consecutive polar angles. If point P is a boundary point, the difference Δ_{i + 1,i} between boundary points P_{i}_{i + 1} is much larger than the difference Δ_{i,i − 1} between boundary point P_{i}_{i − 1}. Therefore, once the difference is greater than a predefined threshold, point P is labeled as a boundary point.

The bounding lines of a tunnel usually contain segments of straight lines, curves and transition curves, which are parameterized as follows:

Straight line model:

Transition curve model:

Curve model:

The bounding line fitting process includes the estimation of multiple models. To ensure the robustness of the fitting, the RANSAC algorithm [

A statistical testing algorithm is proposed to automatically detect the initial models from the extracted boundary points so the proper model is selected to fit each segment of the bounding line. The statistical testing is implemented using straight-line, transition-curve and curve models.

The statistical testing process is implemented using a histogram, which illustrates the distribution of the discrete hypothesis model parameter sets that are computed during different iterations. The degree of convergence of a candidate parameter set is used as a criterion of the statistical testing. This criterion describes how the other sets converge to it and is calculated by dividing the number of converging sets by the total number of parameter sets. We construct vectors with two, four and three dimensions for each set of model parameters. The Euclidian distances between different vectors are computed to describe the deviation between the candidate parameter set and the other hypothesis model parameter sets. If the distance between a hypothesis set and the candidate set is smaller than the predefined threshold, the hypothesis set is considered the converging set of the candidate set, and the degree of convergence of the candidate set increases. In this method, the histogram of the candidate parameter sets is updated during each iteration using the newly calculated hypothesis parameter set. When the degree of convergence of a candidate parameter set reaches a predefined threshold, the candidate parameter set is detected as an initial model to fit the bounding line segment. If the degree of convergence fails to reach the threshold after a predefined number of iterations, we believe that there is no such model.

To visualize the statistical results, we illustrate them as a histogram (

After the initial model is detected, RANSAC is used to robustly estimate the optimized model parameters. Two, four, and three points are used to estimate the model parameters to fit a straight line, a transition curve and a curve, respectively. The criterion used to identify outliers is based on the deviations of the tested points from the fitted model. The inlier bounding points of a certain model are classified as a segment that is used in the following global optimization. The final optimal parameters are computed by the least-squares adjustment using the obtained inlier points.

After fitting the bounding lines, the boundary points are evenly resampled. To extract the central axis of the tunnel, the normal vector _{l}_{l}_{l}_{l}′_{l}′_{l}′_{l}_{l} P_{l}′_{l}′_{l}″_{l}_{l}′_{l}″_{l}_{l}′_{l}′ P_{l}″_{l}_{l}′_{l}″_{r}_{r}

Because the extraction of the segments of the bounding lines and the central axis on the XOY plane using the three models may suffer from noise in the tunnel points, there may be deviations in the overlapping parts of adjacent fitted models (

To maintain consistency between adjacent fitted models, the divided segments overlap each other somewhat, and a global least-squares adjustment is developed to implement the multiple model fitting of all of the segments together by minimizing the deviations in the overlapping parts of adjacent fitted models. Using _{i}_{i}_{j}_{j}_{j}_{j}

The coefficient matrix of the observation and constraint equations of the global least-squares adjustment is derived in _{ij}

As introduced in Section 1, a widely used strategy for cross section extraction is to project the tunnel point subset that forms a sliced body onto the plane of the desired cross section. However, it is almost impossible for this method to find a cross-sectional point that is located exactly at an arbitrary vertical angle from the central axis. Although the cross section is orthogonal to the central axis, any cross-sectional point can be extracted by intersecting a straight line that orthogonally rotates about the central axis with the tunnel point cloud. Therefore, we propose an extraction strategy that uses this idea to compute an arbitrary cross-sectional point within a small subset of the large tunnel point cloud.

To extract the cross-sectional points, the direction of the cross-sectional plane is first determined based on the central axis by assuming that the cross-sectional plane is orthogonal to the central axis. However, even when a fine extraction of the central axis is implemented, the direction of a pseudo cross-sectional plane is still subject to the errors from the fitting processes and the noise and measurement errors of the tunnel point cloud. Therefore, the algorithm proposed by Han

To adjust the cross-sectional plane, a point group _{p}_{n}_{k}S_{n}

As shown in _{1} and _{2} are the bounding lines of point group _{p}_{1} and _{2} on the lines, a final cross-sectional plane _{n}_{1} and _{2}, where the horizontal length of the line segment _{1}_{2} is minimized. The horizontal length of line segment _{1}_{2} is equivalent to the width of the final cross section at the station (

Interpolation is a method to construct new data points within a discrete set of known data points. Therefore, when no cross-sectional point can be directly acquired from the tunnel points, we use the interpolation method to compute the cross-sectional point as accurately as possible. In our strategy, the cross section is calculated within a small subset of the tunnel point cloud so that only laser points in the vicinity of the computed cross-sectional point (highlighted in red in

The form of a quadric parametric surface [

Because our so-called continuous extraction of cross sections may need to interpolate a large number of cross-sectional points, we adopt an improved BaySAC [

A well-regarded technique for robust model fitting of point clouds is the statistical framework of RANSAC (RANdom SAmple Consensus) [

To improve the computational efficiency, this study employs a conditional sampling method based on BaySAC (Bayesian SAmpling Consensus) [

After the quadric parametric surface parameters are determined, a cross-sectional point is interpolated by computing the intersection point of the fitted surface and the radial line along the extraction direction.

The proposed approach was tested on a real dataset (

As proposed in Section 2.1, the tunnel points of the VZ-400 dataset were projected onto the XOY plane as shown in

As presented in Section 2.1.2, the bounding lines of a tunnel may comprise segments of straight lines, curves and transition curves. The proposed statistical testing algorithm was implemented using the three models to automatically detect the corresponding initial model from the extracted boundary points.

Based on the detected initial model (e.g., the transition curve), we fit the bounding lines using RANSAC (

As shown in

To optimize the extraction result, the global least squares adjustment proposed in Section 2.2 was implemented to minimize the deviations in the overlap zones between the adjacent fitted models.

To test the fitting accuracy, we set 24 Y coordinates along the central axis within the overlap zones highlighted in the two yellow boxes. Their corresponding points on the two adjacent segments were computed. The deviations between the corresponding points were then calculated and are shown in

As presented in Section 2.2, the proposed algorithm for the continuous extraction of cross sections must interpolate a large number of cross-sectional points. The interpolation of a cross-sectional point involves the fitting of a quadric parametric surface. Therefore, we implemented the improved BaySAC algorithm described in Section 3.2.2 to improve the computational efficiency and the robustness of the quadric parametric surface fitting. In this section, the performances of the quadric parametric surface fitting using the optimized BaySAC algorithm and the RANSAC framework were compared in terms of their computational efficiencies and fitting accuracies.

Because hypothesis testing is an iterative process, the computational efficiencies of the proposed strategies were evaluated in terms of the number of iterations. The iterations of RANSAC are shown as circular icons in

As shown in

To further test the fitting accuracy, we selected ten inliers as check points. The point-to-surface distances were then calculated and are shown in

As described in Section 3.2, we proposed an algorithm to interpolate cross-sectional points using quadric parametric surface fitting when no cross-sectional point can be directly acquired from the laser points. To evaluate the accuracy of the interpolation, five laser points were chosen, and their interpolated coordinates were computed, as shown in

As mentioned in Section 3, our proposed algorithm can theoretically extract cross sections at any interval from the terrestrial point clouds. We implemented the cross section extraction experiments at a one-meter interval. The average computational cost of the cross section extraction was less than 3 s/section. The candidate cross-sectional points were extracted using the straight line that was orthogonal to the central axis of the tunnel within a vertical plane at a one-degree interval (

To evaluate the accuracy of the cross section fitting, 100 cross sections were fitted from the VZ-400 data. Because the standard shape of the tunnel cross section of interest is a circle, the radii of the fitted circular cross sections were compared with the design radius of 2.75 m.

Moreover, the chord length between every two cross-sectional points was calculated (e.g.,

In this paper, we proposed an efficient algorithm to continuously extract tunnel cross sections. First, the central axis of a subway tunnel was extracted using multiple model fitting based on 2D projections of a tunnel point cloud. This axis was optimized using a global extraction strategy based on segment-wise fitting. After the central axis was determined, the cross section was extracted by intersecting a radial line that orthogonally rotates about the central axis of the tunnel point cloud. An interpolation algorithm based on quadric parametric surface fitting using an improved BaySAC (Bayesian SAmpling Consensus) algorithm was proposed to compute a cross-sectional point when no cross-sectional point could be directly acquired from the tunnel points along the direction of interest. Because the standard shape of the tunnel cross section of interest is a circle, circle fitting was implemented using RANSAC (RANdom SAmple Consensus) to filter the noise.

The proposed algorithm was implemented using a terrestrial laser scanning dataset that was acquired in a subway tunnel. The results of the central-axis extraction show that the proposed algorithm of global extraction based on segment-wise fitting achieved an accuracy of 2 mm, while the accuracy acquired by the fitting based on a single model is 26 mm. The results of the cross section extraction process show that the interpolation of the cross-axial points based on the improved BaySAC algorithm produces a mean deviation of 0.9 mm from the measured points. The millimeter-level accuracies (6 mm, 1.4 mm and 1 mm) of the cross section extraction were demonstrated respectively by comparing the fitted radii with the designed radius of the cross section in the subway and comparing corresponding chord lengths in different cross sections. Moreover, the average computational cost of the cross section extraction method was less than 3 s/section, which demonstrates its high efficiency.

Because variations in the scanning parameters, such as point density, range to the scanner and incidence angle, and can affect the quality of the results, their effects and the optimization of the scanning parameters will be investigated in subsequent studies. Moreover, the deformation tendency of a tunnel can be reflected by the variance of its radius among the fitted circular cross sections and the variance between corresponding chord lengths in different cross sections. Future work will also focus on optimizing the proposed algorithm to detect possible deformation based on so-called continuously extracted cross sections.

This work was supported by the Natural Science Foundation of China under Grant No. 41171358, the Fundamental Research Funds for the Central Universities under Grant No. 2652012103 and the National High Technology Research and Development Program of China (863 Program) with the serial number 2012AA121303.

All authors contributed extensively to the work presented in this paper.

The authors declare no conflict of interest.

Extraction of boundary points using a moving window.

Histogram of hypothesis model parameters.

Determination of the central-axis point.

Segment-wise fitting.

Adjustment of the pseudo cross-sectional plane [

Cross-sectional point estimation.

Extraction of the cross section. (

The experimental dataset.

2D projection of the tunnel points onto the XOY plane.

Statistical test results of the optimized BaySAC algorithm. (

Fitting of the tunnel axis (the transition curve). (

Extracted central axis.

Central-axis fitting with global least squares adjustment.

Computational efficiency.

Fitting results of the laser point subset of a tunnel. (

Cross section extraction. (

Deviations between the fitted radii and the designed radius.

Chords used for comparison. (

Comparison of corresponding chord lengths. (

Description of the point cloud dataset.

RIEGL VZ-400 | 0.046° | 2,686,866 | ±5 mm |

Note:

Comparison of the fitting accuracies.

1 | 0.005 | 0.035 | 13 | 0.001 | 0.056 |

2 | −0.003 | −0.026 | 14 | 0.004 | 0.039 |

3 | 0.002 | 0.019 | 15 | −0.005 | 0.022 |

4 | 0.003 | 0.016 | 16 | 0.003 | 0.047 |

5 | −0.005 | 0.002 | 17 | 0.001 | −0.032 |

6 | 0.001 | 0.001 | 18 | 0.004 | 0.03 |

7 | −0.003 | −0.007 | 19 | 0.001 | 0.038 |

8 | 0.002 | 0.003 | 20 | −0.007 | −0.015 |

9 | −0.001 | 0.008 | 21 | −0.001 | 0.035 |

10 | −0.003 | 0.004 | 22 | 0.003 | 0.002 |

11 | 0.0013 | 0.01 | 23 | 0.001 | −0.025 |

12 | −0.002 | −0.035 | 24 | 0.002 | 0.015 |

RMSE | 0.002 | 0.026 |

Accuracies of quadric parametric surface fitting.

1 | 0.0021 | 0.0009 |

2 | 0.0020 | 0.0009 |

3 | 0.0008 | 0.0002 |

4 | 0.0012 | 0.0005 |

5 | 0.0016 | 0.0007 |

6 | 0.0025 | 0.0015 |

7 | 0.0019 | 0.0010 |

8 | 0.0010 | 0.0007 |

9 | 0.0020 | 0.0015 |

10 | 0.0014 | 0.0009 |

Accuracy of the cross-sectional point interpolation.

1 | −3.171 | 15.578 | 0.051 | −3.1705 | 15.578 | 0.0510 | 0.0005 |

2 | −3.182 | 15.544 | 0.013 | −3.1821 | 15.544 | 0.1330 | 0.0001 |

3 | −2.842 | 15.680 | 1.918 | −2.8409 | 15.680 | 1.9173 | 0.0013 |

4 | −2.864 | 15.759 | 1.875 | −2.8646 | 15.759 | 1.8754 | 0.0007 |

5 | −0.596 | 15.576 | −1.573 | 0.5965 | 15.576 | −1.5744 | 0.0015 |

| |||||||

Mean deviation (m) | 0.0008 |