Fig 1.
The overlap between the fitted axes is much larger than that between the tie points.
Fig 2.
Extraction of boundary points using a moving window.
All of the points within the moving window are considered to be neighboring points of point P. The polar angles of the neighboring points are computed relative to point P (e.g., α1). If point P is a boundary point, the difference between the consecutive polar angles of boundary points Pi and Pi + 1 is much greater than the difference between the polar angles of point Pi and interior point Pi − 1. Therefore, once the difference is greater than a predefined threshold, point P is labeled as a boundary point.
Fig 3.
Histogram of hypothesis model parameters.
The horizontal axis denotes the mean value of the model parameters, and the vertical axis represents the degree of convergence of each cell. A high degree of convergence for a parameter reflects a high probability of finding the initial model.
Fig 4.
Determination of the central-axis point.
To extract the central axis of the tunnel, the normal vector Vl of the left bounding curve at boundary point Pl is determined. A straight line orthogonal to the normal vector reaches the right bounding curve from Pl and generates point Pl'. Theoretically, the radial line from point Pl' that is orthogonal to Vl' reaches the left bounding curve at point Pl, so the extracted central-axis point is the midpoint of line Pl Pl'. However, because the bounding curves are subject to errors that are generated from the fitting process, the radial that is orthogonal to Vl' produces point Pl" instead of point Pl. Ml' and Ml" are the midpoints of Pl Pl' and Pl' Pl", respectively. The extracted central-axis point is determined as Ml, which is the average of points Ml' and Ml". The same process is implemented from boundary point Pr on the right bounding curve to extract the point on the central axis as point Mr. Based on the extracted central-axis points, the presented strategy to fit a bounding line is used to generate the central axis.
Fig 5.
To maintain consistency between adjacent fitted models, the divided segments overlap each other slightly, and a global least-squares adjustment is developed to implement the multiple model fittings of all of the segments together by minimizing the deviations in the overlapping parts of adjacent fitted models.
Fig 6.
We derive the central-axis constraint by adding an additional observation model to the AEKF system. The observation model is established to minimize the deviation that describes how the point on the axis segment in the analyzed scan does not fit the corresponding axis segment model that was determined in the fixed scan when the point is transformed into the global reference frame.
Fig 7.
The proposed approach was tested on real datasets that were acquired by a RIEGL LMS VZ-400 laser scanner in a subway tunnel in Shanghai, China. Twelve scans were captured; the scans had an average shift of 10 m between their centers and cover tunnel segments with different shapes for a distance of 450 m.
Fig 8.
Tie points selected for registration.
(a) Tie points (7 pairs) between Scans 1 and 2; (b) Tie points (8 pairs) between Scans 2 and 3; (c) Tie points (7 pairs) between Scans 3 and 4; (d) Tie points (7 pairs) between Scans 4 and 5; (e) Tie points (6 pairs) between Scans 5 and 6; (f) Tie points (7 pairs) between Scans 6 and 7; (g) Tie points (8 pairs) between Scans 7 and 8; (h) Tie points (6 pairs) between Scans 8 and 9; (i) Tie points (9 pairs) between Scans 9 and 10; (j) Tie points (8 pairs) between Scans 10 and 11; (k) Tie points (6 pairs) between Scans 11 and 12.
Table 1.
Accuracies of the pair-wise registrations.
Fig 9.
Error accumulation of pair-wise registration.
(a) Overview; (b) Scans 1 and 4; (c) Scans 2 and 5; (d) Scans 3 and 6; (e) Scans 7 and 10; (f) Scans 8 and 11; (g) Scans 9 and 12.
Table 2.
Deviations between check point pairs (Scans 1 and 4 to Scans 9 and 12).
Fig 10.
Global registration results (AEKF).
(a) Overview; (b) Scans 1 and 4; (c) Scans 2 and 5; (d) Scans 3 and 6; (e) Scans 7 and 10; (f) Scans 8 and 11; (g) Scans 9 and 12.
Table 3.
Deviations between check point pairs (AEKF).
Fig 11.
2D projections of the tunnel points onto the XOY plane (Scans 1 and 12).
(a) 2D projection of Scan 1; (b) 2D projection of Scan 12.
Fig 12.
Statistical test results of the optimized BaySAC algorithm.
(a) Straight line; (b) Transition curve; (c) Curve.
Fig 13.
Fitting of the tunnel axis (straight line and transition curve).
(a) Central-axis extraction (Scan 12); (b) extracted 3D central axis (Scan 12); (c) central-axis extraction (Scan 1); (d) extracted 3D central axis (Scan 1).
Fig 14.
The central axis that was fit from the tunnel points consists of three segments. The yellow box on the left highlights the overlap between the curve and the transition curve, while the box on the right shows the overlap between the transition curve and the straight line. 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. Large deviations with an RMSE of 26 mm (detailed views) are present within the overlap zones between the segments due to the noise in the tunnel point dataset.
Table 4.
Comparison of the fitting accuracies.
Fig 15.
Central-axis fitting with global least squares adjustment.
To optimize the extraction results, the proposed global least squares adjustment was implemented to minimize the deviations in the overlap zones between the adjacent fitted models. Fig 15 shows that the differences shown in Fig 14 were reduced significantly, and a globally optimized central axis was extracted.
Fig 16.
Global registration results (AEKF + Central-axis constraint).
(a) Overview; (b) Scans 1 and 4; (c) Scans 2 and 5; (d) Scans 3 and 6; (e) Scans 7 and 10; (f) Scans 8 and 11; (g) Scans 9 and 12.
Table 5.
Deviations between check point pairs (AEKF+Central-axis constraint).