2.1. Estimation of the Tunnel Boundary Lines in the X-Y Plane

An entire tunnel is generally a long tube, so it can be scanned along its centerline station by station. The tunnel point clouds are registered in a user-specified coordinate system using sphere reference targets. In this system, the origin is near the tunnel entrance, and the Y axis is oriented along the direction of the initial segment of the tunnel. Several algorithms [22,23,24] have been developed to quickly extract features from LIDAR point clouds based on projection and gridding. Projecting the scanned data onto the X-Y plane can simplify the 3D tube to a long and narrow 2D object, from which the two boundary points’ groups are extracted from the both sides of the 2D object. In order to improve the speed of extraction, an algorithm for extracting the boundary point groups is proposed using a fixed grid.

The projections of the entire tunnel point clouds in the X-Y plane are discretized using a square grid. A grid size that is too large or too small will decrease the computational efficiency or the extraction precision of boundary cells, respectively. The appropriate size of the grid is about one-twentieth of the width of the tunnel. The value of N_ij is used to determine whether points exist or not in the cell ij, N_ij has a value of 1 if points exist and 0 if there are no points. The empty cell (N_ij = 0) is obviously a non-boundary cell; in Figure 2, a 9 × 9 sub-gird consisting of the cell ij and eight neighboring cells is used to determine if the cell ij is a boundary cell or not when N_ij = 1, which is formulated as:

The criterion for determining whether a cell is a boundary or not. (a) Non-boundary; and (b) boundary.

The center points of non-boundary cells will be used instead of all points in them for the further extraction of boundary points (Figure 3).

The simplified point clouds used for the further extraction of boundary points.

As shown in Figure 4, the point of interest P is an arbitrary point in a boundary cell. We use its eight neighbor cells for avoiding the incorrect extraction of the pseudo-boundary points near the bounding rectangle ABCDE. An angle criterion is proposed based on the distribution of neighboring points (the center points of non-boundary cells and all points in the boundary cells) of point P in the 9 × 9 sub-gird. Cartesian coordinates of the neighbor points are converted to polar coordinates, with the pole set at point P and polar axis L oriented along the positive direction of the X axis. The angular coordinates (e.g., α₁) of all neighboring points are sorted by value, and then the differences (e.g., Δα_i−1,i) between two consecutive neighboring coordinates are computed. Point P is a boundary points, if the maximum difference exceeds a pre-specified threshold (T), and a non-boundary point, otherwise. This angle threshold (T) is set to 175°, and can work well even in a curved tunnel, because the radius of curvature of the curve segment is large enough (generally greater than 200 m) to consider that the curved boundary in the area of 9 × 9 sub-gird is nearly straight.

Extracting boundary points from the 9 × 9 sub-girds.

Since the tunnel boundary line can be used to determine the initial directions of cross-sectional planes, a cubic polynomial function is chosen to smooth and represent tunnel boundary points, which is parameterized as follows:

where a_b, b_b, c_b, and d_b are the parameters of a boundary line.

The points belonging to the measurement errors or different structures (i.e., boundary points of refuge recesses) will also be extracted if they meet the angle criterion. The RANSAC algorithm was first proposed by Fischler and Bolles [25] and it is an iterative method to estimate parameters of a mathematical model from a set of observed data which contains outliers. Hence, we adopt this algorithm to find the real boundary points and estimate the parameters of Equation (2).