Extracting planar features from laser altimetry data is usually done over urban areas where many buildings with planar roofs are available. There are many methods in the literature for detecting buildings and identifying roof planes in aerial laser data, e.g., [27–32]. In this study, a region growing segmentation method is used to extract planar features from gabled roofs or dike slopes. The points that belong to each strip in the overlapping area are interpolated into a raster format with a resolution of 50 centimeters using an Inverse Distance Interpolation algorithm based on the weighted average method [33]. Slope and aspect orientation that best define the terrain in the vicinity of each point are derived. Homogenous regions are created by using a segmentation algorithm performed with the slope and aspect data. The size of the created regions is dependent on a scale parameter, which defines the maximum allowed heterogeneity within the resulting regions. The region-growing segmentation starts with single data points as initial regions, and repeatedly merges them into larger regions as long as a homogeneity criterion holds. The homogeneity criterion is controlled by the scale parameter: larger scale parameters result in larger regions, and smaller parameters in smaller regions [34,35]. To avoid overgrown regions, a suitable scale parameter is found experimentally. The segmentation is performed with different scale parameters on a subset of data, and the segmentation results are visually examined. The best scale parameter is selected as one that provides the largest regions which do not overgrow beyond the boundaries of the roofs or dike slopes.

Planar surfaces that represent gabled roofs and dike slopes are selected from the resulting regions by constraining the regions to a minimum area of 6 m² and a slope between 15 and 70 degrees. The boundaries of the corresponding regions in the two strips are combined and intersected. These intersected regions are buffered inwards with 25 cm (half the raster pixel size) to make sure that the points that are within the intersection area are part of the plane. These points are selected for the estimation of the plane parameters. As the segmentation is performed on gridded data, the selection criteria are not applied on the points directly. Therefore outliers in the selected point cloud might still be present after the segmentation process.

The plane parameters are obtained by applying the Principal Component Analysis (PCA) to the selected points [36]. The principal components are the axes of maximum variation of the data. For a set of points that lie on a plane the last principal component is the axis of minimum variation, which is normal to the plane. The principal components are found by the Singular Value Decomposition (SVD) of the mean-centered points. The eigenvector corresponding to the smallest eigenvalue is the last principal component that is the plane normal. The distance of the plane to the origin of the coordinate system is taken as the median of the dot product of the normal vector and the vector of each individual point.

The segmentation procedure does not necessarily always provide regions of points that are perfectly coplanar. In practice the regions might contain outliers, e.g., points on the walls, trees or the ground surface. To deal with the outlying points a robust plane fitting has to be applied to the points in both strips. Then only those points that are identified by the robust fitting algorithm as inliers are used to obtain the point-to-plane distances. The robust plane fitting is performed using the RANSAC algorithm [37]. Considering that each random sample should contain three points (minimum to define a plane), and assuming that 50% of the points are outliers, 35 random samples are needed to ensure, with a probability of 99%, that at least one sample is outlier-free [38]. The relatively small number of the required random samples indicates that the computational cost of the robust plane fitting is easily affordable. Figure 3 demonstrates the robust plane fitting using RANSAC.

The output of the previous step is a set of planes in one strip and their corresponding points in the other strip. The parameters of a transformation between the two overlapping strips are estimated by minimizing the distances between points and their corresponding planes. Let the transformation of a set of points P by a transformation H be expressed by matrix multiplication H P. The condition that the transformed points lie on a plane Л is expressed as [38,39]: (1) Л T HP = 0where Л = (n₁, n₂, n₃, −d)^T represents a plane with normal n = (n₁, n₂, n₃)^T and distance d to the origin, P = (x, y, z, 1)^T denotes the homogenous representation of a point in 3D space. In practice, H can be a homogenous similarity transformation: (2) H = [ R t 0 1 × 3 1 ]in which R is a 3D rotation matrix, t = (t_x, t_y, t_z)^T is a translation vector and a scale factor is neglected since laser strips are normally assumed to have identical scales. However, to make the estimation model linear, we ignore the orthogonality of R and assume that H is an affine transformation. To estimate the transformation Equation (1) is rewritten as: (3) [ n 1 n 2 n 3 − d ] [ r 1 T t x r 2 T t y r 3 T t z 0 1 × 3 1 ] [ p 1 ] = 0where r₁^T, r₂^T, r₃^T are the three rows of R, and p = (x, y, z)^T is the Euclidian notation of a point in 3D space. For one point on one plane Equation (3) reduces to: (4) n 1 ( r 1 T p + t x ) + n 2 ( r 2 T p + t y ) + n 3 ( r 3 T p + t z ) = d

Equation (4) is the essential observation equation that basically expresses the condition that the transformed points in one strip rest on their corresponding plane in the other strip. For m points corresponding to k planes we will have a system of equations as follows: (5) [ n 1 1 p 1 T n 2 1 p 1 T n 3 1 p 1 T n 1 1 n 2 1 n 3 1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ n 1 k p m T n 2 k p m T n 3 k p m T n 1 k n 2 k n 3 k ] [ r 1 r 2 r 3 t x t y t z ] = [ d 1 ⋮ d k ]where superscripts denote the point and plane numbers. The system of Equation (5) is of the form AX = L for which the least-squares solution that minimizes the norm of the squared distances between the transformed points and their corresponding planes, i.e., ||AX − L||, is given in Equation (6): (6) X = ( A T WA ) − 1 A T WLwhere W is a weight matrix. In this paper, the same weight is applied to each observable based on the assumption that all observables have the same precision.

A special case of the strip adjustment model as described above occurs when we expect the transformation between the strips to be only a translation vector. In such a case, the estimation model given in Equation (4) reduces to: (7) n 1 ( x + t x ) + n 2 ( y + t y ) + n 3 ( z + t z ) = dwhich basically expresses that the translated points lie on the plane. The system of equations with m points and k planes becomes: (8) [ n 1 1 n 2 1 n 3 1 ⋮ ⋮ ⋮ n 1 k n 2 k n 3 k ] [ t x t y t z ] = [ d 1 − n 1 1 x 1 − n 2 1 y 1 − n 3 1 z 1 ⋮ d k − n 1 k x m − n 2 k y m − n 3 k z m ]for which a solution that minimizes the distances between the translated points and their corresponding planes can be obtained similar to the general case as given in Equation (6).

It is worth noting that to obtain a solution for the estimation models derived above the design matrices in Equations (5) and (8) have to be of full rank. This requires a minimum of 12 points and 3 non-parallel planes to estimate the affine transformation in Equation (5), and a minimum of 3 points and 3 non-parallel planes to estimate the translation vector in Equation (8). In practice, this is not an issue since usually a large number of point-to-plane distances are available.

The precision of the observables as well as the estimated transformation parameters are derived from the residual vector v that actually contains the remaining point-to-plane distances after the adjustment: (9) v = AX − L

The reference variance obtained from the residual vector v is an indication of the precision of the observables, or the random error of individual points: (10) σ 0 2 = v T     W     v m − uwhere u is the number of unknown parameters in the adjustment. The precision of the estimated transformation parameters is then obtained by: (11) Σ xx = σ 0 2 ( A T   W   A ) − 1

The strip adjustment method as described above was employed to assess the accuracy of the pilot AHN-2 dataset acquired by Fugro-Inpark over Zeeland province in 2007. The area consists of a crop land with large farm buildings and several urban areas, of which the city of Flushing is the largest. The data were acquired by the helicopter-mounted FLI-MAP 400 laser scanner from an altitude of 375 meters. The data consists of the forward-, nadir- and backward-looking scan lines. From this dataset, 15 strips were selected, resulting in 13 overlaps, as shown in Figure 4. Table 1 lists the overlaps and their corresponding strips, number of points and width of each overlap. The overlap areas are of different sizes. Within each overlap also the numbers of points belonging to each strip are slightly different, which is due to the difference in the scanning geometry, the topography and the type of objects present in the overlap area.

To evaluate the performance of the segmentation procedure strip overlap o12 was segmented with different scale parameters. The resulting roof regions were examined and compared with building boundaries from an existing large scale base map and with aerial images of the area. The result is shown in Figure 5. Scale parameters 10 and 15 provide the most reliable roof segments. Scale parameter 15 provides more segments that are also larger in size; therefore, it is chosen as the optimal scale parameter.

Figure 6 shows an example of roof segments created by using three different scale parameters. The segments are shown in yellow outlines. The building on the left has a dormer on the right roof side. It can be seen that by using scale parameter 20 one of the created segments is overgrown and contains points on the dormer. In contrast, scale parameter 15 provides homogeneous segments within the roof boundaries. The irregularity in the shape of the segments is due to the intersection of the segments in the two strips and the applied inwards buffering.

The point-to-plane distances were derived for each strip overlap from the segmentation results. For every region in one strip plane parameters were computed using the robust plane fitting method. Then, distances between this plane and the inlying points in the corresponding region in the other strip were obtained and entered as observables in the estimation model. Two transformations were estimated for each pair of overlapping strips: a 3D translation only and a full affine transformation.

The mean and standard deviation of the point-to-plane distances before and after the adjustment were computed. Assuming that the data are not contaminated by gross errors (outliers), the mean before the adjustment indicates systematic error in the data. After the adjustment, the mean is expected to be very small, and the standard deviation indicates the random error of the observables and the precision of the estimated parameters. Figure 7 shows the mean of the point-to-plane distances before and after the adjustment. Before the adjustment the mean ranges from −3 cm to 2 cm. Adjustment with only a translation reduces the mean in all overlaps, except for overlap o2 (although here the mean is only −1 cm). Adjustment with a full transformation further reduces the mean to values below ±5 mm. Figure 8 shows the standard deviation of the point-to-plane distances before and after the adjustment. Before the adjustment the standard deviations range from 3 cm to 11 cm. These become smaller after the translation (2 cm to 8 cm), and are the smallest after the full transformation where the largest standard deviation is 5 cm.

Figure 9 and Figure 10 show the estimated horizontal offsets respectively in x and y direction together with their precision for all strip overlaps. These values are compared to those obtained by a line-based method [40], which provides only 2D translation parameters. The precision of the estimated parameters is remarkably better in all overlaps when point-to-plane distances are used. Both T_x and T_y are estimated with a precision better than 1 mm. In most of the overlaps, the estimated horizontal offsets are in agreement with those obtained by the line-based method. The discrepancies are in overlaps o6 (T_x), o7 (T_y) and o9 (both T_x and T_y), where differences from 10 cm to 20 cm can be observed. In the case of overlap o9, the precision of both T_x and T_y estimated from the line correspondences is very low, whereas the offsets estimated from the point-to-plane distances using both adjustment models are more precise and very close in magnitude. In fact, both T_x and T_y estimated from point-to-plane distances are within the range of uncertainty of the offsets estimated by the line-based method. In overlap o6 also the T_x values estimated by the plane-based method using both adjustment models are very close and more precise than that estimated by the line-based method. The disagreement in T_y corresponding to overlap o7 may be due to the presence of a small rotation between the strips. Note that the mean and standard deviation of the point-to-plane distances after the adjustment with a translation only and those after the full affine transformation have the largest discrepancy in overlap o7 (see Figure 7 and Figure 8).

Figure 11 shows the vertical offsets T_z estimated from the point-to-plane distances for all overlaps. Here the bars representing the precision of the offsets are magnified by a factor of 10. It can be seen that the vertical offsets resulting from the two transformation models are very close. A difference of about 1 cm can be observed in overlap o7, which again might imply the presence of a small rotation between the strips. The precision of the estimated vertical offsets is better than 2 mm.