Conceptualizing Georeferencing for Terrestrial Laser Scanning and Improving Point Cloud Metadata

Most studies on the topic of georeferencing point clouds are concerned with improving the positional accuracy and reducing efforts required in the field (Scaioni, 2005; Schuhmacher and Böhm, 2005; Alba and Scaioni, 2007; Otepka et al. 2013; Fan et al. 2015; Osada et al. 2017), but the fact that different georeferencing methods result in geometries that are conceptually different from each other in terms of shape and the horizontal scale is not addressed. In this article, we describe the most common georeferencing methods used for terrestrial laser scanning (TLS) and the different types of point clouds they result in. The magnitude of the geometrical differences between the point clouds are analyzed, and metadata parameters capable of describing the most impactful point cloud characteristics are proposed. We also consider the quality of 3D models derived from point clouds and identify limitations in commonly used exchange formats, as these lack the ability to express positional uncertainty in a satisfactory manner.

While this article only addresses georeferencing methods for TLS, the concepts and geometric characteristics discussed are valid for all types of point clouds, including those created by mobile laser scanning and photogrammetry.

This article addresses metadata for geographic information and its role in the digital built environment. The main focus lies on georeferencing point clouds, the conceptual differences caused by choices made during georeferencing, and how these conceptual differences can be described using metadata. The secondary focus is how the quality of geodata affects models of the built environment and how the major exchange formats can express the quality of a model.

The first strict method simply transforms a point cloud from its local coordinate system to the ECEF coordinates $(X,Y,Z)$. This does not require any information regarding the position of the scanner, but it does require at least three non-collinear GCPs, and it results in the type of point cloud shown in Fig. 3(a). The transformation is performed using the Helmert transformation [Eq. (1)], and the transformation parameters are derived from the GCPs. Assuming a laser scanner that is correctly calibrated, the scale matrix $\mathit{S}$ is equal to the identity matrix. This transformation does not introduce any distortions, making the point cloud a $1\text{:}1$ representation of the scanned scene. All axes have the same scale, and vertical lines will not be parallel but instead perpendicular to the curved surface of the Earth. This can be done for a single point cloud or for point clouds that have been registered without plumb line constraints.

A point cloud with ECEF coordinates can be projected to a map projection, which is the second strict georeferencing method [Fig. 3(b)]. Because the planar coordinates $(N,E)$ are a projection of latitude and longitude, the $(X,Y,Z)$ coordinates must be converted to $(\phi ,\lambda ,h)$. The height above the ellipsoid $h$ can be converted to a height above geoid $H$ using a geoid model, which results in a point cloud with the coordinates $(N,E,H)$. In this point cloud, geometries are deformed, and the horizontal scale is not constant and differs from the vertical scale. This means that the shortest path between two points often is a curve and that the sum of the angles in a triangle is not necessarily $\pi$ radians. However, this point cloud can be seamlessly combined with geodata from other sources that are expressed in the same map projection.

The third strict georeferencing method is a direct method that results in the type of point cloud shown in Fig. 3(a). This requires that the position and azimuth of the laser scanner are determined by other surveying techniques and that the scanner is aligned with the local plumb line. The ellipsoidal normal is not the same as the local plumb line, and in order to correctly relate the scanned points to the reference ellipsoid, it is necessary to use the deflection of the vertical (Fig. 4) to account for this difference. For a more detailed explanation of this method, see the study by Osada et al. (2017).

For practical reasons, there are two assumptions that are commonly used when georeferencing point clouds: all vertical lines are parallel, and the map projection is a Euclidean coordinate system. When registering and georeferencing more than one point cloud (batch georeferencing in Fig. 2), we can assume that the vertical axes of all instruments are parallel. This means that if all instruments were leveled prior to scanning, we only need to find, besides the translation, the rotation around the vertical axis for each point cloud. If we can assume that a map projection is a Euclidean coordinate system, it is no longer necessary to transform the coordinates of a point cloud to ECEF and project them to a map projection, and instead, a 2D Helmert or a rigid body transformation can be used.

The first approximate method is a one-step indirect transformation from local coordinates to $(N,E,H)$. This can be done for a single point cloud or for a group of point clouds, but in the latter case, the point clouds are georeferenced individually without registration. The resulting type is shown in Fig. 3(d). For a leveled instrument, five transformation parameters (three translations, one scale factor, and one rotation) must be estimated. This can be done using at least two GCPs, out of which at least one must have a determined height. For an instrument that is not leveled, seven parameters (three translations, one scale factor for the horizontal plane, and three rotations) must be estimated. This can be done using at least three non-collinear GCPs. If this is done for several point clouds, the vertical direction would be close to constant over the entire scene but vary within each individual point cloud. This will be referred to as a semiconstant vertical direction.

The second approximate method is a two-step indirect transformation from local coordinates to $(N,E,H)$. In this method, the point clouds are not only georeferenced but also registered using a plumb line constraint. The two steps can be solved either in sequence (first registration and then Helmert transformation) or simultaneously. The simultaneous solution is more robust and typically yields a better result, but for the purpose of this article, they can be considered to be the same method as the geometric characteristics of the final point cloud are identical. The method requires that all instrument setups be leveled, and five transformation parameters (three translations, one horizontal scale factor, and one rotation) must be estimated for the entire scene. This results in a point cloud of the type shown in Fig. 3(c). This point cloud has a semi-constant vertical direction and a semi-constant scale. The latter means that the scale is close to constant over the entire point cloud but that it varies slightly depending on the height. The transformation applied in this method can either be a 2D Helmert transformation, resulting in a point cloud, which approximates the map scale over the entire area, or a rigid body transformation, which causes the scale to better resemble the scene but also causes larger residuals in the transformation to the map projection.

The third approximate method is approximate direct georeferencing. This requires that the position and azimuth of the instrument be determined and that the instrument is leveled. The points can be given ECEF coordinates [Fig. 3(a)] or cartographic coordinates [Fig. 3(c)]. In the case of ECEF coordinates, the vertical direction of the point cloud will be slightly incorrect as the direction of the local plumb line and the ellipsoidal normal are not identical. This difference is shown in Fig. 4.

The described georeferencing methods will result in point clouds with different geometries from the same input data. These geometries are first described conceptually and through visualizations and then through numerical estimates of their respective magnitudes.

The conceptual differences between the resulting point clouds are shown in Figs. 3 and 4. For convenience, the subplots will be referred to using only the figure number and their bracketed labels, i.e., Fig. 3(a) is referred to as (3a).

In direct georeferencing, the deflection of the vertical is also of importance. Fig. 4 shows the difference between aligning the vertical axis of the instrument with the direction of gravity (plumb line) or with the ellipsoidal normal. This does not affect the internal geometry of the point cloud, but it does change its relation to the global CRS.

Figs. 3 and 4 show the conceptual differences between the outcomes of the different georeferencing methods but without any regard for the magnitude of the differences. To determine this magnitude, several aspects of the geometry of the Earth and the geometry of the point clouds must be considered.

The first aspect considered is the difference in height due to the curvature of the Earth. This is relevant both when comparing curved point clouds (3a) with flat point clouds (3b–3d) and when analyzing the internal height differences present in (3c) and (3d). A simplified model of this effect in which the reference ellipsoid has been replaced by a sphere is shown in Fig. 5.

The difference between the measured height and the height above the sphere can be calculated from the following equations. The height correction $c$ is givenand

The explanations of the quantities in Eqs. (2) and (3) can be derived from Fig. 5. Using these equations, it is possible to estimate the difference between a flat and a curved point cloud based on the point clouds’ horizontal extent. Results from such calculations are shown in Table 1. In these calculations, the approximate Earth radius of 6,731 km was used. Compared to an ellipsoid, a sphere is a less accurate approximation of the physical shape of Earth, but it is sufficient for the purpose of computation of the height corrections.

Table 1 shows that the difference between a sphere and an ellipsoid in the most extreme case ($\phi =0°$, $\beta =0°$) is small relative to the corresponding height correction (vertical distance). This shows that the model presented in Fig. 5 is good enough to be used for these types of estimates.

The difference between the curved point cloud (3a) and the flat point clouds (3b), (3c), and (3d) can be explained using the height differences presented in Table 1. For example, if a 1-km stretch of a road is scanned sequentially, and if the coordinate system of the first instrument setup is used as a basis for registration, the difference in height between corresponding points at the opposite end of the road segment would be 8 cm. If an instrument setup in the middle of the road segment is used for registration, the difference will be 4 cm in both ends. This effect can also be seen internally within (3c) and (3d), where there will be small height undulations in what should be a flat surface. However, because a point cloud created from a single instrument setup rarely exceeds 100 m, this effect will, for most cases, be well below 1 mm.

The differences in scale between (3a) and (3b) depend on the chosen map projection, the location within the map projection, the height above the ellipsoid, and the horizontal extent of the point cloud. This makes it difficult to give general estimates of the magnitude of this effect. To give one example, consider a point cloud that is following the central meridian of a Universal Transverse Mercator projection and that has a height of 0 m above the ellipsoid. The scale difference between (3a) and (3b) would be 400 parts per million (ppm), or 4 cm per 100 m. Using local map projections, this number will be lower. In Sweden, for example, the scale differences using the national projection zones would be limited to roughly 50 ppm (Lantmäteriet 2020). In the case of (3c), where all point clouds are registered before georeferencing, the scale could follow either cartographic or ECEF coordinates. The scale would be semi-constant over the entire area, which means that it would not follow a map projection in the same way as (3b) or (3d). Instead, the scale would be averaged, and depending on the geographic extent, the residuals between the point cloud and the map projection could be significant. In (3c) and (3d), there are also internal scale differences. When compared to (3a), the points in the upper parts of the point clouds have been shifted closer together, and the points in the lower parts have been shifted farther apart. The magnitude of this shift comes from the differences in horizontal distances depending on the height above the reference ellipsoid, which increases with roughly 16 ppm per 100 m (Uggla and Horemuz, 2018). To give an example of what this can mean in practice, consider two adjacent point clouds in the shape of cubes where the sides are 100 m long. The greatest difference in scale would occur in a situation in which they are tangent along their bottom edges and aligned by rotating around that edge. This is shown in Fig. 6. In our example, $d_2$ would be 200 m, and $d_1$ would be 16 ppm longer, which is roughly 3 mm.

The difference between the deflection of the vertical and the ellipsoidal normal, see (4a) and (4b), affects the TLS point clouds that are directly georeferenced. The deflection of the vertical is typically limited to 20 arc seconds (arc/s) in lowland terrain and $70\mathrm{arc}/\mathrm{s}$ in more mountainous areas (Bomford 1980). The georeferencing errors are mainly manifested along the vertical axis, and it depends on the size of the deflection of the vertical, the elevation angle, and the measured slope distance. Osada et al. (2017) performed calculations for $7.5\mathrm{arc}/\mathrm{s}$ (lowlands) and $50\mathrm{arc}/\mathrm{s}$ (mountainous) and estimated the measurement errors due to neglecting the deflection of the vertical to be below 10 and 50 mm, respectively, for a slope distance of 200 m.