STRIP OFFSET ESTIMATION USING LINEAR FEATURES
George Vosselman Department of Geodesy, Delft University of Technology Thijsseweg 11, NL-2629 JA Delft, The Netherlands
[email protected]
KEY WORDS: Laser altimetry, strip adjustment, error estimation, least squares matching, fitting of point clouds.
ABSTRACT: For high precision digital elevation models, systematic errors in point clouds acquired by airborne laser altimetry need to be estimated and removed by strip adjustment procedures. Estimated threedimensional offsets between patches of overlapping strips are required as observations for this procedure. This paper discusses varies approaches to offset estimation. Furthermore, models of linear features such as gable roofs and ditches are introduced for the estimation of offsets. As this model-based approach does not suffer from systematic biases that are to be expected from image matching techniques, high accuracies can be obtained. Examples of offsets estimations in two datasets illustrate the procedure.
1.
INTRODUCTION
Systematic geometric errors in point clouds acquired by airborne laser altimetry may occur for a wide variety of reasons (Huising and Gomes Pereira 1998). They become most evident in overlaps between adjacent strips: at the same planimetric location the strips show different height values. Strip adjustment procedures have been defined to estimate and eliminate these systematic errors (Kilian et al. 1996, Burman 2000, Crombaghs et al. 2000). This paper first reviews different approaches one can take to strip adjustment in section two. The remainder of the paper focuses on methods for the estimation of three-dimensional offsets between overlapping strips. These estimated offsets may serve as observations for a strip adjustment procedure. Pitfalls and good practice of offset estimation in height data are discussed in section three. Section four and five detail the estimation of offsets using linear features like gable roofs and ditches. Observation equations are derived and examples of estimated offsets are presented and analysed.
2.
APPROACHES TO STRIP ADJUSTMENT
In general, observation equations used in adjustments relate observations to the parameters to be estimated. In recent years several approaches to strip adjustment have been suggested. These approaches show differences with respect to both the parameterisation of the errors in the laser data and the kind of observations that are used. 2.1
Parameterisation of errors
Instrumental or calibration errors in one of the components of a laser scanner cause systematic displacements and/or distortions of the acquired point clouds. The first approaches to strip adjustment tried to model the effects of the errors onto the point clouds by straightforward transformations between the laser strip coordinate systems and a reference coordinate system. Kilian et al. (1996) modelled this transformation by constant and time-dependent shifts and rotations between a strip and the reference coordinate system. Crombaghs et al. (2000) even simplified the transformation to a height offset and two tilts, one in flight direction and one across flight direction. Although these transformations are able to absorb many distortions that are caused by the instrumental and calibration errors, some non-linear effects of the errors onto the point clouds are not properly modelled. Crombaghs et al. (2000) discuss the example of a scale error in the scanning angle measurement which results in strips that look like the lower part of a horizontal pipe (sometimes referred to as the smiley error). They show that strip adjustments in laser blocks that exhibit errors that are not completely modelled in the observation equations may even result in a deterioration of the geometric quality of the point clouds. To avoid these problems the effect of instrumental and calibration errors onto the point clouds needs to be modelled explicitly. Such geo-location equations are discussed by Schenk (2001) and Filin (2002). This approach, however, also has a disadvantage which relates to the fact that several error sources may have the same or a very similar effect onto the point clouds. Based on measurements of discrepancies between the point clouds of overlapping strips one can therefore not distinguish the original source of these discrepancies. Including all possible errors as unknown parameters will then result in a (nearly) singular normal equation system due to the high correlation between these parameters. The amount of correlation may also depend on the terrain characteristics. In general, flat terrain will lower the estimability of several errors sources. The usage of geo-location equations for strip adjustment, therefore, always needs to be accompanied by a careful analysis of the correlation between the error parameters to be estimated. In case of high correlation one or more parameters need to be removed from the observation equation or constraints need to be added in order to avoid singularity in the normal equation system. 2.2
Observations
The parameters of the error model need to be estimated based on the observed discrepancies between overlapping strips and between strips and reference objects measured by, e.g., GPS. This task is very similar to the estimation of corresponding image positions by matching techniques. As with image matching one can discern area based and feature based techniques.
In the area based case observation equations will be formulated that minimise the discrepancies between the height data in the strip overlaps (e.g. Burman 2000, 2002). In the feature based case one would first extract features, like distinct points, edges or planar faces, from both strips and use the differences between the locations of those features in both strips as observations. Kilian et al. (1996) first used an area based matching technique on height images to determine corresponding points and then derived observations based on the determined shifts in X-, Y- and Z-direction. Crombaghs et al. (2000) manually select XY-locations in strip overlaps and take the height differences at those locations as observations. In contrast to the Kilian et al., they do not observe planimetric strip offsets.
3.
MATCHING HEIGHT DATA
For stereo matching it is well-known that texture is required in order to obtain a good precision of the estimated disparities. Similarly, when matching height data using an area-based image matching algorithm, height variations are required in order to be able to estimate planimetric offsets. However, there are two restrictions to the kind of height variations that can be used for matching height data: • First, it is well-known that areas that are occluded in one of the overlapping strips should not be used for matching (Kilian et al. 1996, Maas 2000). Usage of such areas will lead to a bias in the estimation of the planimetric offsets towards the occluded area. • Furthermore, the height variations in the patches used for matching need to be smooth. The usage of height jump edges for offset estimation may also lead to biases, similar to biases that are found in edge extraction from binary imagery (Vosselman 2002). Because of this restriction to smooth surfaces, many height variations in urban scenes can not be used for offset estimation with an area-based matching algorithm. Area-based matching algorithms compare the height of a position in one strip to the height in the other strip at the same XY-location. As the points of the overlapping strips usually will not have the same XY-location, interpolation in at least one of the strips is required to determine the height difference. The smoothness condition introduced above is needed to assure that the errors caused by this interpolation in the height data will be small. For this purpose, both the first and second order derivatives need to be continuous. Because the derivatives are reconstructed from the height data, the smoothness condition also implies that the distance between the laser points should be relatively small compared to the slope variations of the terrain features. Area-based matching can e.g. not be Figure 1: Point cloud (top) and generated surface model (bottom) used in the example of figure 1. The of a ditch.
top picture in this figure shows a point cloud of a ditch viewed in longitudinal direction. By interpolation in the triangulated point cloud a surface model is derived that is shown below the point cloud. As the width of the ditch is not much larger than the point distance, the surface model at several places was constructed by interpolation between points on either side of the ditch. Such interpolation errors may lead to biases when using area-based matching algorithms for offset estimations. Still, features like ditches may be required to obtain a sufficient number of offset estimations for a strip adjustment. In particular in flat rural areas small linear features may be the only objects that can be used for the estimation of planimetric shifts between strips. In order to use these kind of features it is, however, necessary to explicitly model their shape. The ditch in figure 1, for example, can be modelled quite well be a cosine function. The position of the ditch can be estimated by fitting this analytical model to the point cloud. By fitting the model to both point clouds separately, the offsets between the strips can be computed from the estimated model locations. This approach does not have the disadvantages of the area based matching algorithm. By using an analytical model of a terrain feature, the required calculations of surface gradients can be inferred from that model. Therefore, no interpolation is required in the point clouds. A further advantage of this approach is that the surface gradients will be noiseless. This leads to a better convergence behaviour and to a more realistic estimation of the accuracy of the determined object parameters. In the next two sections the applicability of analytical models for the purpose of offset estimation between strips will be demonstrated for the cases of gable roofs and ditches.
4.
OFFSET MEASUREMENTS USING GABLE ROOFS
Gable roofs are well-defined shapes that often occur in urban landscapes. They are very suitable for measurements of offsets in the two dimensions of the plane perpendicular to the ridge line. By combining offset estimations of gable roofs with different orientations, three-dimensional shifts between strips can be determined (Maas, 2000). In the next paragraph the mathematical model for fitting gable roofs is formulated. In paragraph 4.2 several results are presented and analysed. 4.1
Observation equations
Let the position of the ridge of a gable roof be described by X cos α + Y sin α = d
(1)
with α as the ridge orientation an d as the distance of the ridge to the origin of the coordinate system. The signed distance u of a point (X, Y) to the ridge is then given by u = X cos α + Y sin α − d
(2)
and the surface height of the gable roof can be defined by H (u ) = H − s u
(3)
with ridge height H and roof slope s. Substitution of (2) into (3) and linearisation results into the observation equation that can be formulated for each laser point (X, Y, Z) E(Z − H (u ) ) = dH − u ds + s
u (X sin α − Y cos α ) dα + s u dd u u
(4)
with incremental changes to the four parameters of the roof model: dH, ds, dα, and dd. 4.2
Experimental results
Ridge lines of gable roofs have been measured with the above observation equations in a data set with a point density of 5-6 points per square meter. Gable roofs in several strip overlaps were selected manually by measuring the ridge line in a height image. These lines are shown in green in the centre picture of figure 2. The bounds of two strips are shown in yellow and red.
Figure 2: Offsets estimated for six gable roofs
The dataset covered a small village in fourteen strips. One of these strips showed a clear planimetric offset in flight direction. For six gable roofs the ridge lines as estimated by fitting the gable roof model to the point clouds are shown on the left and right side of figure 2. The ridge line colours indicate the two different strips. The lines are shown on a background image with a pixel size of one meter. The line patterns clearly show a systematic offset
After correction for the estimated offsets the point clouds show much better alignment, as can be seen in figure 3. All gable roofs were fitted to point clouds consisting of 100-300 points with RMSvalues ranging from 6 to 15 cm. When propagated to the ridge position accuracy, this corresponded to a theoretical accuracy of 0.5 to 2 cm. The offsets between two ridge lines are therefore expected to be estimated with an accuracy of 0.7 to 3 cm. When comparing Figure 3: Point clouds of two gable roofs before (top) and the ridge line offsets of (nearly) parallel after (bottom) alignment. gable roofs, one would expect to find similar values. The measurements for three groups of building orientations in two strip overlaps, however, showed differences between the offsets values of up to 36 cm (Table 1). To a certain extent this is explained by the high correlation between the coordinates of the points on a roof. As a roof is recorded in a small part of a second and the GPS receiver usually measures once or twice per second, the noise in the GPS positioning will be about the same for all points of a roof and will not average out, as assumed in the above error propagation. Still, the differences shown in Table 1 can not be completely attributed to GPS noise. It is assumed that error sources causing non-linear planimetric offsets will have occurred in the analysed strips. Measured offsets in three sets of parallel roofs (m) 0.39 0.71 0.40 0.34 0.37 0.50 0.56 0.63 0.62 0.86 0.79 0.65 0.76 Table 1.
5.
OFFSET MEASUREMENTS USING DITCHES
The same experiments have been conducted using a model for ditches in a rural environment. The next two paragraphs again describe the observation equations and the results of the experiments. 5.1
Observation equations
For the location of the centre line of a ditch, we again take the definition as in equation (1). The height profile perpendicular to the ditch is described by the cosine model as a function of the signed distance u of a point to the centre line 1 2πu H − D 1 + cos 2 W H (u ) = H î
1 u≤ W 2 1 u > W 2
(5)
with terrain height H, ditch depth D and ditch width W. Substituting equation (2) into the above function results into the observation equation: Duπ sin β Dπv sin β Dπ sin β 1 ( β ) α dH − dW − 1 + cos dD + d − dd 2 2 W W W E (Z − H (u ) ) = dH î 2πu with β = and v = − X cos α + Y sin α . W 5.2
1 u ≤ W 2 (6) 1 u > W 2
Experimental results
In our experiments this cosine model appeared to fit quite well to most of the ditches. Based on the RMS-values of the fit the validity of the model has to be verified for each ditch. If a ditch profile is not properly modelled, the partial derivatives in the observation equations will be incorrect. In the case the points are not symmetrically distributed over the ditch, this may lead to a bias in the estimated ditch position.
Figure 4: Offsets estimated for eight ditches.
Similarly as for the gable roofs, several centre lines of ditches in strip overlaps have been measured in a height image (figure 4). The point density in this dataset was about one point per two square meters. Ditch locations were determined in both strips by fitting the cosine model. For eight ditches the fitting results are shown. The pixel size of the images with the estimated ditch positions is 2 m. All ditch models were fitted to point clouds consisting of 150-300 points with RMSvalues ranging from 10 to 23 cm. Only 30 to 60 of those points are actually located in Figure 5: Point clouds of two ditches before (top) and after (bottom) alignment. the ditch. Only those points contribute to the estimation of the ditch location. The remaining points are located on either side of the ditch and only contribute to the estimation of the terrain height. An improvement in the alignment of the point clouds after fitting is visible from figure 5, although it is not as manifest as for the gable roofs. When propagated to the ditch position accuracy, the point noise results in a theoretical location accuracy of 3 to 9 cm. The offsets between two ditches are therefore expected to be estimated with an accuracy of 4 to 13 cm. Considering that the surface of the terrain is expected to be much rougher than the roof faces of the experiments described in the previous section, it is striking that the RMSvalues of the fitting are fairly low. As for the gable roofs, the ditches were grouped by orientation and offsets measurements at parallel ditches are expected to yield comparable results. As can be seen from Table 2, the offsets show consistent values. They differ by a maximum of 26 cm. This is quite acceptable, considering the estimated offset accuracy of 4 to 13 cm and the GPS noise. Measured offsets in two sets of parallel ditches (m) 0.02 0.22 0.65 0.88 0.62 0.82 0.76 Table 2.
6.
CONCLUSIONS
Offset estimations are required as observations for strip adjustment. In particular in flat terrain, relatively small features of the landscape need to be used to obtain these observations. In this paper it was shown that linear features, such as gable roofs and ditches, are suitable for offset estimation if there shape is modelled by an analytical function. Geometrically modelling the feature is essential to avoid interpolation errors that will occur when using image matching techniques. The experiments on several datasets showed that offsets can be measured very accurately with these linear features. In the experiments with gable roofs it was found that offsets within one strip overlap varied much more than the standard deviation of the offset estimation, even when taking into
account the correlation between points due to common GPS observations. It is therefore assumed that non-linear deformations occurred in (at least one of) the strips.
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