After the establishment of geodetic network by means of precise classical terrestrial method and extraction of target centers at each scanner station, the transformation parameters can be estimated to be able to position the point clouds of objects under inspection within the reference frame. Pro-vided the high accuracy of extracted target centers (from the high density point clouds of individual targets) can be assured, this point based positioning approach is to be preferred to other positioning methods (e.g., feature based or iterative closest point, see Vosselman and Mass, 2010) for it offers the possibility to derive the exact and direct correspondences between the extracted target centers and network points (Figure 4). Not using the targets in many deformation monitoring applications,
Figure 4: TLS measurement positioning diagram. Targets represent the link between point clouds and the reference frame. Control points can be used for displacement comparison or may support the determination of representative points, described in section 2.8.2.
the derivation of correspondences from multi-temporal scan data, whether point or feature based, may be hard to derive with sufficient accuracy as well as decide which objects (or parts of objects) in the scene have remained unaffected if these are to be treated as a reference. Since precise posi-tioning of point clouds within the reference frame is one of very important steps especially when approaching the deformation evaluation in the millimeter domain, the use of targets may become unavoidable.
Another popular method, though inferior to target based positioning is the so-called direct georefe-rencingdescribed in detail by Reshetyuk (2009). The method is based on stationing the scanner over a known point, levelling and orientating it to at least one other point with known coordinates in the reference frame. Conceptually this is how measurements are positioned in the traditional surveying.
However, for the precise positioning of point clouds direct georeferencing is less appropriate be-cause it introduces additional instrumental errors, such as centering and levelling errors which may be much larger compared to precise tacheometers even with scanners that use inclination sensors which are typically not accurate enough.
Focusing now on the target based approach, two sets of points denoted herein by Xi, Yi ∈ <3 and
referring to the identical physical entity but different coordinate system can be connected by esti-mating the spatial 7-parametric Helmert transformation in a LSA, minimizing the sum of squares of coordinate differences:
where R is the rotation matrix, t the translation vector and s the scale factor. These unknowns can be estimated on the basis of three or more point pairs. If this is done in the scanner-to-scanner station relation, the process is called relative orientation. In case one set of points has its position in the pre-defined reference frame it is called absolute or exterior orientation.
Translation
To minimize the sum of squares of errors in equation 3 it is useful to first refer all points Xi and Yi
to their centroids Xc= 1nPn
i=1Xi and Yc= n1Pn i=1Yi:
Xi0 = Xi− Xc Yi0 = Yi− Yc (4)
Now equation 3 can be rewritten using equation 4 to produce:
n second term is zero since the points are referred to their centroids (note that Pn
i=1Xi0 = 0 and Pn
i=1Yi0 = 0). Moreover, the first term does not depend on t0and the last cannot be negative. Hence, the sum in equation 5 is minimized when t0 = 0, so the estimated translation vector represents only the difference of the first and the scaled and rotated second centroid:
t = Yc− sR (Xc) (6)
According to Horn (1987), estimating the translation by using all points is to be preferred to one where only one or few selected points are used provided that all are comparable in precision and accuracy. Before the translation vector could be obtained, the scale and the rotation have to be esti-mated.
Estimating the scale
After the introduction of centroids and finding the translation vector, the total error to be minimized following equation 5 becomes: Expanding equation 7 and taking into account the fact that rotation is a linear transformation pre-serving lengths, i.e., in s in equation 9 leads to:
With respect to the scale, this is obviously minimized when the term in the brackets is zero, that is s = SD
In general this asymmetrical scale factor from equation 11 is direction dependent. If the transfor-mation is done in the inverse direction, that is Xi = ˜s ˜R (Yi) + ˜t, it is not likely to expect that ˜s = 1s,
Again, referring to Horn (1987), one of the two scale factors from equations 11 and 12 may be more appropriate when the coordinates in one of the two systems are known with much greater precision than those in the other. This may be taken under consideration in the case of the absolute orientation with the well-defined and very precise classical terrestrial coordinate estimation. On the other hand, if the quality of coordinate estimation in both point sets is similar, using symmetry in scale is more reasonable (e.g., relative orientation). In this case equation 7 has to be slightly modified:
n
Finally, a similar rearrangement of equation 13 to that of equation 7 shows that the minimization with respect to the scale s (only this time symmetric) results in:
s =
The advantage of this symmetrical case is that the scale can be determined without the need to es-timate the rotation beforehand. However, in each case the estimation of rotation is independent of the choice of scale and the remaining error is minimized when D is as large as possible.
Finding the rotation using unit quaternions
Compared to the more familiar matrices, the representation of rotations with Hamilton’s unit quater-nions has a number of advantages. For example, it is much simpler to enforce the constraint that a quaternion has a unit magnitude than it is to ensure that a rotation matrix is orthonormal (Horn, 1987). Furthermore, quaternions are simple to compose, they are numerically more stable and avoid the problem of gimbal lock. A unit quaternion representing a rotation by angle θ around an axis ~u = (ux, uy, uz)T, where k~uk = 1 is:
In the quaternion notation, a point in space can be represented by a pure imaginary quaternion
˙r = 0 + ~r and its position after rotation using equation 15 in the form:
˙r0 = ˙q ˙r ˙q∗ =
where ˙q∗ is the conjugate of a quaternion obtained by negating the imaginary part in ˙q. Expanding equation 16 leads to the Rodrigues’ rotation formula which is exactly what a rotation using angle-axis representation is.
Returning back to the problem of estimating the rotation, knowing that D has to be as large as po-ssible in order to achieve the final minimization in equations 7 or 13, a unit quaternion maximizing:
n
i. Based on the laws of quaternion arithmetic, Horn (1987) has proved that solving equation 17 is about finding the eigenvector corresponding to the most positive eigenvalue of a symmetrical 4 x
4 matrix:
i and so on are sums of products of coordinate components in both systems that have been reduced to their centroids beforehand. The sought-after eigenvector is the unit quaternion ˙q = q0+ qx + qy + qz representing the estimated rotation.
Once this quaternion is determined, the computation of the corresponding 3 x 3 rotation matrix is straightforward:
For the above presented approach of estimating the transformation no approximate values are needed.
The unknowns are determined in a one-step procedure incorporating all the points Xi and Yi and providing the best rigid body transformation between two coordinate systems given coordinates of a set of points are not collinear. The robustness of this method is an important advantage compared to the one where rotation is estimated using orthonormal matrices (Horn, 1988). If needed weights can also be introduced in the process to account for the inhomogeneous precision of Xiand Yi(see Horn, 1988).
After estimating the transformation parameters these have to be applied to the point clouds that are the result of scanning the object under inspection. Besides the error vector (equation 3) or the a posteriori standard deviation σAO(σRO)1providing one measure of the quality of transformation the overlapping areas should be examined to see if points acquired at different scanner stations coincide to a required degree. The transformed point locations will be influenced not only by the quality of coordinate estimation coming from both classical terrestrial and scanner measurements but also on the configuration of targets at each scanner station. Moreover, target and object distance from the scanner should be comparable otherwise small discrepancies between point sets Xi and Yi af-ter applying the transformation may produce large point cloud offsets at the object side. Prior to taking measurements in the field, the study of effects of a particular network configuration on the transformed point locations is possible, for example by using simulations during which each point is assigned a randomly generated noise. The results of such simulations may provide information on sensitivity of the transformed point locations with respect to all influencing factors.
1AO – absolute orientation, RO – relative orientation