Keywords: Control Surface, Laser Range Finder, INSAR
Abstract
With the increased availability of surface-related sensors, the collection of surface information
becomes easier and more straightforward than ever before. Thus, the integration of surface
information into the photogrammetric workflow, the task which has been long time interesting
as well as challenging, is gaining focus again within the photogrammetric community. In this
paper, the author proposes a model in which the surface information is integrated into the aerial
triangulation workflow by hypothesizing plane observations in the object space, the estimated
object points via photo measurements (or matching) together with the adjusted surface points
would provide a better point group describing the surface. Apart from releasing aerial
triangulation from the necessity of identifying control points in the object space, the proposed
algorithms require no special structure of surface points and involve no interpolation process.
The proposed system is proven workable having data collection in the photogrammetric
laboratory.
1. Introduction
More and more surface related sensors, such as airborne laser range finder, INSAR
(INterferometric Synthetic Aperture Radar) with tightly integrated on-board GPS/INS system
became commercially available during the last decade. By that, the booming research and
applications mainly in generating elevation information for the area of interest and scene
analyses, especially for the buildings in residential areas have been, among others, promising an
era of sensors in which the collection of surface information becomes easier and more
straightforward than ever before. Besides, due to the state-of-the-art of the sensor integration
technique [Schwarz, 1995], the accuracy of the analyzed surfaces via airborne laser scanning
system proves competitive with the scenario where a well-controlled data set and careful
measurements by the operator are the necessities for the accuracy typical in traditional
photogrammetric production line. Thus, how would photogrammetrists consider this newly
available technique and its seemingly favorable data set apart from the aforementioned interests?
One of the attractive thoughts follows: Can aerial triangulation by taking photo measurements
benefit from this sensor dominant era and do a better job for the task of the surface
reconstruction and how? These questions led the author into this study.
This very same idea and attempt had been carried out a decade ago at the Technical University
of Munich, Germany, conducted by Ebner [Ebner/Strunz,1988][ Ebner et al.,
1991][ Ebner/Ohlhof, 1994] even when the laser range data did not yet come to applications.
Their algorithms focused mainly on the satisfaction of accuracy for the middle and small scale
photogrammetry by minimizing the differences between the heights of the object-to-be-solved
and the interpolated heights (bilinear interpolation) via the surrounding surface points, DEMs
(Digital Elevation Models) in their case, as constraints. In this study, without interpolation on
the surface points and requiring no special structure of the surface points, the author exploits
different algorithms by hypothesizing planes (also called “control surface” in this paper) and
assessing the uncertainties via checking the fitting planes with local surface points; the
minimization takes distances along the surface normal as the target function when formulating
the surface constraint.
The rest of this paper consists of the following: Section two introduces the surface constraint
with formulating its mathematical as well as stochastic model. The integration of surface
constraint into aerial triangulation is explained in section three. Section four demonstrates the
experimental test in the photogrammetric laboratory together with the accuracy (root mean
square error) report and the analyses. Section five concludes this study by giving some
observations of this research from this author’s perspective.
2. Model Formation of the Surface Constraint
2.1 The First– Order Surface (Plane) Equation:
Mathematically, three points, let’s say and
P
1(X
1, Y
1, Z
1),
P
2(X
2, Y
2, Z
2) and
P
3(X
3, Y
3, Z
3)define a plane.
The plane equation with c b and a, being the function of coordinate components of three
points can be seen as equation (2.1).
aX + bY + cZ + d =0 (2.1)
To calculate the distance along the surface normal between a point
P
i(X
i, Y
i, Z
i) and the plane
aX + bX +cZ + d =0, one could use equation (2.2) shown as below,
2.2 The Surface Constraint:
Analytically, if a point is known to lie on the known plane, equation (2.1) would be
automatically realized. A direct result of that is that the distance calculated by using equation
(2.2) would end up with zero.
Realistically, due to the measuring errors of the employed instrument, the physical truth of the
plane could hardly be found by the collected data. Thus if the points are judged to lie on the
known plane, one would compromise the observations by minimizing the distances along the
normal, shown as equation (2.3), such that the imperfection of the measurements and the overall
registration of the data set could be taken into account in an optimal way towards the solution.
By that, the author proposes an algorithm in which the plane is hypothesized in the object space
and confirmed by the surface points, thus forming the surface constraint into the adjustment for
simultaneously determining the object points and the exterior orientation parameters via aerial
triangulation procedures.
