4. Triangulation
In order to obtain a realistic and geometrically correct 3D model of the Bayon tower, 13 images covering the whole horizon were triangulated as follows:
- Determination of coordinates of fictitious fiducial marks
- Image measurement for triangulation
- Bundle adjustment with self-calibration
4.1 Determination of coordinates of fictitious fiducial marks
Unknown and unstable parameters of interior orientation, film unflatness and the lack of fiducial marks are essential features of non-metric small format cameras. Several methods have been developed to eliminate or reduce these shortcomings. To overcome the lack of fiducials, corners of the frame are commonly calculated as intersections of frame edges.
In this paper, the coordinates of fictitious fiducial marks were determined by indirect measurement in several analog images. On an analytical plotter two points on adjacent edges of the image frame were measured and intersecting points were calculated (Figure 4a). As a result individual sets of coordinates in the machine coordinate system were obtained for all images. An affine transformation was performed in order to relate the different coordinate systems to a unique image coordinate system (Figure 4b).
Figure 4: (a) Indirect measurement of fiducial marks on an analytical plotter (points A - D for the
fiducial mark 1), (b) Image coordinate system with the principal point located by intersection of diagonals.
4.2 Image measurement for triangulation
The image measurements for triangulation were performed on an analytical plotter using the approximate interior orientation (camera constant 35 mm, no lens distortion, zero coordinates of the principal point and estimated coordinates of fiducial marks). The optimal spatial distribution of orientation points was not possible due to incomplete frame fill. Altogether about 170 points were measured in 13 images around the tower. Each object point is imaged in two to six images.
Figure 5: Cartesian object coordinate system and three control points with coordinates given in meters
For the absolute orientation of the whole block two full control points and one depth control point were used. Control points 1 and 2 are defined as marks on a vertical scale-bar. Depth control point 3 is situated on a small decorated stone to the left of the southern tower side. These three control points define the right-handed Cartesian object coordinate system with z-axis going towards the southern projection centers (Figure 4). Since control points are only visible from the southern tower side, the absolute orientation on the analytical plotter was performed in two models only.
4.3 Bundle adjustment with self-calibration
The bundle adjustment of the whole image sequence was performed in a stepwise mode. For that purpose the procedure was started with the first absolutely oriented images number 1, 2 and 3. Consequently additional images were added with manually obtained approximations for exterior orientation elements and new object points. This stepwise mode was necessary because, with the available coarse first approximations, the complete block did not converge simultaneously.
After the establishment of a stable adjusted image block, ten additional parameters (Brown, 1971) were used to model systematic errors: focal length correction, principal point coordinate offsets, five parameters modelling radial and decentering lens distortion and two parameters for a differential scale factor and shear (Beyer, 1992). The radial distortion amounted to 810 microns at an image radius of 20 mm.
5. Image Matching
For the extraction of the 3D surface geometry the images were digitized with a resolution of 1200 dpi (pixelsize 21 microns). Image matching was performed with the commercial software package MATCH-T. However, MATCH-T can only generate the 2.5D model from one stereopair. This is necessarily incomplete for close-range applications because occluded parts of the scene cannot be processed. Therefore four image pairs taken from south, east, north and west (images number 2/3, 5/6, 8/9 and 11/12 in Figure 3) were selected and matching was performed in each model separately. This procedure required a transformation of orientation parameters from the Cartesian object coordinate system to local systems for each geographic direction (so that - like in an aerial case - the z-axis in each model is directed towards the projection centers). As a result of fully automatic matching, four separate surface models with a grid width of 3 cm were constructed. The visualization of matched points has shown that the image matching procedure in MATCH-T works reasonably well in this case. Within the individual models only about 0.2% of the matched points had to be edited manually (Figure 6). In the next step, the four separate point clouds were transformed back to a joint object coordinate system and merged together. In the complete 3D point cloud containing 46 850 points overlapping areas occurred and gaps and outliers between the adjacent models showed up (Figure 7). To achieve a good visualization result, these errors were eliminated automatically in a second editing step (Chapter 6).
Figure 6: Matching result within one model (a detail of the southern face profile)
Figure 7: Mathching result at the connection of two separately processed models withmarked
area needing additional editing
