## Measurements from Stereomodels

Beyond the simple viewing and interpreting of stereopairs, stereomodels also enable various kinds of 3D measurements using photogrammetric techniques. The 3D ground

FIGURE 3-11 Two stereopairs with equal image scale and air base B taken over the same terrain. Focal length and flying height of stereopair B (normalangle lens) are twice those of stereopair A (wide-angle lens), resulting in a halved base-height ratio. Note how the parallactic angles g and the stereo-parallax—the difference between and d2—decrease from stereopair A to stereopair B.

FIGURE 3-11 Two stereopairs with equal image scale and air base B taken over the same terrain. Focal length and flying height of stereopair B (normalangle lens) are twice those of stereopair A (wide-angle lens), resulting in a halved base-height ratio. Note how the parallactic angles g and the stereo-parallax—the difference between and d2—decrease from stereopair A to stereopair B.

coordinate of any given object point can be determined if the corresponding 2D image coordinates in a stereopair and the position of the camera within the ground coordinate system are known. This is illustrated in Figure 3-12, where the position of an object point A in the landscape is reconstructed by tracing the rays from the homologous image points ai and a2 back through the lens. With a single image (the left photo in the figure), no unique solution can be found for the position of A along the reconstructed ray. By adding a second (stereo) image on which A also appears, a second ray intersecting the first can be reconstructed and the position of A can be determined. This method is called a space-forward intersection; it is based on the formulation of collinearity equations describing the straight-line relationship between object point, corresponding image point and exposure station.

Deviating from the schematic situation in Figure 3-12, the differences between the focal length and the flying height for aerial surveys are in reality much greater. For SFAP surveys, the focal length is usually well below 10 cm and the flying height somewhere between 20 m and 500 m; for conventional aerial surveys, the focal length is normally between 9 and 16 cm and the flying height somewhere between 2000 m and 5000 m. Therefore, the accuracy of the reconstructed 3D coordinate of A is highly dependent on the precision with which aj, a.2, and the focal lengths are measured. This is why photogrammetry is a science with many decimal places! To make the extrapolation from the (known) interior of the camera into the (unknown) space beyond as exact as possible, both the values of the interior orientation and the exterior orientation (see above) need to be known precisely.

FIGURE 3-12 Reconstruction of the 3D ground coordinates of an object point in a stereomodel by space-forward intersection. From the image coordinates x', y' of the homologous points a1 and a2, the rays of light are traced back through the exposure center L to the object point A. The interior orientation—the geometric relation of L to the image plane—and the exterior orientation—the position X, Y Z of L and the rotations w, f, k of the image plane with respect to the ground coordinate system—must be known for this reconstruction.

FIGURE 3-12 Reconstruction of the 3D ground coordinates of an object point in a stereomodel by space-forward intersection. From the image coordinates x', y' of the homologous points a1 and a2, the rays of light are traced back through the exposure center L to the object point A. The interior orientation—the geometric relation of L to the image plane—and the exterior orientation—the position X, Y Z of L and the rotations w, f, k of the image plane with respect to the ground coordinate system—must be known for this reconstruction.

As outlined in the previous section, the required exact exterior orientation for SFAP images is not known a priori. However, it can be determined using the same reconstruction method backwards in a space resection. With the known 3D coordinates of three ground control points and their corresponding 2D image coordinates, the position X, Y, Z of the exposure center at the intersection of the three rays and the three rotations of the camera u, f, k relative to the ground coordinate system can be calculated. Afterwards, these exterior orientation parameters can be used in the space-forward intersection in order to calculate any other object point coordinate in the area covered by the stereomodel.

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