Abstract

• This document consists of seven pictures describing the procedure of stereography. The procedure is shown in steps of
• definition of the object,
• orientation of the stereo camera,
• calculation of the rotation matrix between the object and camera coordinate systems,
• rotation of the object coordinates to a system parallel to the camera coordinate system,
• shifting the object coordinate system into the camera coordinate system,
• central projection of the 3-D object to the 2-D image plane, and
• presentation of the stereo images.
• The procedure is a known procedure for every one of us but aims to show this in a descriptive way. It is followed by a numerical example, which may be transacted by an application at <http://ns.foto.hut.fi/publications/paperit/hhaggren/bildteknik_2002/ktorlegard_stereo_photography.zip>.

1 Definition of the object

• The object is defined as a cube having a dimension of 100. The cube locates coincident with the positive octant of the 3-D object coordinate system XYZ.

2 Orientation of the stereo camera

• The stereo camera O1O2 performs the photography corresponding to the normal case of stereo photogrammetry. The camera constant c and the base B define the camera. The camera coordinate system xyz is right-handed having its origin in the projection centre O1 of the left camera, the positive z-axis pointing against the viewing direction, and the positive x-axis coinciding with the base vector O1O2.
• The orientation of the stereo camera is defined by locating the projection centre O1 relative to the object and by aiming the camera into a target point T. The base is supposed to be kept horizontal. This implicitly defines the rotation of the camera through the vertical projection of the aiming vector O1T onto XY-plane. The option of exact vertical photography is here thus excluded, since the rotation would remain ambiguous.
• By the above definition the three points T, O1 and O2 are defined both in the object and camera coordinate systems.

3 Calculation of the rotation matrix

• The elements of the rotation matrix between the object and camera coordinate systems are calculated as the orthogonal unit vectors ijk of O1O2, TO1 x O1O2, and TO1. The idea is adopted from (Kraus, 1997, p. 55-58).

4 Rotation of the object coordinates to a system parallel to the camera coordinate system

• The object coordinates XYZ are rotated to a system (x)(y)(z), which is parallel to the camera coordinate system xyz.

5 Shift of the object coordinate system into the camera coordinate system

• The rotated object coordinates (x)(y)(z) are then shifted to the camera coordinate system xyz.

6 Central projection of the 3-D object to 2-D image plane

• The central projection of the cube onto the image plane is calculated by scaling. As the image planes p' and p'' are defined as z = -c, the corresponding image scales of the object points P will be

• The x''-coordinates of the image points p'' are shifted by the parallax px,

7 Presentation of the stereo images

• Finally, the images are shown here as a stereogram. The image format is outlined and cut to any proper format, here having the size of ± 30 vertically and ± 40 horizontally. The arrangement of stereography is considered normal, and corresponds to the viewing with parallel eyes, if the order is "left on left" and "right on right". In case this is changed to the opposite, the arrangement corresponds to viewing with crossed eyes.

 

Conclusion

• The procedure described here is used for educating stereoscopy during the basic course of photogrammetry. The aim is to provide students with a tool to exercise the behaviour of changes in imaging geometry, i.e. by changing the values of the parameters of camera orientation: c, B, O and T. The application includes also the presentation of stereo images as anaglyph projection (Haggrén, 2002).

References

Haggrén, Henrik, 2002. MS Excel-application for stereo photography, available at  <http://ns.foto.hut.fi/publications/paperit/hhaggren/bildteknik_2002/ktorlegard_stereo_photography.zip>.

Kraus, Karl, 1997. Photogrammetry, Volume 2, Advanced Methods and Applications, Dümmler Verlag, Bonn, 1997, 466 p.