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Bräuer-Burchardt, Voss 
3. PLANE RECTIFICATION TECHNIQUES 
3.1 Rectification with Two Vanishing Points 
Having only two vanishing points, the principal point P should be known to use the PPT for image rectification. Then/and the third 
vanishing point can be calculated according to (2) and (3). If only / is known two solutions for P are obtained one of which holds 
good. The false solution can normally be recognised and removed. Unfortunately, the knowledge of the principal point or the focal 
length cannot be assumed using historical photographs. It is usually unknown, whether the photograph represents the whole negative 
film which would imply the principal point P to be situated near image centre C. Hence the identification of P=C may lead to con 
siderable errors. 
Supposing perpendicularity, rectification of the plane of interest can be achieved by transforming the finite vanishing points into 
infinity. Thus parallel lines of the object become parallel in the image and perpendicularity may be used to rectify the image. But the 
scaling factor between the vertical and horizontal image co-ordinate axis is still unknown. This factor can be determined if one of the 
following conditions (Liebowitz 1998) is satisfied 
• a length ratio of two nonparallel directions is known 
• two length ratios of nonparallel directions are equal 
• an angle (not the right angle between x- and y-axis) is known 
• two angles formed by nonparallel lines are equal 
• a curve is known to represent a circle arc (Hemmleb 1999) 
Figures 4 and 5 show two examples for facade rectification. If no of the discussed conditions is satisfied the remaining scale factor 
cannot be determined. 
Fig. 4: Original image and rectified plane using a circle arc 
Fig. 5: Image rectified by one known window aspect ratio 
3.2 Rectification with Horizontal Camera 
A typical situation is the horizontal adjustment of the optical axis of the camera parallel to the ground plane. All vertical lines of the 
object are also parallel in the image and the vertical vanishing point is near infinity; see the view in Fig.4 of the totally destroyed 
Municipal-theatre of Athens (Greece) which was already analysed for reconstruction (Karras 1999). Although a determination of the 
principal point and focal length from the vanishing point triangle cannot be robust, Karras tried to calculate the vertical vanishing 
point. But without using additional information it would be difficult to estimate the accuracy of the obtained reconstruction result. 
Li (Li 1997) suggested in such cases the assumption of the x-co-ordinate of the principal point to be central. But this might have two 
drawbacks. First, it is usually unknown whether the photograph is only a section of a larger image thereby the difference between the 
image centre and the actual principal point could be considerable. Second, the camera construction might imply a principal point 
shift which is not unusual for historical photographs. 
Supposing that the vanishing points of two perpendicular directions (normal vectors to the facade planes) are detectable, both planes 
can be rectified with the method described in the previous section if one of the conditions holds. If not, we may use symmetries in 
both facade planes. 
Below we give a method to determine the principal point of such images. Assuming that the vertical vanishing point Q\ to be at 
infinity and the other two vanishing points Q 2 and Q 3 of the normal directions of the facade planes can be determined. For this 
method it is essential that the length ratio of two pairs of nonparallel edges, each pair on one of the two facade planes, be equal. Let 
the length ratio defined by the four (or three) coplanar 3D-points be P', P', P', P 4 * and P‘, P', P', P g *, respectively, having their 
image points P\ to P 8 (see Fig. 6). It should hold