Facade Reconstruction of Destroyed Buildings Using Historical Photographs 
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Vi — (si - A 
;?i-y,-/? 
I 
Vfi 
II 
• m - y, -f) 1 
II 
l 
',m-Y, -f) T 
are equal to the unit vectors of the three considered orthogonal directions up to a normalising factor. As it holds 
< Vi, v 2 ) = < V 2 , v 3 ) = < Vj , V! ) = 0 
the effective focal length/can be determined by the principal point P and two of the three vanishing points: 
f = }j- [(- 7 - A)(- 7 - /) + i 7 “ A)( 7 ~./)] ■ ij el,2,3;* * j (2) 
Then the third vanishing point Q k = (£ h Tj k ) can be calculated by: 
4 =X + {f ) % = 7 + “T( 7 “ - v r) k G W (3) 
where p L = (x L ,y L ) is the intersection-point of the perpendicular line from P to the straight line through Q- t and Qj and q is the distance 
between P and p L . 
2.3 Principal Point Transformation 
Using the principal point transformation (PPT) proposed by Kanatani (Kanatani 1988) similarity mapping of any plane perpendicular 
to one of the three directions can be achieved. 
This transformation is performed in such a way that a given image point p=(%,rj) is mapped onto the principal point P. That means 
the ray t through p and the projection centre will be turned into the optical axis. All planes orthogonal to / are parallel to the image 
plane after the transformation. More precisely, they are mapped from their original 3D world co-cordinates by similarity transforma 
tion to the 2D-image plane thereby removing projective and affine distortions. 
Now one of the three vanishing points, e.g. Q={% q , rj q ) is transformed into P. This mapping is a special 2D-2D-transformation, char 
acterised by five parameters £ q , rj q ,X, Y and / only (Kanatani 1988, Brauer-Burchardt and Voss 2001). 
This method yields good results if all the three vanishing points are not very far from the principal point. Otherwise if one of the 
vanishing points is at infinity, the vanishing point calculation becomes erroneous and the calculation of the principal point and the 
focal length becomes numerically unstable. If the conditions are ’’good”, rectification, i.e. similarity mapping of the main planes of 
the object can easily be obtained as demonstrated by Fig. 3. 
Fig. 3: Original image with vanishing point triangle and principal point, rectified facades 
Unfortunately, two main reasons hinder such a straightforward solution. First, there are often only two instead of three perpendicular 
directions represented in the image which means either the principal point or the focal length should be known which cannot be 
otherwise assumed. Second, a lot of historical photographs are taken with horizontal optical axis of the camera, i.e. the vertical van 
ishing point is at infinity which also prevents the calculation of the principal point by the aforesaid method.