THE USE OF NON-METRIC CAMERAS 93 
P. 
  
Fic. 1. Photogrammetric central projection. 
lenses with a uniform scale factor will be referred to as amorphic lenses. Photographs 
produced with amorphic lenses require a single projection distance c for the reconstruction of 
the bundle, while anamorphic lenses require two projection distances c, and c, in planes 
perpendicular to each other. 
Non-metric cameras are usually not equipped with suitable fiducial marks defining the 
image coordinate system. Hence, the direction of both major and minor axis (x and y) of an 
anamorphic lens is, in general, unknown. 
PHOTOGRAMMETRIC CENTRAL PROJECTION 
Let us assume an origin 0 and a plane i (Figure 1), 0 being the origin of a Cartesian spatial 
coordinate system X, Y, Z and 7, with the coordinates x, y, beinga plane parallel to the plane X, 
Y. Each pointp ofthe plane? can be uniquely defined by the ray Op which in turn is defined by 
any point P located on it, as long as P is not identical with 0. The spatial coordinates X, Y, Z of 
point P can therefore be used to determine the location of p in 7 as long as Z is not zero. In 
addition to P, v points P; with X;, Y;, Z; are also located on the ray OP. These points can all be 
derived from P by multiplying X, Y, Z with a suitable scale factor S;, 
Xi 
Yi = S; . Y 
Zi Z 
A rotation of a plane around the origin of the coordinate systems can be described by a 
system of homogeneous linear equations. The coefficients of such a system form an orthogonal 
matrix with the coefficients a;;. If the scale factor S is also introduced, one can write these 
equations as 
x 11 12 13 X 
Ul- 21 22 z is Y 
& Az, 039 433 Z 
In practically all photogrammetric applications it is impossible to arrange for the center of 
the object bundle 0 to be located in the origin of the spatial coordinate system X, Y, Z. Hence, 
the location of 0 with the coordinates X,, Y,, Z, must be introduced into the formulae 
X 
x -K 0% tarte M > 
s-1y-Y]=10,5 017117 
ZZ, 00 $1721]: 
resulting in the pseudo-homogeneous coordinates (X, Y, Z, S). 
Introducing the shift of the origin into the projection equation, one can now re-write the 
transformation equation as