THE USE OF NON-METRIC CAMERAS 95 
can then, after division of all elements by b,,.S, i.e., cj; = s. be used to form the projection 
equations 34 
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which no longer contain the scale factor S as unknown. 
Linearization of the two projection equations leads to normal equations permitting the 
determination of the remaining eleven unknowns. The solution of these equations, however, 
will only yield acceptable results if a sufficient number of object points with suitable distribu- 
tion is available. It is always advisable with the use of non-metric cameras, first to derive the 
correlation matrix, and then to decide on which of the eleven possible parameters should 
actually be used. 
If an amorphic lens was used, for example, « and « cannot be distinguished and c, - c,. 
Often, a strong correlation may also exist between x, and $ (ora13) and/or between y, and w (or 
423). 
Once the object bundle has been re-established and oriented, it is possible to search along 
the object ray for that point which corresponds to the object point, by enforcing a minimum 
distance constraint between ray and object point. Jahn! replaced the rotation a by a shear 
factor e. This has not been introduced here because there is no element in the optical 
projection suggesting the use of such a factor. 
DETERMINATION OF THE LENS DISTORTION 
If sufficient redundant object points are available, the inclusion of parameters into the 
projection equation for the lens distortion could be considered. An amorphic lens will have a 
certain design distortion which is rotationally symmetrical. The assembled lens will have 
additional lens distortion caused by minor imperfections in the production of each lens 
element as well as in the centering of all the elements. This latter distortion, which is 
commonly referred to as decentering distortion, has both radial and tangential components. 
Itis assumed for the following presentation that the lens distortion of an anamorphic lens is 
proportional to the coordinates themselves. 
The rotationally symmetrical lens distortion for an anamorphic lens can then be expressed 
as 
di,-—x(du? + d(r22 "dir v...) 
dg, -9g(du?. dir? t dí?) t...) 
Il 
with r2 = c,2 X? + c,? ÿ? and F and ÿ referenced to the defined center of the photograph. The 
decentering distortion can be expressed as 
dx, — lex 3c? 72 + CUS) +e, 26, EC, yi + 0.72 +e (r2)2 +, . a 
dy, = lex . Qe C) + eq - (c,2x2 + 3c,’ 2)}{1 + esr? T e,(r2)? + } 
with 72, x, and y as before. 
An attempt to calibrate non-metric cameras on camera calibration equipment designed for 
mapping cameras could also be considered. There are, however, several factors which speak 
against such a calibration: 
e Non-metric cameras often lack a well defined image frame permitting the definition of the 
principal point of autocollimation;