106 
photographs taken with metric cameras can 
be used for precision measurements or for 
restitution in analog plotters without addi- 
tional control of the elements of inner and 
relative orientation. The measuring preci- 
sion should be limited only by unavoidable 
errors of the photographic material. Further- 
more the lens distortion should be small 
enough so that it can be neglected for plot- 
ting on analog restitution instruments. 
TOLERANCES FOR THE PRINCIPAL POINT AND 
THE PRINCIPAL DISTANCE 
The tolerances for the inner orientation 
depend on a number of factors such as the 
opening angle of the camera, the size of the 
object, or the type of photographic material to 
be used (roll film, plane film, or glass plates). 
Therefore it is not possible to give figures 
which are applicable under all circum- 
stances. These tolerances will also help to 
judge under which conditions non-metric 
cameras can be used for precision measure- 
ments without any loss of accuracy. 
Such an accuracy evaluation can be per- 
formed by a simulated camera calibration. If 
one assumes that the object to be measured 
includes control points, then the principal 
distance, the coordinates of the principal 
point, and eventually the distortion can be 
computed. The precision of the calculated 
parameters is obtained from the system of 
inverted normal equations. In case the con- 
trol points are unfavorably distributed, the 
unknowns are fairly inaccurate. This does 
not effect the precision of the points to be 
measured within the area defined by the 
control points, provided that the camera cali- 
bration is restricted to the principal distance 
and the coordinates of the principal point. 
The simulated camera calibration has been 
based on the projection equations of 
Hallert? extended for the elements of the 
inner orientation. 
=x. +A’, +c 
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1976 
ordinates of the projection center. The rota- 
tion elements are the swing x, the tip ¢, and 
the tilt w. The lens distortion is taken into 
account by Ar', and Ar',. 
The form of the projection equations and 
the sequence of the axes of the rotation 
elements is immaterial; it is of importance 
only that the physical imaging process is 
approximated sufficiently. The inclusion of 
the radial distortion or the principal point of 
symmetry as unknowns would degrade the 
precision of the estimated parameters con- 
siderably. The increase of the variances is 
then combined with a strong correlation 
between the unknowns. The high correlation 
between the orientation elements means that 
an error of one of these parameters can be 
compensated by a proper choice of the other 
variables. This compensation is possible 
only if at least one of two highly correlated 
values can be chosen freely. 
For the study of the reproducibility of the 
elements of inner orientation it is assumed 
that these parameters are considered con- 
stant and only the elements of the exterior 
orientation are variable. Therefore the accu- 
racy estimations should be performed for 
each orientation element separately. Be- 
cause the correlation between the principal 
point and the principal distance is small, this 
precaution is not necessary for these three 
parameters, but it would not be correct to 
introduce more parameters of the inner ori- 
entation as stochastic variables. 
The size of an object or the size of the test 
field for a calibration is physically limited 
by the characteristics of the camera. Its lat- 
eral extension is restricted by the opening 
angle of the camera, and the depth extension 
in general by the depth of focus. Especially 
for short imaging distances the depth of focus 
can be very narrow. Figure 1 gives a survey of 
the depth of focus for cameras with different 
focal lengths. It has been assumed for the 
(x—x,) (cos¢ cosk — sing sinw sink) —(y—y,) cose sink -(H —h,) (sin cosk * cosó sino sinx) 
  
(x—x,) sin$ coso — (y-y,) sino -(H—h,) cos$ coso 
yy, tA. dc 
(1) 
(x—x,) (cos¢ sink + sing sin w cosk)+(y—y,) cosw cosk+(H—h,) (sing sink —cos¢ sinw cosk) 
  
(x—x,) sind cosw — (y— 
In the formulae x' and y' are the picture co- 
ordinates (measured in the comparator); x, y, 
and H the corresponding coordinates of the 
points in the test field; c the principal dis- 
tance; x', and y', the coordinates of the 
principal point; and x, y,, and h, the co- 
yo) sino — (H —h,) coso coso 
(2) 
computation that the circle of confusion 
should not be larger than 30um and the 
smallest admittable aperture stop has been 
fixed at 1: 16. These limitations might appear 
rather narrow but less severe restrictions 
would cause a serious degradation of the