ACCURACY OF CLOSE-RANGE ANALYTICAL RESTITUTIONS 365 
KararalAbdel Aziz trials. The characteristics of the five cameras (and their prices) are 
listed below: 
Camera and lens 
Kodak Instamatic 154 
Crown Graphic-Graphex f/4.7 
Honeywell Pentax 
Super Takumar f/1.4 
Hasselblad 500-Planar f/2.8 
Hasselblad MK70-Biogon f/5.6 
Focal Image 
length format Approx. 
(mm) (mm) Price ($) 
43 12 x 12 15 
135 120 x 100 300 
50 36 x 24 500 
80 55.x 55 550 
60 55 x 55 4500 
The geometry of the picture taking (Figure 15) was the same for all the cameras (targets in 
two planes, determined with an accuracy better than 0.1 mm). For each camera, five photo- 
graphs (camera hand-held) were taken at both stations (ten plates for each camera). 
At first the authors studied the means to reduce film deformation and lens distortion by 
using different mathematical models (accounting for linear deformations and radial distor- 
tion with polynomials of different degrees, and eventually for asymetrical lens distortion 
due to non-linear film deformation). 
The conclusion is very neat. À sophisticated model is not required, a model with linear 
film deformations and radial distortion represented by Ar =k 3 (r = radial distance), con- 
sidering the number of supplementary unknowns to be introduced in the computations, 
being the most efficient and economical. Others refinements seem to be unnecessary. 
With this model, the average RMS plate residuals o, estimated for each camera from ten 
resections in space are given in Table 5. 
Furthermore, in Table 6, the estimated values for the mean RMS spatial residuals rXYZ, 
rX, rY, and rZ (in micrometers referred to the image plane) in the planes 1 and 2 are given. 
Comparison of experimental and predicted accuracies. We have two predictors at our 
disposal. 
(1) The predictor obtained from simulations, and which concerns the RMS residuals 
(Figures 11 and 12). 
For r = 0.73 (case of the envisaged geometry plane 1), we read on the prediction curves 
that, for o, = 2 um, we can anticipate rXYZ = 5.6 um, rX = 1.8 um, rY = 4.5 um, and rZ = 
2.5 um. 
For any other value o,, we multiply these quantities by 0,/2, so that 
rXYZ = 2.80 0,; rX = 0.9 0,; rY = 2.25 0°; rZ 2 1250, (22) 
(2) The Karara/Abdel Aziz predictor which concerns the central point accuracy: 
oXYZ = VoX2 + oY? + 02” 
guia ld tan a tan 21.9. 
oX = uml tung re uc Ur x OX; 
B 
oZ = V2 1 — tan (a — ¢) tan ÿ 
in which ¢ = 15° and tan a = 5 z 0.365. 
l/cos 
Jos 
Then we obtain for the envisaged geometry (plane 1) 
oXYZ = 2.44 0,; OX = 0.79 09; OY = 2.18 09; 9Z = 0.75 0, (23) 
3 
d 0m -— 
20cm 
  
L.!. Plane 2 
| (targets) 
| 
1 Plane 1 
(17 targets) 
Fic. 15. Geometry for the Karara/Abd el Aziz 
studies. 
TABLE 5. EVALUATION OF ERROR MEASUREMENT 
FROM RESECTIONS IN SPACE (COMPARE WITH 
METRIC CAMERAS: THE RMS PLATE RESIDUAL WAS 
Fouwp TO BE $, — 2.8 um). 
  
  
Camera RMS plate residual (um) 
To 
Kodak 15.2 
Crowngraphic 11.5 
Honeywell Pentax 3.9 
Hasselbled 500 C 6.1 
Hasselbled MK70 5.0