a zero, and a long bar represented a 1. The entire 
control field covered an area of 2.25 m(H) x 2.75 
m(W) x 2.41 m(D). The locations and sizes of the 
targets were designed so as to provide a minimum of 
12 targets of sufficient size and dispersion to 
facilitate the calibration of zoom lenses of the 
entire focal range of 12.5 mm to 75 mm. The three- 
dimensional coordinates of the center of each target 
were determined by triangulation. The average 
estimated standard errors of the target coordinates 
were computed to be: O, - + 0.3 mm, Oy, = + 0.8 mm, 
and 6, - 0.4 mm. The X- and Z- axes lied in a 
vertical plane, with the X-axis being horizontal and 
the Z-axis being in the vertical direction. The Y- 
axis was horizontal and approximately along the 
depth of the target field. 
4. DISTORTION MODEL 
After extensive experimental tests, the following 
model was found to provide excellent representation 
of the distortion characteristics of a vision System 
at a given focal length setting( Wiley and Wong, 
1990; Wong et al, 1991): 
dx s lxr? « [p (r? « 227) * 2p, Xy] 
dx = Lyr? + [2p, xy +p, lr? + 2y°)] 
x = (X = 2) (1 + &k) 
«i 
Il 
< 
l 
v 
r?-xX4y? 
where x and y are image coordinates; and y, are 
image coordinates of the principal point; k is a 
scale factor for the x-coordinates; I4, is the first 
term of symmetric radial distortion; and p, and p; 
are the first two terms of decentering lens 
distortions. 
5. TARGETING ALGORITHM 
An algorithm was developed to automatically identify 
and locate the center of each target in an image. 
It consisted of the following steps: 
1. find the approximate locations and identification 
numbers of all the targets in an image using the 
method reported in Wong et al (1988); 
2. perform sub-pixel edge detection along the 
boundary of each target using local thresholds; 
and 
3. compute the image coordinates of the center of 
each target by least-squares fitting with an 
elliptical template. 
The estimated standard error of the computed 
coordinates of the target centers typically ranged 
between + 0.005 and + 0.02 pixel. There was no 
significant difference in the targeting accuracy of 
the two types of cameras used, in spite of the 
slightly higher resolution of the Pulnix cameras. 
This was largely attributed to the size of the 
targets, which typically had a diameter of more than 
10 pixels in the calibration images. 
6. CALIBRATION IMAGES 
Images of the control field were obtained using the 
following six different combinations of camera and 
zoom lenses: 
   
Combin- Serial Serial 
ation Camera No. Lens No. 
3 Pulnix TM80 001146 Fujinon 987894 
2 Pulnix TM80 001136 Computar 1473508 
3 General TCZ-200 7001009 Computar 1472638 
4 Pulnix TM80 001146 Fujinon 994104 
5 Pulnix TM80 001136 Fujinon 994104 
6 General TCZ-200 6027001 Computar 1473508 
In each case, the camera-lens combination was 
positioned in front of the control field and at a 
distance of approximately 5.5 m from the center of 
the target field. A total of 16 images were 
acquired in sequence for each combination at the 
following nominal focal settings: 12.5, 15, 20, 25, 
30,:35, 40, 45, 50, 55, 60, 65, 70, 75, 12.5, and 15 
mm. 
7. FREE CALIBRATION 
A free calibration was performed for each focal 
setting of each camera-lens combination by a bundle 
adjustment. Only the object-space coordinates of 
the control targets were constrained in an 
adjustment, which yielded the following: six 
exterior orientation parameters of the camera (X, 
VS, Z°, Q, ¢, and x); and five interior orientation 
parameters ( f, k, L., p,, and p,). 
Table 1 lists the root-mean-square errors (RMS) of 
the residuals for all the adjustments. For focal 
length of 35 mm or shorter, the RMS errors were 
between + 0.05 and + 0.09 pixel. The small 
magnitude of the RMS errors for these adjustment 
confirmed the validity of the distortion model as 
well as the accuracy capability of the targeting 
algorithm. For focal lengths greater than 35 mm, 
largely because of the fewer number of control 
points available in each calibration image and the 
degradation in resection geometry, the RMS errors 
were increased to between + .07 and + .17 pixels. 
Figure 2 shows the changes in the interior and 
exterior orientation parameters with respect to the 
focal setting for camera-lens combination 4. Space 
limitation does not permit the inclusion of similar 
plots for the other five cases. As can be expected, 
there were small systematic changes in the exterior 
orientation parameters. Changing the focal setting 
resulted in a small movement of the exposure center 
and, in some cases, small changes in the direction 
of' the optical axis. In all six cases, the changes 
in the interior orientation parameters were also 
highly systematic. 
Of particular interest from the free calibration 
results are that 1) there were large linear shift of 
the principal point, and 2) decentering distortions 
were quite large. For camera-lens combinations Nos. 
2 and 5, which involved the same camera, linear 
shift of the principal point amounted to about 90 
pixels, and decentering distortions amounted to 
about 5 pixels near the edge of the images acquired 
with f- 75 mm. Discussions with A. Burner of NASA 
led to the conclusion that both of these phenomena 
were most likely caused by tilting of the optical 
axis with respect to the focal plane. Burner 
reported that tilts of up to 0.5 degree were not 
uncommon in this type of cameras (Burner et al, 
1990). A linear. shift of the principal point 
amounting to 80 pixels over the zoom range of 12.5 
mm to 75 mm would be equivalent to a tilt of only 1 
degree. 
One obvious approach to zoom lens calibration is to 
simply model these patterns with a separate 
polynomial for each of the parameters. Another 
possible approach is to use these calibration 
results directly in a table look-up scheme. The 
problem with both of these two approaches is that 
corrections for changes in exterior orientation 
parameters must be applied for different focal 
settings. 
Further analysis of the results from free 
calibration showed that the RMS errors for the 
exterior orientation parameters were quite large 
comparing to the magnitude of changes when the focal 
length was varied between 12.5 and 75 mm. Much of 
the problem was due to the small field of view at