calibration parameters with optimum accuracy and 
independence. Calibration using a target array, multiple 
convergent photographs or images and a self-calibrating 
collinearity solution can determine all parameters, but 
does not realise the parameters with a high degree of 
independence. Calibration using straight lines can 
determine only the lens distortions, albeit with a high 
level of independence and accuracy, however knowledge 
of all calibration parameters is required for the successful 
application of Magill's formula at other focus distances. 
Magill's formula has been verified experimentally many 
times, commonly using a straight line calibration of 
conventional film cameras (Brown, 1971; Fryer and 
Brown, 1986). Straight line (Fryer and Mason, 1989) 
and targeted test range (Shortis et al, 1991; Wiley and 
Wong, 1995) calibrations have been successfully applied 
to machine vision and still video cameras, although 
generally no information on adherence to Magill’s 
formula has been included. 
2.2 Variation with Distance 
Simple lens distortion models, derived from a calibration 
or the application of Magill’s formula, are applicable 
only to the plane of best focus. If the camera is focussed 
on a single plane in the object space, or perhaps a very 
narrow depth of field within the object space, then the 
simple lens distortion model is sufficient. 
However, lens distortion does vary within the object 
space, although the magnitude of the variation is 
typically much less than the variation with focus distance 
(Fraser and Shortis, 1992). The variation increases with 
magnification and distance from the plane of best focus, 
and so is most relevant to applications close range 
photogrammetry and machine vision where the object 
extends across a significant depth of field in the object 
space. In particular, applications which require high 
accuracy, such as tool inspection and surface 
characterisation for the aerospace and manufacturing 
industries, require an extended lens model to eliminate 
the systematic error caused by the variation. 
Brown (1971) developed an extended lens model to 
account for the variation of distortion outside of the 
plane of best focus. The model is based on a function of 
the lens distortion at the plane of best focus and a scale 
factor derived from the geometry of the image : 
Örs' (5) 
Örss' = 
' 
where 
Orgg' = radial distortion at an object distance s' 
for a lens focussed at an object distance s 
536 
radial distortion at an object distance s' 
for a lens focussed at an object distance s' 
focus distance for the camera 
distance to the plane of the target point 
The scale factor is given by : 
JepaqGshu og euer f) 
fe $ sb (6) 
where 
cs = principal distance for the focus distance s 
Cs' = principal distance for the focus distance s' 
Brown (1971) verified this extended lens model for 
medium format, glass plate cameras using straight line 
calibrations. However the camera lenses exhibited a 
relatively low magnitude of distortion, and Brown noted 
that there was some variability for different lenses. 
Fraser and Shortis (1992) showed that Brown’s extended 
model is not able to model the variations in distortion 
when the magnitude of the distortion, and therefore the 
gradient of distortion across the format, is very large. An 
alternative model was developed which expresses the 
lens distortion as a function of the distortions at the 
distances of the camera focus and the point of interest : 
Orgs’ = Org + ggg' ( Ors' - Org) (7) 
where 
ss' = a constant value derived empirically 
Org = radial distortion at an object distance s 
for a lens focussed at an object distance s 
Fraser and Shortis (1992) verified this new extended lens 
model for large and medium format film cameras, again 
using straight line calibrations. The lens to lens variation 
was relatively low, although clearly present. Hence the 
empirically derived constant factor gss' could be applied 
to any lens of a specific type and the model error was 
shown to be significantly less than the magnitude of the 
distortion variation. 
3. EXPERIMENT DESIGN 
In order to select an appropriate extended lens model for 
a typical still video camera, an experiment was designed 
to conduct a comprehensive camera calibration and 
accuracy test. The basis for the calibration was a test 
range set up to enable the simultaneous targeted test 
range and straight line calibration of a still video camera. 
The layout of the test range is shown in Figure 1. The 
range comprises the targeted test range, the straight line 
range, lighting for the straight lines and a number of 
relocatable camera station pillars. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
  
  
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