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2. LENS DISTORTION 
There are a variety of ways by which the parameters of lens 
distortion may be determined. These methods range from 
laboratory calibrations using optical arrangements of collimators 
or precise test ranges bristling with targets, to calibrations made. 
“on—the—job” with the photography of control frames or “self— 
calibration" techniques which utilise a multi-station approach 
and a bundle adjustment to extract the details of the camera’s 
lens characteristics. 
In industrial applications where the situation usually demands 
photography from a multi-station situation, the self—calibration 
technique has proven to be most popular in the last decade. 
The other technique worthy of further discussion is the 
analytical plumb-line method which, unlike other methods, 
only solves for the parameters of lens distortion and not for 
principal distance nor offsets of the principal point from the 
intersection of the fiducial axes. 
2. Self-Calibration 
The technique of self-calibration has been briefly described 
above, but mention should be made of some of the features of 
this technique. In the "purest" form of self-calibration one 
would expect approximately four camera stations to be 
convergently arranged around an object to which 50 to 500 
targets may be affixed. Two or more photographs would be 
taken from each station and it is desirable to roll the camera 
through 90? between photographs in order to successfully 
recover xy and yp. If convergent photography is not employed, 
then the object must be non-planar. 
Equations (1) and (2) are solved simultaneously to extract the 
camera calibration data. 
In reality, it may be necessary to alter the focus setting of the 
camera between camera stations and this complication will lead 
to the photogrammetrist nominating which parameters should be 
constant (or block invariant) or variable from station to station 
(block variant). 
À most important feature of a self-calibrating bundle adjustment 
is that the photogrammetrist need not know any values for 
object-space control. In fact object space control is only 
required for the actual object evaluation with the classical 
minimum of two horizontal and three vertical control points 
being known. 
2.2 Analytical Plumb-Line 
The analytical plumb-line method (Brown, 1971) was 
developed as a rapid practical way to compute lens distortion 
parameters at a range of focussed magnifications from 5X to 
20X. The principle of this technique is the axiom that straight 
lines in object space should project through a perfect lens as 
straight lines onto the image plane. Any variations from 
linearity is attributed to radial or decentering distortion. 
Two photographs from one camera station are always taken: 
one referred to as "horizontal" and the other as "vertical" since 
the camera is rolled through 90? between shots. 
The plumb-line technique is presently applied by many 
researchers as their method for estimating the parameters of lens 
distortion. It is a technique well suited to automation as line— 
following is a relatively simple procedure. The Autoset 
automatic monocomparator developed by Geodetic Services Inc 
in Florida (e.g., Fraser, 1986) and the small format Adam 
Technology MPS-2 analytical stereoplotter (Elfick, 1986) have 
both been programmed to utilise this technique. Fryer and 
Mason (1989) used the method for the close-range calibration 
of a video camera with the data capture being via a frame— 
grabber into a personal computer. 
Typically, eight to ten plumb-lines are photographed and, say, 
50 points on each line on each photograph are digitised. 
Approximately 1000 points will then be provided as data 
observations and only the five parameters of radial and 
decentering distortion must be solved for in a least squares 
solution, along with two parameters to define the spatial 
location of each line. 
The extrapolation of the plumb-line technique from a laboratory 
situation at close-focus to the photography of man-made 
straight objects such as long glass panels in multi-storey 
buildings has been reported (Fryer, 1987). The further 
extrapolation to the use of linear features such as railway-lines 
for the calibration of aerial cameras has also been demonstrated 
(Fryer and Goodwin, 1989). Also interesting to note is the 
extension into aerial camera calibration of the close-range self— 
calibration technique described in Section 2.1, where 
convergent aerial photography has been used over a test range 
was established on flat terrain (Merchant and Tudhope, 
). 
It is important to realise what the analytical plumb-line 
technique does not provide: it does not provide a solution for 
the principal distance c, nor the offsets of the principal point xp 
and yp. The relevance of these parameters is discussed further 
in the next Section when a comparison of the self-calibration 
and plumb-line techniques is made. 
Finally it should be remembered that the lines which are to be 
photographed do not have to be “plumb” in any real sense of 
that word. Straightness is the only criterion, the term “plumb- 
lines” being derived from the earliest use of the technique when 
wires with weights attached were used. The vertically of the 
lines is not a feature of the mathematics involved. 
2.3 Comparison of Lens Distortion Techniques 
The importance of including an allowance for the parameters of 
radial and decentering lens distortions has been demonstrated in 
many studies including those conducted by Karara and Abdel- 
Aziz (1974) and Murai et al., (1984). These studies showed the 
effect of radial distortion to be almost an order of magnitude 
larger than decentering distortion. In both these extensive 
studies the rms values of the residual plate errors decreased by 
up to a factor of seven when lens distortions were included. 
Non-metric cameras were shown to be able to approach the 
accuracy of metric cameras. The K term of radial distortion is 
always the most significant, with K2 and K3 usually not 
relevant for lenses in typical small format cameras. 
A study by Fryer and Fraser (1986) compared the self— 
calibration and plumb-line methods of lens calibration on some 
small format cameras which were to be used both in and out of 
water and in a watertight housing with a plane glass port. The 
tests were interesting because of the large range of radial 
distortion present in the situations examined. For example, a 
50 mm Distagon f—4 lens fitted to a Rolleiflex SLX reseau 
camera in an Aquamarin WKD-SLX/6006 submarine housing 
produced radial distortions in air of 4273.2 um at r = 30 mm 
and —1551.0 um under water. These extremely large ranges of 
distortions were recovered by each of the two methods with an 
average difference along the radial distortion profiles of 2 um 
and a maximum difference of 4.5 um at r = 30 mm. Similar 
close agreement occurred with the testing of a 35 mm Nikonos 
V camera fitted with a Nikkor 28 mm f-3.5 lens. 
The magnitude of the decentering distortion profile for the case 
of the Rolleiflex camera in the underwater housing was the 
largest experienced by this author in over ten years of 
examining photogrammetric cameras. The value of 91 um at 
r = 30 mm was undoubtedly a consequence of the 12 mm thick 
plane glass port in the underwater housing not aligning 
perpendicularly to the optical axis of the camera. The 
discrepancy in this value of the decentering distortion profile 
was only 2.1 um between methods, and since each technique 
had a one-sigma error bound of slightly over 1 jum, the result 
was most satisfying.