ANALYTICAL PLUMB LINE CALIBRATION 
Perhaps the most convenient method of calibrating the radial and de- 
centering distortion of a lens is the analytical plumb line method developed in 
Brown (1971). It is based on the consideration that, in the absence of distortions 
of any kind, the photographic image of any straight line in object space is itself 
a straight line; thus any systematic departure from strict linearity of the image of a 
straight line can be attributed to distortion. As the name implies, the analytical 
plumb line method employs exposures of an array of plumb lines in the desired 
object plane to generate the necessary observational material. The plate coordinates 
of a moderate number of points are measured on the images of each of several lines 
well-distributed throughout the format. The set of points measured on the i th line 
generates a set of observational equations that are functionally of the form 
Fx, , LY 4EXprYo Kui Ko, i PPS, e. 0, 294) z0, 1-1,2,...;n,7 
in which 
X,4rŸ,, = measured plate coordinates of sth point on ith line; 
Xp 1 Ys coordinates of principal point; 
Ryo evs; P ‚P,... = coefficients of radial and decentering distortion; 
0,,0, = parameters defining the equation of the undistorted image of the 
i th plumb line; 
number of points measured on ith plumb line. 
It is noteworthy that the observational equations are independent of all six elements 
of exterior orientation x, «w, x,X€ , Yt , Z^ as well as of the principal distance c. 
Each measured plumb line introduces two unknowns 0,,0, in addition to the set 
common to all plumb lines (x, , y, , K, , Ka, .. .;P, , P5, ...) which we shall assume to 
be q in number. If n denotes the total number of points measured on the entire set 
of m plumb lines (i.e., n 2 n, * ng * ...* n,), the total system of observational 
equations will consist of n equations in q + 2m unknowns. No matter how large m 
may be, the system of normal equations generated by the least squares adjustment 
can be reduced to a qxq system by applying the same computational algorithms 
that were referred to earlier in the SMAC calibration. Hence, the plumb line 
method entails a computational effort that tends to increase only linearly with m 
rather than with the cube of m as would otherwise be the case. Accordingly, there 
is no practical limit to the number of plumb lines that can be exercised in the ad- 
justment.