15 
1.3 
1.4 
1.5 
I 
ted 
on. 
ese 
ve- 
gle 
his 
2.1 
I 
ion 
1 
2.2 
2.3 
s a 
m- 
:o- 
dx = fl; 5 r 2 
c a^r 
....) 
(y) 
dy = ~~y- (c a%r 2 + c a 5 r 4 + . . . .) 
l-w 
5.2.4 
5.2.5 
The number of terms in formulas 5.2.4 and 5.2.5 depends on the character of 
the radial distortion. More terms mean a better mathematical approximation, 
but this may introduce correlation a posteriori between the parameters. This 
can in combination with the decreasing number of degrees of freedom cause 
larger confidence intervals for the determined parameters. 
r 
Fig. 4. 
Relation between radial distortion 
and change of camera constant. The 
angle between the camera axis and 
the incident ray is O. The angle of 
the out-going ray deviates from this 
value causing a radial displacement 
dr. If the angles of the incident and 
out-going rays are defined to be 
equal, this causes a variation, dc, of 
the camera constant. 
5.3. TANGENTIAL DISTORTION 
In this chapter we will deal with tangential distortion arising from eccentric 
lens elements. In chapter 7.2 asymmetric radial and so-called tangential dis 
tortion arising from the calibration procedure is treated. 
A slight eccentricity of a lens element changes the location of the image 
points. This change has very often been assumed to be equivalent to the proper 
lens and a thin prism in front of it. This thin prism model has been adopted in 
USA, where tangential distortion has been very much discussed. However, 
Duane Brown [4] refers to a paper by Conrady [6], where another model for 
tangential distortion is given. Brown prefers the Conrady model and gives the 
following formulas for the displacements of the image points