General lens model

General lens model are based on mathematical descriptions of ideal projections. In the radially symmetric models below theta (θ) is the angle between view ray and optical axis. Next to the model name mathematical descriptions in three colums. Firstly the ideal projection, secondly a two parameter polynomial estimation using odd powers of theta only (based on Kannala and Brandt), thirdly a one parameter (catadioptric) model due to Ying and Hu. A rough estimate for the popular Peleng 8mm lens has been added, based on calibration by Kumler and Bauer.

model name	ideal projection	$r=k_{1}\theta +k_{2}\theta ^{3}$	$r={\frac {(l+1)\sin \theta }{l+\cos \theta }}$
perspective projection(pinhole camera )	$r=f\tan \theta$	$r=\theta +{\frac {1}{3}}\theta ^{3}$	$l=0$
stereographic projection	$r=2f\tan(\theta /2)$	$r=\theta +{\frac {1}{12}}\theta ^{3}$	$l=1$
equidistance projection	$r=f$	1	$l=$
Peleng	$r=3f\sin(\theta /3)$	$r=\theta -{\frac {1}{54}}\theta ^{3}$	$l={\frac {19}{8}}$
quisolid angle projection	$r=2f\sin(\theta /2)$	$r=\theta -{\frac {1}{24}}\theta ^{3}$	$l={\frac {527}{259}}$
orthogonal projection	$r=f\sin(\theta )$	$r=\theta -{\frac {1}{6}}\theta ^{3}$	$l=\infty$

A one parameter generic lens model might be handy in optimizing panorama's for example hugin to have an independent initial check based on the control points of multiple images what kind of lens really is used, and providing an initial estimate for multiple parameter models, like the well known a, b, c lens correction model. The table above suggests to set a and c always to zero, and only use b, as a and c represent even powers.

Apart from radial distortion of the ideal model - producing barrel distortion - most lenses will show tangential distortion, or asymmetric distortion, mainly due to not all elements in the lens being exactly aligned with the optical axis.

references

Kannala and Brandt 1 K&B 2, citing Ying and Hu Kumler and Bauer