229 
THE GRID METHOD , A SIMPLE PROCEDURE FOR THE DETERMINATION OF THE LENSE 
RADIAL DISTORTION 
G. Fangi - Ancona University- Italy 
email: fangi@pocsi. unian. it 
Carla Nardinocchi — Parma University — Italy 
Email: nardinoc@dits. ing. uniromal.it 
ISPRS COMMISSION VI WG 3 
KEY WORDS: Photogrammetry, non-metric cameras, calibration, distortion 
Abstract: We propose a simple procedure for the determination of the radial distortion for non-metric cameras. 
Normally the distortion characteristics are estimated together with the other interior orientation parameters both 
with laboratory equipment via goniometer, both by block bundle adjustment with the so-called additional 
parameters. The self-calibration approach is followed for non-metric cameras also, with the difference that any 
orientation is in practice a new calibration. Often adding the parameters for the distortion brings the solution system 
to numerical problems especially when the geometric configuration is weak. It happens with narrow angle of field 
lenses, of poor distribution of control points, leading to the impossibility of the estimation for the distortion. 
Therefore we intend to estimate the main distortion - the radial component - a part from the orientation. The 
proposed approach is intended to be a first step for the full camera calibration, divided in distortion determination 
and interior orientation. We neglect the variation of the distortion with the focussing distance. The distortion is 
deeply linked to the other interior orientation parameters. We will find that particular interior orientation set fitting 
into the computed distortion curve. The ideal distortion-free camera corresponds to the pinhole model. The 
distortion is the systematic difference from the real camera to the pinhole model. The equations of the strict 
projection are the collinearity equations, or, in an implicit form, the Direct Linear Transformations. A particular 
case of the DLT are the equations of the homografic transformations. In order to avoid numerical problems with the 
DLT and additional parameters, we use the special case of DLT, that is the homografic transformation. The 
transformation residuals are the components of the distortion vector. We tested the procedure to estimate the radial 
distortion for a already calibrated semi-metric camera, a Rollei 6008 with a 40mm Distagon lens, using a picture of 
an high accuracy plotted grid. 
1. INTRODUCTION 
The radial distortion of non metric cameras is 
normally corrected by adding additional parameters 
to the perspective transformation or DLT (Abdel- 
Aziz 1971, 1973 ): 
The distortion parameters can be expressed as 
function of the so-called additional parameters, being 
k;, the coefficients of radial distortion, Pj those of the 
asymmetrical distortion, 0 the rotation angle of the 
axis where the asymmetrical distortion, or 
decentering lens distortion, becomes null. There exist 
many distortion models up to the very sophisticated 
24 additional parameters function of Mueller, Bauer, 
Jacobsen (Kruck, 1985). But normally the distortion 
is distinguished in two parts: the radial distortion and 
then tangential one. A complete treatment of lens 
distortion modélisation can be found in (Fryier, 
1986). 
4- a 2 Y + afiZ + a^ 
x + dx = 
(1) 
QqX + ûjqK + d^\Z 1 
with 
dx = dx'+dx"= f\x,y,x M ,yM,k l ,k 2 ,k 3 ,...) + 
+ g'(x,y,x M ,y M ,Q,p h p 2 ,p 3 ,...) 
Adding additional parameters to the projectivity 
equations has several drawbacks: very many control 
points are required, while when the points are not 
well distributed the estimation of the distortion is 
poor, due also to the high correlation between the 
DLT or collinearity equations parameters and those 
of the distortion. The introduction of the distortion 
parameters brings to numerical problems for the 
stability of the solution system (Faig and Shih 1986), 
especially with narrow angle lenses (Fangi, 1990). 
The long focal lenses non-metric cameras are 
impossible to be corrected for radial distortion. 
(2) 
dy = dy'+dy" = f'\x,y,x M ,y M ,k v k 2 , k 3 ,...) + 
+ g" (*, y,x M , y M 9, P\, p 2 > Py>•••) 
where 
• x, y are the comparator co-ordinates 
• X, Y, Z are the terrain co-ordinates 
• aj are the transformation parameters 
• dx ’ and dy ’ are the components of radial 
distortion and 
• dx” and dy” those of the tangential distortion. 
The present research has been financed by Cofin97