2) Closed-Form Solution: parameters values are com 
puted directly through a noniterative algorithm 
based on a closed-form solution. It is a fast proce 
dure but, in general, camera distortion parameters 
cannot be incorporated in the algorithm. 
3) Two-Step Method: this technique involves a direct 
solution for most of the calibration parameters and 
some iterative solution for others. An example is 
the Tsai-Lenz calibration algorithm, where howe 
ver only the radial distortion is taken into account. 
From a short analysis of these calibration methods, one 
can conclude that noniterative procedures involve clo 
sed form solution of linear equations without estima 
ting the distortions parameters, while iterative methods 
allow to evaluate lens distortion through nonlinear opti 
mization procedures, but they require a good initial 
guess of the parameters. 
In order to solve this trade-off, we addressed the came 
ra calibration problem adopting a two-steps method, 
based on a combination of the Tsai-Lenz [3] and 
Cohen-Hemiou [4] calibration techniques. In the first 
step we use the noniterative Tsai-Lenz algorithm to 
directly compute a closed-form solution for all external 
and some major internal parameters of a distortion free 
camera model. In the second step we apply a nonlinear 
optimization based on a camera model that takes into 
account various kind of geometrical lens distortions. 
Because an iterative algorithm is involved, the solution 
of the first step is used as initial guess. 
The main advantages of our method are as follows: 
1) Unlike the Weng-Cohen-Hemiou method, in the 
first step we use the well known Tsai-Lenz calibra 
tion algorithm, in order to get an initial guess of 
internal and external parameters. Being nonitera 
tive, this algorithm is fast and easy to implement; 
2) Compared to Tsai method, in the second step we 
improve the estimate of all camera parameters, 
3) Through the application of a nonlinear optimiza 
tion, we can consider various kind of lens distor 
tion rather than just the radial one; 
The method has been implemented in such a way to 
allow a full camera calibration or a computation of the 
exterior orientation parameters only, using inner orien 
tation and distortion parameters determined from a 
previous full calibration. This approach can be useful if 
only an estimate of new targets positions is required, in 
which the inner and distortion parameters are already 
given. 
In the following sections a detailed description of the 
proposed method, so as a short overview of the imple 
mented calibration software, will be presented. 2 
2. THE CAMERA MODEL 
In every camera calibration procedure, a certain set of 
reference systems are required to define the coordinates 
of target points and of corresponding projections onto 
the image. In our case we adopted the following set 
(see Fig. 1): 
■ I w (O w , X w , Y w , Z w ) target fixed 3D reference 
system, with origin at point O w ; in case of coplanar 
target points the X w and Y w axes are choosen in 
such a way to set Z w =0. 
■ I c (O c , X c , Y c , Z c ) is the 3D camera fixed reference 
system; its origin coincides with the optical center 
of the camera and the Z c axis coincides with the 
optical axis. The (X c , Y c , Z c ) axes form a right- 
hand triplet. 
■ Su, v , 2D image reference system centered at O’, 
the intersection point between optical axis and 
image plane n (the CCD surface). This plane is as 
sumed to be parallel to the (X c , Y c ) plane and at a 
distance / to the origin O c , where / represents the 
effective focal length of the camera. 
■ £ r , c , 2D image reference system where the points 
coordinates are computed according to row and 
column number of corresponding pixel for the dis 
crete image in the frame memory. The origin is lo 
cated at the upper left comer of the image plane n. 
Adopting a pin-hole camera model, the relationships 
between the 3D coordinates of target points and the 
corresponding 2D image coordinates can be defined as 
follows: 
1) Rototraslation, transforming (X w , Y w , Z w ) coor 
dinates of target point P in I w , in the camera coor 
dinates (X c , Y c , Z c ). 
pq 
(X 
w 
Y c 
= R * 
Y w 
UJ 
•T* / 
where R is the rotation matrix defined by roll, pitch 
and yaw angles, while T is the traslation vector 
denoted by (T x ,T y ,T z ).