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 ).
