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a) Non-linear optimization of vector m=(ro,c 0j />,R,T) 
at step 2) is accomplished by iterative Levenberg- 
Marquardt algorithm [4]. As initial guess for m, at 
first iteration we use the value m resulted from 
step 1), while in following iterations we use the 
value rri\ obtained as output of step 2) in previous 
loop. 
b) In the second part of the procedure, where all the 
image points are used to estimate also the distor 
tion parameters, we perform again the non-linear 
estimate of m through L-M algorithm, while for 
vector d we carrie out a linear estimate based on 
solution of (12), with (k b k 2 , pi, p 2 , s b s 2 ) as 
unknows. 
c) After about four loops, all camera parameters are 
estimated together simultaneously by non-linear L- 
M algorithm, obtaining the optimal vector estima 
tes m opt and d opt . 
The overall scheme of the adopted procedure is showed 
in Fig. 2 and 3. 
We have adopted the Levenberg-Marquardt iterative 
algorithm principally because it moves smoothly bet 
ween two others widely used minimization methods, 
the Steepest Descent and the Hessian, combining them 
into one simple equation. In this way the algorithm can 
converge to true minimum, switching at each iteration 
between the two methods on the base of a changing 
threshold value X [4]. We retain that this property can 
be succesfully used in a calibration procedure, since we 
deal with initial guesses about which we don’t know 
how close to the true solution they are. 
Such “switch” behavior can therefore resolve the pro 
blem of the “goodness” of initial estimates in mini 
mizing the objective function F. 
ig. 2 : Flowchart of the estimation procedure of 
external parameters only. 
I 
t 
T 
Fig. 3: Flowchart of calibration procedure of all 
parameters. 
As reported in Fig. 2 and 3, our calibration procedure 
can be applied both to a set of coplanar (Z w =0) and 
non-coplanar target points, so as it can be used to per 
form a full camera calibration or to evaluate only exter 
nal parameters. In the last case the internal and distor 
tion parameters of previous calibration are employed. 
Since the initial linear estimate of vector m is based on 
Tsai-Lenz method, we defer the reader to references for 
more detailed explanations on it, while we provide a 
short overview on the application of Levenberg- 
Marquardt algorithm and on the linear estimate of dis 
tortion parameters. 
Basically a maximum likelihood estimate of the model 
parameters is obtained minimizing the quantity