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International cooperation and technology transfer
Mussio, Luigi

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
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
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.
Fig. 3: Flowchart of calibration procedure of all
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