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14 Autonomous Mobile Robots

2. Nonlinear optimization techniques account for lens distortion in
the camera model through iterative minimization of a determined
function. The minimizing function is usually the distance between
the image points and modeled projections.

In guidance applications, it is common to adopt a two-step technique: use
a linear optimization to compute some of the parameters and, as a second step,
use nonlinear iteration to refine, and compute the rest. Since the result from the
linear optimization is used for the nonlinear iteration, the iteration number
is reduced and the convergence of the optimization is guaranteed [18–20].
Salvi [17] showed that two-step techniques yield the best result in terms of
calibration accuracy.
Calibration should not be a daunting prospect because many software tools
are freely available [21,22]. Much of the literature originated from photo-
grammetry where the requirements are much higher than those in autonomous
navigation. It must be remembered that the effects of some parameters, such as
image skew or the deviation of the principal point, are insignificant in com-
parison to other uncertainties and image noise in field robotics applications.
Generally speaking, lens distortion modeling using a radial model is sufficient
to guarantee high accuracy, while more complicated models may not offer much
improvement.
A pragmatic approach is to carry out much of the calibration off-line in
a controlled setting and to fix (or constrain) certain parameters. During use,
only a limited set of the camera parameters need be adjusted in a calibration
routine. Caution must be employed when calibrating systems in situ because
the information from the calibration routine must be sufficient for the degrees of
freedom of the model. If not, some parameters will be confounded or wander in
response to noise and, later, will give unpredictable results. A common problem
encountered in field applications is attempting a complete calibration off essen-
tially planar data without sufficient and general motion of the camera between
images. An in situ calibration adjustment was adopted for the calibration of the
IR camera used to take the images of Figure 1.1. The lens distortion effects were
severe but were suitably approximated and corrected by a two-coefficient radial
distortion model, in which the coefficients (k 1 , k 2 ) were measured off-line. The
skew was set to zero; the principal point and aspect ratio were fixed in the
calibration matrix. The focal length varied with focus adjustment but a default
value (focused at infinity) was measured. Of the extrinsic parameters, only the
tilt of the camera was an unknown in its application: the other five were set by
the rigid mounting fixtures. Once mounted on the vehicle, the tilt was estimated
from the image of the horizon. This gave an estimate of the camera calibration
which was then improved given extra data. For example, four known points
are sufficient to calculate the homographic mapping from ground plane to the
image. However, a customized calibration routine was used that enforced the

© 2006 by Taylor & Francis Group, LLC

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