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A homography can be decomposed to explicitly reconstruct the pose (the translation
and the rotation in Cartesian space) of the camera [9, 18] and the associated servo
control task undertaken in Cartesian space [25, 16, 2, 24]. Alternatively, the control
task can be defined in both image and Cartesian space; the rotation error is estimated
explicitly and the translation error is expressed in the image [17, 7, 22, 8]. The re-
sulting visual servo control algorithms are stable and robust [16] and do not depend
on tracking of individual image features. Some recent work has been done on direct
servo control of the homography matrix [1], an approach which offers considerable
advantages in situations where the homography decomposition is ill-conditioned. A
key component of this work is the identification of the group of homographies as a
Lie group isomorphic to the special linear group
SL
(3), an observation that has been
known for some time in the computer vision community but had not been exploited
before in the visual servo community.
In all cases, the performance of the closed-loop system depends on the qual-
ity of the homography estimates used as input to controller. In the case of visual
servo control applications, the homographies must be computed in real-time with
minimal computational overhead. Moreover, in such applications the homographies
vary continuously and usually smoothly. It is natural, then, to consider using a dy-
namical observer (or filter) process in the closed-loop system to achieve temporal
smoothing and averaging of the homography measurements. Such a process will
reduce noise in the homography estimates, smoothing resulting closed-loop inputs
and leading to improved performance, especially in visual servo applications. There
has been a surge of interest recently in nonlinear observer design for systems with
certain invariance properties [23, 5, 10, 15, 4] that have mostly been applied to ap-
plications in robotic vehicles [19, 20]. From these foundations there is an emerging
framework for observer design for invariant systems on Lie groups [13, 3, 14]. The
special linear group structure of the homographies [1] makes the homography ob-
server problem an ideal application of these recent developments in observer theory.
In this chapter, we exploit the special linear Lie group structure of the set of all
homographies to develop a dynamic observer to estimate homographies on-line. The
proposed homography observer is based on constant velocity invariant kinematics
on the Lie group. We assume that the velocity is unknown and propose an integral
extension of the nonlinear observer to obtain estimates for both the homography
and the velocity. We prove the existence of a Lyapunov function for the system, and
use this to prove almost global stability and local exponential stability around the
desired equilibrium point. The proposed algorithm provides high quality temporal
smoothing of the homography data along with a smoothed homography velocity
estimate. The estimation algorithm has been extensively tested in simulation and on
real data.
The chapter is organized into five sections followed by a short conclusion. The
present introduction section is followed by Section 8.2 that provides a recap of the
Lie group structure of the set of homographies. The main contribution of the chapter
is given in Section 8.3. Sections 8.4 and 8.5 provide an experimental study with
simulated and real data.
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