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Fig. 1.1 PCS experimental
setup composed of a pinhole
camera mounted on the end-
effector of a manipulator
and two planar mirrors
Planar mirrors
Pinhole camera
several works in the robotics and computer vision literature have recently proposed
the use of catadioptric vision sensors [18, 4, 21, 15], which, combining both re-
fracting (lens) and reflecting (mirrors) elements, naturally enlarge the field of view
of classic pinhole cameras.
By using mirror reflections of a scene, stereo images can be captured with a
single camera (catadioptric stereo). This chapter focuses on a special class of sin-
gle camera stereo systems called PCS sensors, consisting of a pinhole camera and
two, or more, planar mirrors [8, 19]. Single camera stereo has several advantages
over traditional two-camera stereo. First of all, only a single set of internal camera
calibration parameters needs to be determined. Second, no additional software or
hardware is needed for exact camera synchronization [10]. The necessity of pos-
sibly large and unwieldy planar mirrors makes PCS systems not suited for mobile
robotics. However, these sensors find a natural application in industrial robotics; in
fact multiple planar mirrors disposed in the workspace can be used to improve the
pose accuracy of an eye-in-hand robotic arm performing high precision assembling
and close manipulation tasks.
Several PCS sensors have been designed and implemented in the last few years
(see, e.g. , [20, 3]) and preliminary results on the geometry, calibration and 3D scene
reconstruction from reflective symmetry have been presented in [5, 6, 17]. The set
of constraints to be satisfied in order to obtain rectified stereo images from a cata-
dioptric system has been studied in [7]. An affine approximation for epipolar-based
PCS rectification has been recently proposed in [22].
From the previous overview, we see that although some research has been done in
this field, the specific properties of PCS systems have been addressed only sparsely
in the literature and a systematic theory on PCS sensors has yet to be established.
In particular, the case of a moving camera in a fixed mirrors scenario and the
multiple-view geometry associated with it, have not been explored so far.
1.1.2
Contributions
The original contribution of this chapter is threefold:
we provide a rigorous analytical treatment of the imaging geometry of PCS sen-
sors composed of a pinhole camera and two planar mirrors (see Figure 1.1) and
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