Catadioptric Stereo with Planar Mirrors: Multiple-view Geometry and Camera Localization - Visual Servoing via Advanced Numerical Methods - page 3

introduce new algebraic results on the multiple-view geometry for the case of
static and moving cameras (multi-view PCS);

•

on the grounds of these novel results, we address the image-based camera local-
ization problem and present original methods for mirror calibration ( i.e. ,com-
putation of the angle between the mirrors); and

• we present extensive simulation and real-data experiments conducted with an

eye-in-hand robot, in order to illustrate the theory and show the effectiveness of
the proposed designs in real scenarios.
A preliminary version of this chapter appeared in [16], compared to which
we provide here new theoretical results as well as a more extensive experimental
validation.

1.1.3 Organization

The rest of this chapter is organized as follows. Section 1.2 reviews the basic theory
related to perspective projection through planar mirrors and introduces the reflec-
tive epipolar geometry. Section 1.3 deals with the single and multiple-view geom-
etry for PCS sensors. Section 1.4 addresses a solution to the mirror calibration and
image-based camera localization problems. Simulation and real-data experiments
are reported in Section 1.5. In Section 1.6, the main contributions of the chapter are
summarized and possible avenues of future research are highlighted.

1.2 Planar Mirrors and Perspective Projection

In this section we review the imaging properties of catadioptric systems with a sin-
gle planar mirror [5]. The basic concepts of virtual point , reflection transformation
and virtual camera projection equivalence are introduced. The original notion of
reflective epipolar geometry is presented at the end of the section.

1.2.1 The Virtual Point and the Reflection Transformation

Let us consider the setup reported in Figure 1.2(a) where a perspective camera c

[ xyz ] T
is supposed to be in front of the mirror as well ( X indicates its extension in ho-
mogeneous coordinates). For the sake of clearness, hereafter we will refer to the
simplified setup in Figure 1.2(a): however, the results of this section are valid for
generic camera-mirror arrangements. Note that the perspective image u (pixels) of
X after its reflection by the planar mirror

is in front of a planar mirror Π with normal vector n π . A 3D point X

Π can be calculated as the direct projection

of the so-called virtual point X [π] .
Proposition 1.1 (Perspective projection). Let us consider the setup of Figure 1.2(a)
in which a planar mirror with normal vector n π is distant d π from the camera

on c

c .

[ uv 1] T Then the perspective projection u (pixels) of a generic 3D point X that is

mirrored by Π is given by