Catadioptric Stereo with Planar Mirrors: Multiple-view Geometry and Camera Localization - Visual Servoing via Advanced Numerical Methods - page 7

Since F [π] is skew-symmetric (in fact we have, F [π] +( F [π] ) T = 2 d π ( K − T [ n π ] × K − 1
−

K − T [ n π ] × K − 1 )= 0 ), the left and right null-space of F [π] are equal. As such, the

epipoles are equal:

e = e π .

Note that since F [π]

has only 2 degrees of freedom (that correspond to the position

of the epipole e ), at least 2 pairs of corresponding points are necessary to determine

F [π] .Since

e π the epipolar lines in the real and virtual view coincide and the
epipole can be regarded as a vanishing point , being the intersection of 3D parallel
lines (the lines joining the real and virtual points) projected onto the image plane.

e =

Remark 1.1. Note that the imaging geometry relating cameras corre-

sponds (in terms of essential matrix) to that existing between two cameras undergo-
ing a pure translational motion.

c and v

1.3 Single-view and Multiple-view Geometry for PCS Sensors

In this section we introduce the single-view and multiple-view geometry for PCS
sensors. These results play a relevant role in the subsequent derivations.

1.3.1 Single-view Geometry

In this section we assume that a camera observes a set of 3D points reflected by

two planar mirrors (see Figure 1.4). In this case two corresponding virtual cameras

c

can
be defined. The generalization to multiple mirrors is straightforward and it will be
not discussed herein. Let be given the image points u [1]
i

v 1 and v 2 exist and suitable geometries relating c with both v 1 and v 2

, u [2]
i

∈{ ,..., }

, i 1 n in c ,

projections of a set of n 8 3D points X i reflected onto the mirrors n 1 and n 2 ,

respectively. Note that while the subscript i is the point index, the superscript inside
the brackets will always refer to the mirror number through which that vector is
reflected (for the sake of simplicity, we will henceforth neglect the subscript

≥

in
the mirrors parameters). Given the two-mirror setup reported in Figure 1.4, let D [1]
and D [2] be the corresponding reflection transformations. The following expression
holds true:

π

D [2] D [1] = R D t D
0 T

1

where

I + 4( n 1 n 2 ) n 1 n 2 − 2 n 1 n 1 − 2 n 2 n 2 , R D

2( d 1 ( n 1 n 2 )+ d 2 ) n 2 . t D 2 d 1 n 1 −

Note that ( R D , t D ) only depends on the mirror setup ( i.e. , n 1 , n 2 and d 1 , d 2 ), and

not on the observed scene. On the other hand, let H v v 1 be the homogeneous transfor-
mation matrix representing the rigid body motion between the frames v 1 and v 2 .

It is easily found that (see Figure 1.4)