Catadioptric Stereo with Planar Mirrors: Multiple-view Geometry and Camera Localization - Visual Servoing via Advanced Numerical Methods - page 6

Fig. 1.3 Reflective epipolar
geometry

v

Π

u π

e π

X v

d π

X

e

n π

u r

c

Proof. Let X and X v be the 3D coordinates of a point in the camera frames c

and v , (see Figure 1.3). X and X v are related by the rigid-body transformation

X v = S [π] X + 2 d π n π (1.9)

readily derived by substituting (1.7) into (1.5). Since by hypothesis the calibration
matrix K is the identity, then X v = λ π u π , X = λ r u r and (1.9) can be rewritten as

λ r S [π]

λ π u π =

u r + 2 d π n π

(1.10)

+ are unknown depths. Simple matrix manipulations on (1.10) lead
directly to the epipolar constraint

∈ R

where λ π , λ r

u T
π

(2 d π [ n π ] × S [π] )

u r = 0 .

By definition, E [π] 2 d π [ n π ] × S [π] = 2 d π [ n π ] × ( I 2 n π n T

− )=2 d π [ n π ] × .

π

Note that the vector n π can be readily recovered (up to a scale factor), from the right
null-space of E [π] . If the camera calibration matrix

⎡
⎣

⎤
⎦

f x su 0
0 f y v 0
001

K =

where f x , f y (pixels) denote the focal length of the camera along the x and y direc-
tions, s is the skew factor and ( u 0 , v 0 ) (pixels) are the principal point coordinates

of the charge-coupled device (CCD), we can introduce the reflective fundamental
matrix

F [π] K − T E [π] K − 1 . (1.11)