Catadioptric Stereo with Planar Mirrors: Multiple-view Geometry and Camera Localization - Visual Servoing via Advanced Numerical Methods - page 4

v

X [π]

X [π]

Π

Π

u [π]

d

n π

X

X v

d π

u

u

c c

(a) (b)

and the virtual point X [π] ; and (b) the

Fig. 1.2 (a) Reflection through the planar mirror Π

virtual camera v

X with

u = K [ I0 ] D [π]

+

λ λ ∈ R (1.1)

where K is the calibration matrix of the camera and D [π] is the reflection transfor-

mation about the mirror n π , defined as

S [π]

2 d π n π

D [π] =

(1.2)

0 T 1

where S [π] = I 2 n π n T − .

π

Proof. The perspective projection u of the virtual point X [π] is given by

u = K [ I0 ] X [π] λ . (1.3)

From geometrical considerations in Figure 1.2(a), we see that

X [π] = X + 2 d n π . (1.4)

X T n π , then (1.4) can be rewritten as

Since d = d π −

X [π] =( I 2 n π n T − ) X + 2 d π n π . (1.5)

π

By introducing the matrix D [π] defined in (1.2), (1.5) becomes

X [π] = D [π] X

. (1.6)