Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 25

1.4.6.2 Metric Self-calibration

This section describes how a metric self-calibration can be obtained based on the
projective reconstruction of the scene, following the presentation by Hartley and
Zisserman ( 2003 ). In a metric coordinate system, the cameras are calibrated and the
scene structure is represented in a Euclidean world coordinate system. Each of the
m cameras is defined by its projection matrix P (M)
i

x (M)
i

, for which a scene point W ˜

x (M)
i

P (M)
i

x (M)

S i W

yields the image point

i . The index M denotes that the pro-
jection matrices as well as the scene and image points, although given in homoge-
neous coordinates, are represented in Euclidean coordinate systems. The projection
matrices can be written as P (M)
i

˜ = ˜

1 ,...,m . The projective re-
construction according to Sect. 1.2.2 yields projection matrices P i from which the
corresponding Euclidean matrices P (M)
i

= A i [ R i | t i ] for i =

are obtained by

P (M)
i

= P i H (1.57)

for i 4 projective transformation. According to Hartley

and Zisserman ( 2003 ), the aim of metric self-calibration is the determination of H
in ( 1.57 ).
Hartley and Zisserman ( 2003 ) assume that the world coordinate system is iden-
tical to the coordinate system of camera 1, i.e. R 1 = I and t 1 =

= 1 ,...,m , with H as a 4 ×

0 . The matrices R i
and the translation vectors t i denote the rotation and translation of camera i with
respect to camera 1. Furthermore, we have P (M)
1

= A 1 [ I | 0 ] . The projection matrix

P 1 is set to its canonical form P 1 =[ I | 0 ] .If H is written as

B .

t

H = (1.58)

v T k

Equation ( 1.57 ) reduces to the simplified form [ A 1 | 0 ]=[ I | 0 ] H , from which it can

be readily inferred that B 0 . To prevent the matrix H from becoming

singular, it is required that its element H 44 =

= A 1 and t =

k = 0. A favourable choice is to set

H 44 = k = 1, which yields

A 1 .

0

H = (1.59)

v T

1

Under these conditions, the plane at infinity corresponds to

⎛ ⎞ ⎛ ⎞

0
0
0
1

0
0
0
1

A − T
1

A − T
1

⎝ ⎠ = ⎝ ⎠ =

A − T
1

− v

− v

H − T π ˜

∞ =

(1.60)

1

0 T

1

and hence v T p T A with the upper triangular matrix A

A 1 denoting the intrin-
sic calibration of the first camera (Hartley and Zisserman, 2003 ). Equation ( 1.59 )
then becomes

=− ≡

A .

0

=

H (1.61)

p T A

−

1