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

Fig. 1.1 The pinhole model. A scene point C x defined in the camera coordinate system is projected
into the image point I x located in the image plane

the camera coordinate system C is denoted by C x , where the origin of C corresponds
to the principal point. Similarly, a transformation of a point in the world coordinate
system W into the camera coordinate system C is denoted by a transformation W T ,
where the lower index defines the original coordinate system and the upper index
the coordinate system into which the point is transformed. The transformation W T
corresponds to an arbitrary rotation and translation. In this notation, the transforma-
tion is given by C x

(x,y,z) T defined in the camera
coordinate system C is projected on the image plane into the point I x , defined in
the image coordinate system I , such that the scene point C x , the optical centre,
and the image point I x are connected by a straight line in three-dimensional space
(Fig. 1.1 ). Obviously, all scene points situated on this straight line are projected into
the same point in the image plane, such that the original depth information z is lost.
Elementary geometrical considerations yield for the point I x

C

W T W x . A scene point C x

= =

= ( u, ˆ v) in the image ˆ

coordinate system the relations

u
b =

ˆ x

z

(1.1)

v
b =

ˆ y

z

ˆ ˆ

(Horn, 1986 ). The coordinates v in the image plane are measured in the same

metric units as x , y , z , and b . The principal point is given in the image plane by
u = v =

u and

0. In contrast, pixel coordinates in the coordinate system of the camera
sensor are denoted by u and v .
While it may be useful to regard the camera coordinate system C as identical
to the world coordinate system W for a single camera, it is favourable to explic-
itly define a world coordinate system as soon as multiple cameras are involved. The
orientation and translation of each camera i with respect to this world coordinate
system is then expressed by

C i

W x from the world coordi-

W T , transforming a point