Camera Calibration for 3D Reconstruction and View Transformation - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 73

Linear Coordinate Transformation

The relation between x o and the corresponding x w can be expressed as:

w w

o

o w

o

x = R x + t , (1)

( )

w
o

where the rotation matrix R = R φ , ϕ , ψ and the translation vector

[ ]

T

t determine the pose (including position and orientation) of
the OCS in the WCS.

w
o

w
o

w
o

w
o

= x y z

is defined as the product of three separate rotations
around the respective axes, that is,

R

w
o

() () ()

w
o . Equation 1 is
a rigid body transformation, in which only rotation and translation are permitted,
but scaling is not allowed ( Euclidean geometry ). This kind of transformation is
called Euclidean transformation .
A similar relation exists between x w and the corresponding x c :

R = R ψ ⋅ R ϕ ⋅ R φ

z y x

()( )

T

x

w

= R

w
c

x

c

+ t

w
c

or

c w

c

w w

c

(2)

x = R x − t

( )

w
c

where the rotation matrix R = R α , β , γ and the translation vector

[ ]

T

= t determine the pose of the CCS with respect to the WCS.
They actually represent the extrinsic parameters of the camera. And they have
in total six degrees of freedom (DOFs).

w
c

w
c

w
c

w
c

x y z

Figure 1. The pinhole camera model.

(b) The principle of a pinhole
projection

(a) Relation between a CCS and
the corresponding PCS

(a) Relation between a CCS
and the corresponding PCS.

(b) The principle of a
pinhole projection.