Camera Calibration for 3D Reconstruction and View Transformation - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 75

Thus, three transform matrices can be identified to project a 3-D world point onto
its 2-D correspondence in an image. These three matrices are termed the
intrinsic transform matrix K (ITM, encoding all intrinsic camera parameters),
the extrinsic transform matrix M (ETM, encoding all extrinsic camera
parameters), and the projection matrix P (PM, encoding all linear camera

parameters), and are given by:

− f 0 x

x 0

K

() ()

= 0

001

− f y

T T

w w w

MR = − Rt ⋅

, , and

PKM =

. (4)

y 0

c c c

T

x

im im im

Thus, for the projective coordinates 1 = x y 1 and

T

w w w w

x = xyz 1 , we have:

. (5)

im w

xPx ≅⋅

Through this projection, a 3-D straight line will map as a 2-D straight line. From
this observation, this pure pinhole modeling is called linear modeling.

Modeling Nonlinear Components

The perfect pinhole model is only an approximation of the real camera system.
It is, therefore, not valid when high accuracy is required. The nonlinear
components (skew and distortion) of the model need to be taken into account in
order to compensate for the mismatch between the perfect pinhole model and the
real situation.
In applications for which highly accurate calibration is not necessary, distortion
is not considered. Instead, in this case a parameter u characterizing the skew
(Faugeras, 1993) of the image is defined and computed, resulting in an ITM as:

− f ux

f

x 0

=

im im

x x ˆ

ˆ

K = 0

001

− y

and . (6)

y 0

im im

y y