Camera Calibration for 3D Reconstruction and View Transformation - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 95

Now, by assuming that the camera distortion is subject to the reconstruction-
distortion model, and by taking equation 12 into equation 31, we obtain:

x u

, (33)

w

ˆ

im

ˆ

im

where D is a matrix with 3 rows and 36 columns. is a column vector with

()()

i j

36 elements in the form ( i , j = 0 7 and i + j ≤ 7 ).

ˆ

im

ˆ

im

x y

ˆ depend on the distortion (coefficients)
considered. Several examples are shown in Table 2, where the last case is the
same linear situation that was just handled in the previous section.
If enough pairs of x w and corresponding

im

In fact, the dimensions of D and

ˆ ) are available, D can
be calculated from equation 33 in the same way as illustrated. Subsequently,
from D , the line of sight of any image pixel in the current camera can be
computed. Furthermore, assume that in another camera, which has transforma-
tion matrix D' , a point

ˆ

im im

(and thus

u

x

im im

' (and thus ' ) can be located which is the projection

ˆ

im

of the same world point x w as the point . Then:

xDu

. (34)

w

≅ ' ˆ '

im

By combining equations 33 and 34, one can easily reconstruct x w (Faugeras,
1993).

Table 2. The dimensions of the transformation matrix and vector used in
DLT considering distortion.

ˆ

Considered distortion coefficients Dimension of D

Dimension of

im

Re
1

Re
2

Re
3

Re Re Re Re

2

k , k , k ,

P

,

P

,

s

, s 3 × 36 36 × 1

Re
1

Re
2

Re Re Re Re

2

k , k ,

P

,

P

,

s

, s 3 × 21 21 × 1

Re
1

Re Re Re Re

2

k ,

P

,

P

,

s

, s 3 × 10 10 × 1

None 3 × 3 3 × 1