Camera Calibration for 3D Reconstruction and View Transformation - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 104

()

[ ]

T

r (

where

w
c

w
o

( j = 1 3 ) is the j th column of the

R = R ⋅ R = r r r

1 2 3

matrix R ) and

()( )

T

.

t = R

w
c

⋅ t

w
o

− t

w
c

Thus:

[ ]

[ ]

~

~ ~ ~

where . (46)

x

im

≅ H ⋅ x

o

y

o

1 H = K ⋅ r r t

1 2

By applying the simple DLT method introduced, with at least four pairs of
corresponding

[ ] [ ]

, ~ can be determined up to a non-

T T

and

x

im

y

im

x

o

y

o

~

zero factor as H , which means .

H ≅ H

~ −

of ~

1

As the inverse K exists (ref. equation 32), equation 46 can be rewritten

as:

[ ]

~ ~ ~

− 1 − 1

. (47)

r r t = K ⋅ H ≅ K ⋅ H

1 2

Thus:

~ ~

r K

− 1

h and r K

− 1

h ,

= ρ ⋅ = ρ ⋅

1 1 2 2

where is a non-zero factor and h j ( j = 1...3) is the j th column of matrix H .

Because ( r 1 ) T r 1 = ( r 2 ) T r 2 = 1, it turns out that:

ρ

() (

)

() (

)

~ ~ ~ ~

T T

− T − 1 − T − 1

, h ⋅ K K ⋅ h = h ⋅ K K ⋅ h (48)

1 1 2 2

() (

)

~ ~

T

, (49)

− T − 1

h ⋅ K K ⋅ h = 0

1 2