Camera Calibration for 3D Reconstruction and View Transformation - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 103

Preferably, all camera parameters (including extrinsic and intrinsic parameters,
and distortion coefficients) should be optimized simultaneously. To do so, usually
a certain iterative technique called bundle adjustment is adopted (Triggs et al.,
1999). Among them, the Levenberg-Marquardt method is probably the most
extensively employed, due to its robustness.

Conclusions

As it combines the linear initialization and the nonlinear full-scale optimization,
the iterative two-phase strategy can provide very accurate calibration results
with reasonable speed. It is now employed extensively. There exist many
variations of it aiming at different compromises between accuracy and effi-
ciency as described above. However, for a complete passive calibration system,
the design of the calibration object also plays a quite important role. The next
section will introduce a simple but effective calibration object.

Planar Pattern Based Calibration

Various 2-D planar patterns have been used as calibration targets. Compared
with 3-D calibration objects, 2-D planar patterns can be more accurately
manufactured and fit easier into the view volume of a camera. With known
absolute or relative poses, planar patterns are a special type of 3-D calibration
object. In this case, traditional non-co-planar calibration techniques can be
applied directly or with very little modification (Tsai, 1987). More often, a single
planar pattern is put at several unknown poses to calibrate a camera (Zhang,
2000). Each pose of the planar pattern is called a frame . It has been demon-
strated that this is already adequate for calibrating a camera. The iterative two-
phase strategy discussed can still be applied here. For planar patterns, only phase
1 is different and it is discussed below.

Recovering of linear geometry

[ ]

T

Assume a linear camera model. For an arbitrary point in the

x

o

= x

o

y

o

0

w
o

w
o

calibration plane with orientation R and position t , we obtain from equation 5

() () () ()

~ ~ ~ ~

[ ]

~

[ ]

~ ~ ~ ~

T T T T

x

im

≅ P ⋅ x

w

= K ⋅ M ⋅ x

w

= K ⋅ R

w
c

− R

w
c

⋅ t

w
c

⋅ x

w

= K ⋅ R

w
c

⋅ x

w

− R

w
c

⋅ t

w
c

[ ] [ ]

~

()( ) ()

~

() ()( )

~

T T T T

= K ⋅ R

w
c

⋅ R

w
o

x

o

+ t

w
o

− R

w
c

⋅ t

w
c

= K ⋅ R

w
c

⋅ R

w
o

R

w
c

⋅ t

w
o

- t

w
c

⋅ x

o

, (45)

~

] [ ]

~

] [ ] [ [

o o o o

= K ⋅ R t ⋅ x y 0 1 = K ⋅ r r t ⋅ x y 1

1 2