Matchmoving - Computer Vision for Visual Effects - page 220

. 8 Since we know the
form of the camera calibration matrix in terms of the internal parameters from
Equation ( 6.3 ), we can verify that

we can see that Equation ( 6.22 ) is linear in the elements of ω

1
α

x 0
α

0 −

x x

y 0
α

1
α

0 −

ω = (6.24)

y y

x 0
α

y 0
α

x 0
α

y 0
α

− −

x +

+ 1

x y y

, and each
projective transformation H i puts two linear constraints on them via Equation ( 6.22 ).
Thus, we need at least three images of the plane in different positions to estimate

That is, there are five unique parameters (ω

11 ,

ω

13 ,

ω

22 ,

ω

23 ,

ω ) in ω

33

. 9 ω

Since these linear equations are of the form

ω 33 ] =

A i [ ω 11 ,

ω 13 , ω 22 , ω 23 ,

0 (6.25)

we can again use the Direct Linear Transform to estimate the values of up to a scalar

multiple. Finally, we can recover the actual internal parameters by taking ratios of
elements of

ω

, namely: 10 ω

=− ω

13
ω 11
y 0 =− ω

x 0

23
ω 22

2

(6.26)

13 23

ω ω ω − ω ω − ω ω

11 22 33 22 11

α =

y

2
22

ω ω

11

y ω

1
2

22
ω 11

α = α

x

This algorithm was proposed by Zhang [ 575 ] and independently by Sturm and
Maybank [ 473 ], who also analyzed planar configurations where the method fails.
The solution obtained from the linear estimate can be used as the starting point for
subsequent nonlinear estimation of the camera parameters with respect to image
reprojection error (see Section 6.5.3 ). Once the camera calibration matrix K is deter-
mined, we can determine the parameters r 1 , r 2 , r 3 , t i corresponding to the i th position
of the camera (see Problem 6.9 ). If the images of the plane suffer from lens distortion,
we can alternate the estimation of the internal parameters using Zhang's algorithm
with the estimation of the lens distortion parameters

using Equation ( 6.7 ).
Figure 6.7 shows an example camera calibration result using this approach with
nine images of a checkerboard (four of these images are shown in Figure 6.6 ). In this
example, the focal length was computed as 552 pixels (corresponding to 3.1mm for
this camera), the principal point was found to be at the center of the image, and the
pixels were found to be square ( d y

κ

/ d x = 1).

8 For reasons we won't go into here,

is also called the image of the absolute conic . See Section 6.8 .
9 These positions must be non-coplanar to avoid linear dependence of the equations.
10 These equations can be viewed as the explicit solution of the Cholesky decomposition

ω

ω − 1 KK . =