Camera Calibration for 3D Reconstruction and View Transformation - 3D Modeling and Animation: Synthesis and Analysis Techniques for the Human Body - page 108

plane lie on the same line in the image; this line is called the vanishing line (Wang
& Tsai, 1991); 2) Given the vanishing points of three sets of mutually orthogonal
lines, the orthocenter of the triangle with the three vanishing points as vertices
is the principal point; 3) Given the vanishing points

[ ] [ ]

T T

x y and x y of

1 1 2 2

2

two sets of orthogonal lines, it can be shown that: xx (Guillou,

Meneveaux, Maisel & Bouatouch, 2000); 4) If the camera moves, the motion of
the vanishing points in the image plane depends only on the camera rotation, not
on the camera translation. The vanishing points of three non-co-planar sets of
parallel lines fully determine the rotation matrix (Guillou et al., 2000); and 5)
Given the vanishing points

+ y y + f = 0

1 2 1 2

[ ] [ ] [ ]

T

x 3 of three sets of
mutually orthogonal lines, from equation 5 it can be verified immediately that
(Cipolla, Drummond, & Robertson, 1999):

T

,

T

,

x y x y y

1 1 2 2

1 0 0

λ x λ x λ x

1 1 2 2 3 3

0 1 0

~ ~

()

T

λ y λ y λ y = P ⋅ = K R

w
c 1 1 2 2 3 3

0 0 1

,

λ λ λ

1 2 3

0 0 0

where λ 1 , λ 2 , and λ 3 are unknown scaling factors.

Special shape calibration objects

Under linear perspective projection, the image of a circle is a sphere, whose
major axis is on a line passing through the principal point. The eccentricity of the
ellipse is a function of the focal length f , the distance of the center of the ellipse
from the principal point, and the length of the major axis. The principal point can
thus be located by intersecting major axes of several spheres in the image space.
The focal length is then calculated from the eccentricity. Subsequently, other
intrinsic and extrinsic parameters can be recovered. Most recently, concentric
circles were employed to get the correct projected circle center (Kim & Kweon,
2001).
Taking the camera distortion into consideration, the sphere is curved. Assuming
only k 1 not being zero, the curved sphere is a fourth order polynomial, from which
the aspect ratio s y / s x can be computed directly. Besides circles, parallelepipeds
have also been used (Wilczkowiak, Boyer & Sturm, 2001).