ISPRS Commission III, Vol.34, Part 3A »Photogrammetric Computer Vision“, Graz, 2002
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APPENDIX A: PROOF OF THEOREM 1
II
P
z
m
n m’
C e C
T
Figure 8. Homography for Plane II
Homography matrix for plane II is determined by geometric
relationship between the stereo cameras (C and C') and the
plane II:
= , T 2l
Hp —Q.M (R+ zn ny JM (23)
where M and M' are matrices of interior orientation parameters
of C and C', R is the rotation matrix between C and C', T is
the translation vector from C' to C in C' 's coordinate system,
np; is a normal vector to the plane II in C 's coordinate system,
zy is the depth from the camera C to plane IT in the direction
of ng , and a is a scale factor which adjust the lower right ele-
ment of the matrix to 1. The matrix of interior orientation M isa
3 x 3 matrix:
F/s, Qu
M = 0 F/s, v (24)
0 07-1
where F is focal length, s, is the size of pixel in x direction, s
is the size of pixel in y direction, and (u, v) is coordinates of the
principal point.
Assume that two homography matrices Hg, and Hg, for two
planes IT, and IT, are given:
(25)
If ng nmn (i.e., two planes are parallel) and
Y = 0/0, , we can define another homography matrix Hg, by
a linear combination of Hg. and Hg, with parameter D :
Hy, = (-Bu *B y: Hy
ent mor t
=o, M|R +
0 Zn Z
IT, IT,
ap (26)
Now another homography matrix H(D) with parameter D :
H(D) = (-D)-Hy "DH (27)
Equation (27) can be deformed into:
H(D) = o((1- B)Hg, +B-Y- Hp) Q8)
a= ((1-D)y+D)/y
B=D/((1-D)y+D)
Il
Comparison between equation (28) and equation (26) reveals
that H(D) is a homography matrix for a plane parallel to IT, .
A - 233
