ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002
homography). Therefore, the function H in equation (4) is pro-
jective transformation. Projective transformation has eight inde-
pendent parameters. Projective transformation between
homogeneous coordinates (x,, y,, 1) and (X4, y, 1) can be
expressed with the following equation:
y3| = |84 8s àg| |y, (8)
1 a, à, l||1
where = means equality up to scale factor, that is:
[X Y.Zig[X Y, 2/] X33 :Z 9X :Y 2: .. (9)
The matrix on the right side of equation (8) is called homogra-
phy matrix.
Figure 3. Homography H for Plane A
Two linear equations can be derived by substituting a pair of cor-
responding points between I, and L, into equation (8). This
means that 8 parameters of the projective transformation can be
determined with four pairs of corresponding points. Three of
those can be given by control pairs that are used in the definition
of the parameter D in equation (7). Next we show that epipolar
geometry provides the fourth pair of corresponding points; epi-
poles.
E, (Epipole on Ij)
| O, (Optical Center of I)
O, (Optical Center of I.)
E; (Epipole on I;)
Oz NU Lue
(Intersection betwee — b
Line OO, And Plane 4»
Figure 4. Epipoles
A - 230
Epipoles E, and E, are intersection points between image
planes and the straight line O,O, which goes through two opti-
cal center O, and O, as shown in Figure 4. Note that epipoles
are the projection of the point Q, , where O,O, intersects with
plane À . In other words, these epipoles are also corresponding
points for plane A .
Epipoles are known to have the properties below.
1) Epipoles are independent from geometry of plane A and
determined only by epipolar geometry. Epipoles can be com-
puted by substituting optical centers for P in equation (1):
E, = F,(0,)
E, = F,(04) (10)
2) All epipolar lines go through epipoles.
3) When an image plane is parallel to O0, , epipolar lines are
also parallel and epipole is at infinite distance.
4.1 Determination of H with D and Epipoles
From three control pairs and a pair of epipoles gives eight linear
equations for eight unknown parameters of homography matrix.
These equations can be written in matrix style as follows:
M.B-V (11)
t . :
where B = (a), a,, a5, A4, as, ag, a7, 8g], M isa 8 xX 8 matrix,
and V is 8 dimensional vector.
Homogeneous expression of projective transformation allows
handling epipoles in infinite distance. For example, when all epi-
polar lines are parallel to x axis both in I, and I, , homogeneous
coordinates of epipoles are [1, 0, 0] . Substitution of this coordi-
nates to both side of equation (8) leads to:
à, = 0
a, = 0 (12)
5. OPTIMIZATION WITH LEAST SQUARE METHOD
We have adopted Gauss-Newton method for non-linear least
square optimization. This method revises target parameters D
with AD :
D+AD=D (13)
AD can be computed by the following equation:
t
As cd ee (14)
