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It will be shown now how one can compute the Homographic relationship be-
tween two central views of points. Consider two positions
F m of the cen-
tral camera (see Figure 16.1). Those frames are related by the rotation matrix R and
the translation vector t .Let(
F
m and
F m by the vector
π
) a 3D reference plane given in
π =[ n
d ],where n is its unitary normal in
F m and d is the distance from
F m .
(
π
) to the origin of
be a 3D point with coordinates X =[ XYZ ] with respect to
Let
X
F
m and
with coordinates X =[ X Y Z ] with respect to
F m . Its projection in the unit
sphere for the two camera positions is given by the coordinates X s =
ρ 1 X and
ρ 1 X . The distance d (
X s =
X , π
) from the world point
X
to the plane (
π
) is
given by the scalar product [ X 1]
· π :
ρ n X s
d .
d (
X , π
)=
(16.5)
F m can
The relationship between the coordinates of
X
with respect to
F m and
be written as a function of their spherical coordinates:
ρ RX s + t
ρ
X s =
.
(16.6)
By multiplying and dividing the translation vector by the distance d and accord-
ing to (16.5), the expression (16.6) can be rewritten as
ρ HX s +
ρ
X s =
α
t
,
(16.7)
t
d
d (
X , π
)
n and
with H = R +
. H is the Euclidean homography matrix
written as a function of the camera displacement and of the plane coordinates with
respect to
α
=
d
F m . It has the same form as in the conventional perspective case (it can
be decomposed into a rotation matrix and a rank 1 matrix). If the world point
X
belongs to the reference plane (
π
) ( i.e.
α
= 0) then (16.7) becomes
HX s .
X s
The homography matrix H related to the plane (
π
) can be estimated up to a
HX s = 0(where
scale factor by solving the linear equation X s
denotes the
cross-product) using, at least, four couples of coordinates ( X s k ; X s k ) (where k =
1
···
n with n
4), corresponding to the spherical projection of world points
X k
belonging to (
) are available then at least five
supplementary points are necessary to estimate the homography matrix by using for
example the linear algorithm proposed in [15].
From the estimated homography matrix, the camera motion parameters (that is
the rotation R and the scaled translation t d = d t ) and the structure of the observed
scene (for example the vector n ) can thus be determined (refer to [8, 28]). It can
also be shown that the ratio
π
). If only three points belonging to (
π
= ρ
ρ
σ
can be computed as
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