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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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