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interpolated and the corresponding layers in the two original frames of an input se-
quence, it is useful to express the flow in terms of parameters of induced 3D motion.
Hence, model parameter interpolation for the eight-parameter homography mapping
involves in decomposition of the homography matrix into structure and motion ele-
ments of the captured scene.
Homography model is appropriate for compactly expressing the induced 2D mo-
tion caused by a moving planar scene or a rotating (and zooming) camera captur-
ing an arbitrary scene. Suppose that the observed world points corresponding to an
estimated motion layer lie on a plane. Then, in homogeneous coordinates, two cor-
responding points
x
t
−
1
and
x
t
on consecutive frames
F
t
−
1
and
F
t
, respectively, are
related by
x
t
−
1
=
sP
t
x
t
,
(2)
where
P
t
is the projective homography matrix of the backward motion field and
s
is a
scale factor. The projective homography can be expressed in terms of the Euclidean
homography matrix
H
t
and the calibration matrices
K
t
−
1
and
K
t
corresponding to
time instants
t
−
1and
t
, respectively as follows:
P
t
=
K
t
−
1
H
t
(
K
t
)
−
1
.
(3)
=(
n
T
,
d
),where
n
is the
unit plane normal and
d
is the orthogonal distance of the plane from the camera
center at time
t
. The Euclidean homography matrix can then be decomposed into
structure and motion elements as follows [32]:
Suppose that the observed world plane has coordinates
π
H
t
=
R
tn
T
,
−
(4)
where
R
is the rotation matrix,
t
is the translation vector of the relative camera
motion
t
d
normalized with the distance
d
(See Fig. 3 for an illustration). Although
several approaches exist to estimate
R
,
t
,and
n
from a given homography ma-
trix [33] - [36], the decomposition increases computational complexity and requires
the internal calibration matrices to be available. It will be shown in the following
paragraphs that the decomposition can be avoided by a series of reasonable assump-
tions. For the time being, let the decomposed parameters be
R
,
t
,and
n
for the
rotation matrix, the translation vector and the surface normal, respectively.
The rotation matrix can be expressed in the angle-axis representation with a ro-
tation angle
θ
about a unit axis vector
a
[32]:
R
=
I
+
sin
(
))
[a]
x
,
θ
)
[a]
x
+(1
−
cos
(
θ
(5)
where
[a]
x
is the skew-symmetric matrix of
a
and
I
is identity matrix. The unit axis
vector
a
can be found by solving (
R
I
)
a
=
0
(i.e., finding null space of
R
I
)and
the rotation angle can be computed using a two argument (full range) arctangent
function [32]:
−
−
