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Thanks to this mathematical representation, it is very easy, from a mathematical
point of view, to estimate the homography parameters from a set of point cor-
respondences taken from two different images representing the same real-world
location at different image coordinates. Unfortunately, even the best feature
space based methods have performance limitations and in practical cases it is
not always trivial to detect and perfectly associate points from two images that
correspond to the same real-world location. It is therefore common practice to
use the RANSAC algorithm [1] to sort out putative point correspondences ob-
tained by the mean of some feature space methods. This algorithm delivers both
an estimation of the homography and a set of point correspondences which are
consistent with this estimate. Even if RANSAC is known to be very robust, it
can possibly fail and lead to results which do not reflect a rigid-body transforma-
tion. This was the motivation for the investigation reported in this paper, which
led us to clarify the fact that a homography cannot be reduced to a rigid-body
transformation and that RANSAC is not always able to reject non-rigid-body
transformations. Finally, we present a lightweight rigidity constraint that not
only allows RANSAC to avoid some degenerated homographies, but also speeds
up the whole process in unexpected proportions.
2 Backgrounds of Homography Estimation
2.1 Homography Estimation and Image Registration
A
homography
is a
projective transformation
also called
projectivity
or
col lineation
defined by an invertible mapping
h
from the projective plane
2
P
to itself that
2
maps lines to lines [2,3]. Points in
P
are described by column 3-vectors of
x
1
,x
2
,x
3
)
defining their so-called
homogeneous coordinates
.
In homogeneous coordinates, given a non-zero constant
k
, the set of vectors
(
the form
p
=(
x
3
)
describes the same point of
2
. A representation of an ar-
k
·
x
1
,k
·
x
2
,k
·
P
x
1
,x
2
,x
3
)
from
2
in the Euclidean plane defined in
2
can be
bitrary point
(
P
R
1)
of the homogeneous co-
ordinates which leads to the Euclidean coordinates
obtained by the usual normalization
(
x
1
/x
3
,x
2
/x
3
,
x
1
/x
3
,x
2
/x
3
)
.
In the same way, a point from the Euclidean plane defined by a column 2-vector
(
)
=(
(
x, y
)
in
1)
.Itisimportant
2
can be represented in
2
by the 3-vector
x, y
R
P
(
x, y,
1)
is not the unique representation of
)
in
to remember that
(
x, y,
(
x, y
P
2
)
. Hence,
as it is by definition equivalent to the set of 3-vectors
(
k
·
x, k
·
y, k
homography matrix
H
and two points
p
and
p
, the projective
transformation which maps
p
to
p
is written:
p
=
given a
3
×
3
H
·
p.
(1)
Using the homogeneous coordinates of
p
and
p
this projective transformation
can be expressed in the matrix form as:
⎛
⎞
⎛
⎞
⎛
⎞
x
x
x
3
h
1
h
2
h
3
h
4
h
5
h
6
h
7
h
8
h
9
x
x
x
3
⎝
⎠
=
⎝
⎠
·
⎝
⎠
,
(2)
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