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Then the definition of the cross product
V
n
×
V
n
+1
can also be expressed as the
determinant of the following matrix, where
i
,
j
,
k
are the unit vectors of the
standard basis:
⎛
⎞
i
j
k
V
x
n
V
y
n
0
V
x
n
+1
V
y
n
+1
0
⎝
⎠
,
V
n
×
V
n
+1
=
det
(10)
which gives:
V
n
+1
=
V
x
n
·
V
x
n
+1
·
k.
V
n
×
V
y
n
+1
−
V
y
n
·
(11)
Based on this formulation, the final expression of the rigidity constraint (8) can
be written as:
sign
V
x
n
·
sign
V
x
n
·
V
x
n
+1
=
V
y
n
+1
−
V
y
n
·
V
x
n
+1
V
y
n
+1
−
V
y
n
·
∀
n
=1
..
4
,
V
1
.
(12)
This rigidity constraint has to be added to the
is_not_degenerated()
test
function of Algorithm 1. It must be noticed that the collinearity test commonly
suggested can also be performed by evaluating the cross product
V
n
×
with
V
5
=
V
1
and
V
5
=
V
n
+1
.If
it equals zero, the three consecutive points are collinear and if it is very small,
they are quasi-collinear. Thus, the rigidity constraint we are proposing does not
requier any additional processing cost.
Even if the rigidity constraint proposed here has been investigated following
our own approach for the purpose of homography estimation, it can be related
to an extension of the geometrical constraint proposed in [6] for ane homo-
graphies. Those simplified homographies have only six degrees of freedom and
are obtained from only three point correspondences [3]. Our approach should
hopefully strengthen the assumption made in [6] concerning an extension of the
geometrical constraint for ane homographies to the case of projective transfor-
mations.
Additionally, when the chain codes identified in the first row are compared
to those immediately below them in the second row of figure 3, it clearly ap-
pears that they represent a mirrored version of the same oriented path and that
they only differ by their signs. This makes it quite trivial to loosen the rigidity
constraint in order to allow the registration of images even if a number of them
are mirrored images. This can be useful in cases where it is necessary to register
images from film negatives or slides, the correct side of which is unknown. The
rigidity constraint has then to be adapted; it simply consists of changing the
sign of the elements of one of the chain codes to be compared, if and only if, the
first elements tested in the two chain codes differ.
4 Evaluation Tests
In the context of image registration, it is clear that if point correspondences are
perfectly chosen and therefore are all inliers, the resulting homography will be
a rigid-body transformation. Besides, if all the point correspondences are inliers
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