Image Processing Reference
In-Depth Information
(9)
where
L
a
is the total affine arc length of the considered curve, ∧ represents the cross product
between two vectors and, ||·|| denotes the Euclidean norm.
2.2.2 Contours alignment using geometrical affine parameters estimation
We consider two closed curves
O
1
and
O
2
which define the same shape.
O
1
and
O
2
are said to be
related by an affine transformation if and only if:
(10)
where
B
is a translation vector,
A
is a linear transformation,
l
0
is the shift value,
f
and
h
are the
affine reparametrization of two contours having the same affine shape. In the Fourier space
we get:
(11)
where
U
k
(
h
) and
U
k
(
f
) are, respectively, the Fourier coefficients of
f
and
h
.
So estimation of affine motion can be resumed to the estimation of its three parameters: the
affine matrix
A
, the shift value
l
0
, and finally the scale factor
α
if we consider a normalization
under translation.
Estimation of the scale factor α
The scale factor can be estimated using the following formula:
(12)
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