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)

Search WWH ::

Custom Search