Image Processing Reference
In-Depth Information
That is, under dilation, all the coefficients are multiplied by the scale factor except
a
x
0
and
a
y
0
which remain invariant.
Rotation can be defined in a similar way to Equation 7.59. If ρ
represents the rotation
angle, then we have that
a
a
a
b
xt
yt
()
cos(
)
sin(
)
cos(
kt
kt
)
()
=
1
x
0
xk
xk
(7.61)
2
+
- sin(
)
cos(
)
a
b
k
=1
sin(
)
y
0
yk
yk
This equation can be obtained by translating the curve to the origin, rotating it and then
returning it to its original location. By comparing Equation 7.55 and Equation 7.61, we
have that
aa
=
cos(
) +
a
sin(
)
bb
=
cos(
) +
b
sin(
)
xk
xk
yk
xk
xk
yk
b
= -
b
sin(
) +
b
cos (
)
(7.62)
a
= -
a
sin(
) +
a
cos(
)
yk
x k
yk
yk
x k
yk
= =
That is, under translation, the coefficients are defined by a linear combination dependent
on the rotation angle, except for
a
x
0
and
a
y
0
which remain invariant. It is important to notice
that rotation relationships are also applied for a change in the starting point of the curve.
Equations 7.58, 7.60 and 7.62 define how the elliptic Fourier coefficients change when
the curve is translated, scaled or rotated, respectively. We can combine these results to
define the changes when the curve undergoes the three transformations. In this case,
transformations are applied in succession. Thus,
aaaa
x
0
x
0
y
0
y
0
as a
= (
cos(
) +
a
sin(
))
bs b
= (
cos (
) +
b
sin (
))
xk
xk
yk
xk
xk
yk
′
as
= (-
a
sin(
) +
a
cos(
))
bs
= (-
b
sin (
) +
b
cos (
))
(7.63)
yk
x k
yk
yk
x k
yk
= + 2 = + 2
Based on this result we can define alternative invariant descriptors. In order to achieve
invariance to translation, when defining the descriptors the coefficient for
k
= 0 is not used.
In Granlund (1972) invariant descriptors are defined based on the complex form of the
coefficients. Alternatively, invariant descriptors can be simply defined as
aa t
aa t
x
0
x
0
x
y
0
y
0
y
|
A
A
|
|
+
|
B
B
|
k
k
(7.64)
|
|
|
1
1
The advantage of these descriptors with respect to the definition in Granlund (1972) is that
they do not involve negative frequencies and that we avoid multiplication by higher frequencies
that are more prone to noise. By considering the definitions in Equations 7.51 and 7.63 we
can prove that,
2
2
2
2
aa
+
bb
+
|
A
A
|
and
|
B
B
|
xk
yk
xk
yk
k
k
|
=
|
=
(7.65)
|
|
aa
2
+
2
bb
2
+
2
1
1
x
y
x
y
These equations contain neither the
scale
factor,
s
, nor the
rotation
,
. Thus, they are
invariant
. Notice that if the square roots are removed then invariance properties are still
maintained. However, high-order frequencies can have undesirable effects.
The function
EllipticDescrp
in Code
7.3
computes the elliptic Fourier descriptors
