Image Processing Reference
In-Depth Information
7.2.3.7 Invariance
As in the case of angular Fourier descriptors, elliptic Fourier descriptors can be defined
such that they remain invariant to geometric transformations. In order to show these definitions
we must first study how geometric changes in a shape modify the form of the Fourier
coefficients. Transformations can be formulated by using both the exponential or trigonometric
form. We will consider changes in translation, rotation and scale using the trigonometric
definition in Equation 7.54.
Let us denote c ′ ( t ) = x ′ ( t ) + jy ′ ( t ) as the transformed contour. This contour is defined as
a
a
a
b
xt
yt
()
cos(
kt
kt
)
() = 1
x
0
xk
xk
2
+
(7.55)
a
b
k
=1
sin(
)
y
0
yk
yk
If the contour is translated by t x and t y along the real and the imaginary axes, respectively,
we have that
a
a
a
b
t
t
xt
yt
()
cos(
kt
kt
)
() = 1
x
0
xk
xk
x
(7.56)
2
+
+
a
b
sin(
)
k
=1
y
0
yk
yk
y
That is,
a
+ 2
+ 2
t
a
b
xt
yt
()
cos(
kt
kt
)
() = 1
x
0
x
xk
xk
(7.57)
2
+
a
t
a
b
k
=1
sin(
)
y
0
y
yk
yk
Thus, by comparing Equation 7.55 and Equation 7.57, we have that the relationship between
the coefficients of the transformed and original curves is given by
aa bbaabb
=
=
=
=
for
k
0
xk
xk
xk
xk
yk
yk
yk
yk
(7.58)
aa t
=
+ 2
aa t
=
+ 2
x
0
x
0
x
y
0
y
0
y
Accordingly, all the coefficients remain invariant under translation except a x 0 and a y 0 . This
result can be intuitively derived by considering that these two coefficients represent the
position of the centre of gravity of the contour of the shape and translation changes only
the position of the curve.
The change in scale of a contour c ( t ) can be modelled as the dilation from its centre of
gravity. That is, we need to translate the curve to the origin, scale it and then return it to its
original location. If s represents the scale factor, then these transformations define the
curve as,
a
a
a
b
xt
yt
()
cos(
kt
kt
)
() = 1
x
0
xk
xk
2
+
s
(7.59)
a
b
k
=1
sin(
)
y
0
yk
yk
Notice that in this equation the scale factor does not modify the coefficients a x 0 and a y 0
since the curve is expanded with respect to its centre. In order to define the relationships
between the curve and its scaled version, we compare Equation 7.55 and Equation 7.59.
Thus,
asa bsbasabsb
=
=
=
=
for
k
0
xk
xk
xk
xk
yk
yk
yk
yk
(7.60)
aaaa
x
=
=
0
x
0
y
0
y
0
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