Image Processing Reference
In-Depth Information
m
2
k
2
k
1
a
*
=
sin
s
- sin
s
(7.38)
k
i
i
i
-1
k
L
L
i
=1
m
2
k
2
k
= -
2
+
1
b
*
cos
s
- cos
s
i
i
i
-1
k
k
k
L
L
i
=1
A further simplification can be obtained by considering that Equation 7.28 can be expressed
in discrete form as
i
(7.39)
=
κ τ
-
i
rr
0
r
=1
where
κ
r
is the curvature (i.e. the difference of the angular function) at the
r
th point. Thus,
m
= -2 -
2
*
a
s
ii
-1
L
i
=1
2
k
m
1
*
a
= -
κ τ
sin
s
(7.40)
k
ii
i
-1
k
L
i
=1
m
2
k
m
= -
2
-
1
1
b
*
κ τ
cos
s
+
κ τ
ii
i
-1
ii
k
k
k
L
k
i
=1
i
=1
Since
m
Σ
i
=1
κ τ
= 2
(7.41)
ii
thus,
m
= -2 -
2
a
*
s
ii
-1
L
i
=1
m
2
k
1
*
(7.42)
a
= -
κ τ
sin
s
k
ii
i
-1
k
L
i
=1
2
k
m
1
*
b
= -
κ τ
cos
s
k
ii
i
-1
k
L
i
=1
These equations were originally presented in Zahn (1972) and are algebraically equivalent
to Equation 7.35. However, they express the Fourier coefficients in terms of increments in
the angular function rather than in terms of the cumulative angular function. In practice,
both implementations (Equations 7.35 and 7.40) produce equivalent Fourier descriptors.
It is important to notice that the parameterisation in Equation 7.21 does not depend on
the position of the pixels, but only on the change in angular information. That is, shapes in
different position and with different scale will be represented by the same curve γ *(
t
).
Thus, the Fourier descriptors obtained are
scale
and
translation
invariant.
Rotation
invariant
descriptors can be obtained by considering the shift invariant property of the coefficients'
amplitude. Rotating a curve in an image produces a shift in the angular function. This is
because the rotation changes the starting point in the curve description. Thus, according to
Section 7.2.3.2, the values
