Image Processing Reference
In-Depth Information
2.3.2
P
ARAMETRIC
M
ETHODS
Parametric image models often take the following form:
∑
ρ
()
x
=
c
x
nn
ϕ
( ,
(2.9)
n
ϕ
n
(
x
) are the basis functions used to absorb any
a priori
information and
c
n
are the series coefficients chosen to match the measured data.
Selecting a set of “good” basis functions is essential for the model in Equation 2.9.
A particular set of basis functions is given in the form of weighted complex
sinusoids [5,6]:
where
ϕ
=
Cxe
()
i
2
π
n x
∆
,
(2.10)
n
where
C
(
x
) is a nonnegative function incorporating
a priori
information. With
this set of basis functions, the model, known as the
generalized series
(GS)
model
[5,6], becomes
∑
ρ
()
xCx ce
n
=
()
i
2
π
n x
∆
.
(2.11)
n
This model has several useful properties. Specifically, when no nontrivial
a priori
information is available, namely,
C
(
x
)
1, Equation 2.11 automatically reduces to
the conventional Fourier series model. This is desirable because the Fourier series
model is indeed optimal in this case. On the other hand, if
C
(
x
)
=
(
x
), the multi-
plicative Fourier series factor will be forced to unity by the data-consistency
constraint, and a perfect reconstruction will result. In general, if
C
(
x
) is properly
chosen, the new basis functions given in Equation 2.10 enable the GS model to
converge faster than the Fourier series model. Therefore, within a certain error
bound, fewer terms can be used to represent an image function than are required
by the Fourier series method, leading to a reduction of the truncation artifact.
The optimality of the GS model in Equation 2.11 can also be justified from the
minimum cross entropy principle [7].
Selection of the weighting function
C
(
x
) is application dependent. For the
limited data reconstruction problem, it was suggested [8] that
C
(
x
) be chosen to
be a summation of boxcar functions as
=
ρ
M
x
−
ββ
ββ
1
2
(
+
)
∑
∏
Cx
()
=
a
mm
+
1
,
(2.12)
"
m
−
!
m
+
1
m
m
=
1
where
M
represents the number of boxcar functions in the model, and
β
m
and
a
m
are the edge locations and amplitude of the
m
th boxcar, respectively. This function
is particularly suitable for image functions containing sharp edges because they
are explicitly built into the basis functions.
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