Image Processing Reference
In-Depth Information
discrepancy between the simulated and the measured scattered field ampli-
tude. Harada et al. (1995) defined this functional as
M
N
11
(
)
(
) 2
XV
[
( )]
r
=
fV
();
r
φφ φφ
() ()
m
;
f
()
m
;
s n
(()
n
(7. 5)
est
est
inc
inc
s
m
==
n
where f is the simulated scattered field amplitude which is calculated using an
estimated object function V est ( r ), f is the measured scattered field amplitude
and X [ V ( r )] is the norm of the difference between the measured and simulated
scattered field amplitude. The goal of the optimization method, which in this
case is the conjugate gradient, is to find the ideal object function V est ( r ), which
minimizes the functional X [ V ( r )]. The gradient direction of the functional is
found by using the Fréchet derivative (Harada et al., 1995; Lobel et al., 1996;
Takenaka et al., 1992). Similar to the Born iterative method, the forward scat-
tering problem is solved using the method of moments (Richmond, 1965) with
the pulse basis functions and point matching, which transforms the integral
equations into matrix equations. The conjugate gradient method shows good
immunity to noise levels, and its convergence rate can be increased by using
a priori information about the outer boundary of the object.
7.5 pRIoR dISCRete FouRIeR tRAnSFoRM (pdFt)
The PDFT is a processing step that all methods could potentially benefit from.
If we consider function V Ψ in Equation 7.1, in practice, the analyticity of the
estimate for V Ψ is assured because of the fact that the data available are always
limited in k -space. However, processing this assumes a good estimate of the
separable function V Ψ, since it is necessary to separate V from Ψ. Since V is
assumed to be of finite extent, then V Ψ should also be of compact support;
hence, the data in k -space should be analytic. In principle, one may use analyti-
cal continuation of the scattered field data in k -space to obtain a better estimate
of V Ψ as a product rather than a low pass filtered function and hence a band-
limited function. However, it has been observed by several authors (Habashy
and Wolf, 1994; Remis and van den Berg, 2000; Roger, 1981) that analytic con-
tinuation is not practical because of its instability and sensitivity to noise. The
inverse scattering algorithms are Fourier based in nature, and interpolating
and extrapolating the Fourier data lying on semicircular arcs in k -space can
accomplish improvement in the image quality of the recovered object.
A very stable (regularizable) spectral estimation method known as the
PDFT (Byrne et  al., 1983; Shieh and Fiddy, 2006), which gives a minimum
norm solution of the estimate for V Ψ using properly designed Hilbert spaces,
is described here. The success of the PDFT algorithm relies on its flexibility
and effective encoding of prior knowledge about the object. The PDFT is a
Fourier-based estimator, so it is easy to incorporate it into most signal process-
ing methods. More details of the PDFT algorithm can be found in Shieh and
Fiddy (2006), Darling et  al. (1983), and the appendices, but in summary, the
PDFT assumes that Fourier information about the product of V Ψ is available,
which is precisely our measured data in k -space. Let
f
()
r
= Ψ
V
() ()
r
r
(7. 6)
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