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)

Search WWH ::

Custom Search