Image Processing Reference

In-Depth Information

change induced by the target or scattering object must be very small in terms

of the unit wavelength. Equation 4.46 implies that this approximation begins

to break down as
V
(
r
) approaches zero. A global condition for the validity of

the Rytov approximation is

∫

∫
∇

kV

()

r

′

Ψ

() (

r

′

G

rr r

,

′

)

d

′

[

Φ

()

r

]

Ψ

() (

r

′

G

r rr

,

′

)

d

2

2

(4.47)

inc

0

s

inc

0

D

D

So, when this criterion is satisfied and the Rytov approximation is valid,

the complex scattered phase can be expressed as

ik

2

∫

Φ

()

r

=

V

()

r

′

Ψ

() (

r

′

G

rr r

,

′

)

d

′

(4.48)

s

Ψ

()

r

inc

0

inc

D

This can now be substituted back into Equation 4.40 to compute the total

field as follows:

�

∫

e
i k

(

2

/

Ψ

(

r

))

V

()

r

′

Ψ

(

r

′

)(

G

r rr

,

′

)

d

′

(4.49)

·

inc

inc

0

Ψ

(

rr

,

)

=

Ψ

( )

r

D

inc

inc

If the argument of the exponent is now isolated by dividing by the incident

field and the logarithm applied, the resulting form of this equation is

·

Ψ

Ψ

(

r r

rr

,

)

·

�
=−
k
D

Ψ

(

rr

,

)

ln

inc

2

()

r

′

Ψ

() (

r

′

G
0

r rr

,

′

)

d

′

(4.50)

inc

inc

(

,

·

)

in

c

inc

inc

This equation is comparable to Equation 4.35 from the Born approxima-

tion analysis in that it basically defines an inverse Fourier relationship or

procedure to recover
V
(
r
). Equation 4.50 can be very difficult to evaluate due

to the nature and challenges of dealing with the multivalued issues of the

natural logarithm (Fiddy et al., 2004). We will encounter the same difficulty

in Chapter 8.

Also, comparable with the Born approximation, when the conditions for

the Rytov approximation are not valid,
V
RA
(
r
) is recovered in lieu of
V
(
r
) where

1

2

·

V

(

rr

,

)

=

V

( )

r

−

[

∇

Φ

()

r

]

2

(4.51)

RA

inc

s

k

ReFeRenCeS

Avish, C. K. and Slaney, M. 1988.
Principles of Computerized Tomographic

Imaging.
New York: IEEE Press.

Chew, W. C. 1995.
Waves and Fields in Inhomogeneous Media.
Piscataway:

IEEE Press.

Darling, A. M. 1984.
Digital Object Reconstruction from Limited Data

Incorporating Prior Information.
Thesis, University of London.

Search WWH ::

Custom Search