Image Processing Reference

In-Depth Information

where
r
inc
is the unit vector that specifies the direction of the incident field

An expression for the total field Ψ(
r
) in terms of the coordinate system
r
can be

obtained by looking at the interaction of the incident field, Ψ
inc
(
r
), and the tar-

get
V
(
r
). This relationship can be expressed as an inhomogeneous Helmholtz

equation (Avish and Slaney, 1988) as follows:

(

∇+

2

k

2

)

Ψ

()

r

=−

k

2

V

() ()

r

Ψ

r

(4.5)

The total field in Equation 4.5 can now be expressed generally as the sum of

the incident field and the scattered field as follows:

ΨΨ Ψ

()

r

=

inc
()

r

+

()

r

(4.6)

s

where Ψ
s
is the scattered field All fields in Equation 4.6 are expressed in

terms of the position as defined by the point
r
. Now the total field, Ψ(
r
), can be

expressed in terms of an inhomogeneous Fredholm integral equation of the

first kind (Morse and Feshbach, 1953) as follows:

kV

·

·

Ψ

(

rr

,

)

=

Ψ

()

r

−

2

(

r rr rr

′

)

Ψ

(

′

,

)

G

(

,

′

)

d

r

′

(4.7)

inc

inc

inc

0

D

where
G
0
(
r
,
r
′) is (free space) Green's function which is the solution of the sca-

lar Helmholtz equation (Equation 4.4), and it satisfies the differential equation

(Morse and Feshbach, 1953)

(

∇+ =−

2

kG
rr rr

2

)

(

,

′

)

δ

(

′

)

(4.8)

0

Green's function basically gives the field amplitude at any point
r
, generated

by any given point source located at
r
′. Since the modeling space in this case

is homogenous and rotationally symmetrical, Green's function can be solved

in spherical coordinates with
r
′ at the origin as follows:

(

∇+

2

kG

2

)

()

r

=− =−

δ

()

r

δ δδ

()()()

xyz

(4.9)

0

For homogenous, spherically symmetrical partial differential equations

(PDEs), the solution for Green's function in free space can be written as

Ce

r

ikr

Ue

r

−

ikr

(4.10)

G

0
(
r
=

+

Utilizing the radiation boundary condition, which is simply that the

sources cannot be present at infinity, translates to this: that only the outgoing

wave solution(s) exist(s) and therefore
U
must be equal to 0 (
U
= 0). This sim-

plifies Equation 4.10 as follows:

C

e
ikr

(4.11)

G

0
(
r
=

r

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