Image Processing Reference

In-Depth Information

=
∑

ikxkykzt

(

++−

ω

)

Er

()

E

(, )

kke

x

y

z

(2.91)

0

xy

kk

,

xy

where

k

=

ω
2

/

c

2

−

kk

2

−

2

(2.92)

z

0

x

y

However,
k
z
takes only real values if

ωπ

λ

2

2

=>+

kk

x

2

2

(2.93)

y

c

2

0

Those waves or spatial frequencies with imaginary propagation constants

carry the highest resolution information about the scattering object, but decay

exponentially away from the object. Near-field measurements, that is, gather-

ing data within a distance of ~λ from the surface of the scatterer, can capture

some of this information. More typically, the scattered field is measured many

wavelengths from the scatterer, and only the propagating scattered waves con-

tribute to the signal. This limits the resolution available about the scatter-

ing target to ~λ/2. As will be discussed in later chapters, many methods to

improve the resolution of an image exist and most implicitly involve using

some kind of cost function to extract a single solution from a space of solu-

tions which minimizes (or maximizes) some meaningful quantity such as the

total energy or entropy of a data consistent (or almost consistent) image. The

incorporation of prior knowledge (when available) about the anticipated shape

or structure of the scattering object can greatly help here, but this is frequently

only known in special cases such as medical or nondestructive testing appli-

cations of inverse scattering.

Propagation from the near to the far field is easily done by using a plane

wave expansion of the propagating wave and recognizing that propagation

in a homogeneous medium is a convolution which can be executed by taking

the Fourier transform of the wave in one plane, applying a phase shift to each

plane wave component corresponding to the distance the wave is to be propa-

gated, and then inverse transforming the results (see Appendix C).

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