Image Processing Reference

In-Depth Information

Eight

Homomorphic (Cepstral) Filtering

8.1 CepStRAl FIlteRIng

As previously discussed in Section 4.3, the Born approximation seems to gen-

erally perform well when the target is a “weak” scatterer, since one can replace

the total field inside the target by the known incident field, and this leads

naturally to the Fourier transform relationship described earlier. However,

as the permittivity of the target increases, the performance of “Born” algo-

rithm tends to decrease. This is not unexpected due to the fact that less of

the incident wave might be penetrating and propagating through the target,

but is reflected off the surface of the target as well as being scattered multiple

times from inhomogeneities that exist inside the target. The reflected wave

emerging from more highly structured scattered field components needs to be

interpreted as carrying information about
V
(
r
) (see Equations 4.7 and 4.28) or,

depending on how the scattered field is processed, as noise-like terms arising

from strong scattering that one might be able to remove. In the latter case, if

the noise-like terms can be identified and removed or attenuated, then this

can ideally reduce a strongly scattering target to one that can be imaged more

like a weakly scattering one. In the cepstral method, the total field estimated

within the target volume is regarded as a form of spatial noise to be removed.

When this is justified, the reconstructed image of the target based on assum-

ing the first Born approximation can be expressed as

·

Ψ

Ψ

(,

rr

rr

)

·

inc

V

(

rr

,

)

≈

V

( )

r

(8.1)

BA

inc

·

(

,

)

in
c

inc

·

·

inc inc inc
is a symbolic representation for a complex

and noise-like term with a characteristic range of spatial frequencies dom-

inated by the average local wavelength of the source. The problem is now

reduced to a complex filtering problem in which the multiplicative term

〈

where 〈

Ψ

(,

rr

)/

Ψ

(,

rr

)

〉

·

·

inc inc inc
needs to be filtered out of the data.

There are a number of ways to approach this problem, but one of the more

recent methods is a well-known and documented method based on cepstral

filtering, a technique originally developed to eliminate multiplicative noise

(Childers et al., 1977; Raghuramireddy and Unbehauen, 1985). The homomor-

phic filtering method uses the log operation to convert a multiplicative modu-

lation relationship into an additive relationship. This is demonstrated here by

taking the complex logarithm of Equation 8.1 as follows:

Ψ

(,

rr

)/

Ψ

(,

rr

)

〉

103

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