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| Titel |
A novel regularization scheme for electromagnetic inverse problems |
| VerfasserIn |
Matteo Pastorino, Claudio Estatico, Andrea Randazzo |
| Konferenz |
EGU General Assembly 2011
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 13 (2011) |
| Datensatznummer |
250058072
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| Zusammenfassung |
| In the last years, there has been an ever growing interest in the development of
electromagnetic non-destructive testing and evaluation techniques in civil engineering as well
as in several other areas [1]. Radar-based methods, such as Ground Penetrating Radar (GPR),
are now considered powerful tools for inspecting materials and structures. However, the
inversion of scattered data is still considered a difficult task in many inverse-scattering-based
reconstruction methods.
Formally, the inverse scattering problem can be modeled by an operator equation
Ax = y, where AÂ : Â XÂ - Â Y is an operator which maps the space X of the
dielectric distributions of the investigation domain with the space Y of the scattered
fields. In particular, by the knowledge of the measured scattered field y - Y (i.e.,
the known data), the aim is to recover the related dielectric distribution x - X
(i.e., the unknown solution). Many classical resolution algorithms involving the
linearization Gauss-Newton scheme allows one to approximate the solution x by means
of the minimization of the least squares cost functional Φ2(x) = ||Ax - y||L2,
where ||-
||L2 is the classical Euclidean norm of the L2 functional Hilbert space
[1]. Iterative regularization algorithms for the minimization of Φ2 give in general
oversmoothed solutions, which does not allow a well reconstruction of the discontinuities
arising in natural dielectric distributions of different objects located in the domain of
investigation.
In this work, we propose to solve the inverse scattering problem by means of the
minimization of a different cost functional Φp(x) = ||Ax-y||Lp, where ||-
||Lp is the norm
of the Lp functional Banach space, defined as ||w||Lpp = -«
|w|pdw. This way, the “size” of
the residual r(x) = Ax - y is measured by means of the metric of the Banach space Lp,
which emphasizes, for values of the constant 1 < p < 2, the points where the residual is
small with respect to the classical L2 norm. This choice gives rise to a substantial reduction
of the over-smoothing in the restored solution x. Indeed the weight of the small values of the
residual in the restoration process is enlarged, by allowing a further reduction of such a small
values. A description of the developed inversion approach and some numerical results will be
presented.
[1] M. Pastorino, Microwave Imaging. Hoboken, NJ: John Wiley & Sons, 2010. |
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