In this work, selected results from extensive numerical simulations will be presented to assess the performances of an integrated inversion scheme based on the CSI method and the Fuzzy Logic Systems. The results will also highlight the interaction effects between the fuzzy regularization coefficients and the regularization approach based on the total variation for both single and multiple-frequency data sets

Abubakar, A.

Casagranda, Aronne

Franceschini, Davide

Massa, Andrea

van den Berg, P.M.

English

Unitn-eprints Research

       UNIVERSITY OF TRENTO  DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE  38123 Povo – Trento (Italy), Via Sommarive 14 http://www.disi.unitn.it            INTEGRATION OF A FUZZY LOGIC SYSTEM WITH THE CONTRAST SOURCE INVERSION METHOD - A NUMERICAL VALIDATION  A. Casagranda, D. Franceschini, A. Massa, P. M. van den Berg, and A.Abubakar   January 2011  Technical Report # DISI-11-255                                                                  Integration of a Fuzzy Logic System with the ContrastSource Inversion Method - A Numerical ValidationA. Casagranda1, D. Franceschini1, A. Massa1, P.M. van den Berg2 and A. Abubakar31University of Trento, Italy2Delft University of Technology, The Netherlands3Schlumberger-Doll Research, USAThe electric properties of an unknown region cannot be determined with arbitrary accuracy whenthe scenario is probed by means of electromagnetic waves. In fact, different corrupting factors (i.e.,the measurement errors and the environmental noise) and the intrinsic ill-posedness of the problemhighly affect the performances of whatever reconstruction technique. Moreover, the limited amount ofinformation on the scenario under test collectable from scattering experiments is an additional limitingfactor to be carefully taken into account addressing practical applications like sub-surface inspection,medical imaging or non-destructive testing. In order to overcome such intrinsic drawbacks, the use ofsuitable strategies is mandatory. As far as the presence of noise in the data and the measurement errorsare concerned, a possible countermeasure has been proposed and validated in [1], where an innovativefuzzy-logic-based [2] strategy has been presented. The fuzzy system (FLS) is able to provide two seriesof coefficients used as weighting parameters in the cost function, defined when recasting the inversescattering problem in term of an optimization one. Thanks to the FLS, a direct (and expensive)estimation of the noise or a priori assumptions on the level of noise are avoided, but the corruptionsand the errors in the scattering data are taken into account in an unsupervised fashion.Concerning the mathematical formulation, in [1] the problem has been stated in terms of contrastand field unknowns. In this contribution, maintaining the same FL approach for estimating the noiseimpact, the data and state equations have been redefined in terms of contrast function and contrastsources. Such a formulation has been proposed in [3] (Contrast Source Inversion method, CSI) and it isbased on the modified gradient method and on the source-type integral equation method. It has beenproved to be a very powerful algorithm, showing robustness to noise and computational efficiency.Moreover, in [4], a regularization term based on the minimization of the total variation (TV) of thescenario has been proposed to further enhance the quality of the reconstructions. Successively, to avoidthe calibration of any artificial parameter, the regularization has also been proposed as a multiplicativefactor of the original cost functional.In this work, selected results from extensive numerical simulations will be presented to assess theperformances of an integrated inversion scheme based on the CSI method and the Fuzzy Logic Systems.The results will also highlight the interaction effects between the fuzzy regularization coefficients andthe regularization approach based on the total variation for both single and multiple-frequency datasets.REFERENCES1. R. Azaro, A. Casagranda, D. Franceschini, and A. Massa, “An innovative fuzzy-logic-based strat-egy for an effective exploitation of noisy inverse scattering data,” Progress in ElectromagneticsResearch, PIERS 54, pp. 283-302, 2005.2. L. A. Zadeh, “Fuzzy sets,” Inf. Contr., vol. 8, pp. 338-341, 1965.3. P.M. van den Berg and R.E. Kleinman, “A contrast source inversion method,” Inverse Problems,vol. 13, pp. 211-277, 1997.4. P.M. van den Berg and A. Abubakar, “Contrast source inversion method: state of the art,” Progressin Electromagnetics Research, PIER 34, pp. 189-218, 2001.

Integration of a Fuzzy Logic System With the Contrast Source Inversion Method: A Numerical Validation

University of Trento : UNITN-eprints

  
 
 
 
 
 
UNIVERSITY 
OF TRENTO 
 DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE
  
38123 Povo – Trento (Italy), Via Sommarive 14 
http://www.disi.unitn.it 
 
 
 
 
 
 
 
 
 
 
 
INTEGRATION OF A FUZZY LOGIC SYSTEM WITH THE 
CONTRAST SOURCE INVERSION METHOD - A NUMERICAL 
VALIDATION 
 
A. Casagranda, D. Franceschini, A. Massa, P. M. van den Berg, and 
A.Abubakar 
 
 
January 2011 
 
Technical Report # DISI-11-255 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Integration of a Fuzzy Logic System with the Contrast
Source Inversion Method - A Numerical Validation
A. Casagranda1, D. Franceschini1, A. Massa1, P.M. van den Berg2 and A. Abubakar3
1University of Trento, Italy
2Delft University of Technology, The Netherlands
3Schlumberger-Doll Research, USA
The electric properties of an unknown region cannot be determined with arbitrary accuracy when
the scenario is probed by means of electromagnetic waves. In fact, different corrupting factors (i.e.,
the measurement errors and the environmental noise) and the intrinsic ill-posedness of the problem
highly affect the performances of whatever reconstruction technique. Moreover, the limited amount of
information on the scenario under test collectable from scattering experiments is an additional limiting
factor to be carefully taken into account addressing practical applications like sub-surface inspection,
medical imaging or non-destructive testing. In order to overcome such intrinsic drawbacks, the use of
suitable strategies is mandatory. As far as the presence of noise in the data and the measurement errors
are concerned, a possible countermeasure has been proposed and validated in [1], where an innovative
fuzzy-logic-based [2] strategy has been presented. The fuzzy system (FLS) is able to provide two series
of coefficients used as weighting parameters in the cost function, defined when recasting the inverse
scattering problem in term of an optimization one. Thanks to the FLS, a direct (and expensive)
estimation of the noise or a priori assumptions on the level of noise are avoided, but the corruptions
and the errors in the scattering data are taken into account in an unsupervised fashion.
Concerning the mathematical formulation, in [1] the problem has been stated in terms of contrast
and field unknowns. In this contribution, maintaining the same FL approach for estimating the noise
impact, the data and state equations have been redefined in terms of contrast function and contrast
sources. Such a formulation has been proposed in [3] (Contrast Source Inversion method, CSI) and it is
based on the modified gradient method and on the source-type integral equation method. It has been
proved to be a very powerful algorithm, showing robustness to noise and computational efficiency.
Moreover, in [4], a regularization term based on the minimization of the total variation (TV) of the
scenario has been proposed to further enhance the quality of the reconstructions. Successively, to avoid
the calibration of any artificial parameter, the regularization has also been proposed as a multiplicative
factor of the original cost functional.
In this work, selected results from extensive numerical simulations will be presented to assess the
performances of an integrated inversion scheme based on the CSI method and the Fuzzy Logic Systems.
The results will also highlight the interaction effects between the fuzzy regularization coefficients and
the regularization approach based on the total variation for both single and multiple-frequency data
sets.
REFERENCES
1. R. Azaro, A. Casagranda, D. Franceschini, and A. Massa, “An innovative fuzzy-logic-based strat-
egy for an effective exploitation of noisy inverse scattering data,” Progress in Electromagnetics
Research, PIERS 54, pp. 283-302, 2005.
2. L. A. Zadeh, “Fuzzy sets,” Inf. Contr., vol. 8, pp. 338-341, 1965.
3. P.M. van den Berg and R.E. Kleinman, “A contrast source inversion method,” Inverse Problems,
vol. 13, pp. 211-277, 1997.
4. P.M. van den Berg and A. Abubakar, “Contrast source inversion method: state of the art,” Progress
in Electromagnetics Research, PIER 34, pp. 189-218, 2001.


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Integration of a Fuzzy Logic System With the Contrast Source Inversion Method: A Numerical Validation

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