scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A reconstruction procedure for microwave nondestructive evaluation based on a numerically computed green's function

20 May 2003-Vol. 53, Iss: 4, pp 987-992
TL;DR: A new microwave diagnostic tool for nondestructive evaluation based on the numerical computation of the inhomogeneous Green's function in order to fully exploit all the available a priori information of the domain under test.
Abstract: This paper describes a new microwave diagnostic tool for nondestructive evaluation. The approach, developed in the spatial domain, is based on the numerical computation of the inhomogeneous Green's function in order to fully exploit all the available a priori information of the domain under test. The heavy reduction of the computational complexity of the proposed procedure (with respect to standard procedures based on the free-space Green's function) is also achieved by means of a customized hybrid-coded genetic algorithm. In order to assess the effectiveness of the method, the results of several simulations are presented and discussed.

Summary (2 min read)

A Reconstruction Procedure for Microwave Nondestructive Evaluation based on a Numerically Computed Green's Function

  • In the framework of NDE/NDT, this paper presents a new microwave diagnostic tool.
  • Keywords: Imaging Systems, Imaging Processing, Material Characterization, Green’s Function, Genetic Algorithms I. INTRODUCTION.
  • In very few cases, the objects under test are weak enough to allow the practical use of simplified [8][9] or closed-form solutions [10].
  • Recently, the development of reconstruction procedures for microwave tomography has been addressed by resorting to the numerical discretization of the integral equations of the inverse scattering problem.
  • Section III gives a description of the optimization procedure based on a customized genetic algorithm pointing out the keypoints of its application to the NDE/NDT framework.

II. MATHEMATICAL FORMULATION

  • These operators are defined in details in [11]-[13] for the case in which the kernel is the free space Green's function [14].
  • A numerically computed Green's function for the unperturbed configuration (the configuration without the defect) is considered.
  • Equation (3) can be solved off-line and once for all by means of the moment method.
  • As an example of these computations, Figure 1 shows the amplitudes of the inhomogeneous Green’s function for a point source located at the center of the investigation domain and in correspondence with different host medium configurations.
  • After discretization of the continuous model, ( ) ( ){ }rEr toti,γℑ (equation (2)) is minimized by means of a suitable GA-based procedure [15] able to efficiently exploit the features of the proposed approach.

III. GA-BASED PROCEDURE

  • Binary tournament selection [16] and real/binary double point crossover [15][17] are used for the selection and the crossover, respectively.
  • The mutation is performed with probability mP on an individual and consists in perturbing one element of its genetic sequence.

IV. NUMERICAL VALIDATION

  • In order to assess the effectiveness of the proposed approach (in the following indicated by IGA) pointing out its efficiency by a computational point of view, some numerical simulations have been performed.
  • Figure 3 shows the images of the reconstructed distributions at different iterations.
  • Finally, in order to quantify the computational effectiveness of the proposed approach, the following parameter is evaluated: 100* ** ×−=∆ FGA IGAFGA conv K KK (7) and the obtained results are reported in Figure 5.
  • As can be observed, it results that, on an average, an IGA iteration took approximately 52 of the time necessary for the FGA iterative step.

V. CONCLUSIONS AND FUTURE DEVELOPMENTS

  • An innovative approach for the crack detection in known host medium has been presented.
  • In order to fully exploit the knowledge of the scenario under test, a new formulation based on the numerical computation of the Green’s function of the unperturbed configuration has been proposed.
  • This fact, confirmed by several numerical simulations, clearly indicates a possibility for the quasi real-time implementation of the proposed technique in real-word monitoring of industrial processes.
  • To this end, further improvements and generalizations are mandatory.
  • Let us consider the extension to more general crack shapes (and consequently the need for more complete and complicated crack parameterizations), or the possibility of dealing with multiple defects in the same host medium, or (in some specific industrial applications) the increase of the resolution capabilities.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

UNIVERSITY
OF TRENTO
DEPARTMENT OF INFORMATION AND COMMUNICATION TECHNOLOGY
38050 Povo – Trento (Italy), Via Sommarive 14
http://www.dit.unitn.it
A RECONTSRUCTION PROCEDURE FOR MICROWAVE
NONDESTRUCTIVE EVALUATION BASED ON A NUMERICALLY
COMPUTED GREENS FUNCTION
Salvatore Caorsi, Andrea Massa, Matteo Pastorino, Andrea
Randazzo, and Andrea Rosani
August 2004
Technical Report DIT-04-078

.

IMTC-7319
A Reconstruction Procedure for Microwave Nondestructive Evaluation based on a
Numerically Computed Green's Function
S. Caorsi,
1
A. Massa,
2
M. Pastorino,
3
A. Randazzo,
3
and A. Rosani
2
1
Department of Electronics, University of Pavia
Via Ferrata 1, I-27100 Pavia, Italy.
Phone: + 39 0382505661, Fax: +39 0382422583, Email: caorsi@ele.unipv.it
2
Department of Information and Communication Technologies, University of Trento
Via Sommarive 14, I-38050 Trento, Italy
Phone: +39 0461882057, Fax: +39 0461882093, E-mail: andrea.massa@ing.unitn.it
3
Department of Biophysical and Electronic Engineering, University of Genoa
Via Opera Pia 11A, 16145, Genova, Italy
Phone:+39 010 3532242, Fax:+39 0103532245, E-mail: pastorino@dibe.unige.it
AbstractIn the framework of NDE/NDT, this paper presents a new microwave diagnostic tool. The approach, developed in the spatial
domain, is based on the numerical computation of the inhomogeneous Green’s function in order to fully exploit all the available a-priori
information of the domain under test. The heavy reduction of the computational complexity of the proposed procedure (with respect to
standard free-space Green’s function based procedures) is also achieved by means of a customized hybrid-coded genetic algorithm. In
order to assess the effectiveness of the method, the results of several realistic simulations are presented and deeply discussed.
Keywords:
Imaging Systems, Imaging Processing, Material Characterization, Green’s Function, Genetic Algorithms
I. INTRODUCTION
In many tomographic approaches, in which electromagnetic waves are used to inspect dielectrics, the inversion procedure is
developed in the spatial domain (as opposed to spectral domain). In practical applications, the main interest is usually
represented by the inspection of inhomogeneous scatterers of arbitrary (bounded) cross sections (e.g., in microwave
nondestructive testing and evaluation (NDT/NDE) [1]-[7].) In very few cases, the objects under test are weak enough to allow
the practical use of simplified [8][9] or closed-form solutions [10].
Recently, the development of reconstruction procedures for microwave tomography has been addressed by resorting to the
numerical discretization of the integral equations of the inverse scattering problem. The Fredholm equation of the first kind
(i.e., the data equation governing the relation among the scattering potential, the total electric field inside the body, and the
scattered electric field at the receivers) results, as it is well known, in a highly nonlinear and ill-posed inverse problem. The
discretized version of this equation is affected by a severe ill-conditioning. The problem solution is generally addressed by
associating to the data equation the so-called state equation, i.e. the equation relating the incident and total fields inside the
scatterer. A suitable functional is constructed (often arbitrarily), whose minimization correspond to the attempt of fulfilling as
much as possible the state and data equations. To this end, present authors proposed in [11]-[13] the application of global
optimization approaches, and, in particular, the use of a hybrid-coded genetic algorithm (GA) [11]. Due to its flexibility, a GA
is able to deal with integer as well as real variables at the same time and does not require neither differentiability nor continuity

IMTC-7319
of the cost function to be minimized. However, as it well known, the main drawback of such very appealing methods (simple to
be implemented, robust, and insensitive to details of the cost function) is the high computational load. Although the continuous
increasing in the computational power of new computers tends to alleviate this problem, a pixel representation of the cross
section of an unknown complex object is still a difficult task.
However, the GA presents other advantages over several deterministic techniques. It allows the simple and straightforward
insertion of a-priori information into the model. The exploitation of a-priori information is very important in practical
applications allowing a reduction of the search space sampled by the optimization procedure and, consequently, an increase of
the convergence rate of the iterative process.
An example is represented by the NDT/NDE problem considered in the present paper. In this case, the object to be detected
is only a defect in an otherwise known object. Consequently, the inverse scattering problem is notably simplified and the use of
a GA for the retrieval of some characteristic parameters of the defect (position, dimensions, orientation, etc.) is convenient. In
this framework, the main novelty of the proposed approach lies in the use of the Green's function for the unperturbed geometry,
which can be numerically computed off-line and once for all. As a result, the only space region occupied by the defect is the
"investigation area" considered during the minimization process and the chromosomes of the GA (coding the unknowns of the
addressed problem) result greatly shortened allowing a significant reduction and of the computational burden.
The paper is organized as follows: in Section II, the mathematical formulation of the proposed approach is presented.
Section III gives a description of the optimization procedure based on a customized genetic algorithm pointing out the key-
points of its application to the NDE/NDT framework. Finally, in section IV, selected numerical results, concerning both
noiseless and noisy environments as well as lossy and lossless investigation domains, are reported in order to show the
capabilities and current limitations of the method in providing accurate defect localizations and reconstructions.
II. MATHEMATICAL FORMULATION
Let us consider an investigation area S modeled by a scattering potential,
γ
, given by
()
()
()
[]
()
[]
{}
1
1
0
+
= rrr
εωεσσωµγ
e
j
e
j
(1)
where
()
ε
σ
,
and
(
)
ee
ε
σ
,
are the conductivity and relative permittivity inside and outside S, respectively. The region S is
illuminated by a set of transverse magnetic (TM) incident fields,
()
Ii
inc
i
,...,1, =rE
. The scattered data are collected in M
measurement points (arranged around the object under test),
MjIi
scat
ij
j
,...,1,,...,1),( ==rE
. The inverse problem can be
recast as an optimization problem, where a functional is to be minimized [11]:
() ()
{
}
() ()
{
}
() ()
{}
rr
rrrr
tot
i
E
State
to
t
i
E
Data
to
t
i
E
,
,,
γ+
γ=γ
(2)
being

IMTC-7319
() ()
{}
() ()
{}
2
2
,
,
=
j
scat
ij
j
scat
ij
tot
i
tot
i
E
Data
rE
rErEr
rr
γ
γ
2
L
and
() ()
{}
() ()
{}
()
()
2
2
,
,
rE
rErEr
rr
inc
i
inc
i
tot
i
tot
i
E
Data
+γ
=γ
1
L
where
() ()
{
rEr
tot
i
,γ
1
L
and
() ()
{
}
rEr
tot
i
,γ
2
L
are nonlinear operators whose unknown functions are
() (){}
rr
σεγγ
,= (which,
in NDE applications, contains the information on the unknown defect) and
()
Ii
tot
i
,...,1, =rE
. These operators are defined in
details in [11]-[13] for the case in which the kernel is the free space Green's function [14]. In this paper, a numerically
computed Green's function for the unperturbed configuration (the configuration without the defect) is considered. The Green's
function satisfies the following equation:
() ()
()
()()
∫∫
Γ
Γ+
Γ=
Γ
S
d
I
x
x
r
r
x
x
r
r
r
r
00
γ
(3)
where
()
r
r
Γ
is the inhomogeneous Green's function,
()
r
r
Γ
0
is the free space Green's function, and
()
()
()
[]
()
[]
{}
1
1
0
+
= rrr
Ie
j
eI
j
I
εωεσσωµγ
is the scattering potential of the unperturbed geometry. Equation (3) can be
solved off-line and once for all by means of the moment method. As an example of these computations, Figure 1 shows the
amplitudes of the inhomogeneous Green’s function for a point source located at the center of the investigation domain and in
correspondence with different host medium configurations.
After discretization of the continuous model,
() ()
{}
rEr
to
t
i
,γ
(equation (2)) is minimized by means of a suitable GA-based
procedure [15] able to efficiently exploit the features of the proposed approach.
III. GA-BASED PROCEDURE
Thanks to the numerical knowledge of the Green’s function for the unperturbed scenario, let us model the defect by means
of a differential scattering potential,
γ
~
,
defined as
() () ()
rrr
I
γγγ
=
~
, which completely describes the dielectric profile of
the investigation domain and whose support is limited to the crack area (on the contrary, in [11] the whole dielectric
configuration of the investigation domain is an unknown). Then, by defining a suitable parameterization of the defect shape,
the set of unknown crack parameters can be suitably represented by means of a small subset of discrete variables
()
),...,1,(
~
Jj
j
=
γ
r
, being
j
the j-th defect discrete descriptor. Consequently, the arising unknown array results in a
variable-length hybrid-encoded “individual” obtained by concatenating the code of discrete and real-valued parameters [11]:

Citations
More filters
Journal ArticleDOI
TL;DR: An innovative inversion procedure based on the use of a genetic algorithm and on the Sherman-Morrison-Woodbury matrix inversion method is presented to reduce computational costs and hence allow a quasi real-time processing of microwave-imaging techniques.
Abstract: Industrial and biomedical applications of microwave-imaging techniques based on inverse scattering integral relations become more and more important. In order to reduce computational costs and hence allow a quasi real-time processing, an innovative inversion procedure based on the use of a genetic algorithm and on the Sherman-Morrison-Woodbury matrix inversion method is presented. Selected numerical results concerning various scenarios and scatterer dimensions are presented in order to give some indications on the effectiveness and also current limitations of the proposed approach

86 citations

Journal ArticleDOI
TL;DR: A new microwave tomography prototype for the inspection of wood materials is described and experimentally tested in this paper, composed of a set of microwave antennas, used for both illuminating the sample and collecting field measurements, a custom RF hardware, and a hybrid tomographic inversion scheme.
Abstract: A new microwave tomography prototype for the inspection of wood materials is described and experimentally tested in this paper. The system prototype is composed by a set of microwave antennas, used for both illuminating the sample and collecting field measurements, a custom RF hardware, and a hybrid tomographic inversion scheme. The antennas are placed in contact with the outer structure of the sample, e.g., a tree trunk, and are optimized to radiate inside the inspected region. A switching matrix connects the set of antennas with a custom vector network analyzer, which is used for acquiring multistatic and multiview field measurements at multiple frequencies. The measured data are processed by means of a combined qualitative/quantitative inversion method. In particular, in a first step, the field scattered by the inner inhomogeneities or inclusions is estimated, producing also a qualitative image of the sample. Then, these data are utilized by a deterministic inversion algorithm in order to obtain a quantitative reconstruction of the dielectric properties of the structure under test.

59 citations

Journal ArticleDOI
TL;DR: In this paper, a prototype of microwave tomographic system is presented and several experimental validation confirming its suitability for the use in the wood and forest product industry are reported, which is very important for the wood industry.
Abstract: Nondestructive testing and evaluation techniques able to extract information about the internal structure of the samples under test are very important in the wood industry. Microwave imaging systems have been considered for a long time promising apparatuses for this task. In this framework, approaches exploiting the full scattering phenomena for creating images of the distributions of the dielectric properties of the targets have been developed in the last few years. In this paper, a prototype of microwave tomographic system is presented and several experimental validation confirming its suitability for the use in the wood and forest product industry are reported.

36 citations


Cites background from "A reconstruction procedure for micr..."

  • ...Microwave tomographic approaches have also been considered for several other applications in industrial fields as well as in medical fields (Li and Hagness 2001, Caorsi et al. 2004, Bozza et al. 2007, Kharkovsky and Zoughi 2007b)....

    [...]

Journal ArticleDOI
TL;DR: Preliminary inversion results are obtained by applying an imaging procedure based on an iterative Gauss-Newton scheme to a realistic model of the human head related to the determination of the presence of a hemorrhagic brain stroke by retrieving the distributions of the dielectric parameters of thehuman tissues inside a slice of the head model.
Abstract: In this paper, a numerical study devoted to evaluate the application of a microwave imaging method for brain stroke detection is described. First of all, suitable operating conditions for the imaging system are defined by solving the forward electromagnetic scattering problem with respect to simplified configurations and analyzing the interactions between an illuminating electromagnetic wave at microwave frequencies and the biological tissues inside the head. Then, preliminary inversion results are obtained by applying an imaging procedure based on an iterative Gauss-Newton scheme to a realistic model of the human head. The proposed imaging algorithm is able to deal with the nonlinear and ill-posed problem associated to the integral equations describing the inverse scattering problem. The aim of the inversion procedure is related to the determination of the presence of a hemorrhagic brain stroke by retrieving the distributions of the dielectric parameters of the human tissues inside a slice of the head model.

30 citations

Journal ArticleDOI
TL;DR: In this article, a fast solution for microwave through wall imaging (TWI) with nonlinear inversion is proposed to reconstruct the unknown targets embedded in an inhomogeneous background medium.
Abstract: In this paper, a fast solution for microwave through wall imaging (TWI) with nonlinear inversion is proposed to reconstruct the unknown targets embedded in an inhomogeneous background medium. We treat inhomogeneous background, i.e., the wall around bounded in a finite domain as a known scatterer, which has the advantage of avoiding the time-consuming calculation of inhomogeneous background Green’s function. Under this scheme, a new approach under the framework of difference integral equation model, i.e., difference Lippmann–Schwinger integral equation, with modified enhanced Levenberg–Marquardt algorithm is proposed. In particular, we used a hybrid regularized technique, i.e., generalized cross-validation and truncated singular value decomposition, to stabilize the inversion. It is shown that the proposed method runs fast and is stable in presence of noise. Also, it is able to alleviate the nonlinearity and reconstruct unknown scatterers of high contrast with respect to the background. Both the numerical and experimental TWI tests validate the efficiency of the proposed inversion method.

28 citations

References
More filters
Book
01 Sep 1988
TL;DR: In this article, the authors present the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields, including computer programming and mathematics.
Abstract: From the Publisher: This book brings together - in an informal and tutorial fashion - the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields Major concepts are illustrated with running examples, and major algorithms are illustrated by Pascal computer programs No prior knowledge of GAs or genetics is assumed, and only a minimum of computer programming and mathematics background is required

52,797 citations

Book
01 Jan 2002

17,039 citations

Book
01 Jan 1992
TL;DR: Inverse Medium Problem (IMP) as discussed by the authors is a generalization of the Helmholtz Equation for direct acoustical obstacle scattering in an Inhomogeneous Medium (IMM).
Abstract: Introduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- III-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle Scattering.- Acoustic Waves in an Inhomogeneous Medium.- Electromagnetic Waves in an Inhomogeneous Medium.- The Inverse Medium Problem.-References.- Index

5,126 citations

Journal ArticleDOI
TL;DR: The distorted Born iterative method (DBIM) is used to solve two-dimensional inverse scattering problems, thereby providing another general method to solve the two- dimensional imaging problem when the Born and the Rytov approximations break down.
Abstract: The distorted Born iterative method (DBIM) is used to solve two-dimensional inverse scattering problems, thereby providing another general method to solve the two-dimensional imaging problem when the Born and the Rytov approximations break down. Numerical simulations are performed using the DBIM and the method proposed previously by the authors (Int. J. Imaging Syst. Technol., vol.1, no.1, p.100-8, 1989) called the Born iterative method (BIM) for several cases in which the conditions for the first-order Born approximation are not satisfied. The results show that each method has its advantages; the DBIM shows faster convergence rate compared to the BIM, while the BIM is more robust to noise contamination compared to the DBIM. >

1,026 citations


"A reconstruction procedure for micr..." refers background in this paper

  • ...In very few cases, the objects under test are weak enough to allow the practical use of simplified [8], [9] or closed-form solutions [10]....

    [...]

Journal ArticleDOI
TL;DR: This paper presents a tutorial and overview of genetic algorithms for electromagnetic optimization, showing genetic-algorithm optimization to be suitable for optimizing a broad class of problems of interest to the electromagnetic community.
Abstract: This paper presents a tutorial and overview of genetic algorithms for electromagnetic optimization. Genetic-algorithm (GA) optimizers are robust, stochastic search methods modeled on the concepts of natural selection and evolution. The relationship between traditional optimization techniques and the GA is discussed. Step-by-step implementation aspects of the GA are detailed, through an example with the objective of providing useful guidelines for the potential user. Extensive use is made of sidebars and graphical presentation to facilitate understanding. The tutorial is followed by a discussion of several electromagnetic applications in which the GA has proven useful. The applications discussed include the design of lightweight, broadband microwave absorbers, the reduction of array sidelobes in thinned arrays, the design of shaped-beam antenna arrays, the extraction of natural resonance modes of radar targets from backscattered response data, and the design of broadband patch antennas. Genetic-algorithm optimization is shown to be suitable for optimizing a broad class of problems of interest to the electromagnetic community. A comprehensive list of key references, organized by application category, is also provided.

855 citations


"A reconstruction procedure for micr..." refers methods in this paper

  • ...Binary tournament selection [16] and real/binary double-point crossover [15], [17] are used for the selection and the crossover, respectively....

    [...]

Frequently Asked Questions (9)
Q1. What are the contributions in "A reconstruction procedure for microwave nondestructive evaluation based on a numerically computed green's function" ?

In the framework of NDE/NDT, this paper presents a new microwave diagnostic tool. 

This fact, confirmed by several numerical simulations, clearly indicates a possibility for the quasi real-time implementation of the proposed technique in real-word monitoring of industrial processes. Let us consider the extension to more general crack shapes ( and consequently the need for more complete and complicated crack parameterizations ), or the possibility of dealing with multiple defects in the same host medium, or ( in some specific industrial applications ) the increase of the resolution capabilities. To this end, further improvements and generalizations are mandatory. 

The exploitation of a-priori information is very important in practical applications allowing a reduction of the search space sampled by the optimization procedure and, consequently, an increase of the convergence rate of the iterative process. 

Due to its flexibility, a GA is able to deal with integer as well as real variables at the same time and does not require neither differentiability nor continuityof the cost function to be minimized. 

the development of reconstruction procedures for microwave tomography has been addressed by resorting to the numerical discretization of the integral equations of the inverse scattering problem. 

In this framework, the main novelty of the proposed approach lies in the use of the Green's function for the unperturbed geometry, which can be numerically computed off-line and once for all. 

Although the continuous increasing in the computational power of new computers tends to alleviate this problem, a pixel representation of the cross section of an unknown complex object is still a difficult task. 

As can be observed, it results that, on an average, an IGA iteration took approximately 52 of the timenecessary for the FGA iterative step. 

In this Section, selected numerical results are presented in order to demonstrate the two main features of the approach:(a) the accuracy in the crack detection and estimation; (b) the reduction of the computational load with respect to the use of the method (namely the FGA) presented in [11].