RECONSTRUCTION OF 3D FLAWS IN EDDY CURRENT
QUANTITATIVE NON DESTRUCTIVE EVALUATION
A. Baussard, D. Premel, and J. M. Decitre
Laboratoire Systemes et Applications des Technologies de 1'Information et de 1'Energie
(SATIE - CNRS UMR 8029), 61 av. du President Wilson, 94235 Cachan, France.
ABSTRACT. The goal of this paper is to localize and reconstruct the shape of 3D flaws in
conductive materials using a particular eddy current setup. To solve the non linear and ill-posed
inverse problem, a bayesian estimation framework has been considered. In order to compensate the
lack of information due to the band-pass behavior of the forward operator, a priori knowledge is
introduced by using the beta prior law and the weak membrane model. To show the capabilities of the
proposed approach, some 3D reconstructions from simulated data are presented.
INTRODUCTION
The examination of conductive materials using eddy current non destructive
evaluation techniques is a central problem in the inspection of structures or products. In
this paper, the problem of evaluating a volumetric flaw from observed data is investigated.
This problem requires to design and model a suitable system for generating eddy currents
and collecting the data. From these data, the inverse problem is solved in order to
reconstruct the spatial variations of the conductivity and thus the corresponding flaws. In
this contribution, the emission and reception functions of the probe are considered
separately. The response of flaws is observed by the changes in the normal component of
the magnetic field which is measured by a pick-up coil moved above the workpiece, the
driving coil being fixed at an appropriated position. Then, each position of the driving coil
provides a set of observed data like a tomographic imaging system.
The forward model of the considered device is based on a set of coupled integral
equations using Green's dyads. In this case, the damaged zone has been considered as an
isotropic and homogeneous half-space conductor. The flawed region is then discretized
into volumetric cells using a variant of the moment method [3] called the Galerkin method.
In order to solve the associated non linear and ill-posed inverse problem, a bayesian
estimation framework has been considered. This approach allows to introduce a priori
knowledge on the object to be reconstructed in order to compensate the lack of information
due to the band-pass behaviour of the forward operator. In this contribution, we propose to
combine two different a priori knowledge by considering the beta law and the weak
membrane model. Finally, the inverse problem is solved by minimizing a criterion which
combines a quadratic data fitting term and the two regularizing energies. Although the
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
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convergence of the algorithm can not be rigorously demonstrated, the numerical results
show the potential of the proposed approach.
The paper is organized as follows. Section 2 describes the forward problem.
Section 3 presents the inverse problem and the regularization scheme. In section 4, some
numerical results are presented. Finally, section 5 gives some conclusive remarks.
THE FORWARD PROBLEM
The defects in the flat conductive material are detected and located by means of
eddy currents. They are localized by local variations of an electrical parameter which
differs from those of the host material. Therefore, a normalized contrast function is
defined:
f(r) =
<J 0 -cr(r)
(1)
in order to characterize them. The host material is isotropic, homogeneous, non magnetic
and its conductivity GO is assumed to be known. The quantity o(r) denotes the conductivity
variations of the damaged zone.
Figure 1 shows the considered device. It consists of a driving coil, fixed at a
particular position, and an auxiliary pick-up coil which scans the interest area. The driving
coil creates the electrical incident field E$(r) within the workpiece. Induced currents in the
domain D interact with the flaw and let E2(r) be the total electrical field inside D. Then the
state equation is defined as follows:
(2)
The dyadic Green's functions Gn, where the subscripts denote the coupled regions
(observation-source), are of electrical type and they are computed for a conductive half
space [2]. Contrary to typical devices which measure the perturbed probe impedance, in
this contribution, we have chosen to measure the e.m.f which is forwardly related to the
normal component of the magnetic field inside the air region.
Region
Driving coil
FIGURE 1. Device: a driving coil creates eddy currents in the flat conductive workpiece, a pick-up coil is
scanning the domain D representative of the volumetric flaw.
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Defining the position of the pick-up coil by the measurement point rm(xm,ym,Zni), the
e.w./induced in a turn of the pick-up coil is defined by:
(r)dl
(3)
where E\ is the observed electrical field in R\ defined by:
£, (r) = /«//0(T0 jG12(r,r')/(r')£2 (r')dr'
(4)
D
where GU is the electric-electric dyadic Green's function, the components of the electric
field are observed at point r in region R\ whereas the point source is located in region R2.
Then, the set of data, related to the contrast function is:
e.m.f(rm) = \Tn(rm,r')f(r')E2
(r')dr'
(5)
D
where the dyadic function F^ is defined as:
12 (r,r')£//.
(6)
The discretization of the two coupled equation (2) and (5) is performed by
subdividing the damaged zone into a regular mesh of N=NxxNyxNz cells. Then, using the
Galerkin method, a variant of the moment method, to discretize the system, the two
coupled integral equations can be written as a system of two coupled matrix equations:
eQ=e2+G22Fe2
where CQ is the 3N (three components x, y, z of the N cells) incident field vector and €2 is
the 3N total field vector. The observation vector y containing the M values resulting from
the pick-up coil scan. TU and Gn are the [Mx3N] and [3Nx3N] matrix derived from the
Green functions and F is a diagonal matrix which contains the values of the contrast
function.
Finally, this model can then be written as a non linear equation relating contrast
function and measured data :
)-^e0.
(8)
THE INVERSE PROBLEM
Bavesian Approach
The goal of the inverse problem is to retrieve the shape of the flaw from the
observed data. In order to solve this non linear and ill-posed problem, a bayesian
estimation framework has been considered. This approach allows to introduce a priori
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knowledge on the object function as a prior probability distribution function. Looking for
piecewise continuous images, the a priori object model is introduced thanks to the GibbsMarkov model:
p(/)ocexp[-£/(/)]
(9)
where U(j) is the energy function of the prior law on the object.
Considering the solution in the sense of the maximum a posteriori (MAP) estimate leads
to maximise the a posteriori density:
where, from the Bayes' rules, the posterior law is given by:
(ID
where p(y) is a normalization constant, p(f/y) is the likelihood of the data to be estimated.
Assuming a zero mean, white and Gaussian noise with known variance, the estimate of/
leads to the minimization of the following criterion:
where Q(/) = ||y-G12F(7 + G22F)~1e0 1 is the data fitting term and y is a weighting
parameter.
Beta Law
The sought object function to be retrieve corresponds to a relative conductivity (see
equation (1)). When the function corresponds to a flaw,/fr) is equal to 1 and within the
homogenous area, f(r) is equal to 0. Then, f(r) takes its values in [0,1] with a strong
probability to be close to one or zero. This strong a priori knowledge can be modeled [5]
by the beta law:
P(ft ) ^ fi"(!"/• )
with/ e [£,!-£] and 8, a, /?>0.
(13)
Assuming independent cells, the energy associated to this prior law becomes:
U, (/) = cste + aj^ ln(/. ) + ^ ln(l - ft ).
/=i
1=1
(14)
Then, the criterion can be written as follows:
', (/)
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(15)
Weak Membrane Model
The a priori knowledge introduced by the beta law does not take into account the
local correlation between pixels whereas the relative conductivity of one pixel is very
likely to be the same as those of its neighbours, excepted to boundaries of flaws. In the
Markov Random field, the weak membrane [1] is known to nicely model piecewise
continuous signals [4]. This energy is defined by:
where, the functional h is defined by:
,
\a2t2 if \t\<T
na,n—\
[//
•
Ai ' '
otherwise
involves an implicit contour process, a is a scale parameter which defines the spreading
out of the homogeneous zone while // fixes the prior discontinuity detection threshold T.
This functional is not differentiate at \t\=T, so a relaxation scheme has been carried out.
The functional is then modified [1] as follows:
a2/2
if \t\<q
if q<\t\<r
(18)
if \t\ > r
where ^ is the relaxation parameter such as \\m.ha^(.(t) = ha^(i) (in practice few
relaxations are necessary and used). The threshold expressions of r and q are:
9=4a r
09)
(/)
(20)
The resulting criterion J$) defined by:
is then minimized by successive sequences (depending on Q of local minimization using a
conjugate gradient method.
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TABLE 1. Parameters of the probe.
Receiver coil parameters
Driving coil parameters
Inner radius r/
Outer radius r2
Height h
Lift-off /o
Number of turns winding
Frequency
1 2 mm
16 mm
0.5 mm
0.5mm
75
8 kHz
Inner radius rml
Outer radius rm2
Height h
Number of turns
0.1 mm
0.2 mm
0.3 mm
40
Joint Criterion
In this contribution, the two previous energy functions have been considered to
construct the proposed criterion. The, final criterion combines the quadratic data fitting
term and the two a priori knowledge:
(21)
where y\ and 72 are the weighting parameters.
In order to reduce the computational time, the convolutional property of the
equations has been judiciously exploited herein.
NUMERICAL RESULTS
In this section, in order to show the potential of the proposed algorithm, some
reconstructions using simulated data are presented. The mesh of the damaged zone is
[16x16x4], with sampling steps (0.2mmx0.2mnix0.1mm) while the data are collected
according to a [41x21] grid, with sampling steps (0.5mmx0.25mm). The parameters for
the two considered simulations are presented in table 1, the operation frequency been set to
8 kHz. Moreover, a back propagation solution has been considered to initialize the
algorithm.
In what follows, the results are presented in a figure which bring the results
obtained without regularization (column 2), with the beta law (column 3), with the weak
membrane (column 4) and using the compound criteria (column 5). The first column
shows the considered simulated flaw, each row corresponding to a certain depth zone (z'=
-0.05mm, -0.15mm, -0.25mm; -0.35mm).
0.6mm
0.4mm
0.6 mm
1 mm
0.4 mm
0.4mm
(a)
(b)
FIGURE 2. Size of the two considered volumetric flaws.
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0J1
• I
FIGURE 3. Reconstruction of a cluster of three parallelepipedal defects. Grey level maps (from 0 to 1) of
the contrast function: original in the first column, without regularization in the second column, using the beta
law in the third column, with the weak membrane model in the fourth column and using the joint criterion in
the last column. Each row corresponds to a certain depth zone (z - -0.05, -0.15, -0.25, -0;35 mm).
In this contribution two simulated defects have been considered. A cluster of two
large parallelepipedal defects (0.6mmx0.6mmx0.2mm) sized and a smaller
(0.4mmx0.6mmx0.2mm) sized (see figure 2 a). Then, a 'Z' shaped flaw on two depth
zones has been considered (see figure 2 b). Figure 3 shows the results obtained for the
cluster of defects and figure 4 shows the reconstruction for the 'Z' shaped defect.
From these results one can note that the least square solution leads to smoothness
solutions with low contrast. The beta law leads to enhance the contrast of the obtained
reconstructions, and using the weak membrane model leads to more homogeneous results.
Finally the joint criterion leads to significant improvement of the reconstructions.
However, the reconstructions suffer from the attenuation phenomenon.
CONCLUSION
In this contribution, quantitative eddy current non destructive evaluation of
volumetric flaws has been investigated. A particular device has been described and, in
order to solve the non linear and ill-posed inverse problem, a bayesian estimation
framework has been considered. To compensate the lack of information, we have proposed
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1
r
FIGURE 4. Same as figure 3 with a 'Z' shaped defect.
to join two kinds of a priori information by using the beta law and the weak membrane
model. The combination of these two regularizing energies leads to a significant
improvement of the final reconstruction. Future works might take into account the skin
effect in the choice of the weighting parameters and to test this algorithm using
experimental data.
REFERENCES
1. Blake, A. and Zisserman, A., Visual Reconstruction, MIT Press, Cambridge, MA,
1987.
2. Chew, W. C., Waves and fields in inhomogeneous media, IEEE Press.
3. Harrington, F.R., Field computation by moment methods, Macmillan, New York, 1984.
4. Nikolova, M., Idier, J. and Mohammad-Djafari, A., Inversion of large-support ill-posed
linear operators using a Markov model with line process, IEEE Trans. On Image
Processing 7(4) p 571-585,1998.
5. Premel, D., Mohammad-Djafari, A. and David, B., Imagerie des milieux conducteurs
par courantsdGFoucsiult, proceeding of GRETSIp 105-108, 1991.
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