Inverse Problems in Magnetic Induction Tomography of Low Conductivity Materials Ryszard Palka, Stanislaw Gratkowski, Piotor Baniukiewicz, Mieczyslaw Komorowski, and Kreysztof Stawicki
Abstract The paper deals with the computational problems typical for magnetic induction tomography (MIT) of low conductivity materials. The forward problem is solved in order to define and verify the solution of the inverse problem. The paper focuses on the formulations and solutions of both. Different formulations and simplifications are discussed, depending on the physical properties of the exciter and examined object.
1 Introduction Magnetic induction tomography (MIT) is a relatively new, non-invasive method which uses eddy currents phenomenon for reconstructing spatial distribution of the electrical conductivity in the examined object. It can be used for diagnostics of objects with wide spectra of physical properties, ranging from molten metals and other conductive fluids, through magnetic environments with low electrical conductivity, to non-magnetic weak-conducting objects like saline solutions representing certain body tissues [1–3]. The measurement system consists of the set of exciters and receivers, sensing the changes in the excited field, due to the different distribution of conductivity in the body. In certain implementations of such systems there may be only one exciter and (or) receiver, which is (are) rotated consecutively around the object. The system discussed in the paper is presented schematically in Fig. 1. Figure 2 shows the picture of the measurement system built by the authors. The exciter is driven by a sinusoidal current of frequency 100 kHz. The exciter consists of a coil with a ferrite core and conducting shield. The conducting shield has been optimized in order to shield sensitive electronic equipment as well as focus the main part of the magnetic flux in the vicinity of the receiver, which ensures sufficient values R. Palka, S. Gratkowski, P. Baniukiewicz, M. Komorowski, and K. Stawicki Szczecin University of Technology, Sikorskiego 37, 70-313 Szczecin, Poland
[email protected] R. Palka et al.: Inverse Problems in Magnetic Induction Tomography of Low Conductivity Materials, Studies in Computational Intelligence (SCI) 119, 163–170 (2008) c Springer-Verlag Berlin Heidelberg 2008 www.springerlink.com
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Fig. 1 Schematic diagram of the measurement system and the excitation unit in details (on the right)
Fig. 2 The measurement system
of the measured signals. Eddy currents induced in the object are the source of the secondary magnetic field, opposing the source field and therefore modifying the measurements taken from the receivers. There is no need for physical contact with the surface of the object being tested. The object can be scanned either manually or with the aid of a mechanical device, or can be rotated as shown in Fig. 1. In general the magnetic field produced by the eddy currents excited in a uniformly conductive object and the field produced by the excitation coil represent unwanted signals. These signals can be eliminated by using differentially connected signal coils.
2 Forward Problem Different formulations can be used to solve the forward problem, i.e., to find the eddy current distribution for a given conductivity distribution and then the magnetic field outside the object in the plane of the signal coil. We have used the Φ–A
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formulation. In a linear, nonmagnetic, isotropic conductive medium one can find the following equation for these two potentials (provided that the condition ∇ · A = 0 is adopted) [4]: (1) ∇ · [(σ + ε jω )∇Φ] = − jω A · ∇(σ + ε jω ), where: σ – conductivity, ε – permittivity, ω – angular frequency. If ωε σ, (1) can be written as follows: (2) ∇ · (σ ∇Φ) = − jω A · ∇σ . Under the assumption that the object does not influence the excitation field, the vector magnetic potential in (2) may be replaced by the primary magnetic potential Ap , computed without the presence of the object. In this case the final equation and the relevant boundary condition have the forms:
∂Φ = − jω Ap · n. ∂n
∇ · (σ ∇Φ) = − jω Ap · ∇σ ,
(3)
The primary magnetic vector potential can easily be calculated for a given configuration of the exciter by using the standard axi-symmetrical FEM. After solving the above equations for the scalar potential, the induced current density in the conductive object can be calculated. Formulation of the forward problem continues by applying Biot–Savart’s law. The object is divided into polyhedrons of any shape with a uniform current density Je (r ). It can be shown that the magnetic field at the point r is given by (Sk refers to the facets of the polyhedron e) [5]: B(r) =
µ0 4π
∑ ∑ Je ×nk e
k
Sk
dS & & &r − r & .
(4)
The finite element mesh and exemplary results of calculations of the magnetic flux density distribution (secondary field only) in the plane of the signal coil are shown in Fig. 3.
a
b
Fig. 3 The finite element mesh and the x-component of the magnetic flux density in the plane of the signal coil; conductivity of the inclusion is equal to 100 S m−1 while the conductivity of the background equals 0.8 S m−1
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3 Application of An Artificial Neural Network for Solving the Inverse Problem For solving the inverse field problem of reconstruction of the conductivity distribution, an artificial neural network (ANN) has been applied. The ANN is a system composed of artificial neurons organized in layers which uses a mathematical or computational model for information processing based on a connectionist approach to computation. The inverse problems, which can be met in the MIT are usually ill-posed and have large number of unknowns to be determined. The ANN, thanks to their flexibility and possibility to model very complicated systems, can be successfully used for solving such problems. The ANN approximates the whole measurement system, which can be treated as a “black box” that transforms the spatial distribution of measured quantity (e.g., conductivity) to the output signal values (e.g., voltage or current) or vice versa. Fitting the ANN to the measurement system, a transfer function is determined during a learning stage. The learning of the ANN is carried out using a “knowledge base” that contains signals obtained from measurements performed on the calibration specimens. These results should cover the whole input space of possible shapes and properties of inclusions. The main advantages of proposed formulation of the problem are that the sensor parameters, excitation parameters and the geometry of the system are not important in point of view ANN-based algorithm. In the approach to the inverse problem proposed by the authors, the artificial neural network was used for modeling the whole measurement system (Fig. 4). The system is identified by input–output data pairs, where the output stands for the spatial distribution of the conductivity, whereas the input stands for the measurement results obtained as a result of scanning the specimen in y − z plane (Fig 5). The signals obtained during measurements are used as the ANN input. The main assumption is that the sensor is only sensitive to the inclusion located close to it. Thus, it is not necessary to analyze at the same time the signals
Fig. 4 The block scheme of the measurement signal inversion using ANN. IL – input layer, OL – output layer, N – neurons, k – number of output space discretization points
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Window
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Fig. 5 The ANN system used for the conductivity identification
measured over the whole specimen. This leads to the idea of rectangular window moved through the measurement plane. The window size corresponds to the distance from which the inclusion affects the sensor. Figure 5 shows the basis of proposed method. The specimen is discretized and approximated by cubic elements with constant conductivity. The window moves in the y − z plane with a defined step depending on the measurement accuracy. The distribution of the conductivity along the x-axis is computed by the ANN for each position of the window. The number of elements, which approximate the specimen along the x-axis, equals to the number of ANN outputs. Thus, particular network output responds to the conductivity of one cubic element. The wire cube stands for the estimated conductivity of the specimen, whereas the solid cube stands for the estimated conductivity of the inclusion.
4 Reconstruction of the Conductivity Distribution Via Biot–Savart’s Law The reconstruction of the conductivity distribution within the object under investigation by the use of the external field values in the close neighborhood of the specimen, known from scanning measurements, constitutes a typical inverse problem and belongs to the class of the improperly posed tasks [6, 7]. This problem can be replaced by the equivalent problem of the determination of the current distribution. Figure 6 shows the region under consideration where the external flux densities can
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Fig. 6 Object under consideration and the measurement plane V0
be defined/measured (the possible region of measurements V0 has been indicated). In order to define the mathematical model for the inverse task, the whole conducting region has been discretized into N identical sub-cubes with three individual current sheet values Jk surrounding each cube. The magnetic field generated by each cube in any point of the region under consideration can be symbolically expressed in the following form (the Biot–Savart’s law): B(x, y, z) = f(x, y, z)Jk1 + g(x, y, z)Jk2 + h(x, y, z)Jk3 .
(5)
Detailed formulas describing all components of the magnetic field generated by current sheets are given in [7]. The resulting magnetic field in any point of the external region can be determined as the sum of the magnetic field of all sub-domains. The mapping: current density distribution → magnetic field distribution leads to an overdetermined, linear and ill-conditioned set of 3P equations (P – number of evaluation points) with 3N unknown surface current densities (P >> N): WJ = B,
(6)
where W is a real m × n matrix (m = 3P, n = 3N). The minimal least squares solution of (6) is the vector of the minimum Euclidean length which minimizes the length of the residual vector R = B − WJ [6, 7]. The matrix W in (6) can be factorised by the singular value decomposition as: W = QDPT ,
(7)
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where Q is an m × m orthogonal matrix, P is an n × n orthogonal matrix and D is n × n diagonal matrix with singular values of W arranged in decreasing order. Finally the minimal least squares solution of this problem is given by J = PS−1 QT BT ,
(8)
where S = diag(sv1 , sv2 , . . . , svk ), with svk being the last singular value greater than zero. The practical implementation of the above algorithm can be very difficult because of the possible instabilities. Therefore the matrix W should be scaled in order to prevent the least squares problem being unnecessarily sensitive to both measurement and numerical calculation inaccuracies. The numerical algorithm has been intensively tested for different external field distribution (both – measured and calculated) and for extremely large equation systems. As the convergence criterion the total error between the primary (measured) BM and final (calculated) field distribution BC has been used:
ε=
V0
(BM − BC )dV .
(9)
The value of above quality criterion together with the calculation time, rank of the equation matrix and their singular values enable the proper choice of the meshing of the conducting region and the size, position and discretization of the measurements regions. An example of the reconstruction of the current density region has been shown in Fig. 7.
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Fig. 7 Identified current density regions with the flux density distribution
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5 Conclusions In this paper the forward and inverse problems of magnetic induction tomography are discussed. In order to find the scalar potential distribution for an object of arbitrary conductivity distribution a 3-D finite element formulation was employed. The secondary magnetic field, due to the induced currents in the object was calculated in the plane of the signal coil using the Biot–Savart law. Next, the inverse problems were formulated. Two algorithms have been proposed and tested: first using the artificial neural network and second using the inversion of the Biot–Savart law. Acknowledgement This work was supported by the Ministry of Education and Science, Poland, under grant 3 T10A 033 30 (2006–2009).
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