The measuring principle is based on the phenomenon whereby the impedance of a coil generating a time-varying magnetic field changes in the presence of a conductive material as eddy currents are induced in the material. Therefore the idea of a contactless electrical conductivity imaging system (CECIS) is based on a sensor composed of two coils. One of the coils, designated the transmitter coil, is driven by an alternating current and generates an oscillating magnetic field that induces currents inside the object to be tested. These induced currents are related to the conductivity distribution of the body. Another coil, the receiver coil, measures the total magnetic field. One component of the magnetic field is related to the primary imposed currents and the other component is related to the induced eddy currents. As the signal originated from deeper voxel significantly attenuates before it reaches the receiving coil the information gets easily corrupted by measurement noise. By incorporating into the design a real time digital signal processor able to perform advanced signal processing techniques it is possible to increase significantly the range of the sensor. Two solutions are envisaged to accumulate data either by relocating the probe several times using a XY scanning mechanism or by using an array of sensors, i. e., several probes in the vicinity of the object to be tested.

The mathematical basis of this methodology is developed in two parts. In the first part, the physics of the induction system is simulated in the forward problem either using a numerical method like the finite element method (FEM) or an artificial intelligent technique (ANN). The simulated system is used to preview the correspondence between first order variations in the secondary magnetic field to conductive perturbations in the conductive body. A set of equations in the discrete form can be formulated: Sx=b, where x is the unknown conductivity perturbation vector, b is the magnetic field measurement vector and S is the sensitivity matrix.
The second part of this study is the formulation of the inverse relation of this correspondence in order to reconstruct the conductivity images for a three dimensional sample object: b=S x + y, where y represents unknown measurement noise present in the real scenario. Since the signal from a deeper voxel is smaller than surface voxel, measurement noise corrupts more the information from deeper voxels. However, the relation between the magnetic measurements and the conductivity distribution is nonlinear. Thus, one way to estimate the solution is to linearize the forward problem around an initial conductivity distribution and update the conductivity distribution using a suitable algorithm by comparing the calculated fields with the measurements. Several algorithms can be used to minimize the norm of ||S x - b|| when solving for x. These algorithms must be investigated in order to select the one that provides the best performance for the problem to be evaluated. The mathematical relation between the conductivity distribution and the measurements is the backbone of the imaging algorithm.

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