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Image Reconstruction

Following electrical measurement from the electrodes via data acquisition hardware, image reconstruction is undertaken to determine the distribution of conductivity or permittivity within a process vessel. However, as the field lines in EIT are affected by the electrical properties of the material under investigation, significant material variations within a vessel can cause related changes in the electrical field. The image reconstruction process therefore requires a higher degree of complexity compared to hard field tomography [1] due to the degree of signal distortion caused by the unknown relative positions of the pixels in relation to the sensing electrodes.

Linear back-projection method (LBP) is the simplest and fastest algorithm via a process of transposed sensitivity matrix multiplication. However, limited accuracy, image distortion and a failure to provide adequate image quality/resolution are features commonly encountered. Iterative methods, e.g. the direct and indirect methods are available for image reconstruction. Newton-Raphson method provides many researchers with a direct solution, while indirect inverse solution algorithms include conjugate gradients methods based inverse solutions (SCG). The SCG method [2,3] searches for the minimised residual vector, and has the ability to reconstruct images with improved accuracy over other indirect methods. The majority of image reconstruction techniques used in the past however are based upon LBP [4].

Direct and indirect methods are available for image reconstruction. Newton-Raphson method provides many researchers with a direct solution, while indirect inverse solution algorithms include; a). conjugate gradients methods based inverse solutions (SCG), b). single step SCG and c). linear back-projection method (LBP). The SCG method [2,3] searches for the minimised residual vector, and has the ability to reconstruct images with improved accuracy over other indirect methods. The majority of image reconstruction techniques used in the past however are based upon LBP [4].
For high contrast materials (oil/gas mixtures) LBP can provide adequate imaging results. However, limited accuracy, image distortion and a failure to provide adequate image quality/resolution are features commonly encountered when imaging material with lower contrast ratios (oil/water mixtures). The Newton-Raphson method also suffers from drawbacks in that artificial errors can be introduced by means of a regularisation method.

###### Conjugate gradients imaging algorithm

A multi-step inverse solution for two-dimensional electric field distribution was developed in order to deal with the non-linear inverse problem of the electric field distribution in relation to its boundary condition and the problem of divergence due to errors introduced by the ill-conditioned sensitivity matrix and the noise produced from electrode modelling and instrumentation[2]. The research develops an algebraic solution of the linear equations at each inverse step, using a generalized conjugate gradients method. Limiting the number of iterations in the generalized conjugate gradients method controls the artificial errors introduced by the linear assumption and the ill-conditioned sensitivity matrix. The solution of the non-linear problem is approached with a multi-step inversion.

The reconstruction process for a single image showing 5 fingers is demonstrated Figure 1.
Based on the 15 2D images shown in Figure 2 (middle part), a 3D human hand was reconstructed and is shown in Figure 2 (right part).

*Figure 1*. 2-D ERT images reconstruction of a human hand.

*Figure 2*. 3-D hand images interpolated from the 2-D ERT images

Referencs:

*Figure 2*. 3-D hand images interpolated from the 2-D ERT images

[1] Q. Marashdeh, W. Warsito, L.-S. Fan, F. Teixeira, “Nonlinear forward problem solution for electrical capacitance tomography using feed-forward neural network,” *IEEE Sens*., vol. 6, pp. 441-449, 2006.

[2] M. Wang, “Inverse solutions for electrical impedance tomography based on conjugate gradients methods,” *Meas. Sci. Technol*., vol. 13, pp. 101-117, 2002.

[3] Y. Ma, N. Holliday, Y. Dai, M. Wang, R.A. Williams and G. Lucas, “A high performance online data processing EIT system,” in Proceedings *of 3rd World Congress on Process Tomography, VCIPT*, 2003, pp. 27-32.

[4] M. Wang, “Impedance mapping of particulate multiphase flows,” *Flow Measurement and Instrumentation*, vol. 16, pp. 183-189, 2005.