Three-dimensional electrical capacitance tomography

Three-dimensional electrical capacitance tomography (3D ECT) also known as electrical capacitance volume tomography (ECVT) is a non-invasive 3D imaging technology applied primarily to multiphase flows. It was introduced in the early 2000s as an extension of the conventional two-dimensional ECT. In conventional electrical capacitance tomography, sensor plates are distributed around a surface of interest. Measured capacitance between plate combinations is used to reconstruct 2D images (tomograms) of material distribution. Because the ECT sensor plates are required to have lengths on the order of the domain cross-section, 2D ECT does not provide the required resolution in the axial dimension. In ECT, the fringing field from the edges of the plates is viewed as a source of distortion to the final reconstructed image and is thus mitigated by guard electrodes. 3D ECT exploits this fringing field and expands it through 3D sensor designs that deliberately establish an electric field variation in all three dimensions. In 3D tomography, the data are acquired in 3D geometry, and the reconstruction algorithm produces the three-dimensional image directly, in contrast to 2D tomography, where 3D information might be obtained by stacking 2D slices reconstructed individually. The image reconstruction algorithms are similar in nature to ECT; nevertheless, the reconstruction problem in 3D ECT is more complicated. The sensitivity matrix of an 3D sensor is more ill-conditioned, and the overall reconstruction problem is more ill-posed compared to ECT. The 3D ECT approach to sensor design allows direct 3D imaging of the outrounded geometry. The second commonly used name electrical capacitance volume tomography (ECVT) was introduced by W. Warsito, Q. Marashdeh, and L.-S. Fan in 2007.

Principles

Capacitance and Field Equations in 3D ECT Two metal electrodes held at different electric potential V {\displaystyle V} and separated by a finite distance will induce an electric field E {\displaystyle E} in the region between and surrounding them. The field distribution is determined by the geometry of the problem and the constitutive medium properties such as permittivity ε {\displaystyle \varepsilon } and conductivity σ {\displaystyle \sigma } . Assuming a static or quasi-static regime and the presence of a lossless dielectric medium, such as a perfect insulator, in the region between the plates, the field obeys the following equation:

∇ . ( ε ∇ φ ) = 0 {\displaystyle \nabla .(\varepsilon \nabla \varphi )=0}

where φ {\displaystyle \varphi } denotes the electric potential distribution. In a homogeneous medium with uniform ε {\displaystyle \varepsilon } , this equation reduces to the Laplace equation. In a lossy medium with finite conductivity, such as water, the field obeys the generalized Ampere equation,

∇ × H = σ E + j ω ε E {\displaystyle \nabla \times H=\sigma E+j\omega \varepsilon E}

By taking divergence of this equation and using the fact that E = − ∇ φ {\displaystyle E=-\nabla \varphi } , it follows:

∇ . ( ( σ + j ω ε ) ∇ φ ) = 0 {\displaystyle \nabla .((\sigma +j\omega \varepsilon )\nabla \varphi )=0}

when the plates are excited by a time-harmonic voltage potential with frequency ω {\displaystyle \omega } . The capacitance C {\displaystyle C} is a measure of electric energy W {\displaystyle W} stored in the medium, which can be quantified via the following relation:

W = 1 2 ∫

ε E 2 d v = 1 2 C V 2 {\displaystyle W={\frac {1}{2}}\int _{}^{}\varepsilon E^{2}\,dv={\frac {1}{2}}CV^{2}}

where E 2 {\displaystyle E^{2}} is the square magnitude of the electric field. The capacitance changes as a nonlinear function of the dielectric permittivity ε {\displaystyle \varepsilon } because the electric field distribution in the above integral is also a function of ε {\displaystyle \varepsilon } .

Soft-Field Tomography Soft-field tomography refers to a set of imaging modalities such as electrical capacitance tomography (ECT), electrical impedance tomography (EIT), electrical resistivity tomography (ERT), etc., wherein electric (or magnetic) field lines undergo changes in the presence of a perturbation in the medium. This is in contrast to hard-field tomography, such as X-ray CT, where the electric field lines do not change in the presence of a test subject. A fundamental characteristic of soft-field tomography is its ill-posedness. This contributes for making the reconstruction more challenging to achieve good spatial resolution in soft-field tomography as compared to hard-field tomography. A number of techniques, such Tikhonov regularization, can be used to alleviate the ill-posed problem. The figure at the right shows a comparison in image resolution between 3D ECT and MRI.

3D ECT Measurement Acquisition Systems

… excerpt ends here. Continue reading the full article.