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 and separated by a finite distance will induce an electric field 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 and conductivity . 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:
where denotes the electric potential distribution. In a homogeneous medium with uniform , this equation reduces to the Laplace equation. In a lossy medium with finite conductivity, such as water, the field obeys the generalized Ampere equation,
By taking divergence of this equation and using the fact that , it follows:
when the plates are excited by a time-harmonic voltage potential with frequency .
The capacitance is a measure of electric energy stored in the medium, which can be quantified via the following relation:
where is the square magnitude of the electric field. The capacitance changes as a nonlinear function of the dielectric permittivity because the electric field distribution in the above integral is also a function of .
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
The hardware of 3D ECT systems consists of sensing electrode plates, the data acquisition circuitry, and the computer to control the overall system and process the data. ECT is a non-intrusive and non-invasive imaging modality due to its contactless operation. Prior to the actual measurements, a calibration and normalization procedure is necessary to cancel out the effects of stray capacitance and any insulating wall between the electrodes and the region of interest to be imaged. After calibration and normalization, the measurements can be divided into a sequence of acquisitions where two separate electrodes are involved: one electrode (TX) is excited with AC voltage source in the quasi-electrostatic regime, typically below 10 MHz (AC method) or with pulse signal, typically lasting a few microseconds (pulse method) while a second electrode (RX) is placed at the ground potential used for measuring the resultant current. The remaining electrodes are also placed at ground potential.
This process is repeated for all possible electrode pairs. Note that reversing the roles of TX and RX electrodes would result in the same mutual capacitance due to the reciprocity. As a result, for 3D ECT systems with N number of plates, the number of independent measurement is equal to N(N-1)/2. This process is typically automated through data acquisition circuitry. Depending on the operation frequency, number of plates and frame rate per second of the measurement system, one full measurement cycle can vary; however, this is in the order of few seconds or less. One of the most critical parts of three-dimensional systems is sensor design. As the previous discussion suggests, increasing the number of electrodes also increases the amount of independent information about the region of interests. However this results in smaller electrode sizes which in turn results in low signal to noise ratio. Increasing the electrode size, on the other hand, does not result in non-uniform charge distribution over the plates, which may exacerbate the ill-posedness of the problem. The sensor dimension is also limited by the gaps between the sensing electrodes. These are important due to fringe effects. The use of guard plates between electrodes have been shown to reduce these effects. Based on the intended application, tomographic sensors can be composed of single or more layers along the axial direction. The three-dimensional tomography is not obtained from merging of 2D scans but rather from 3D discretized voxels sensitivities.
The design of the electrodes is also dictated by the shape of the domain under investigation. Some domains can be relatively simple geometries (cylindrical, rectangular prism, etc.) where symmetrical electrode placement can be used. However, complex geometries (corner joints, T-shaped domains, etc.) require specially designed electrodes to properly surround the domain. The flexibility of ECT makes it very useful for field applications where the sensing plates cannot be placed symmetrically. Since the Laplace equation lacks a characteristic length (such as the wavelength in Helmholtz equation), the fundamental physics of the 3D ECT problem is scalable in size as long as quasi-static regime properties are preserved.
Image Reconstruction Methods for 3D ECT
Reconstruction methods address the inverse problem of 3D ECT imaging, i.e. to determine the volumetric permittivity distribution form the mutual capacitance measurements. Traditionally, the inverse problem is handled through the linearization of the (nonlinear) relationship between the capacitance and the material permittivity equation using the Born approximation. Typically, this approximation is only valid for small permittivity contrasts. For other cases, the nonlinear nature of the electric field distribution poses a challenge for both 2D and 3D image reconstruction, making the reconstruction methods an active research area for better image quality. Reconstruction methods for ECT can be categorized as iterative and non-iterative (single step) methods. The examples of non-iterative methods are linear back projection (LBP), and direct method based on singular value decomposition and Tikhonov regularization. These algorithms are computationally inexpensive; however, their tradeoff is less accurate images without quantitative information. Iterative methods can be roughly classified into projection-based and optimization-based methods. Some of the linear projection iterative algorithms used for 3D ECT include Newton-Raphson, Landweber iteration and steepest descent algebraic reconstruction and simultaneous reconstruction techniques, and model-based iteration. Similar to single step methods, these algorithms also use linearized sensitivity matrix for the projections to obtain the permittivity distribution inside the domain. Projection-based iterative methods typically provide better images than non-iterative algorithms yet require more computational resources. The second type of iterative reconstruction methods are optimization-based reconstruction algorithms such as neural network optimization. These methods need more computational resources than the previously mentioned methods along with added complexity for the implementation. Optimization reconstruction methods employ multiple objective functions and use iterative process to minimize them. The resultant images contain less artifacts from the nonlinear nature and tend to be more reliable for quantitative applications.
Displacement-Current Phase Tomography (DCPT)
Displacement-Current Phase Tomography is an imaging modality that relies on the same hardware as ECT. 3D ECT does not make use of the real part (conductance component) of the obtained mutual admittance measurements. This component of the measurement is related with the material losses in the region of interest (conductivity and/or dielectric losses). DCPT utilizes the full admittance information by means of the small angle phase component of this complex valued data. DCPT can only be used when the electrodes are excited with AC voltage. It applies only to domains that include material losses, otherwise the measured phase will be zero (real part of the admittance will be zero). DCPT is designed to be used with the same reconstruction algorithms designed for 3D ECT. Therefore, DCPT can be used simultaneously with 3D ECT to image the spatial tangent loss distribution of the medium along with its spatial relative permittivity distribution from ECT.
Multi-Frequency 3D ECT Operation