Rotational electrical impedance tomography

IOP PUBLISHING

MEASUREMENT SCIENCE AND TECHNOLOGY

doi:10.1088/0957-0233/18/9/028

Meas. Sci. Technol. 18 (2007) 2958–2966

Rotational electrical impedance tomography Cheng-Ning Huang1, Fang-Ming Yu2 and Hung-Yuan Chung1 1

Department of Electrical Engineering, National Central University, No. 300, Jhongda Rd, Jhongli City, 32001 Taoyuan, Taiwan, Republic of China 2 Department of Computer Science and Information Engineering, St John’s University, 499, Sec. 4, Tam King Road Tamsui, Taipei, Taiwan, Republic of China E-mail: [email protected]

Received 11 December 2006, in final form 22 June 2007 Published 10 August 2007 Online at stacks.iop.org/MST/18/2958 Abstract A high performance rotational EIT (REIT) system capable of producing better quality EIT images is developed by expanding the independent measurements. In this system, electrodes are attached to a rotational phantom tank which is driven by a microstepping motor. The measurement site of the electrode pairs can be precisely changed. We increase the independent measurements by moving electrodes from the original location to a subsequent location. Increasing the number of independent measurements enhances the resolution of the impedance image and improves the quality arising from the ill-posed condition. The experimental results clearly show the improvement of the REIT image. It is believed that this improvement will provide help in the field of electrical impedance tomography. Keywords: rotational electrodes, electrical impedance tomography, number of electrodes

(Some figures in this article are in colour only in the electronic version)

1. Introduction Electrical impedance tomography (EIT) is a powerful tool for mapping the electrical properties of objects. The crosssectional distribution of the object’s electrical impedance can be accessed from measurements made on its surface. Electrical currents are applied to the object through electrodes, and the resulting potential at the electrodes is measured. Since the relation between the measurement data and the impedance distribution is an inverse problem, the impedance distribution inside the object can be calculated by solving the inverse problem. Electrical impedance tomography is a low cost, real time and portable imaging technique. Because of the above potential advantages, the EIT technique has already been developed for many industrial applications, such as in the fields of process tomography, non-destructive testing, geological studies and medical imaging (Djamdji et al 1996, Dickin and Wang 1996, Szczepanik and Rucki 2000, York 2001, Huang et al 2003). The EIT technique also has several advantages 0957-0233/07/092958+09$30.00

© 2007 IOP Publishing Ltd

over the current medical imaging methods for biomedical applications. For example, the procedure for measuring impedance does not produce any harmful radiation, so it presents no known hazards to the subject. It should therefore be possible to use an EIT system to monitor physiological functioning in the long term. Furthermore, the EIT system can be entirely implemented using electrical techniques, which makes it a relatively cheap system from which data can be collected very rapidly. The impedance image has been applied in many clinical applications, but there are few commercial medical EIT systems available. This is because the essential spatial resolution of the impedance image cannot yet be achieved, and this technique also has an intrinsically poor signal-to-noise ratio (Brown 2001). Thus the image quality of EIT is not comparable with that produced by established medical imaging systems, such as computerized tomography (CT) and magnetic resonance imaging (MRI). The quality of an impedance image is limited by the ill-posed problem of EIT and the number of electrodes (Hou and Mo 2002). An ill-posed problem is one where

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Rotational electrical impedance tomography

a small error in measurement will result in a large error in the reconstructed image. Ill-posed problems occur due to the nature of impedance image reconstruction, so are unavoidable. However we still can improve the performance of the EIT system by increasing the number of electrodes. For a conventional N electrode EIT system, we could obtain at most N(N − 1)/2 individual measurement data to reconstruct the impedance distribution. By acquiring more measurement data, a finer mesh is applied to get a closer approximation without impairing the ill-posed condition. This means that the resolution of the EIT image can be improved by increasing the number of electrodes. Nevertheless, increasing the electrodes is not a final solution for improving spatial resolution. The total number of electrodes is restricted by the volume of the object and the size of the electrodes. In order to attach more electrodes to a phantom tank, we need to either reduce the size of the electrodes or expand the area of the measurement site. However, the area of the measurement site is invariable, and reducing the size of the electrodes could create severe contact impedance which limits the number of electrodes in the EIT system. In order to enhance the quality of the EIT image, a high performance rotational EIT (REIT) system is developed for expanding the independent measurements. The rotating scheme could have advantages in several fields. In the field of medical imaging, rotating electrodes can provide more measurement data to improve the quality of a reconstructed image. The process mixing applications offer an opportunity for movement of the EIT electrodes without an extra rotating mechanism. In general, impellers are utilized during mixing process applications for agitation. Electrodes can be attached to such an impeller to provide a readymade rotating host that can be used to provide additional electrode positions. Frounchi and Bazzazi (2003) developed a new rotating ECT system which employed pairs of rotating electrodes, in order to increase the number of measurements. Murphy and York (2006) increased the number of independent EIT observations by utilizing the rotation of a mixing impeller. In previous works on rotating schemes, however, only a few electrodes were implied (e.g., four electrodes) in the EIT system. In order to further improve the spatial resolution of the EIT, we constructed an EIT system with 16 movable electrodes. We hope this research can provide more experience and reference material to those who are interested in the field of rotational EIT. The concept of movable electrodes is described in the following section.

2. System design In this work, a conventional EIT system is expanded to a rotational EIT (REIT). The rotational EIT is equipped with movable electrode pairs attached to electrodes that move to a new measurement site so as to acquire more data. The REIT system can be divided into three subsystems: movement scheme, switching network and measurement system. Figure 1 shows a block diagram of a REIT system. All three subsystems are controlled by a host computer. The movement scheme is designed so as to obtain more independent measurements. The movement scheme includes a phantom equipped with movable

Figure 1. Block diagram of a REIT system.

electrodes and a stepping motor to drive the electrodes. The switching network is constructed from several solid-state relays. These switches are used to change the current path to different electrodes and to pick up the voltage measured from each electrode. For the measurement subsystem, we adopt a four-electrode system to measure the impedance of the sample. The four-electrode method can eliminate the influence of contact impedance. A constant current source injects the fixed current (Ic) into the estimated object at a specific frequency. The lock-in amplifier picks up the potential (Vm), which is a response to the injecting current, and filters out unwanted components from the measured signal. 2.1. Movement scheme and phantom design The phantom of the REIT system is constructed from a plastic cylinder 110 mm in height and 180 mm in diameter. There are 16 compound electrode lines placed on the inner surface of the plastic cylinder. Each compound electrode includes both a voltage electrode and a current electrode. Since the electrodes have an effect on the accuracy of measurements, it is necessary to pay some attention to their design, especially their size and shape (Dickin and Wang 1996, Hua et al 1993). To ensure that a uniform current density is generated within the tank, a large surface area is required for the current-injecting electrodes while for the voltage electrodes a small surface area is optimal so as to avoid the influence from other equipotentials. For this reason, the compound electrode is constructed from one rectangular copper slice (current electrode) and one copper bar (voltage electrode) isolated by insulating tape. The dimensions of the copper slice electrode are 20 mm in width, 90 mm in high and 0.5 mm in thickness. In order to avoid overlapping measurements, the dimensions of the copper bar are 3 mm in width, 50 mm high and 0.5 mm in thickness. These electrode dimensions ensure that the proposed rotational EIT can obtain truly independent data within ten rotations. The compound electrode applied in this study is illustrated in figure 2. The movement scheme is the most important part in a REIT system. The moving electrode scheme is composed of 16 compound electrodes which are fixed on a movable ring frame. The electrodes are distributed around the inner surface of the ring frame. Figure 3(a) shows a vertical view of the phantom tank with an electrode array, and figure 3(b) shows a three-dimensional view of the phantom tank. The driving stepping motor adopted in this study is an Oriental Motor CSK microstepping motor. The CSK Series motor consists of a high-performance stepping motor and a compact, low vibration microstepping driver which offers a smooth position 2959

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Figure 2. Design of compound electrode: (a) front view and (b) lateral view.

function. The microstepping motor combines a compact hightorque type motor and high-resolution one, geared together to achieve high positioning accuracy. The ring frame is driven by a stepping-motor with a minimal rotating angle of 0.018◦ . This means that the electrode pair measurement site can be changed precisely. The neighborhood data acquisition procedure is utilized. First we obtain 16 × (16 − 3)/2 = 104 independent impedance measurements. Next, the angular orientation of the scanning device is incremented by 1 stepping angle (Ex. 0.018◦ ), and then another 104 impedance measurements (second data acquisition procedure) are acquired. In this way, by moving electrodes from the original to the next electrode location, we can increase the number of independent measurements further. In a 16-electrode REIT system, there are 16 electrodes distributed around the cylinder. The angle between adjacent electrodes is 22.5◦ . This means that we can repeat the data acquisition procedure by 0.018◦ increments through an arc of 22.5◦ . In this manner, we can change the measurement sites 22.5/0.018 = 1250 times and accumulate 104 × 1250 = 130 000 electrical impedance measurements. These measurements are transmitted to a computer and reconstructed as an impedance image with a mathematical package. However, a large number of measurements can also render the image reconstruction process difficult. Due to concerns with practicality, in this design, the rotating step angle chosen was 4.5◦ and the angle between the adjacent electrodes 22.5◦ . To completely scan the whole circumference of the phantom tank, the stepping motor needed to drive the electrodes (step angle of 4.5◦ ) five times. Eventually, the total number of measurement data increased five times (104 × 5 = 520). Figure 4 shows an enlargement of the region within the dotted square in figure 3. Figure 4 shows the motion of the electrodes during the scanning procedure. The rotational EIT was constructed as shown in figure 5. In figure 5(a), the 16 compound electrodes attached to the inner wall of the moving ring can be seen. A microstepping motor is also set on the moving ring to drive the electrodes. 2960

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Figure 3. Design of a rotational phantom: (a) vertical view of phantom tank and (b) 3D view of phantom tank.

Figure 4. Detail of the scanning motions.

Figure 5(b) shows the inner view of the phantom tank. The components shown in figures 5(a) and (b) were combined to assemble the rotational EIT in figure 5(c). 2.2. Switching networks and the measurement system In this work, we adopted the four-point method to measure the impedance. The four-point measurement technique can overcome many undesired effects. For each measurement,


(a)

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we require two electrode pairs, i.e., the driving pair and the receiving pair. The driving electrode pair injects the current into the object and the receiving electrode pair measures the boundary voltage. For four-point measurement, four electrodes are attached to the measured sample. A current source forces a constant current through the ends of the sample bar. The injecting current (Ic) is constant without changing with load impedance. A voltmeter simultaneously measures the voltage (Vm) generated across the sample. Because of the decrease of the contact impedance when measuring the conductivity, the four-point measurement system can reduce the effect of time-varying, pressure-sensitive contact impedance (Hart et al 1988). It is necessary to provide high output impedance and to have strong capability of noise rejection for a constant current source (Boone and Holder 1996). A modified-floating current source was applied in this work. In the experimental REIT, we use a sinusoidal current with a 20 kHz frequency to inject a constant current into the phantom tank and then measure the resulting potentials. Since we are not interested in the carrier, but only the amplitude attenuation and phase shift of the injected sine wave signal, we demodulate the carrier signal to extract the amplitude and phase information from the measured signal. In this work, we applied a lock-in amplifier to estimate the amplitude and phase of the measured signal. A lock-in amplifier can act as a narrow band-pass filter (with a pass bandwidth almost equal to 1 mHz) around the reference signal frequency (Frerichs 2000, Min et al 2000). Thus with the lock-in amplifier, we could recover the original signal from the measured signals that had been corrupted by external disturbances thousands of times stronger than the signal of interest. The demodulated data from the lock-in amplifier are sent to a computer via a data acquisition card (NI-DAQ 6251, National Instruments) where they are stored on a disk for further analysis. A switching network (multiplexers) is required in a single current source system or in those systems that share voltage measurement circuitry between multiple electrodes such as for the set of electrodes attached to the inner surface of the phantom tank. The switching network, shown in figure 1, is divided into current and voltage switches. The current switch transmits the excitation current from the current source to different driving pairs. The voltage switch passes the measurement voltage from the receiving pair to the data acquisition card. These switching networks are adopted to achieve the four-point method to deal with the unknown contact impedance problem.

3. The image reconstruction method 3.1. EIDORS

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Figure 5. View of the assembled rotational EIT: (a) 16 compound electrodes attached to the moving ring, (b) inner view of the phantom tank and (c) rotational EIT driven by microstepping motor.

The Matlab package is applied to reconstruct impedance images from the measurement data (Polydorides and Lionheart 2002). The objective of the EIDORS project (Electrical Impedance and Diffuse Optical Reconstruction Software) was to develop freely available software to deal with nonlinear and ill-posed problems from boundary measurements. Nonlinear and ill-posed problems such as electrical impedance problems are typically approached by using a finite-element model for 2961

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sample placed in a cylindrical tank filled with a saline solution. The conductivity of the saline solution was 11 mS cm−1. A copper rod with a diameter of 30 mm is put into the phantom tank. There are 16 compound electrodes spaced at average intervals around the circumference of the tank. Figure 7 depicts the experimental setup and an overhead view of the REIT system. 4.1. Degree of ill-conditioning

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Singular decomposition is a valuable tool in studying an ill-posed problem. The eigenvalues indicate how well the basis image amplitudes are defined by the data; basis images with smaller eigenvalues are more sensitive to data noise. In order to illustrate the degree of ill-conditioning, the singular values of a Jacobian are plotted. Figure 8 shows the singular value of rotational EIT and conventional EIT values. A comparison of the results makes it clear that the values obtained with the rotational EIT are more and greater than the singular one. 4.2. Spatial resolution

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Figure 6. FEM model: (a) for conventional EIT (16 elements in circumference) and (b) for rotational EIT (80 elements in circumference).

the forward calculations to obtain a unique and stable inverse solution. The finite-element method (FEM) is well suited to the task for general geometries and inhomogeneous materials. The finite-element solver included in the EIDORS toolkit can be applied to the complete electrode model to compute approximate solutions. 3.2. Design of the FEM model Owing to the fact that the rotational data acquisition scheme provides additional independent measurements, a more finely meshed experimental model can be established. A finer mesh provides both better accuracy and greater spatial resolution. The tank of the FEM model is illustrated in figure 6. Figure 6(a) shows the FEM model for the conventional EIT. There are 16 elements around the circumference of the model. Each element indicates an electrode. In this study, however, the REIT system rotates 16 compound electrodes five times, which is equal to 80 electrodes attached to the inner surface of the phantom tank. A model of the resultant rotational configuration, including 80 elements around the circumference, is shown in figure 6(b).

4. Experimental result In order to assess the performance of the proposed system, the REIT is applied to reconstruct an impedance image of a 2962

The edge response and position dependence of the reconstructed image are plotted, and a comparison of the spatial resolution with the conventional and rotational EIT scheme is made. Figures 9(a) and 10(a) show the reconstructed impedance image of a conducting cylinder obtained using both conventional EIT and rotational EIT. A 3.5 cm diameter conducting cylinder (copper rod) is placed 2 cm away from the boundary electrode. The dashed lines enclose the true locations of the objects. Figures 9(b) and 10(b) show the cross-sections of conventional EIT and rotational EIT images. The cross-sectional profile is measured along the line across the sample and the center of the phantom tank. The red lines show the profile of the impedance image and the black lines show the true locations of the measured sample. From the results, it is clear that the rotational EIT has a sharper edge near the boundary electrode. This is because increasing the number of measurements improves the spatial resolution in the peripheral area. However, the improvement of spatial resolution in the center of the impedance image is limited. In order to provide a quantitative improvement of spatial resolution, the blur radius (Br) is defined as a measure of the resolution. Az rz = , (1) Br = ro Ao where ro and Ao are the radius and area of the original sample, respectively; and rz and Az are the radius and area of the zone containing half the magnitude of the reconstructed image from a point of contrast, respectively. The blur radius in figure 9 is 1.5571 and the blur radius in figure 10 is 1.3857. 4.3. Noise distribution measurements The noise distributions of both the measured voltages and the reconstructed image pixels were determined by calculating the root-mean-square noise (defined as the standard deviation) and the signal-to-noise ratio (defined as mean/standard deviation) (Wang et al 1994).


Figure 7. Experimental setup of the REIT system.

Figure 8. Log scale of the singular Jacobian values.

The noise distribution of the measurement was calculated over 1040 frames. It was found that the RMS of the measurement noise was about 7 dB. The SNR of the impedance image reconstructed from conventional EIT (fixed, 208 measurements) was 4.89 dB, and the SNR of the impedance image reconstructed from the proposed rotational EIT (5 rotations, 1040 measurements) was 5.55 dB. This result shows that rotational EIT could improve the SNR image. Increasing the angular resolution of REIT may provide greater improvement of the image quality. However, when we reconstructed the impedance image, the huge number of

data caused ‘out of memory’ problems even though there was enough physical memory. This was because the total memory space required for high angular resolution exceeded the capability of the 32-bit application (about 2 GB). The result was that we could not reconstruct REIT impedance images when the electrodes rotated more than six steps. 4.4. Reconstructed impedance image To demonstrate the improvement of resolution in a rotational EIT system, the impedance images reconstructed from both 2963

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Red line : profile of impedance image Black line: position of measured sample

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Normalized profile

Red line : profile of impedance image Black line: position of measured sample

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Figure 9. The profile of impedance image reconstructed by conventional EIT: (a) reconstructed impedance image (red: higher conductivity; blue: lower conductivity) and (b) image profile.

conventional EIT and REIT are compared. The phantom tank with a metal object is illustrated in figure 11. The impedance image reconstructed by applying the conventional EIT measurement configuration is shown in figure 12(a). The image reconstructed from the REIT configuration is shown in figure 12(b). Because the REIT collected more measurement data, the REIT system could reconstruct an image with a higher resolution FEM model making it easier to identify the shape and the position of a metal object in the phantom tank; see figure 12(b). In order to make a fair comparison between the EIT and the REIT, the same FEM model is applied to reconstruct the impedance image again. In this trial, the mesh model of figure 6(b) was applied simultaneously to the EIT and REIT images. The EIT image with higher density mesh is shown in figure 13. It is observed that the results suffer from a serious ill-posed problem. This is because insufficient measurement data are applied to reconstruct the high density mesh, so the ill-posed condition generally worsens. 2964

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Figure 10. Profile of impedance image reconstructed by rotational EIT: (a) reconstructed impedance image (red: higher conductivity; blue: lower conductivity) and (b) image profile.

Figure 11. Phantom tank with metal object.


Table 1. Comparison of the MEIT and the rotational EIT system.

Electrode number Minimal step angle Maximum data Diameter of phantom Shape of electrodes Type of electrodes

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Figure 12. Reconstructed images from the EIT and the REIT systems: (a) EIT image and (b) REIT image.

Figure 13. EIT image with higher density mesh.

5. Conclusion and summary In this study, a REIT system which could increase measurement data is developed by the application of a scanning motion in an EIT system. Compared with the EIT system, the REIT offers several significant improvements. (1) It substantially increases the number of independent measurement data which would improve the resolution of the impedance image and the ill-posed condition. (2) Applying the scanning scheme can decrease the channel number of the measurement system. This could reduce the complexity and the cost of the EIT system. (3) The scanning scheme also provides flexibility in terms of imaging resolution and time consumed. The authors have previously proposed a simple movable EIT (MEIT) architecture (eight electrodes, small phantom) (Chang et al 2005). In this work, the performance of the aforementioned MEIT to achieve a high quality impedance image has been improved. Table 1 lists the difference between the MEIT and the REIT. Although this approach yields sample impedance images of higher resolution, considerable time is still required for data accumulation. In this research, a microstepping motor

MEIT system

Rotational EIT

8 7.5◦ 120 100 mm Bar (small area) Common electrode

16 0.018◦ 130 000 180 mm Flat (large area) Compound electrode

is used to expand the measurement sites. The angle between adjoining electrodes is 22.5◦ . The minimal stepping angle of the microstepping motor is 0.018◦ . By applying 0.018◦ increments through an arc of 22.5◦ , we could have at most 1250 measurement sites. The speed of the microstepping motor is set as 0.18◦ s–1 so the time for driving the electrodes to scan all measurement sites is about 125 s. In this work, the data acquisition time for conventional EIT (208 measurements) is 90 s. In the rotational EIT (five turns), the time for acquiring data is 125 + 90 × 5 = 575 s (about 10 min). To shorten the acquisition time, a parallel configuration can be used to acquire voltages at the same time. In this way, data accumulation time could be shortened to 1 s. The positioning of the electrodes is important, since the reconstruction algorithm assumes that the electrodes are located at precisely defined intervals. The REIT utilizes a microstepping motor to locate the position of electrodes, so the scanning motion can eliminate the position error. The large number of measurement data obtained from the REIT however can cause serious problems during image reconstruction. It is necessary to develop a fast reconstruction algorithm capable of dealing with the huge numbers of measurement data. Applying larger current electrodes for the EIT measurement could eliminate the effect of contact impedance and improve the image quality. However, when we increase the area of the current electrodes, a near-metal boundary is constructed. For conductive boundary measurement, the equipotential lines would produce vanishingly small potential differences and a large proportion of the probing current would be shorted out of the imaged region (Record et al 1995). Even though the current supply electrodes are not used in voltage measurement, they still present a conductor to the perimeter either side of the voltage sense electrode which will upset the boundary voltages. Therefore, applying compound electrodes will decrease electrode–skin contact impedance but it also reduces the current density in the interior of the phantom tank. The trade-off between sensitivity and size of the compound electrode should be further explored. Many studies also try to optimize the structure size of the compound electrode (Wang et al 2001). In this paper, the width of the current electrodes is 20 mm and the distance between two adjacent electrodes is 10 mm. It is found that this configuration can provide maximal boundary potential without exceeding the input limit of the data acquisition card. The concept of improving spatial resolution has always been associated with the modification of electrodes. Pinheiro et al (1998) suggested the idea that, to increase the resolution, the driving electrodes should have a large contact surface, while the measurement electrodes should be as small as possible. The concept of compound electrodes was introduced 2965

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by Hua et al (1993), who noted that the injection of currents and the measurement of voltages could be carried out through different compartments of the same electrode. Those improvements in spatial resolution have been limited. Increasing the number of electrodes is the most significant way to improve spatial resolution. However, Polydorides and McCann (2002) concluded that increasing the number of electrodes is a rather inefficient option, as the improvement in the spatial resolution of the image is obtained at the cost of more sophisticated and expensive hardware, slower data acquisition cycles, and higher computational workload. The method of rotational electrical impedance tomography proposed in this paper is simply a modification of the present fixed electrode system designed to improve impedance performance. For hardware assembly, the electrodes are attached to a ring driven by a microstepping motor. In the reconstruction algorithm, the rotational electrical impedance tomography is treated as an extended electrode system. In adapting the current EIT system, we can collect more measurement data for image reconstruction, without constructing a more complicated hardware system. It is believed that this primary work can attract more researchers to put more effort into this field.

References Boone K G and Holder D S 1996 Current approaches to analogue instrumentation design in electrical impedance tomography Physiol. Meas. 17 229–47 Brown B H 2001 Medical impedance tomography and process impedance tomography: a brief review Meas. Sci. Technol. 12 991–6 Chang F-W, Huang C-N, Yu F-M, Hsieh T-C and Chung H-Y 2005 A high-performance movable EIT system BMES 2005: (Taiwan) Dickin F and Wang M 1996 Electrical resistance tomography for process applications Meas. Sci. Technol. 7 247–60 Djamdji F, Gorvin A C, Freeston I L, Tozer R C, Mayes I C and Blight S R 1996 Electrical impedance tomography applied to semiconductor wafer characterization Meas. Sci. Technol. 7 391–5 Frerichs I 2000 Electrical impedance tomography (EIT) in applications related to lung and ventilation: a review of

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experimental and clinical activities Physiol. Meas. 21 R1–21 Frounchi J and Bazzazi A 2003 High resolution rotary capacitance tomography system Proc. 3rd World Congress on IPT (Banff, Canada) pp 858–63 Hart L W, Ko H W, Meyer J H Jr, Vasholz D P and Joseph R I 1988 A noninvasive electromagnetic conductivity sensor for biomedical applications IEEE Trans. Biomed. Eng. 35 1011–22 Hou W D and Mo Y L 2002 Increasing image resolution in electrical impedance tomography Electron. Lett. 38 701–2 Hua P, Woo E J, Webster J G and Tompkins W J 1993 Using compound electrodes in electrical impedance tomography IEEE Trans. Biomed. Eng. 40 29–34 Huang Z, Wang B and Li H 2003 Application of electrical capacitance tomography to the void fraction measurement of two-phase flow IEEE Trans. Instrum. Meas. 52 7–12 Min M, Martens O and Parve T 2000 Lock-in measurement of bio-impedance variations Measurement 27 21–8 Murphy S C and York T A 2006 Electrical impedance tomography with non-stationary electrodes Meas. Sci. Technol. 17 3042–52 Pinheiro P A T, Loh W W and Dickin F J 1998 Optimal sized electrodes for electrical resistance tomography Electron. Lett. 34 69–70 Polydorides N and Lionheart W R B 2002 A Matlab toolkit for three-dimensional electrical impedance tomography: a contribution to the Electrical Impedance and Diffuse Optical Reconstruction Software project Meas. Sci. Technol. 13 1871–83 Polydorides N and McCann H 2002 Electrode configurations for improved spatial resolution in electrical impedance tomography Meas. Sci. Technol. 13 1862–70 Record P, Wang M and Dickin F 1995 Conducting boundary strategy: a new technique for medical EIT Physiol. Meas. 16 A249–55 Szczepanik Z and Rucki Z 2000 Frequency analysis of electrical impedance tomography system IEEE Trans. Instrum. Meas. 49 844–51 Wang H, Wang C and Yin W 2001 Optimum design of the structure of the electrode for a medical EIT system Meas. Sci. Technol. 12 1020–3 Wang W, Brown B H, Leathard A D and Lu L 1994 Noise equalization within EIT images Physiol. Meas. 15 A211–6 York T 2001 Status of electrical tomography in industrial applications J. Electron. Imaging 10 608–19