Fig. 10: (a) DRP of the images of Fig. 8, (b) DRP of the images of Fig. 9.
Fig. 8 Table Imaging 3: with CNR, simulated PCR, COC, data IR in Max EIDORS and IR Mean (a) of Original reconstructed object (simulated images in EIDORS phantom with with STR, inhomogeneity LMR and near PEPR electrode as shown no. in 3), Fig. 8 (b) image with STR, (c) image with LMR, (d) image with PEPR.
In the Eq. 12, the regularization parameter λ v aries in the range from 0.01/3 to 0.01/2. It includes two primary considerations [81]: (1) due to the ill-posed characteristic of the inverse problem, the regularization should not be reduced with iterations to too small a value. (2) the regularization value should be quantitatively similar to the diagonal of the matrix J T J (calculated by the MATLAB code: max(max( J T J )) ).
In EIT, the response matrix ( J J ) is formed by the first derivative ( J ) of the forward solution (FS) and I is formed as the approximation of the Hessian [61, 72] which is, actually, the second derivative of the FS. Generally, the order of magnitude of Hessian is less than the J T J due to the higher order derivative and hence, in Eq. 9, I acts as the quadratic term which is formed with the maximum value of J T J . As a rule, the magnitude of the diagonal elements of λ I should be less than the maximum value of J T J . Therefore, as the identity matrix I is formed with the diagonal values equal to the maximum value of the matrix J T J , λ is taken as, in general, a constant less than 1. It is observed that the projection error in LMR and PEPR becomes minimum at λ = 0.1 and Ψ = 0.01 respectively [47-48]. That is why in EIDORS with STR method, λ is taken as 0.1 throughout the reconstruction process whereas in LMR λ is taken as 0.1 first iteration then it is modified as 0.1/k, where k is the number of iterations. On the other hand, in EIDORS with PEPR λ is calculated with Ψ = 0.01 in Eq. 12. For further analysis the resistivity reconstruction is also conducted with a range of regularization parameters [47- 48]. Furthermore, for better understanding the regularization effects of both the methods, λ is calculated in PEPR technique (using the Eq. 12) with Ψ = 0.01. The result is compared with the reconstruction obtained in STR method (with λ = 0.01) and LMR technique in which the iteration starts with λ = 0.01 and then it is decreased by a factor of 10 for the other iterations [47-48]. In MoBIIR algorithm with STR regularization, a constant regularization parameter ( λ = λ c * η ) is used where the η is a constant calculated as the largest element of the response matrix J T J (max(max(J T J))). Hence, in STR method, the λ remains unchanged for all the iterations. On the other hand, in MoBIIR with LMR method, η is taken as the largest element of the response matrix J T J to form the λ ( λ = λ c * η ) in first iteration. But for all the other iterations λ is gradually decreased by a factor of 10 . Hence, in both STR and LMR methods, regularization parameter ( λ ) remain small constant number suitably calculated (as stated above) from the response matrix (J T J) in each of the iterations of the reconstruction process. Therefore, the local or regional physiological information of the nodal points in the domain under imaging are not taken into account in STR and LMR methods. Hence, in both STR and LMR methods, a constant regularization parameter is been used at each iteration and as a result the local or regional physiological attributes are not taken into account in them. Furthermore, due to the addition of constant regularization parameter to the matrix A , it perturbs the original system of equations and these perturbations are not based on the spectral information. Hence this perturbation may introduce a significant error in solution as well as some unwanted solution could be obtained. Conductivity reconstruction algorithms (MoBIIR and the EIDORS) with different regularization methods are represented by the following flow charts (Fig. 1). Simulated boundary data are generated for a circular domain (150 mm diameter) simulated with a circular inhomogeneity of different diameters near different electrodes. Boundary data are generated with simulated phantoms developed with a low resistive background (resistivity = 1.71 Ω m) and high resistive circular inhomogeneities (resistivity = 50.05 Ω m) at different electrode positions. Practical phantoms [82-88] are essential for studying the resistivity imaging to assess the reconstruction algorithm. Different practical phantoms are developed using biological and non-biological materials (biological and non-biological) and the resistivity imaging is conducted with different regularization techniques. A non-biological phantom called NaCl-Nylon phantom is developed with 0.9 % (w/v) NaCl solution as the bathing medium [82] and nylon cylinders as the inhomogeneity [85]. A sixteen electrode practical phantoms is developed with a shallow glass tank (150 mm diameter) and sixteen stainless steel electrodes [85] equally spaced on the tank inner wall. PEPR method is also studied with the boundary data collected from the chicken muscle tissue paste (CMTP) phantoms (Fig. 2a-2b) [87] and chicken muscle tissue block (CMTB) phantoms (Fig. 2c-2d) [88]. CMTP phantoms are developed with the chicken muscle tissue past (Fig. 2a) as the background medium and the fat tissues as the inhomogeneities (Fig. 2b). The CMTB phantoms are developed with the chicken muscle tissue blocks (Fig. 2c) as the background medium and the fat tissues as the inhomogeneities (Fig. 2b). The height of the background medium in both the CMTP and CMTB phantoms in the phantom tanks is kept as 10 mm so that an effective electrode area becomes 10 mm ×10 mm. Cylindrical stainless steel rods (25 mm dia.) are placed at the phantom center (Fig. 2a and 2b) to act as a common mode electrodes (CME) [86] which are connected to the ground point of the EIT electronics to reduce the common mode error [89]. A resistivity measurement setup [85] is used to measure the chicken muscle tissue resistivity and chicken fat tissue resistivity using an impedance analyzer (QuadTech 7600, QuadTech Inc.USA) with a test signal of 1mA, 50 kHz [90]. A USB based high speed data acquisition system developed with a NI USB-6251 DAQ card is used for boundary data acquisition through a multiplexer board connected to the surface electrodes. Data acquisition software is written in LabVIEW and interfaced with the DAQ card. LabVIEW based EIT instrumentation [6] consisting of constant current injector, signal conditioner and data acquisition system is used for boundary data collection. Constant current injector is developed with a sinusoidal signal generator and a constant current source [6, 91-93]. Signal conditioner block is developed with a 50 Hz notch filter and a narrow band pass filter with a center frequency of 50 kHz. In electrical impedance tomography a low frequency constant sinusoidal current is injected to the domain under test through the electrodes called current electrodes. The boundary potentials developed at the surface electrodes due to the current flux generated for current injection are collected for the image reconstruction. RMS potentials on all the electrodes (excluding the electrode connected to the positive terminal of current source) are measured for sixteen current projections and the complete voltage data set are saved as a .txt file in PC for computation. The boundary potentials are measured on the electrodes called voltage electrodes. In general, the differential potentials are measured across the different electrodes to avoid the error due to the contact impedance [94]. In the present study, however, in spite of the problem of skin impedance, to obtain the greatest sensitivity to changes in the resistivity of the body, voltages from current carrying electrodes should also be measured as reported by Cheng et al. [95]. To study the PEPR and other (STR and LMR) methods, the impedance images are reconstructed from the boundary potential data measured at the surface electrodes attached to the domain boundary. Boundary data are collected from different practical phantom by injecting a constant sinusoidal current of constant amplitude using different current patterns. 1 mA, 50 kHz constant sinusoidal current is injected to the NaCl-Nylon phantom, chicken tissue paste phantom and chicken tissue block phantom and the boundary potential data are collected using opposite (Fi.g-4a-4b) and neighbouring current injection (Fi.g-4c- 4d) patterns. Boundary data are collected from different phantoms containing different configurations with single and multiple inhomogeneity and different current patterns. Boundary voltage data collected for different phantom configurations are analysed to evaluate the system performance. The boundary potentials of all the symmetric electrode pairs (SEP) [6, 91] are collected and analyzed. Boundary potential data are collected repetitively from the phantom without inhomogeneity and the standard deviation (SD BP ) [6] and algebraic mean (BP ) [6] of the boundary potential data are calculated. Resistivity images are reconstructed from the simulated boundary data as well as the data collected from the real tissue phantoms using in MoBIIR and EIDORS. Images obtained from PEPR method are compared with the images obtained from STR and LMR methods. Resistivity imaging is studied with the boundary data collected from biological and non-biological phantoms containing single and multiple inhomogeneity at different electrode positions. Imaging study is also conducted for opposite and neighbouring current patterns for better understanding the effect of PEPR technique. Resistivity images are also reconstructed from the boundary data combined with random noise [47-48] of different percentages. Noisy data are used for reconstruction with the STR, LMR and PEPR methods and the images are compared. To analyze the proposed method, the normalized projection error (error due to the voltage mismatch), E V [47-48] and the normalized solution error norm (E ρ ) [47-48] are calculated. All the images are analyzed with their CNR [47-48, 85, 96], PCR [47-48, 96], COC [47- 48, 96] and DRP [47-48, 96] to evaluate their elemental resistivity profiles. Simulating a current injection in forward solver boundary potential data are collected for sixteen current projections and compared with the data reported by Ider et al [97]. The differential potential is calculated with the phantom data provided by Ider et al using our forward solver. The deviation between the data generated through our forward solver and that provided by Ider et al is only 0.08% to 0.80% [86]. Calculated or estimated boundary potential (V c ) obtained by the forward solver shows that (Fig. 3a) the maximum and minimum values of V c are 65 mV and 4.93 mV respectively for each projections. On the other hand the maximum and minimum values of average of the measured data (V m ) are 61.31 mV and 5.9 mV. The boundary potentials of the phantom with homogeneous chicken paste (without fat inhomogeneity) are measured repetitively and the standard deviation and SNR are calculated. It is observed that the average SNR and standard deviation of measured boundary potentials are 31.65 dB and 0.65 mV respectively (Fig. 3b). The breaking points of the SNR curve in Fig. 3b represent the measurement points with zero standard deviation because of the same measurement data at those points. Potential of the SEPs (Fig. 4a) shows that the experimental phantom is symmetric with respect to the current electrode axis. It is observed that the calculated voltage of the SEP8 is 4.98 mV where as the experimental voltage of the SEP8 is found as 4.5 mV. The SEP potentials are calculated both for the mean (average of all projections) value of estimated potentials (V c ) and the measured potentials (V m ). Results show that the SEP potentials are well matched both for the estimated potentials (Fig. 4b) and the measured potentials (Fig. 5a). It is observed that the standard deviations (STDV) in all the calculated SEP potentials points are zero (Fig. 5b). STDV of all the calculated SEP data points are zero and hence the SNR is very high (mathematically infinity as the SNR is the ratio of mean to standard deviation). On the other hand, the standard deviations in all the experimental SEP potentials points are nonzero (Fig. 5b) except at SEP8 (potential of the electrode connected to the ground point of VCCS). STDV of all the experimental SEP data points varies from 0.03 to 0.62 (Fig. 5b) and hence the SNR of all the experimental SEP points varies from 39.98 dB to 63.09 dB. Obviously the SNR of the SEP8 data point is found very high. Hence the boundary potential studies show that the phantom and its electrode system are symmetric, stable and suitable for EIT imaging. The PEPR technique is studied with the boundary data generated from the simulated phantom (Fig. 6a) with circular inhomogeneity (diameter = 35 mm, resistivity = 50.05 Ω m) at electrode no. 3. The boundary data generated from the simulated phantom (Fig. 6a) is found suitable for resistivity reconstruction with MoBIIR algorithm. Results show that the CNRs of the resistivity images with STR (Fig. 6b) and LMR (Fig. 6c) methods are 3.09 and 3.29 respectively whereas in the PEPR (Fig. 6d) it is 3.56 (Table 1). It is also observed that the PCRs of the resistivity images with STR and LMR techniques are 64.15 % and 60.62 % respectively where as in the PEPR it is 66.36 % (Table 1). COC in the PEPR method (4.04) is slightly higher than the PCR in STR methods (4.00) but it is much higher than the LMR method (Table 1). The boundary data collected from the chicken phantom (Fig. 7a) is also found suitable for resistivity imaging in MoBIIR. Resistivity images (Fig. 7b-7d) show that the CNRs of the resistivity images with STR (Fig. 7b) and LMR (Fig. 7c) methods are 2.71 and 3.04 respectively where as in PEPR method (Fig. 7d) CNR is 3.15 (Table 2). It is also observed that the PCRs of the reconstructed images are 61.89 % and 61.29 % in STR and LMR methods respectively where as in PEPR method it is 62.27 % (Table 2). Also, COCs in the PEPR method is found as 3.49 where as in STR and LMR methods COCs are 3.39 and 3.32 respectively (Table 2). The PEPR technique is studied with the boundary data generated from the simulated phantom (Fig. 8a) with circular inhomogeneity (diameter = 60 mm, resistivity = 50.05 Ω m) at electrode no. 3. The boundary data generated from the simulated phantom (Fig. 8a) is found suitable for resistivity reconstruction with EIDORS algorithm. Resistivity images reconstructed with STR (Fig. 8b) and LMR (Fig. 8c) technique have their CNRs as 3.15 and 3.38 respectively whereas the CNR of the image with PEPR (Fig. 8d) is 3.75 (Table 3). It is also noticed that the PCRs of the resistivity image with STR and LMR techniques with the same phantom are 29.86 % and 35.72 %respectively whereas the PCR is 54.41 % with PEPR (Table 3). Reconstructed results (Table 3) also show that the COCs of the resistivity images are 2.02 and 2.22 in STR and LMR methods respectively whereas the COC in PEPR is 2.79. Maximum inhomogeneity resistivity (IR Max ) and IR Mean obtained in PEPR technique are found more appropriate compared to the STR and LMR methods (Table 3). The PEPR technique is also studied with the boundary data generated from a similar simulated phantom (Fig. 9a) with circular inhomogeneity (diameter = 60 mm, resistivity = 50.05 Ω m) at electrode no. 5. Resistivity images (Fig. 9b-9d) reconstructed from the boundary data show that the CNR, PCR and COC with STR (Fig. 9b) and LMR (Fig. 9c) technique are lower than the PEPR (Fig. 9d) method (Table 4). IR Max and IR Mean obtained in PEPR are also found more appropriate compared to the STR and LMR methods (Table 4). DRP of the resistivity images shown in Fig. 8 and Fig. 8 are shown in Fig. 10a and Fig. 10b respectively. DRPs in Fig. 10a-10b show that the resistivity profiles of the reconstructed background and inhomogeneity are better in PEPR method compared to the STR and LMR techniques. Results show that the DRP of the resistivity image with PEPR is more similar to the DRP of the original object whereas they did not follow the original DRP in STR and LMR method (Fig. 10) properly. The resistivity profiles of the images of simulated phantom of Fig. 8a show that the elemental resistivity did not reach above 36.24 Ω m and 41.29 in STR and LMR method respectively (Fig. 10a). On the other hand, for PEPR technique, the maximum resistivity of the inhomogeneity reaches up to 57.07 Ω m (Fig. 10a). It is also observed that, for the images of simulated phantom of Fig. 9a, the elemental resistivity did not reach above 35.82 Ω m 40.58 Ω m in STR and LMR method respectively (Fig. 10b) whereas, for PEPR technique, the maximum resistivity of the inhomogeneity reaches up to 56.81 Ω m (Fig. 10b). Result also demonstrate that, for the images of simulated phantom of Fig. 9a, the elemental resistivity did not reach above 45.90 Ω m in STR and LMR method (Fig. 10b) whereas, for PEPR technique, the maximum inhomogeneity resistivity reaches up to 56.51 Ω m (Fig. 10b). Boundary data of the simulated phantom (Fig. 11a) are mixed with the 25 % random noise and the resistivity images are reconstructed from the noisy data in EIDORS with STR, LMR and PEPR methods. It is observed that the PEPR method reconstructs the better resistivity image from the noisy boundary data for all the inhomogeneity positions arbitrarily chosen for noise analysis study. Result show that, for the phantom with inhomogeneity at electrode number 1 (Fig. 11a), the CNRs of the resistivity image with STR (Fig. 11b) and LMR (Fig. 11c) methods are 2.16 and 2.41 respectively. On the other hand, for the same phantom, CNR of the image of with PEPR (Fig. 11d) technique is 3.69 (Table 5). The PCRs of the resistivity images with STR and LMR methods are 22.00 % and 26.75 % respectively whereas it is 54.02 % in PEPR technique (Table 5). It is also noticed that, the COC of the resistivity images in STR and LMR methods are 2.11 and 2.35 respectively but the PCR with PEPR is 3.50 (Table 5). IR Max and IR Mean obtained in PEPR method are also found more appropriate compared to the STR and LMR methods (Table 5). Results obtained for the simulated phantoms with inhomogeneity at electrode number 3 (Fig. 12a) and 5 (Fig. 12a) show that the resistivity images and their image parameters obtained with STR and LMR techniques are poor compared to the PEPR method (Table 6 and 7). It is observed that, for the phantom with inhomogeneity at electrode number 3 (Fig. 12a), the CNR, PCR and COC with STR (Fig. 12b) and LMR (Fig. 12c) technique are lower than the PEPR (Fig. 12d) method (Table 6). Similarly, for the phantom with inhomogeneity at electrode number 5 (Fig. 13a), the CNR, PCR and COC with STR (Fig. 13b) and LMR (Fig. 13c) technique are lower than the PEPR (Fig. 13d) method (Table 7). IR Max and IR Mean obtained in PEPR technique are found also more appropriate compared to the STR and LMR methods for both the phantom configurations (Table 6 and 7). Image reconstruction studies with noisy boundary data show that the PEPR technique improves quality of the image reconstructed from noisy boundary data more effectively. This is because in noisy environment the boundary data mismatch ( Δ V=V -V ) is more and hence the regularization parameter ( λ ) is defined as per the reconstruction requirement. A large mismatch in boundary data produces a large regularization parameter to give the proper image reconstruction. In STR method regularization parameter is constant throughout the reconstruction and it is independent of mismatch ( Δ V ). LMR technique uses a regularization parameter which gradually reduces as the iteration goes on and hence it, also, fails to counteract the increased Δ V . On the contrary, PEPR method is a function of Δ V and hence the regularization is automatically adjusted according to the boundary data mismatch. Hence, as the Δ V increases (in noisy data), λ increases accordingly to counteract the large mismatch. As a result, during the reconstruction with noisy data in PEPR, the λ is automatically set to a large value required to restrict the solution domain of the EIT and gives a better image reconstruction. DRP of the resistivity images shown in Fig. 11, Fig. 12 and Fig. 13 are shown in Fig. 14a, Fig. 14b and Fig. 14c respectively. DRPs in Fig. 14a-14c show that the resistivity profile of the reconstructed background and inhomogeneity is better in PEPR method compared to the STR and LMR techniques. Results show that the DRP of the resistivity image with PEPR is more similar to the DRP of the original object whereas they did not follow the original DRP properly in STR and LMR method (Fig. 14). The resistivity profiles of the images of simulated phantom of Fig. 13a show that the elemental resistivity did not reach above 26.81 Ω m and 31.97 in STR and LMR methods respectively (Fig. 14a). On the other hand, for PEPR technique, the maximum resistivity of the inhomogeneity reaches up to 57.00 Ω m (Fig. 14a). It is also observed that, for the images of simulated phantom of Fig. 12a, the elemental resistivity did not reach above 41.34 Ω m in STR and LMR methods (Fig. 14b) whereas, for PEPR technique, the maximum resistivity of the inhomogeneity reaches up to 56.53 Ω m (Fig. 14b). For the simulated phantom of Fig. 13a, the DRP obtained with PEPR method is also found better compared to the STR and LMR methods (Fig. 14c). It is observed that the elemental resistivity did not reach above 44.82 Ω m and 48.75 Ω m in STR and LMR methods (Fig. 14b) respectively whereas, for PEPR technique, the maximum resistivity of the inhomogeneity reaches up to 55.82 Ω m (Fig. 14b). Resistivity reconstruction is also studied with the boundary data collected from the NaCl-Nylon phantom (Fig. 15a) in EIDORS with all three regularization techniques. Results show that the CNRs of the resistivity images with STR (Fig. 15b) and LMR (Fig. 15c) are 2.28 and 2.28 respectively (Table 8) whereas in PEPR (Fig. 15d), the CNR is found as 2.79 (Table 8). It is also noticed that (Table 8) the PCRs of the resistivity image with STR and LMR techniques with the same phantom are 56.72 % and 56.72 % respectively whereas the PCR is 73.46 % with PEPR. Reconstructed results show that the COCs of the resistivity image with STR and LMR methods are same (3.19) whereas the COC in PEPR is 3.88 (Table 8). It is observed that, for the boundary data collected from NaCl-Nylon phantom, the best resistivity image is reconstructed in the first iteration in EIDORS. Therefore, the regularization parameters in STR and LMR become identical and hence their effects on image reconstruction become same. As a result the image parameters (CNR, PCR, COC, IR Max and IR Mean ) of the resistivity images with STR and LMR become same. IR Max and IR Mean obtained in PEPR technique are found also more appropriate compared to the STR and LMR methods (Table 8). Resistivity reconstruction is also studied with the boundary data collected from the same chicken phantom (Fig. 16a) in EIDORS with all three regularization techniques. Results show that the CNRs of the resistivity images with STR (Fig. 16b) and LMR (Fig. 16c) are 1.88 and 1.88 respectively (Table 9) whereas in PEPR (Fig. 16d), the CNR is found as 2.29 (Table 9). It is also noticed that (Table 9) the PCRs of the resistivity image with STR and LMR techniques with the same phantom are 57.39 % and 57.39 % respectively whereas the PCR is 72.79 % with PEPR. Reconstructed results show that the COCs of the resistivity image with STR and LMR methods are same (2.84) whereas the COC in PEPR is 3.52 (Table 9). It is observed that, for the boundary data collected from chicken phantom, the best resistivity image is reconstructed in the first iteration in EIDORS. Therefore, the regularization parameters in STR and LMR become identical and hence their effects on image reconstruction become same. As a result the image parameters (CNR, PCR, COC, IR Max and IR Mean ) of the resistivity images with STR and LMR become same. IR Max and IR Mean obtained in PEPR technique are found also more appropriate compared to the STR and LMR methods (Table 9). Resistivity reconstruction is also studied with the boundary data collected from the chicken tissue block phantom (Fig. 17a) in EIDORS with all three regularization techniques. Results show that the CNRs of the resistivity images with STR (Fig. 17b) and LMR (Fig. 17c) are 1.79 and 2.33 respectively (Table 10) whereas in PEPR (Fig. 17d), the CNR is found as 2.68 (Table 10). It is also noticed that (Table 10) the PCRs of the resistivity image with STR and LMR techniques with the same phantom are 47.51 % and 61.48 % respectively whereas the PCR is 72.86 % with PEPR. It is also noticed that, the COC of the resistivity images in STR and LMR methods are 2.45 and 2.84 respectively but the PCR with PEPR is 3.34 (Table 10). It is observed that, the image parameters (CNR, PCR, COC, IR Max and IR Mean ) of the resistivity images with STR and LMR are found poor compared to the PEPR method. IR Max and IR Mean obtained in PEPR technique are found also more appropriate compared to the STR and LMR methods (Table 10). DRP of the resistivity images shown in Fig. 15, Fig. 16 and Fig. 17 are shown in Fig. 18a, Fig. 18b and Fig. 18c respectively. DRPs in Fig. 18a-18c show that the resistivity profiles of the reconstructed background and inhomogeneity are better in PEPR method compared to the STR and LMR techniques. Results show that the DRP of the resistivity image with PEPR is more similar to the DRP of the original object whereas they did not follow the original DRP in STR and LMR method (Fig. 18) properly. The resistivity profiles of the images of NaCl-Nylon phantom of Fig. 15a show that the elemental resistivity did not reach above 5.01 Ω m and 5.01 in STR and LMR method respectively (Fig. 18a). On the other hand, for PEPR technique, the average resistivity obtained is 5.35 Ω m with a maximum resistivity of the inhomogeneity of 6.44 Ω m (Fig. 18a). It is also observed that, for the images of simulated phantom of Fig. 16a, the elemental resistivity did not reach above 45.9 Ω m 45.9 Ω m in STR and LMR method respectively (Fig. 18b) whereas, for PEPR technique, the average resistivity obtained is 49.18 Ω m with a maximum resistivity of the inhomogeneity of 56.51 Ω m (Fig. 18b). Result also demonstrate that, for the images of simulated phantom of Fig. 17a, the elemental resistivity did not reach above 41.74 Ω m and 51.33 Ω m in STR and LMR method (Fig. 18c) respectively, whereas, for PEPR technique, the average resistivity obtained is 50.25 Ω m with a maximum resistivity of the inhomogeneity of 58.08 Ω m (Fig. 18c). Resistivity images are reconstructed from the boundary data collected from the practical phantoms with different current injection methods and the PEPR regularization is studied. Practical phantoms are developed with NaCl solution and the nylon cylinders (40 mm diameter) are used as the inhomogeneities. Boundary data are collected from the NaCl-Nylon phantoms with opposite and neighbouring current injection methods and the resistivity images are reconstructed with STR, LMR and PEPR methods. The results obtained with PEPR method are compared with the STR and LMR methods for both the current injection methods. Image reconstruction is also studied with the boundary data collected from the NaCl-Nylon phantom (Fig. 19a) in EIDORS with all three regularization techniques in opposite current injection method. Results show that, for opposite current injection method, the CNRs of the resistivity images with STR (Fig. 19b) and LMR (Fig. 19c) are 2.72 and 2.72 respectively (Table 11) whereas in PEPR (Fig. 19d), the CNR is found as 3.59 (Table 11). It is also noticed that (Table 11) the PCRs of the resistivity image with STR and LMR techniques with the same phantom are 53.46 % and 53.46 % respectively whereas the PCR is 85.63 % with PEPR. It is also noticed that, the COC of the resistivity images in STR and LMR methods are 3.38 and 3.38 respectively but the PCR with PEPR is 5.24 (Table 11). It is observed that, for the boundary data collected from chicken phantom, the best resistivity image is reconstructed in the second iteration in EIDORS. Therefore, the regularization parameters in STR and LMR become identical and hence their effects on image reconstruction become same. As a result the image parameters (CNR, PCR, COC, IR Max and IR Mean ) of the resistivity images with STR and LMR become same. IR Max and IR Mean obtained in PEPR technique are found also more appropriate compared to the STR and LMR methods (Table 11). Resistivity reconstruction is also studied with the boundary data collected from the NaCl-Nylon phantom (Fig. 20a) in EIDORS with STR, LMR and PEPR in neighbouring current injection method. Results show that, for neighbouring current injection method, the CNRs of the resistivity images with STR (Fig. 20b) and LMR (Fig. 22c) are 3.34 and 4.07 respectively (Table 12) whereas in PEPR (Fig. 20d), the CNR is found as 4.60. It is also noticed that (Table 12) the PCRs of the resistivity image with STR and LMR techniques with the same phantom are 34.42 % and 51.83 % respectively whereas the PCR is 62.62 % with PEPR. It is also noticed that, the COC of the resistivity images in STR and LMR methods are 2.03 and 2.54 respectively but the PCR with PEPR is 2.84 (Table 12). It is observed that, the image parameters (CNR, PCR, COC, IR Max and IR Mean ) of the resistivity images with STR and LMR are found poor compared to the PEPR method. IR Max and IR Mean obtained in PEPR technique are found also more appropriate compared to the STR and LMR methods (Table 12). Resistivity images are also reconstructed from the boundary data collected from the NaCl-Nylon phantoms with multiple inhomogeneities (40 mm diameter) using different current injection methods and the PEPR regularization is studied. Multiple inhomogeneity phantoms are developed with NaCl solution and the nylon cylinders and the boundary data are collected with opposite and neighbouring current injection methods. Resistivity images are reconstructed with STR, LMR and PEPR methods. Results obtained with PEPR method are compared with the STR and LMR methods for the reconstructions obtained with both the current patterns. DRP of the resistivity images shown in Fig. 19 and Fig. 20 are shown in Fig. 21a and Fig. 21b respectively. Results show that the reconstructed resistivity profiles of the phantom background and inhomogeneity obtained with opposite current injection method are better in PEPR compared to STR and LMR methods (Fig. 21a). It is also observed that, for the resistivity images of the practical phantom (Fig 20) obtained with the neighbouring current injection method, the reconstructed background profile and reconstructed inhomogeneity profile are better in PEPR compared to STR and LMR methods (Fig. 21b). Results show that, for opposite current injection method, the PEPR regularization technique produces the better resistivity images compared to the STR and LMR methods for all the phantom configurations as shown in the Fig. 22a-22d. It is observed that (Fig. 22), for opposite current injection method, the background noise is reduced in PEPR method and consequently, the image quality of the resistivity images is improved. Results show that, for neighbouring current injection method, the PEPR regularization technique produces the better image compared to the STR and LMR methods for all the phantom configurations as shown in the Fig. 23a-23d. Results also show that (Fig. 23), for neighbouring current injection method, the background noise is reduced in PEPR method and consequently, the image quality of the resistivity images is improved. A novel Projection Error Propagation-based Regularization (PEPR) method is proposed to improve the image quality in electrical impedance tomography (EIT). PEPR technique is implemented successfully in MoBIIR and EIDORS algorithms to regularize the solution domain in resistivity reconstruction in EIT and the impedance reconstruction is improved. Resistivity images are reconstructed from the simulated and practical phantom data in MoBIIR algorithm as well as with EIDORS with PEPR and the results are compared with the single step regularization (STR) and Modified Levenberg Regularization (LMR) techniques. The performance of the PEPR method is studied the simulated boundary data, simulated boundary data with random noise and the boundary data collected from the NaCl phantoms and real tissue phantoms with different inhomogeneity configurations and the results are compared. The PEPR method is also studied with boundary data collected from the practical phantoms with multiple inhomogeneities and the practical phantom data collected with different current injection patterns. Results demonstrate that, for both all the simulated data, PEPR technique reduced the projection error and solution error more effectively and efficiently in each iterations and improved the image quality in MoBIIR and EIDORS compared to STR and LMR methods. Noise analysis show that, the quality of the resistivity images reconstructed from noisy boundary data is remarkably improved with PEPR method for its better selection of regularization parameter in noisy environment. The resistivity imaging studies conducted with NaCl phantoms and the chicken tissue phantoms show that the PEPR method improves the reconstruction compared to STR and LMR. On the contrary, in STR and LMR method the quality of the resistivity image are found low for noisy data because of their constant and decreasing regularization parameters respectively all for simulated data, noisy simulated data, NaCl phantom data and chicken tissue phantom data. Resistivity imaging with multiple inhomogeneity phantoms shows that, the PEPR technique improves the resolution and image quality for multiple inhomogeneity phantoms both for with opposite and neighbouring current injection methods. Image analysis of all the reconstructed images obtained with simulated and practical phantom data shows that the resistivity images obtained with PEPR method are found with higher contrast parameters and better resistivity profiles both in MoBIIR and EIDORS. The entire study on the resistivity imaging with PEPR method is conducted in two dimensions though the similar result could be easily obtained in three dimensions with the availability of boundary data of a 3-D system. Furthermore the same system can be modified to a 3-D system with a linear shift (in Z-direction) of the electrode array. Hence the PEPR technique can be suitably used for 2D as well as 3D EIT systems (like real measurements, such as, on the human body) for enhancing the image quality. The simulated and experimental result of PEPR technique applied on 3D system will be presented in future communications. The authors acknowledge the support of the Indian Institute of Science Bangalore, India, to carry out the above investigations.Join ResearchGate to access over 30 million figures and 100+ million publications – all in one place.Copy referenceCopy captionEmbed figurePublished in
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