# Quantitative Evaluations with 2d Electrical Resistance Tomography in the Low-Conductivity Solutions Using 3d-Printed Phantoms and Sucrose Crystal Agglomerate Assessments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. ERT Imaging

#### 2.1. Modeling and Simulation Studies in ERT

#### 2.2. Reconstruction Methods

^{N X LK}, where K is the number of current patterns, L is the number of electrodes, and N is the number of parameters to be estimated. The normalized changes in the electrical conductivities can be computed as

_{n}= diag (V

_{1}

^{1}, V

_{2}

^{1}, V

_{3}

^{1}, …, V

_{L}

^{K})J,

#### 2.3. ERT and Quantitative Spatial Evaluations

## 3. Experimental Design

#### 3.1. Experimental Setup and Sensor Design

^{2}. Reactor measurement sizes are shown in Table 2.

#### 3.2. Phantom Design and 3D Printing

#### 3.3. Sucrose Crystal Agglomerate Assesments, Experimental Progression, and Test Objectives

## 4. Results

#### 4.1. Differences Due to Reconstruction Methods

#### 4.2. Varying the Iterations in the TV Reconstruction

#### 4.3. Analysing Segmentation Methods and Morphological Image Processing

_{P}) with phantom diameters (PD) and the expected percentage area of phantoms are shown in Table A1 in the Appendix A. The effect of applying erosion as morphological processing to obtain is shown in Figure A2a–e in the Appendix A. The E0 to E30 signifies the morphological image processing erosion applied. E0 stands for no erosion applied, and E10, E20, and E30 stands for incremental erosion applied using ‘disk’ operation in MATLAB image processing toolbox with 10, 20, and 30 as the radius, respectively.

#### 4.4. Towards Quantitative Estimations Using a Combination of Image Processing Methods to Achieve the Expected Area Estimation

#### 4.4.1. Contrast Profile Assessment at Various Iteration Levels

#### 4.4.2. Evaluation of the Area Covered by Phantoms at Various Iterations

#### 4.4.3. Evaluation of the Area Covered by Phantoms at Various Threshold Levels

#### 4.5. Experimental Industrial Application in Assessment of Sucrose Crystals in Demineralized Water

_{C}. It was observed that the G-Channel segmentation provides better results compared to the Otsu method. The area of the 2D region visualized inside the reactor is compared.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**a**) Contrast profile plot for 50 mm phantom R1, reconstruction: TV, iterations: 2–12; channel: green. (

**b**) Contrast profile plot for 40 mm phantom R2, reconstruction: TV, iterations: 2–12; channel: green. (

**c**) Contrast profile plot for 30 mm phantom R3, reconstruction: TV, iterations: 2–12; channel: green. (

**d**) Contrast profile plot for 20 mm phantom R4, reconstruction: TV, iterations: 2–12; channel: green. (

**e**) Contrast profile plot for 10 mm phantom R5, reconstruction: TV, iterations: 2–12; channel: green. (

**f**) Contrast profile plot for phantom R6-L1 (2 × 10 mm), reconstruction: TV, iterations: 2–12; channel: green, location: 1.

**Figure A2.**Comparison of the area percentage for Otsu and G-Channel segmentation using TV, LBP, and GN reconstruct (

**a**) R1 (

**b**) R2, (

**c**) R3, (

**d**) R4, and (

**e**) R5.

**Table A1.**Table showing correlation coefficient of the evaluated percentage area with the phantom diameter and expected area.

Corr. PD | Corr. A_{P} | |||||||
---|---|---|---|---|---|---|---|---|

Phantom diameter (PD) | 10 | 20 | 30 | 40 | 50 | 0.9811 | ||

Expected area (A_{P}) | 1.45 | 5.81 | 13.06 | 23.23 | 36.29 | 0.9811 | ||

TV | G-Channel | 14.99 | 17.44 | 22.10 | 26.48 | 38.38 | 0.9563 | 0.9904 |

Otsu | 13.06 | 17.44 | 20.69 | 25.23 | 35.39 | 0.9714 | 0.9920 | |

LBP | G-Channel | 16.76 | 18.53 | 19.66 | 21.21 | 26.16 | 0.9499 | 0.9811 |

Otsu | 10.29 | 12.22 | 12.13 | 13.90 | 17.76 | 0.9314 | 0.9662 | |

GN | G-Channel | 7.44 | 8.28 | 9.34 | 10.71 | 15.38 | 0.9255 | 0.9740 |

Otsu | 4.60 | 4.70 | 5.57 | 6.49 | 9.23 | 0.9196 | 0.9748 | |

stdevA | 4.63 | 5.69 | 6.84 | 8.18 | 11.59 | |||

˄ (%) | 319.14 | 97.95 | 52.33 | 35.20 | 31.95 |

_{P}with the phantom diameters. Corr. EA shows the correlation coefficient of the evaluated percentage area with the expected percentage area. The correlation coefficient was computed using MATLAB function corrcoef(). Factor ˄ was evaluated using the following Equation (A1). Higher ˄ signifies the higher deviations from the expected percentage area A

_{P}.

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**Figure 1.**Schematic of the electrical resistance tomography (ERT) data acquisition and data processing system.

**Figure 3.**Factors affecting quantitative measurements using ERT as an imaging modality for the crystallization process.

**Figure 4.**(

**a**,

**b**) Setup of the laboratory-based batch reactor with sensor and signal conditioning unit mounted on the reactor; (

**c**) signal conditioning unit mounted on the 3D-printed frame.

**Figure 5.**(

**a**) Design of phantoms R1–R5; (

**b**) 3D-printed acrylonitrile butadiene styrene (ABS) phantoms; (

**c**) printing of sensor mounting unit and phantom R6; (

**d**) design and print of phantom R6.

**Figure 6.**Current detection at various electrodes in the reactor for (

**a**) industrial-grade saturated sucrose solution and tap water, and (

**b**) demineralized water.

**Figure 7.**(

**R1**–

**R5**) phantom reference; (

**a1**–

**a5**) Gauss–Newton (GN) reconstructions; (

**b1**–

**b5**) linear back projection (LBP) reconstructions; and (

**c1**–

**c5**) total variation (TV) reconstructions at 10 iterations for tap water.

**Figure 8.**(

**R1**–

**R5**) phantom reference; (

**a1**–

**a5**) Gauss–Newton reconstructions; (

**b1**–

**b5**) LBP reconstructions; and (

**c1**–

**c5**) TV reconstructions at 10 iterations for industrial grade saturated sucrose solution.

**Figure 9.**(

**R1**–

**R5**) phantom reference; (

**a1**–

**a5**) Gauss–Newton reconstructions; (

**b1**–

**b5**) LBP reconstructions; and (

**c1**–

**c5**) TV reconstructions at 10 iterations for demineralized water.

**Figure 10.**(

**a**) Changes in current at various electrodes after phantom placement; (

**b**) detailed view from electrode 2 to 14.

**Figure 11.**(

**a**) Tap water; (

**b**) industrial-grade saturated sucrose solution; (

**c**) demineralized water: (R5, R6-L1, R6-L2) phantom reference; (

**a1**–

**a3**) Gauss–Newton reconstructions; (

**b1**–

**b3**) LBP reconstructions; (

**c1**–

**c3**) TV reconstructions at 10 iterations.

**Figure 12.**(

**a0**–

**a5**) phantom R5 reference and TV iteration 10,8,6,4,2; (

**b1**–

**b5**) surface plot of the reconstructed images for phantom R5; (

**c0**–

**c5**) phantom R6-L1 reference and TV iteration 10,8,6,4,2; (

**d1**–

**d5**) surface plot of the reconstructed images for phantom R6-L1; (

**e0**–

**e5**) phantom R6-L2 reference and TV iteration 10,8,6,4,2; (

**f1**–

**f5**) surface plot of the reconstructed images for phantom R6-L2.

**Figure 13.**Various image segmentation methods for phantom R5. Solution: demineralized water, reconstruction method: total variation, iterations: 2.

**Figure 14.**Comparison of the area percentage for Otsu and G-Channel segmentation using TV, LBP, and GN reconstructions for (a) R1, (b) R2, (c) R3, (d) R4, and (e) R5.

**Figure 15.**Various image segmentation methods for R6-L1. Reconstruction method: total variation, iterations: 2, solution: demineralized water.

**Figure 17.**Contrast profile plot for phantom R6-L2 (2 × 10 mm), reconstruction: TV, iterations: 2–12; channel: green, location: 2.

**Figure 18.**Percentage area covered by phantoms at constant imaging threshold level and the varying number of iterations and erosion factors. Reconstruction: TV, iterations: 2–12; channel: green; for the phantoms (

**a**)R1; (

**b**) R2; (

**c**) R3; (

**d**) R4; (

**e**) R5.

**Figure 19.**Percentage area covered by phantoms at constant imaging thresholds and erosion factor and a varying number of iterations: combined view. Reconstruction: TV, iterations: 2–12; channel: green.

**Figure 20.**Percentage area covered by phantoms at a constant number of iterations and varying imaging thresholds and erosion levels. Reconstruction: TV, iterations: 2; channel: green; for the phantoms (

**a**) R1; (

**b**) R2; (

**c**) R3; (

**d**) R4; (

**e**) R5.

**Figure 21.**Percentage area covered by 2 × 10 mm (

**a**) phantom 1 and (

**b**) phantom 2 at various threshold levels. Reconstruction: TV, iterations: 2, channel: green.

**Figure 23.**Visualization of the sugar crystals in the demineralized solution at various frames representing time points. Reconstruction: TV, segmentation: Otsu and G-Channel at threshold 0.6.

Experimental Variable | Count | |
---|---|---|

1 | Number of solutions with varied conductivities | 3 |

2 | Number of phantoms | 6 |

3 | Number of reconstruction methods compared | 3 |

4 | Number of segmentation methods | 4 |

5 | Number of electrodes | 16 |

6 | Number of planes | 1 |

7 | Minimum accuracy tested | 1.5% of the beaker area |

8 | Location of object | Central and incremental, separability |

Object | Measured Values mm | |
---|---|---|

1 | Batch reactor’s inner diameter | 83 |

2 | Electrode tail diameter | 5 |

3 | Electrode head diameter | 12 |

Phantom | Measured Values mm | Expected Percentage Area of the Phantom Region (A_{P})% | |
---|---|---|---|

1 | Phantom R1 | 50 ± 0.1 | 36.28 |

2 | Phantom R2 | 40 ± 0.1 | 23.22 |

3 | Phantom R3 | 30 ± 0.1 | 13.06 |

4 | Phantom R4 | 20 ± 0.1 | 5.8 |

5 | Phantom R5 | 10 ± 0.1 | 1.45 |

6 | Phantom R6 | 2 × 10 ± 0.1 | 1.45 and 1.45 |

5 | Diameter of the base of phantoms | 50 | |

7 | Distance between centers of phantom R6 | 40 |

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## Share and Cite

**MDPI and ACS Style**

Rao, G.; Sattar, M.A.; Wajman, R.; Jackowska-Strumiłło, L.
Quantitative Evaluations with 2d Electrical Resistance Tomography in the Low-Conductivity Solutions Using 3d-Printed Phantoms and Sucrose Crystal Agglomerate Assessments. *Sensors* **2021**, *21*, 564.
https://doi.org/10.3390/s21020564

**AMA Style**

Rao G, Sattar MA, Wajman R, Jackowska-Strumiłło L.
Quantitative Evaluations with 2d Electrical Resistance Tomography in the Low-Conductivity Solutions Using 3d-Printed Phantoms and Sucrose Crystal Agglomerate Assessments. *Sensors*. 2021; 21(2):564.
https://doi.org/10.3390/s21020564

**Chicago/Turabian Style**

Rao, Guruprasad, Muhammad Awais Sattar, Radosław Wajman, and Lidia Jackowska-Strumiłło.
2021. "Quantitative Evaluations with 2d Electrical Resistance Tomography in the Low-Conductivity Solutions Using 3d-Printed Phantoms and Sucrose Crystal Agglomerate Assessments" *Sensors* 21, no. 2: 564.
https://doi.org/10.3390/s21020564