# Calibration Alignment Sensitivity in Corneal Terahertz Imaging

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## Abstract

**:**

## 1. Introduction

## 2. Theory Based on Fourier Optic and Vector Spherical Harmonics

## 3. Cornea Calibration

#### 3.1. Correctly Calibrated Cornea

#### 3.2. Perturbed Calibrated Cornea

## 4. Extraction of Corneal Features

#### 4.1. PSO Analysis for Correctly Calibrated Cornea

#### 4.2. PSO Analysis in Case of Perturbation

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AWGN | Additive White Gaussian Noise |

CAWC | Corneal Anterior Water Content |

CCT | Corneal Central Thickness |

COC | Center of Curvature |

CPWC | Corneal Posterior Water Content |

PSO | Particle Swarm Optimization |

RMSD | Root-Mean-Square Deviation |

ROC | Radius of Curvature |

SNR | Signal to Noise Ratio |

## References

- Topfer, F.; Oberhammer, J. Millimeter-Wave Tissue Diagnosis: The Most Promising Fields for Medical Applications. IEEE Microw. Mag.
**2015**, 16, 97–113. [Google Scholar] [CrossRef] - Sun, Q.; Stantchev, R.I.; Wang, J.; Parrott, E.P.; Cottenden, A.; Chiu, T.W.; Ahuja, A.T.; Pickwell-MacPherson, E. In vivo estimation of water diffusivity in occluded human skin using terahertz reflection spectroscopy. J. Biophotonics
**2019**, 12, e201800145. [Google Scholar] [CrossRef] [PubMed] - Taylor, Z.D.; Garritano, J.; Sung, S.; Bajwa, N.; Bennett, D.B.; Nowroozi, B.; Tewari, P.; Sayre, J.; Hubschman, J.P.; Deng, S.; et al. THz and mm-wave sensing of corneal tissue water content: Electromagnetic modeling and analysis. IEEE Trans. Terahertz Sci. Technol.
**2015**, 5, 170–183. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Taylor, Z.D.; Garritano, J.; Sung, S.; Bajwa, N.; Bennett, D.B.; Nowroozi, B.; Tewari, P.; Sayre, J.W.; Hubschman, J.; Deng, S.X.; et al. THz and mm-Wave Sensing of Corneal Tissue Water Content: In Vivo Sensing and Imaging Results. IEEE Trans. Terahertz Sci. Technol.
**2015**, 5, 184–196. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sung, S.; Selvin, S.; Bajwa, N.; Chantra, S.; Nowroozi, B.; Garritano, J.; Goell, J.; Li, A.D.; Deng, S.X.; Brown, E.R.; et al. THz Imaging System for in vivo Human Cornea. IEEE Trans. Terahertz Sci. Technol.
**2018**, 8, 27–37. [Google Scholar] [CrossRef] [PubMed] - Sung, S.; Selvin, S.; Bajwa, N.; Chantra, S.; Nowroozi, B.; Garritano, J.; Goell, J.; Li, A.D.; Deng, S.X.; Brown, E.R.; et al. Optical System Design for Noncontact, Normal Incidence, THz Imaging of in vivo Human Cornea. IEEE Trans. Terahertz Sci. Technol.
**2018**, 8, 1–12. [Google Scholar] [CrossRef] [PubMed] - Castoro, J.A.; Bettelheim, A.A.; Bettelheim, F.A. Water gradients across bovine cornea. IOVS
**1998**, 29, 963–968. [Google Scholar] - Orfanidis, S.J. Electromagnetic Waves and Antennas; Rutgers University New Brunswick: New Brunswick, NJ, USA, 2002. [Google Scholar]
- Zarrinkhat, F.; Lamberg, J.; Baggio, M.; Tamminen, A.; Ala-Laurinaho, J.; Khaled, E.E.M.; Rius, J.M.; Robert, J.R.; Taylor, Z. Experimental exploration of longitudinal modes in spherical shells at 220 GHz–330 GHz: Applications to corneal sensing. In Proceedings of the CLEO/Europe-EQEC, Virtual, 21–25 June 2021; p. 1. [Google Scholar]
- Tamminen, A.; Pälli, S.V.; Ala-Laurinaho, J.; Salkola, M.; Räisänen, A.V.; Taylor, Z.D. Quasioptical System for Corneal Sensing at 220-330 GHz: Design, Evaluation, and Ex Vivo Cornea Parameter Extraction. IEEE Trans. Terahertz Sci. Technol.
**2021**, 11, 135–149. [Google Scholar] [CrossRef] - Zarrinkhat, F.; Lamberg, J.; Tamminen, A.; Baggio, M.; Ala-Laurinaho, J.; Khaled, E.E.M.; Rius, J.M.; Robert, J.R.; Taylor, Z. Fourier Analysis of Submillimeter-Wave Scattering from the Human Cornea. In Proceedings of the 15th European Conference on Antennas and Propagation, Düsseldorf, Germany, 22–26 March 2021; pp. 1–5. [Google Scholar]
- Zarrinkhat, F.; Lamberg, J.; Tamminen, A.; Baggio, M.; Ala-Laurinaho, J.; Khaled, E.E.M.; Rius, J.; Romeu, J.; Taylor, Z. Coupling to Longitudinal Modes in Spherical Thin Shells Illuminated by Submillimeter Wave Gaussian Beam: Applications to Corneal Sensing. arXiv
**2021**, arXiv:2112.00096. [Google Scholar] - Baggio, M.; Tamminen, A.; Presnyakov, S.; Kravchenko, N.P.; Nefedova, I.; Ala-Laurinaho, J.; Brown, E.; Deng, S.; Wallace, V.; Taylor, Z.D. Investigation of optimal THz band for corneal water content quantification. In Proceedings of the CLEO/Europe-EQEC, Virtual, 21–25 June 2021. [Google Scholar]
- Tamminen, A.; Baggio, M.; Nefedova, I.I.; Sun, Q.; Presnyakov, S.A.; Ala-Laurinaho, J.; Brown, E.R.; Wallace, V.P.; Pickwell-MacPherson, E.; Maloney, T.; et al. Extraction of Thickness and Water-Content Gradients in Hydrogel-Based Water-Backed Corneal Phantoms Via Submillimeter-Wave Reflectometry. IEEE Trans. Terahertz Sci. Technol.
**2021**, 11, 647–659. [Google Scholar] [CrossRef] - Khaled, E.E.M.; Hill, S.C.; Barber, P.W. Scattered and internal intensity of a sphere illuminated with a Gaussian beam. IEEE Trans. Terahertz Sci. Technol.
**1993**, 41, 295–303. [Google Scholar] [CrossRef] - Khaled, E.E.M.; Hill, S.C.; Barber, P.W. Light scattering by a coated sphere illuminated with a Gaussian beam. Appl. Opt.
**1994**, 33, 3308–3314. [Google Scholar] [CrossRef] [PubMed] - Barber, P.W.; Hill, S.C. Light Scattering by Particles: Computational Methods; World Scientific: Singapore, 1990. [Google Scholar]
- Bohren, C.; Huffman, D.R. Absorption and Scattering of Light by Small Particles; Wiley Science Paperback Series; Wiley: Hoboken, NJ, USA, 1998. [Google Scholar]
- Peña, O.; Pal, U. Scattering of electromagnetic radiation by a multilayered sphere. IEEE Trans. Antennas Propag.
**2015**, 57, 69–116. [Google Scholar] [CrossRef] - Xu, J.; Plaxco, K.; Allen, S.; Bjarnason, J.; Brown, E. 0.15–3.72 THz absorption of aqueous salts and saline solutions. Appl. Phys.
**2007**, 90, 031908. [Google Scholar] - Tamminen, A.; Baggio, M.; Nefedova, I.; Sun, Q.; Anttila, J.; Ala-Laurinaho, J.; Brown, E.R.; Wallace, V.P.; Pickwell-MacPherson, E.; Maloney, T.; et al. Submillimeter-Wave Permittivity Measurements of Bound Water in Collagen Hydrogels via Frequency Domain Spectroscopy. IEEE Trans. Terahertz Sci. Technol.
**2021**, 11, 538–547. [Google Scholar] [CrossRef] - Hespanha, J.P. Topics in Undergraduate Control Systems Design. Available online: https://web.ece.ucsb.edu/~hespanha/published/allugtopics-20210326.pdf (accessed on 26 March 2021).
- Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN’95, Perth, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948. [Google Scholar]

**Figure 1.**The cornea is illuminated by a Gaussian beam in a way that the beam focus on the sub-confocal point. The cornea radius is $7.5$ mm. Corneal phantom thickness is either 580 μm or 680 μm discretized to 29 and 34 equally distanced 20 μm layers, respectively.

**Figure 2.**Correctly calibrated cornea coupling efficiency (

**a**) magnitude and (

**b**) phase are compared with stratified medium theory (shown in black-dashed line) at frequency range of 100–600 GHz.

**Figure 3.**Different perturbations in the calibration sphere are shown: (

**a**,

**b**) calibration ROC variation, fixed COC, (

**c**,

**d**) calibration ROC variation, fixed apex, (

**e**,

**f**) calibration transverse misalignment, and (

**g**,

**h**) calibration axial misalignment. Correct calibration (Cal${}_{0}$) is plotted with a dashed grey line and compared with different perturbation scenarios. Its COC and ROC coincide with the cornea.

**Figure 4.**(

**a**) Magnitude and (

**b**) phase of cornea 1 calibrated by 8 different perturbation scenarios compared to correctly calibrated cornea, Cal${}_{0}$, at frequency range of 100–600 GHz.

**Figure 5.**PSO analysis for extracting CPWC, CAWC, and CCT of six corneas at frequency band of WR $5.1$ (140–220 GHz) considering SNR of (

**a**,

**d**) 40 dB, (

**b**,

**e**) 50 dB, and (

**c**,

**f**) 60 dB. Nominal values are shown with yellow dot and each cornea properties is plotted with different color.

**Figure 6.**PSO analysis comparison for (

**a**,

**c**) WR-$3.4$ (220–330 GHz), and (

**b**,

**d**) WR-$2.2$ (330–500 GHz), considering SNR 60 dB for six correctly calibrated noisy corneas. Nominal values are shown with yellow dot and each cornea properties is plotted with different color.

Cornea | Thickness | ACWC | PCWC |
---|---|---|---|

Cornea 1 | 580 μm | 40 | 70 |

Cornea 2 | 580 μm | 40 | 80 |

Cornea 3 | 580 μm | 40 | 90 |

Cornea 4 | 680 μm | 40 | 70 |

Cornea 5 | 680 μm | 40 | 80 |

Cornea 6 | 680 μm | 40 | 90 |

**Table 2.**RMSD of estimated CCT, CAWC, and CPWC for six noisy cornea calibrated with various noisy perturbed PEC sphere at frequency band of WR $5.1$. The SNR 60 dB is considered.

Ca1${}_{1}$ | Ca1${}_{2}$ | Ca1${}_{3}$ | Ca1${}_{4}$ | Ca1${}_{5}$ | Ca1${}_{6}$ | Ca1${}_{7}$ | Ca1${}_{8}$ | |
---|---|---|---|---|---|---|---|---|

CCT${}_{1}$ | $22.2$ μm | $12.7$ μm | $26.0$ μm | $24.7$ μm | $16.6$ μm | $14.9$ μm | $19.9$ μm | $22.5$ μm |

CAWC${}_{1}$ | $2.5$% | $6.1$% | $3.4$% | $9.7$% | $10.0$% | $10.0$% | $9.7$% | $3.8$% |

CPWC${}_{1}$ | $10.5$% | $9.7$% | $14.7$% | $13.4$% | $29.5$% | $29.3$% | $12.2$% | $12.7$% |

CCT${}_{2}$ | $27.3$ μm | $25.2$ μm | $32.8$ μm | $29.1$ μm | $18.7$ μm | $14.9$ μm | $28.4$ μm | $33.9$ μm |

CAWC${}_{2}$ | $3.0$% | $6.7$% | $2.6$% | $9.9$% | $10.0$% | $10.0$% | $9.8$% | $4.9$% |

CPWC${}_{2}$ | $7.2$% | $9.6$% | $8.2$% | $11.0$% | $19.3$% | $19.4$% | $10.6$% | $10.4$% |

CCT${}_{3}$ | $33.2$ μm | $39.3$ μm | $36.7$ μm | $44.0$ μm | $35.2$ μm | $32.1$ μm | $42.0$ μm | $36.1$ μm |

CAWC${}_{3}$ | $3.0$% | $6.7$% | $2.8$% | $9.8$% | 10% | 10% | $9.9$% | $3.7$% |

CPWC${}_{3}$ | $8.8$% | $10.3$% | $9.7$% | $6.3$% | $9.3$% | $8.9$% | $7.6$% | $11.9$% |

CCT${}_{4}$ | $15.5$ μm | $12.3$ μm | $17.4$ μm | $15.2$ μm | $49.9$ μm | $49.9$ μm | $12.2$ μm | $15.6$ μm |

CAWC${}_{4}$ | $1.9$% | $6.5$% | $0.9$% | $10.0$% | $10.0$% | $10.0$% | $9.9$% | $2.1$% |

CPWC${}_{4}$ | $8.8$% | $7.7$% | $8.3$% | $8.1$% | $29.7$% | $29.2$% | $7.7$% | $7.3$% |

CCT${}_{5}$ | $16.9$ μm | $20.0$ μm | $23.4$ μm | $22.8$ μm | $49.9$ μm | $49.7$ μm | $17.3$ μm | $22.1$ μm |

CAWC${}_{5}$ | $1.7$% | $6.6$% | $0.7$% | $9.8$% | $10.0$% | $10.0$% | $9.9$% | $2.0$% |

CPWC${}_{5}$ | $6.3$% | $9.9$% | $8.4$% | $8.9$% | $19.8$% | $19.8$% | $5.6$% | $7.5$% |

CCT${}_{6}$ | $36.9$ μm | $41.8$ μm | $40.9$ μm | $43.5$ μm | $50.0$ μm | $50.0$ μm | $42.8$ μm | $40.0$ μm |

CAWC${}_{6}$ | $1.44$% | $6.6$% | $0.84$% | $9.9$% | $10.0$% | $10.0$% | $10.0$% | $1.86$% |

CPWC${}_{6}$ | $6.2$% | $4.3$% | $7.3$% | $5.6$% | $10.3$% | $10.1$% | $4.7$% | $7.9$% |

**Table 3.**RMSD of estimated CCT, CAWC, and CPWC for six noisy cornea calibrated with various noisy perturbed PEC sphere at frequency band of WR $3.4$. The SNR 60 dB is considered.

Ca1${}_{1}$ | Ca1${}_{2}$ | Ca1${}_{3}$ | Ca1${}_{4}$ | Ca1${}_{5}$ | Ca1${}_{6}$ | Ca1${}_{7}$ | Ca1${}_{8}$ | |
---|---|---|---|---|---|---|---|---|

CCT${}_{1}$ | $19.4$ μm | $23.5$ μm | $24.1$ μm | $25.2$ μm | $48.7$ μm | $48.4$ μm | $21.1$ μm | $21.6$ μm |

CAWC${}_{1}$ | $2.0$ % | $7.1$ % | $6.0$ % | $9.9$ % | $10.0$ % | $10.0$ % | $9.9$ % | $3.3$ % |

CPWC${}_{1}$ | $11.1$ % | $13.5$ % | $13.6$ % | $15.6$ % | $25.5$ % | $23.4$ % | $13.3$ % | $12.7$ % |

CCT${}_{2}$ | $30.3$ μm | $27.5$ μm | $27.3$ μm | $30.0$ μm | $48.7$ μm | $49.5$ μm | $30.2$ μm | $25.0$ μm |

CAWC${}_{2}$ | $2.1$ % | $7.4$ % | $6.1$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.9$ % | $3.3$ % |

CPWC${}_{2}$ | $12.5$ % | $11.3$ % | $13.7$ % | $12.2$ % | $16.8$ % | $16.8$ % | $12.3$ % | $11.9$ % |

CCT${}_{3}$ | $40.9$ μm | $44.3$ μm | $40.1$ μm | $47.3$ μm | $49.5$ μm | $49.4$ μm | $45.3$ μm | $39.1$ μm |

CAWC${}_{3}$ | $1.9$ % | $6.9$ % | $5.8$ % | $9.9$ % | $10.0$ % | $10.0$ % | $9.9$ % | $3.1$ % |

CPWC${}_{3}$ | $6.4$ % | $5.1$ % | $5.7$ % | $4.1$ % | $11.0$ % | $10.6$ % | $4.5$ % | $5.9$ % |

CCT${}_{4}$ | $31.1$ μm | $33.5$ μm | $32.0$ μm | $32.3$ μm | $49.3$ μm | $49.1$ μm | $32.5$ μm | $35.3$ μm |

CAWC${}_{4}$ | $2.1$ % | $7.5$ % | $5.9$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $3.4$ % |

CPWC${}_{4}$ | $16.1$ % | $17.3$ % | $13.3$ % | $12.2$ % | $14.3$ % | $14.0$ % | $15.2$ % | $15.4$ % |

CCT${}_{5}$ | $35.5$ μm | $37.1$ μm | $36.5$ μm | $34.7$ μm | $49.8$ μm | $49.6$ μm | $37.3$ μm | $39.0$ μm |

CAWC${}_{5}$ | $2.2$ % | $7.5$ % | $6.2$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.9$ % | $3.3$ % |

CPWC${}_{5}$ | $12.4$ % | $12.7$ % | $11.4$ % | $10.6$ % | $5.3$ % | $5.2$ % | $9.5$ % | $12.7$ % |

CCT${}_{6}$ | $44.4$ μm | $42.7$ μm | $46.0$ μm | $43.7$ μm | $49.9$ μm | $49.9$ μm | $40.2$ μm | $43.7$ μm |

CAWC${}_{6}$ | $1.9$ % | $7.4$ % | $6.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.9$ % | $3.3$ % |

CPWC${}_{6}$ | $10.5$ % | $10.8$ % | $9.7$ % | $9.9$ % | $6.1$ % | $5.5$ % | $9.2$ % | $9.3$ % |

**Table 4.**RMSD of estimated CCT, CAWC, and CPWC for six noisy cornea calibrated with various noisy perturbed PEC sphere at frequency band of WR $2.2$. The SNR 60 dB is considered.

Ca1${}_{1}$ | Ca1${}_{2}$ | Ca1${}_{3}$ | Ca1${}_{4}$ | Ca1${}_{5}$ | Ca1${}_{6}$ | Ca1${}_{7}$ | Ca1${}_{8}$ | |
---|---|---|---|---|---|---|---|---|

CCT${}_{1}$ | $33.6$ μm | $35.7$ μm | $33.1$ μm | $32.2$ μm | $22.1$ μm | $17.4$ μm | $29.8$ μm | $34.6$ μm |

CAWC${}_{1}$ | $7.8$ % | $9.4$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.7$ % |

CPWC${}_{1}$ | $18.7$ % | $18.0$ % | $18.5$ % | $18.9$ % | $23.5$ % | $23.0$ % | $17.2$ % | $15.8$ % |

CCT${}_{2}$ | $34.4$ μm | $37.8$ μm | $36.1$ μm | $31.6$ μm | $20.7$ μm | $21.2$ μm | $35.4$ μm | $31.6$ μm |

CAWC${}_{2}$ | $7.7$ % | $9.4$ % | $9.9$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.6$ % |

CPWC${}_{2}$ | $15.9$ % | $14.1$ % | $13.7$ % | $13.4$ % | $14.1$ % | $14.0$ % | $14.3$ % | $15.4$ % |

CCT${}_{3}$ | $36.6$ μm | $39.8$ μm | $38.2$ μm | $35.7$ μm | $18.5$ μm | $16.8$ μm | $36.0$ μm | $34.9$ μm |

CAWC${}_{3}$ | $7.8$ % | $9.4$ % | $9.9$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.8$ % |

CPWC${}_{3}$ | $15.8$ % | $17.4$ % | $16.8$ % | $11.1$ % | $6.2$ % | $5.7$ % | $13.6$ % | $17.2$ % |

CCT${}_{4}$ | $36.0$ μm | $37.9$ μm | $36.2$ μm | $37.1$ μm | $38.9$ μm | $35.8$ μm | $38.2$ μm | $35.6$ μm |

CAWC${}_{4}$ | $7.7$ % | $9.3$ % | 10 % | 10 % | 10 % | 10 % | 10 % | $9.7$ % |

CPWC${}_{4}$ | $17.3$ % | $18.4$ % | $17.6$ % | $20.2$ % | $28.4$ % | $29.0$ % | $20.7$ % | $16.7$ % |

CCT${}_{5}$ | $37.7$ μm | $34.0$ μm | $35.8$ μm | $39.5$ μm | $36.2$ μm | $34.4$ μm | $37.5$ μm | $40.8$ μm |

CAWC${}_{5}$ | $7.6$ % | $9.3$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.8$ % |

CPWC${}_{5}$ | $15.5$ % | $14.0$ % | $15.0$ % | $14.0$ % | $19.4$ % | $19.1$ % | $14.5$ % | $15.5$ % |

CCT${}_{6}$ | $35.7$ μm | $39.1$ μm | $38.1$ μm | $38.5$ μm | $36.8$ μm | $36.4$ μm | $34.3$ μm | $35.2$ μm |

CAWC${}_{6}$ | $7.7$ % | $9.3$ % | $9.9$ % | $10.0$ % | $10.0$ % | $10.0$ % | $10.0$ % | $9.7$ % |

CPWC${}_{6}$ | $17.6$ % | $17.1$ % | $15.9$ % | $13.1$ % | $9.4$ % | $9.6$ % | $14.6$ % | $16.9$ % |

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**MDPI and ACS Style**

Zarrinkhat, F.; Baggio, M.; Lamberg, J.; Tamminen, A.; Nefedova, I.; Ala-Laurinaho, J.; Khaled, E.E.M.; Rius, J.M.; Romeu, J.; Taylor, Z.
Calibration Alignment Sensitivity in Corneal Terahertz Imaging. *Sensors* **2022**, *22*, 3237.
https://doi.org/10.3390/s22093237

**AMA Style**

Zarrinkhat F, Baggio M, Lamberg J, Tamminen A, Nefedova I, Ala-Laurinaho J, Khaled EEM, Rius JM, Romeu J, Taylor Z.
Calibration Alignment Sensitivity in Corneal Terahertz Imaging. *Sensors*. 2022; 22(9):3237.
https://doi.org/10.3390/s22093237

**Chicago/Turabian Style**

Zarrinkhat, Faezeh, Mariangela Baggio, Joel Lamberg, Aleksi Tamminen, Irina Nefedova, Juha Ala-Laurinaho, Elsayed E. M. Khaled, Juan M. Rius, Jordi Romeu, and Zachary Taylor.
2022. "Calibration Alignment Sensitivity in Corneal Terahertz Imaging" *Sensors* 22, no. 9: 3237.
https://doi.org/10.3390/s22093237