# Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. Comparison of Jacobian Matrix Row Distributions with That of Known Reconstruction Algorithm

#### 3.2. Computation Time

#### 3.3. Experimental Validation

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DDA | Discrete Dipole Approximation |

FE | Finite Element |

FDTD | Finite-Difference Time-Domain |

## References

- Preece, A.W.; Craddock, I.; Shere, M.; Jones, L.; Winton, H.L. MARIA M4: Clinical evaluation of a prototype ultrawideband radar scanner for breast cancer detection. J. Med. Imaging
**2016**, 3, 033502. [Google Scholar] [CrossRef] [PubMed] - Meaney, P.M.; Kaufman, P.A.; Muffly, L.S.; Click, M.; Poplack, S.P.; Wells, W.A.; Schwartz, G.N.; di Florio-Alexander, R.M.; Tosteson, T.D.; Li, Z.; et al. Microwave imaging for neoadjuvant chemotherapy monitoring: Initial clinical experience. Breast Cancer Res.
**2013**, 15, R35. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gilmore, C.; Mojabi, P.; Zakaria, A.; Ostadrahimi, M.; Kaye, C.; Noghanian, S.; Shafai, L.; Pistorius, S.; LoVetri, J. A wideband microwave tomography system with a novel frequency selection procedure. IEEE Trans. Biomed. Eng.
**2009**, 57, 894–904. [Google Scholar] [CrossRef] [PubMed] - Fear, E.C.; Stuchly, M.A. Microwave system for breast tumor detection. IEEE Microw. Guid. Wave Lett.
**1999**, 9, 470–472. [Google Scholar] [CrossRef] - Semenov, S.Y.; Corfield, D.R. Microwave tomography for brain imaging: Feasibility assessment for stroke detection. Int. J. Antennas Propag.
**2008**, 2008, 254830. [Google Scholar] [CrossRef] [Green Version] - Persson, M.; Fhager, A.; Trefná, H.D.; Yu, Y.; McKelvey, T.; Pegenius, G.; Karlsson, J.E.; Elam, M. Microwave-based stroke diagnosis making global prehospital thrombolytic treatment possible. IEEE Trans. Biomed. Eng.
**2014**, 61, 2806–2817. [Google Scholar] [CrossRef] [Green Version] - Meaney, P.M.; Zhou, T.; Goodwin, D.; Golnabi, A.; Attardo, E.A.; Paulsen, K.D. Bone dielectric property variation as a function of mineralization at microwave frequencies. Int. J. Biomed. Imaging
**2012**, 2012, 649612. [Google Scholar] [CrossRef] [Green Version] - Sugitani, T.; Kubota, S.I.; Kuroki, S.I.; Sogo, K.; Arihiro, K.; Okada, M.; Kadoya, T.; Hide, M.; Oda, M.; Kikkawa, T. Complex permittivities of breast tumor tissues obtained from cancer surgeries. Appl. Phys. Lett.
**2014**, 104, 253702. [Google Scholar] [CrossRef] - Kaufman, Z.; Paran, H.; Haas, I.; Malinger, P.; Zehavi, T.; Karni, T.; Pappo, I.; Sandbank, J.; Diment, J.; Allweis, T. Mapping breast tissue types by miniature radio-frequency near-field spectroscopy sensor in ex-vivo freshly excised specimens. BMC Med Imaging
**2016**, 16, 57. [Google Scholar] [CrossRef] [Green Version] - Cheng, Y.; Fu, M. Dielectric properties for non-invasive detection of normal, benign, and malignant breast tissues using microwave theories. Thorac. Cancer
**2018**, 9, 459–465. [Google Scholar] [CrossRef] - Gabriel, S.; Lau, R.; Gabriel, C. The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. Phys. Med. Biol.
**1996**, 41, 2271. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Meaney, P.; Fanning, M.; Paulsen, K.; Li, D.; Pendergrass, S.; Fang, Q.; Moodie, K. Microwave thermal imaging: Initial in vivo experience with a single heating zone. Int. J. Hyperth.
**2003**, 19, 617–641. [Google Scholar] [CrossRef] [PubMed] - Haynes, M.; Stang, J.; Moghaddam, M. Real-time microwave imaging of differential temperature for thermal therapy monitoring. IEEE Trans. Biomed. Eng.
**2014**, 61, 1787–1797. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ley, S.; Schilling, S.; Fiser, O.; Vrba, J.; Sachs, J.; Helbig, M. Ultra-wideband temperature dependent dielectric spectroscopy of porcine tissue and blood in the microwave frequency range. Sensors
**2019**, 19, 1707. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Klemm, M.; Craddock, I.J.; Leendertz, J.A.; Preece, A.; Benjamin, R. Radar-based breast cancer detection using a hemispherical antenna array—Experimental results. IEEE Trans. Antennas Propag.
**2009**, 57, 1692–1704. [Google Scholar] [CrossRef] [Green Version] - Ravan, M.; Amineh, R.K.; Nikolova, N.K. Two-dimensional near-field microwave holography. Inverse Probl.
**2010**, 26, 055011. [Google Scholar] [CrossRef] - Tajik, D.; Foroutan, F.; Shumakov, D.S.; Pitcher, A.D.; Nikolova, N.K. Real-Time Microwave Imaging of a Compressed Breast Phantom with Planar Scanning. IEEE J. Electromagn. RF Microwaves Med. Biol.
**2018**, 2, 154–162. [Google Scholar] [CrossRef] - Shea, J.D.; Kosmas, P.; Hagness, S.C.; Van Veen, B.D. Three-dimensional microwave imaging of realistic numerical breast phantoms via a multiple-frequency inverse scattering technique. Med. Phys.
**2010**, 37, 4210–4226. [Google Scholar] [CrossRef] - Catapano, I.; Crocco, L.; D’Urso, M.; Isernia, T. 3D microwave imaging via preliminary support reconstruction: Testing on the Fresnel 2008 database. Inverse Probl.
**2009**, 25, 024002. [Google Scholar] [CrossRef] - Scapaticci, R.; Kosmas, P.; Crocco, L. Wavelet-based regularization for robust microwave imaging in medical applications. IEEE Trans. Biomed. Eng.
**2015**, 62, 1195–1202. [Google Scholar] [CrossRef] [Green Version] - Fhager, A.; Persson, M. Using a priori data to improve the reconstruction of small objects in microwave tomography. IEEE Trans. Microw. Theory Tech.
**2007**, 55, 2454–2462. [Google Scholar] [CrossRef] - Burfeindt, M.J.; Colgan, T.J.; Mays, R.O.; Shea, J.D.; Behdad, N.; Van Veen, B.D.; Hagness, S.C. MRI-derived 3-D-printed breast phantom for microwave breast imaging validation. IEEE Antennas Wirel. Propag. Lett.
**2012**, 11, 1610–1613. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Karadima, O.; Rahman, M.; Sotiriou, I.; Ghavami, N.; Lu, P.; Ahsan, S.; Kosmas, P. Experimental Validation of Microwave Tomography with the DBIM-TwIST Algorithm for Brain Stroke Detection and Classification. Sensors
**2020**, 20, 840. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Semenov, S.Y.; Bulyshev, A.E.; Abubakar, A.; Posukh, V.G.; Sizov, Y.E.; Souvorov, A.E.; van den Berg, P.M.; Williams, T.C. Microwave-tomographic imaging of the high dielectric-contrast objects using different image-reconstruction approaches. IEEE Trans. Microw. Theory Tech.
**2005**, 53, 2284–2294. [Google Scholar] [CrossRef] - Gilmore, C.; Mojabi, P.; LoVetri, J. Comparison of an enhanced distorted born iterative method and the multiplicative-regularized contrast source inversion method. IEEE Trans. Antennas Propag.
**2009**, 57, 2341–2351. [Google Scholar] [CrossRef] - Meaney, P.M.; Paulsen, K.D.; Chang, J.T. Near-field microwave imaging of biologically-based materials using a monopole transceiver system. IEEE Trans. Microw. Theory Tech.
**1998**, 46, 31–45. [Google Scholar] [CrossRef] - Meaney, P.; Geimer, S.; Paulsen, K. Two-step inversion in microwave imaging with a logarithmic transformation. Med. Phys
**2017**, 44, 4239–4251. [Google Scholar] [CrossRef] - Semenov, S.Y. Electromagnetic Tomography for Human Brain Imaging; IEEE CAMA: Vasteras, Sweden, 2018. [Google Scholar]
- Isernia, T.; Pascazio, V.; Pierri, R. On the local minima in a tomographic imaging technique. IEEE Trans. Geosci. Remote Sens.
**2001**, 39, 1596–1607. [Google Scholar] [CrossRef] - Catapano, I.; Crocco, L.; D’Urso, M.; Isernia, T. On the effect of support estimation and of a new model in 2-D inverse scattering problems. IEEE Trans. Antennas Propag.
**2007**, 55, 1895–1899. [Google Scholar] [CrossRef] - Poltschak, S.; Freilinger, M.; Feger, R.; Stelzer, A.; Hamidipour, A.; Henriksson, T.; Hopfer, M.; Planas, R.; Semenov, S. A multiport vector network analyzer with high-precision and realtime capabilities for brain imaging and stroke detection. Int. J. Microw. Wirel. Technol.
**2018**, 10, 605–612. [Google Scholar] [CrossRef] [Green Version] - Moore, G.E. Cramming more components onto integrated circuits. Proc. IEEE
**1998**, 86, 82–85. [Google Scholar] [CrossRef] - Kaltenbacher, B. Some Newton-type methods for the regularization of nonlinear ill-posed problems. Inverse Probl.
**1997**, 13, 729. [Google Scholar] [CrossRef] - Souvorov, A.E.; Bulyshev, A.E.; Semenov, S.Y.; Svenson, R.H.; Nazarov, A.G.; Sizov, Y.E.; Tatsis, G.P. Microwave tomography: A two-dimensional Newton iterative scheme. IEEE Trans. Microw. Theory Tech.
**1998**, 46, 1654–1659. [Google Scholar] [CrossRef] - Cui, T.J.; Chew, W.C.; Aydiner, A.A.; Chen, S. Inverse scattering of two-dimensional dielectric objects buried in a lossy earth using the distorted Born iterative method. IEEE Trans. Geosci. Remote Sens.
**2001**, 39, 339–346. [Google Scholar] - Zakaria, A.; Gilmore, C.; LoVetri, J. Finite-element contrast source inversion method for microwave imaging. Inverse Probl.
**2010**, 26, 115010. [Google Scholar] [CrossRef] [Green Version] - Van Den Berg, P.M.; Kleinman, R.E. A contrast source inversion method. Inverse Probl.
**1997**, 13, 1607. [Google Scholar] [CrossRef] - Rubæk, T.; Meaney, P.M.; Meincke, P.; Paulsen, K.D. Nonlinear microwave imaging for breast-cancer screening using Gauss–Newton’s method and the CGLS inversion algorithm. IEEE Trans. Antennas Propag.
**2007**, 55, 2320–2331. [Google Scholar] [CrossRef] [Green Version] - Meaney, P.M.; Paulsen, K.D.; Pogue, B.W.; Miga, M.I. Microwave image reconstruction utilizing log-magnitude and unwrapped phase to improve high-contrast object recovery. IEEE Trans. Med. Imaging
**2001**, 20, 104–116. [Google Scholar] [CrossRef] - Joachimowicz, N.; Pichot, C.; Hugonin, J.P. Inverse scattering: An iterative numerical method for electromagnetic imaging. IEEE Trans. Antennas Propag.
**1991**, 39, 1742–1753. [Google Scholar] [CrossRef] - Bindu, G.; Semenov, S. 2D Fused image reconstruction approach for microwave tomography: A theoretical assessment using the FDTD model. Comput. Methods Biomech. Biomed. Eng. Imaging Vis.
**2013**, 1, 147–154. [Google Scholar] [CrossRef] - Hosseinzadegan, S.; Fhager, A.; Persson, M.; Meaney, P. A Discrete Dipole Approximation Solver Based on the COCG-FFT Algorithm and Its Application to Microwave Breast Imaging. Int. J. Antennas Propag.
**2019**, 2019, 9014969. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fang, Q.; Meaney, P.M.; Paulsen, K.D. Viable three-dimensional medical microwave tomography: Theory and numerical experiments. IEEE Trans. Antennas Propag.
**2010**, 58, 449–458. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Arridge, S.R.; Schweiger, M. Photon-measurement density functions. Part 2: Finite-element-method calculations. Appl. Opt.
**1995**, 34, 8026–8037. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Polydorides, N.; Lionheart, W.R. A Matlab toolkit for three-dimensional electrical impedance tomography: A contribution to the Electrical Impedance and Diffuse Optical Reconstruction Software project. Meas. Sci. Technol.
**2002**, 13, 1871. [Google Scholar] [CrossRef] - Fang, Q.; Meaney, P.M.; Geimer, S.D.; Streltsov, A.V.; Paulsen, K.D. Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation. IEEE Trans. Med. Imaging
**2004**, 23, 475–484. [Google Scholar] [CrossRef] - Dehghani, H.; Eames, M.E.; Yalavarthy, P.K.; Davis, S.C.; Srinivasan, S.; Carpenter, C.M.; Pogue, B.W.; Paulsen, K.D. Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction. Commun. Numer. Methods Eng.
**2009**, 25, 711–732. [Google Scholar] [CrossRef] - Halter, R.J.; Hartov, A.; Poplack, S.P.; Wells, W.A.; Rosenkranz, K.M.; Barth, R.J.; Kaufman, P.A.; Paulsen, K.D. Real-time electrical impedance variations in women with and without breast cancer. IEEE Trans. Med. Imaging
**2014**, 34, 38–48. [Google Scholar] [CrossRef] [Green Version] - Lynch, D.R. Numerical Partial Differential Equations for Environmental Scientists and Engineers: A First Practical Course; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Meaney, P.M.; Paulsen, K.D.; Ryan, T.P. Two-dimensional hybrid element image reconstruction for TM illumination. IEEE Trans. Antennas Propag.
**1995**, 43, 239–247. [Google Scholar] [CrossRef] - Hosseinzadegan, S.; Fhager, A.; Persson, M.; Meaney, P.M. Application of Two-Dimensional Discrete Dipole Approximation in Simulating Electric Field of a Microwave Breast Imaging System. IEEE J. Electromagn. RF Microwaves Med. Biol.
**2019**, 3, 80–87. [Google Scholar] [CrossRef] - Meaney, P.M.; Fang, Q.; Rubaek, T.; Demidenko, E.; Paulsen, K.D. Log transformation benefits parameter estimation in microwave tomographic imaging. Med. Phys.
**2007**, 34, 2014–2023. [Google Scholar] [CrossRef] [Green Version] - Rydholm, T.; Fhager, A.; Persson, M.; Meaney, P.M. A First Evaluation of the Realistic Supelec-Breast Phantom. IEEE J. Electromagn. RF Microwaves Med. Biol.
**2017**, 1, 59–65. [Google Scholar] [CrossRef] - Ostadrahimi, M.; Zakaria, A.; LoVetri, J.; Shafai, L. A near-field dual polarized (TE–TM) microwave imaging system. IEEE Trans. Microw. Theory Tech.
**2013**, 61, 1376–1384. [Google Scholar] [CrossRef]

**Figure 1.**Overlapping forward solution and parameter meshes corresponding to the nodes and elements.

**Figure 2.**Forward solution mesh nodes; example of node n and corresponding region ${\mathsf{\Omega}}_{n}$ is sketched.

**Figure 3.**Parameter mesh nodes; example of node $\tau $ and corresponding region ${\mathsf{\Omega}}_{\tau}$ is sketched.

**Figure 6.**Representation of imagining domain with respect to the meshes used for forward solutions (

**a**,

**b**) and property reconstruction (

**a**,

**c**).

**Figure 7.**Plots of the effective log magnitude (

**left**) and phase (

**right**) distributions for various rows of Jacobian matrix (m${}^{2}$), corresponding to multiple transmitter/receiver pairs, for the discrete dipole approximation (DDA)-based algorithm. The distributions are presented on the DDA grid. Note that the plotting algorithm smoothed the imaging zone shape with triangles around the sharp edges of the reconstruction grid.

**Figure 8.**Plots of the effective log magnitude (

**left**) and phase (

**right**) distributions for various rows of Jacobian matrix (m${}^{2}$), corresponding to multiple transmitter/receiver pairs, for the finite element (FE)-based algorithm. The distributions are presented on the coarse FE-mesh.

**Figure 9.**Plots of the effective log magnitude (

**left**) and phase (

**right**) distributions for ratios of Jacobian matrices calculated with DDA and FE-based schemes. The distributions are presented on the DDA grid.

**Figure 10.**The photograph of the measurement setup for the case that a cylindrical inclusion is positioned close to antennas $\#6$ and $\#7$.

**Figure 11.**Images of reconstructed relative permittivity (

**top row**) and conductivity (S/m) (

**bottom row**) at 1500 MHz as a function of iteration number.

**Figure 12.**Comparison of calculated magnitude and phase projections of the signals transmitted from Antenna 1 for multiple iterations as a function of receiver number. The actual measurement projection is also shown.

**Figure 13.**Computed relative $L2$ norm error of projections for the experimental study as a function of number of iterations.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hosseinzadegan, S.; Fhager, A.; Persson, M.; Geimer, S.; Meaney, P.
Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix. *Sensors* **2021**, *21*, 729.
https://doi.org/10.3390/s21030729

**AMA Style**

Hosseinzadegan S, Fhager A, Persson M, Geimer S, Meaney P.
Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix. *Sensors*. 2021; 21(3):729.
https://doi.org/10.3390/s21030729

**Chicago/Turabian Style**

Hosseinzadegan, Samar, Andreas Fhager, Mikael Persson, Shireen Geimer, and Paul Meaney.
2021. "Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix" *Sensors* 21, no. 3: 729.
https://doi.org/10.3390/s21030729