# Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains

## Abstract

In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to integration of the parallel and perpendicular projection of the vector field along the integration lines and are called the longitudinal and transverse measurements, respectively. In most cases, however, the transverse measurements cannot be physically acquired. Therefore, the VT methods are typically used to reconstruct divergence-free (or source-free) velocity and flow fields that can be reconstructed solely from the longitudinal measurements. In this paper, we show how vector fields with non-zero divergence in a bounded domain can also be reconstructed from the longitudinal measurements without the need of explicitly evaluating the transverse measurements. To the best of our knowledge, VT has not previously been used for this purpose. In particular, we study low-frequency, time-harmonic electric fields generated by dipole sources in convex bounded domains which arise, for example, in electroencephalography (EEG) source imaging. We explain in detail the theoretical background, the derivation of the electric field inverse problem and the numerical approximation of the line integrals. We show that fields with non-zero divergencemore »

- Authors:

- Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, D-48149 Münster (Germany)
- (United Kingdom)
- Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT (United Kingdom)
- Institute for Biomagnetism and Biosignalanalysis, University of Münster, Malmedyweg 15, D-48149 Münster (Germany)
- (New Zealand)

- Publication Date:

- OSTI Identifier:
- 22622238

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 329; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; APPROXIMATIONS; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; DIPOLES; ELECTRIC FIELDS; ELECTROENCEPHALOGRAPHY; HARMONICS; LAPLACIAN; LIMITING VALUES; MATHEMATICAL MODELS; MATHEMATICAL SOLUTIONS; RADON; TOMOGRAPHY; TWO-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL SYSTEMS; VECTOR FIELDS

### Citation Formats

```
Koulouri, Alexandra, E-mail: koulouri@uni-muenster.de, Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT, Brookes, Mike, Rimpiläinen, Ville, and Department of Mathematics, University of Auckland, Private bag 92019, Auckland 1142.
```*Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2016.10.037.

```
Koulouri, Alexandra, E-mail: koulouri@uni-muenster.de, Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT, Brookes, Mike, Rimpiläinen, Ville, & Department of Mathematics, University of Auckland, Private bag 92019, Auckland 1142.
```*Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains*. United States. doi:10.1016/J.JCP.2016.10.037.

```
Koulouri, Alexandra, E-mail: koulouri@uni-muenster.de, Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT, Brookes, Mike, Rimpiläinen, Ville, and Department of Mathematics, University of Auckland, Private bag 92019, Auckland 1142. Sun .
"Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains". United States.
doi:10.1016/J.JCP.2016.10.037.
```

```
@article{osti_22622238,
```

title = {Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains},

author = {Koulouri, Alexandra, E-mail: koulouri@uni-muenster.de and Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT and Brookes, Mike and Rimpiläinen, Ville and Department of Mathematics, University of Auckland, Private bag 92019, Auckland 1142},

abstractNote = {In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to integration of the parallel and perpendicular projection of the vector field along the integration lines and are called the longitudinal and transverse measurements, respectively. In most cases, however, the transverse measurements cannot be physically acquired. Therefore, the VT methods are typically used to reconstruct divergence-free (or source-free) velocity and flow fields that can be reconstructed solely from the longitudinal measurements. In this paper, we show how vector fields with non-zero divergence in a bounded domain can also be reconstructed from the longitudinal measurements without the need of explicitly evaluating the transverse measurements. To the best of our knowledge, VT has not previously been used for this purpose. In particular, we study low-frequency, time-harmonic electric fields generated by dipole sources in convex bounded domains which arise, for example, in electroencephalography (EEG) source imaging. We explain in detail the theoretical background, the derivation of the electric field inverse problem and the numerical approximation of the line integrals. We show that fields with non-zero divergence can be reconstructed from the longitudinal measurements with the help of two sparsity constraints that are constructed from the transverse measurements and the vector Laplace operator. As a comparison to EEG source imaging, we note that VT does not require mathematical modeling of the sources. By numerical simulations, we show that the pattern of the electric field can be correctly estimated using VT and the location of the source activity can be determined accurately from the reconstructed magnitudes of the field. - Highlights: • Vector tomography is used to reconstruct electric fields generated by dipole sources. • Inverse solutions are based on longitudinal and transverse line integral measurements. • Transverse line integral measurements are used as a sparsity constraint. • Numerical procedure to approximate the line integrals is described in detail. • Patterns of the studied electric fields are correctly estimated.},

doi = {10.1016/J.JCP.2016.10.037},

journal = {Journal of Computational Physics},

number = ,

volume = 329,

place = {United States},

year = {Sun Jan 15 00:00:00 EST 2017},

month = {Sun Jan 15 00:00:00 EST 2017}

}