Electrical impedance tomography: 3D reconstructions using scattering transforms

Abstract

In three dimensions the Calderón problem was addressed and solved in theory in the 1980s. The main ingredients in the solution of the problem are complex geometrical optics solutions to the conductivity equation and a (non-physical) scattering transform. The resulting reconstruction algorithm is in principle direct and addresses the full non-linear problem immediately. In this paper a new simplification of the algorithm is suggested. The method is based on solving a boundary integral equation for the complex geometrical optics solutions, and the method is implemented numerically using a Nyström method. Convergence estimates are obtained using hyperinterpolation operators. We compare the method numerically to two other approximations by evaluation on two numerical examples. In addition, a moment method for the numerical solution of the forward problem is given.

Keywords:

• Calderón problem
• electrical impedance tomography
• reconstruction algorithm
• numerical solution
• moment method
• hyperinterpolation

AMS Subject Classifications::

• 35R30
• 65N21
• 65R32

Acknowledgements

Fabrice Delbary was supported by a post doctoral fellowship from the Villum Kann Rasmussen Foundation.