Janne P. Tamminen

Maarten de Hoop, Matti Lassas, Matteo Santacesaria et al.

A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -- in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between $10^{-5}$ and $5$, smaller values giving better results.

Samuli Siltanen, Janne P. Tamminen

The aim of electrical impedance tomography is to form an image of the conductivity distribution inside an unknown body using electric boundary measurements. The computation of the image from measurement data is a non-linear ill-posed inverse problem and calls for a special regularized algorithm. One such algorithm, the so-called D-bar method, is improved in this work by introducing new computational steps that remove the so far necessary requirement that the conductivity should be constant near the boundary. The numerical experiments presented suggest two conclusions. First, for most conductivities arising in medical imaging, it seems the previous approach of using a best possible constant near the boundary is sufficient. Second, for conductivities that have high contrast features at the boundary, the new approach produces reconstructions with smaller quantitative error and with better visual quality.

Janne P. Tamminen, Tanja Tarvainen, Samuli Siltanen

The D-bar method at negative energy is numerically implemented. Using the method we are able to numerically reconstruct potentials and investigate exceptional points at negative energy. Subsequently, applying the method to Diffusive Optical Tomography, a new way of reconstructing the diffusion coefficient from the associated Complex Geometrics Optics solution is suggested and numerically validated.

Nuutti Hyvönen, Lassi Päivärinta, Janne P. Tamminen

We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies to the open upper half-plane. Second, in the unit disk we may choose a region of interest that is magnified using a suitable Möbius transform. To facilitate the efficient use of conformal maps, we introduce input current patterns that are named conformally transformed truncated Fourier basis; in practice, their use corresponds to positioning the available electrodes close to the region of interest. These ideas are numerically tested using simulated continuum data in bounded domains and simulated point electrode data in the half-plane. The connections to practical electrode measurements are also discussed.