Direct electrical impedance tomography for nonsmooth conductivities
Kari Astala Jennifer L. Mueller Lassi Päivärinta Allan Perämäki Samuli Siltanen
Inverse Problems & Imaging 2011, 5(3): 531-549 doi: 10.3934/ipi.2011.5.531
A new reconstruction algorithm is presented for eit in dimension two, based on the constructive uniqueness proof given by Astala and Päivärinta in [Ann. of Math. 163 (2006)]. The method is non-iterative, provides a noise-robust solution of the full nonlinear eit problem, and applies to more general conductivities than previous approaches. In particular, the new algorithm applies to piecewise smooth conductivities. Reconstructions from noisy and non-noisy simulated data from conductivity distributions representing a cross-sections of a chest and a layered medium such as stratified flow in a pipeline are presented. The results suggest that the new method can recover useful and reasonably accurate eit images from data corrupted by realistic amounts of measurement noise. In particular, the dynamic range in medium-contrast conductivities is reconstructed remarkably well.
keywords: Electrical impedance tomography. Nonlinear Fourier transform Complex geometrical optics solution Beltrami equation Conductivity equation Inverse conductivity problem Inverse problem Quasiconformal map Numerical solver
Recovery of jumps and singularities in the multidimensional Schrodinger operator from limited data
Lassi Päivärinta Valery Serov
Inverse Problems & Imaging 2007, 1(3): 525-535 doi: 10.3934/ipi.2007.1.525
The inverse scattering problem for multidimensional Schrödinger operator is studied. More exactly we prove a new formula for the first nonlinear term to estimate more accurately this term. This estimate allows to conclude that all singularities and jumps of the unknown potential can be recovered from the Born approximation. Especially, we show for the potentials in $L^p$ for certain values of $p$ that the approximation agrees with the true potential up to the continuous function.% Text of abstract
keywords: Born approximation. Inverse problems Schrödinger operator
The interior transmission problem
David Colton Lassi Päivärinta John Sylvester
Inverse Problems & Imaging 2007, 1(1): 13-28 doi: 10.3934/ipi.2007.1.13
The interior transmission problem is a boundary value problem that plays a basic role in inverse scattering theory but unfortunately does not seem to be included in any existing theory in partial differential equations.This paper presents old and new results for the interior transmission problem ,in particular its relation to inverse scattering theory and new results on the spectral theory associated with this class of boundary value problems.
keywords: Interior Transmission Problem.
Lassi Päivärinta Matti Lassas Jackie (Jianhong) Shen
Inverse Problems & Imaging 2007, 1(1): i-iii doi: 10.3934/ipi.2007.1.1i
The fields of inverse problems and imaging are new and flourishing branches of both pure and applied mathematics. In particular, these areas are concerned with recovering information about an object from indirect, incomplete or noisy observations and have become one of the most important and topical fields of modern applied mathematics.
    The modern study of inverse problems and imaging applies a wide range of geometric and analytic methods which in turn creates new connections to various fields of mathematics, ranging from geometry, microlocal analysis and control theory to mathematical physics, stochastics and numerical analysis. Research in inverse problems has shown that many results of pure mathematics are in fact crucial components of practical algorithms. For example,a theoretical understanding of the structures that ideal measurements should reveal, or of the non-uniqueness of solutions,can lead to a dramatic increase in the quality of imaging applications. On the other hand,inverse problems have also raised many new mathematical problems. For example, the invention of the inverse spectral method to solve the Korteweg-de Vries equation gave rise to the field of integrable systems and the mathematical theory of solitons.

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Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps
Nuutti Hyvönen Lassi Päivärinta Janne P. Tamminen
Inverse Problems & Imaging 2018, 12(2): 373-400 doi: 10.3934/ipi.2018017

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.

keywords: Calderón problem D-bar method conformal mapping region of interest half-plane point electrodes

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