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*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.

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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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.

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