Regularized D-bar method for the inverse conductivity problem
Kim Knudsen Matti Lassas Jennifer L. Mueller Samuli Siltanen
Inverse Problems & Imaging 2009, 3(4): 599-624 doi: 10.3934/ipi.2009.3.599
A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.
keywords: ill-posed problem electrical impedance tomography inverse problem regularization. inverse conductivity problem
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
Recovering boundary shape and conductivity in electrical impedance tomography
Ville Kolehmainen Matti Lassas Petri Ola Samuli Siltanen
Inverse Problems & Imaging 2013, 7(1): 217-242 doi: 10.3934/ipi.2013.7.217
Electrical impedance tomography (EIT) aims to reconstruct the electric conductivity inside a physical body from current-to-voltage measurements at the boundary of the body. In practical EIT one often lacks exact knowledge of the domain boundary, and inaccurate modeling of the boundary causes artifacts in the reconstructions. A novel method is presented for recovering the boundary shape and an isotropic conductivity from EIT data. The first step is to determine the minimally anisotropic conductivity in a model domain reproducing the measured EIT data. Second, a Beltrami equation is solved, providing shape-deforming reconstruction. The algorithm is applied to simulated noisy data from a realistic electrode model, demonstrating that approximate recovery of the boundary shape and conductivity is feasible.
keywords: conformal deformation. electrical impedance tomography minimally anisotropic conductivity quasiconformal maps Inverse conductivity problem
Three-dimensional dental X-ray imaging by combination of panoramic and projection data
Nuutti Hyvönen Martti Kalke Matti Lassas Henri Setälä Samuli Siltanen
Inverse Problems & Imaging 2010, 4(2): 257-271 doi: 10.3934/ipi.2010.4.257
A novel three-dimensional dental X-ray imaging method is introduced, based on hybrid data collected with a dental panoramic device. Such a device uses geometric movement of the X-ray source and detector around the head of a patient to produce a panoramic image, where all teeth are in sharp focus and details at a distance from the dental arc are blurred. A digital panoramic device is reprogrammed to collect two-dimensional projection radiographs. Two complementary types of data are measured from a region of interest: projection data with a limited angle of view, and a panoramic image. Tikhonov regularization is applied to these data in order to produce three-dimensional reconstructions. The algorithm is tested with simulated data and real-world in vitro measurements from a dry mandible. Reconstructions from limited-angle projection data alone do provide the dentist with three-dimensional information useful for dental implant planning. Furthermore, adding panoramic data to the process improves the reconstruction precision in the direction of the dental arc. The presented imaging modality can be seen as a cost-effective alternative to a full-angle CT scanner.
keywords: limited angle data three-dimensional reconstruction. panoramic image X-ray imaging hybrid data Tikhonov regularization
Estimation of conductivity changes in a region of interest with electrical impedance tomography
Dong liu Ville Kolehmainen Samuli Siltanen Anne-maria Laukkanen Aku Seppänen
Inverse Problems & Imaging 2015, 9(1): 211-229 doi: 10.3934/ipi.2015.9.211
This paper proposes a novel approach to reconstruct changes in a target conductivity from electrical impedance tomography measurements. As in the conventional difference imaging, the reconstruction of the conductivity change is based on electrical potential measurements from the exterior boundary of the target before and after the change. In this paper, however, images of the conductivity before and after the change are reconstructed simultaneously based on the two data sets. The key feature of the approach is that the conductivity after the change is parameterized as a linear combination of the initial state and the change. This allows for modeling independently the spatial characteristics of the background conductivity and the change of the conductivity - by separate regularization functionals. The approach also allows in a straightforward way the restriction of the conductivity change to a localized region of interest inside the domain. While conventional difference imaging reconstruction is based on a global linearization of the observation model, the proposed approach amounts to solving a non-linear inverse problem. The feasibility of the proposed reconstruction method is tested experimentally and with a simulation which demonstrates a potential new medical application of electrical impedance tomography: imaging of vocal folds in voice loading studies.
keywords: absolute imaging imaging of vocal folds. region of interest electrical impedance tomography Inverse problem
Sarah Hamilton Kim Knudsen Samuli Siltanen Gunther Uhlmann
Inverse Problems & Imaging 2014, 8(4): i-ii doi: 10.3934/ipi.2014.8.4i
Complex Geometrical Optics (CGO) solutions have, for almost three decades, played a large role in the rigorous analysis of nonlinear inverse problems. They have the added bonus of also being useful in practical reconstruction algorithms. The main benefit of CGO solutions is to provide solutions in the form of almost-exponential functions that can be used in a variety of ways, for example for defining tailor-made nonlinear Fourier transforms to study the unique solvability of a nonlinear inverse problem.

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Recovering conductivity at the boundary in three-dimensional electrical impedance tomography
Gen Nakamura Päivi Ronkanen Samuli Siltanen Kazumi Tanuma
Inverse Problems & Imaging 2011, 5(2): 485-510 doi: 10.3934/ipi.2011.5.485
The aim of electrical impedance tomography (EIT) is to reconstruct the conductivity values inside a conductive object from electric measurements performed at the boundary of the object. EIT has applications in medical imaging, nondestructive testing, geological remote sensing and subsurface monitoring. Recovering the conductivity and its normal derivative at the boundary is a preliminary step in many EIT algorithms; Nakamura and Tanuma introduced formulae for recovering them approximately from localized voltage-to-current measurements in [Recent Development in Theories & Numerics, International Conference on Inverse Problems 2003]. The present study extends that work both theoretically and computationally. As a theoretical contribution, reconstruction formulas are proved in a more general setting. On the computational side, numerical implementation of the reconstruction formulae is presented in three-dimensional cylindrical geometry. These experiments, based on simulated noisy EIT data, suggest that the conductivity at the boundary can be recovered with reasonable accuracy using practically realizable measurements. Further, the normal derivative of the conductivity can also be recovered in a similar fashion if measurements from a homogeneous conductor (dummy load) are available for use in a calibration step.
keywords: localized Dirichlet to Neumann map boundary determination inverse conductivity problem. Electrical impedance tomography
Iterative time-reversal control for inverse problems
Kenrick Bingham Yaroslav Kurylev Matti Lassas Samuli Siltanen
Inverse Problems & Imaging 2008, 2(1): 63-81 doi: 10.3934/ipi.2008.2.63
A novel method to solve inverse problems for the wave equation is introduced. The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave equation using focusing waves in unknown medium. The approach is computationally effective since the iteration lets the medium do most of the processing of the data.
    The iterative time reversal scheme also gives an algorithm for approximating a given wave in a subset of the domain without knowing the coefficients of the wave equation.
keywords: Inverse problems wave equation control time reversal.
Probing for inclusions in heat conductive bodies
Patricia Gaitan Hiroshi Isozaki Olivier Poisson Samuli Siltanen Janne Tamminen
Inverse Problems & Imaging 2012, 6(3): 423-446 doi: 10.3934/ipi.2012.6.423
This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1--3 for probing the discontinuous part of the conductivity from local temperature and heat flow measurements at the boundary. The approach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is possible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering locations of discontinuities approximately from noisy data.
keywords: Inverse problem heat equation interface reconstruction.
Discretization-invariant Bayesian inversion and Besov space priors
Matti Lassas Eero Saksman Samuli Siltanen
Inverse Problems & Imaging 2009, 3(1): 87-122 doi: 10.3934/ipi.2009.3.87
Bayesian solution of an inverse problem for indirect measurement $M = AU + $ε is considered, where $U$ is a function on a domain of $\R^d$. Here $A$ is a smoothing linear operator and ε is Gaussian white noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k$ε , where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_k A U_n + P_k$ε . Bayes formula gives then the posterior distribution

$\pi_{kn}(u_n\|\m_{kn})$~ Π n $(u_n)\exp(-\frac{1}{2}$||$\m_{kn} - P_kA u_n$||$\_2^2)$

in $\R^d$, and the mean $\u_{kn}$:$=\int u_n \ \pi_{kn}(u_n\|\m_k)\ du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior distributions Π n for all $n\geq n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Π n represent the same a priori information for all $n$ and that the mean $\u_{kn}$ converges to a limit estimate as $k,n$→$\infty$. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_11$ prior is related to penalizing the $\l^1$ norm of the wavelet coefficients of $U$.

keywords: wavelet discretization invariance Inverse problem statistical inversion Besov space. Bayesian inversion reconstruction

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