Papers by Jennifer Mueller
Inverse Problems, 2020
A direct reconstruction algorithm based on Calderón’s linearization method for the reconstruction... more A direct reconstruction algorithm based on Calderón’s linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known a priori, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.
Inverse Problems, 2020
Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed ... more Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed using harmless electric currents. The currents are fed through electrodes placed on the surface of the target, and the data consists of voltages measured at the electrodes resulting from a linearly independent set of current injection patterns. EIT aims to recover the internal distribution of electrical conductivity inside the target. The inverse problem underlying the EIT image formation task is nonlinear and severely ill-posed, and hence sensitive to modeling errors and measurement noise. Therefore, the inversion process needs to be regularized. However, traditional variational regularization methods, based on optimization, often suffer from local minima because of nonlinearity. This is what makes regularized direct (non-iterative) methods attractive for EIT. The most developed direct EIT algorithm is the D-bar method, based on Complex Geometric Optics solutions and a nonlinear Fourier transform. Variants and recent developments of D-bar methods are reviewed, and their practical numerical implementation is explained.
Journal of Computational and Applied Mathematics, 2019
A method of including dynamic spatial priors in the 2-D D-bar reconstruction algorithm is present... more A method of including dynamic spatial priors in the 2-D D-bar reconstruction algorithm is presented for use on time-difference reconstructions of human subject thoracic data. The conductivity values for the prior are updated at each frame in the reconstruction using an optimization method applied to the scattering transform. The updates of the dynamic spatial priors are guided by a principle component analysis of the data to determine the timepoint in the ventilatory (or cardiac) cycle. The effectiveness of the method is demonstrated on human subject ventilatory data.
Inverse Problems & Imaging, 2018
The aim of this paper is to demonstrate the feasibility of using spatial a priori information in ... more The aim of this paper is to demonstrate the feasibility of using spatial a priori information in the 2-D D-bar method to improve the spatial resolution of EIT reconstructions of experimentally collected data. The prior consists of imperfectly known information about the spatial locations of inclusions and the assumption that the conductivity is a mollified piecewise constant function. The conductivity values for the prior are constructed using a novel method in which a nonlinear constrained optimization routine is used to select the values for the piecewise constant function that give the best fit to the scattering transform computed from the measured data in a disk. The prior is then included in the high-frequency components of the scattering transform and in the computation of the solution of the D-bar equation, with weights to control the influence of the prior. In addition, a new technique is described for selecting regularization parameters to truncate the measured scattering data, in which complex scattering frequencies for which the values of the scattering transform differ greatly from those in the scattering prior are omitted. The effectiveness of the method is demonstrated on EIT data collected on saline-filled tanks with agar heart and lungs with various added inhomogeneities.
IEEE transactions on medical imaging, Feb 1, 2017
Electrical Impedance Tomography (EIT) aims to recover the internal conductivity and permittivity ... more Electrical Impedance Tomography (EIT) aims to recover the internal conductivity and permittivity distributions of a body from electrical measurements taken on electrodes on the surface of the body. The reconstruction task is a severely ill-posed nonlinear inverse problem that is highly sensitive to measurement noise and modeling errors. Regularized D-bar methods have shown great promise in producing noise-robust algorithms by employing a low-pass filtering of nonlinear (nonphysical) Fourier transform data specific to the EIT problem. Including prior data with the approximate locations of major organ boundaries in the scattering transform provides a means of extending the radius of the low-pass filter to include higher frequency components in the reconstruction, in particular, features that are known with high confidence. This information is additionally included in the system of D-bar equations with an independent regularization parameter from that of the extended scattering transfo...
Nonlinear Wave Equations, 2015
We review recent progress in theory and computation for the Novikov-Veselov (NV) equation with po... more We review recent progress in theory and computation for the Novikov-Veselov (NV) equation with potentials decaying at infinity, focusing mainly on the zero-energy case. The inverse scattering method for the zeroenergy NV equation is presented in the context of Manakov triples, treating initial data of conductivity type rigorously. Special closed-form solutions are presented, including multisolitons, ring solitons, and breathers. The computational inverse scattering method is used to study zero-energy exceptional points and the relationship between supercritical, critical, and subcritical potentials. Contents 1. Introduction 2. Background for the Zero-Energy NV Equation 2.1.
Inverse Problems & Imaging, 2014
The aim of this paper is to show the feasibility of the D-bar method for real-time 2-D EIT recons... more The aim of this paper is to show the feasibility of the D-bar method for real-time 2-D EIT reconstructions. A fast implementation of the D-bar method for reconstructing conductivity changes on a 2-D chest-shaped domain is described. Cross-sectional difference images from the chest of a healthy human subject are presented, demonstrating what can be achieved in real time. The images constitute the first D-bar images from EIT data on a human subject collected on a pairwise current injection system.
Inverse Problems and Imaging, 2011
Inverse Problems & Imaging, 2009
A strategy for regularizing the inversion procedure for the twodimensional D-bar reconstruction a... more A strategy for regularizing the inversion procedure for the twodimensional 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.
2D EIT reconstructions using Calderon's method
Inverse Problems & Imaging, 2008
ABSTRACT The pioneering work “On an inverse boundary value problem” by A. Calderón [in: Semin. Nu... more ABSTRACT The pioneering work “On an inverse boundary value problem” by A. Calderón [in: Semin. Num. Analysis Appl. Cont. Physics, Soc. Bras. Mat., 65–73 (1980)] has inspired a multitude of research, both theoretical and numerical, on the inverse conductivity problem (ICP). The problem has an important application in a medical imaging technique known as electrical impedance tomography (EIT) in which currents are applied on electrodes on the surface of a body, the resulting voltages are measured, and the ICP is solved to determine the conductivity distribution in the interior of the body, which is then displayed to form an image. In this article, the reconstruction method proposed by Calderón is implemented in 2D for both simulated and experimental data including perfusion data collected on a human chest.
SIAM Journal on Scientific Computing, 2003
The problem of reconstructing an unknown electric conductivity from boundary measurements has app... more The problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging, geophysics, and nondestructive testing. A. Nachman [Ann. of Math. (2), 143 (1996), pp. 71-96.] proved global uniqueness for the two-dimensional inverse conductivity problem using a constructive method of proof. Based on this proof, Siltanen, Mueller, and Isaacson [Inverse Problems, 16 (2000), pp. 681-699] presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration. The method was verified with nonnoisy rotationally symmetric examples. In this paper the method is extended by introducing a new regularization scheme, which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise.
SIAM Journal on Applied Mathematics, 2007
The effects of truncating the (approximate) scattering transform in the D-bar reconstruction meth... more The effects of truncating the (approximate) scattering transform in the D-bar reconstruction method for 2-D electrical impedance tomography are studied. The method is based on Nachman's uniqueness proof [Ann. of Math. 143 (1996)] that applies to twice differentiable conductivities. However, the reconstruction algorithm has been successfully applied to experimental data, which can be characterized as piecewise smooth conductivities. The truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity. Thus the truncation can be interpreted as regularization of the D-bar method. Numerical reconstructions are presented demonstrating that features of discontinuous high contrast conductivities can be recovered using the D-bar method. Further, a new connection between Calderón's linearization method and the D-bar method is established, and the two methods are compared numerically and analytically.
Physiological Measurement, 2006
A practical D-bar algorithm for reconstructing conductivity changes from EIT data taken on electr... more A practical D-bar algorithm for reconstructing conductivity changes from EIT data taken on electrodes in a 2D geometry is described. The algorithm is based on the global uniqueness proof of Nachman (1996 Ann. Math. 143 71-96) for the 2D inverse conductivity problem. Results are shown for reconstructions from data collected on electrodes placed around the circumference of a human chest to reconstruct a 2D cross-section of the torso. The images show changes in conductivity during a cardiac cycle.
Nonlinearity, 2012
The Novikov-Veselov (NV) equation is a (2 + 1)-dimensional nonlinear evolution equation generaliz... more The Novikov-Veselov (NV) equation is a (2 + 1)-dimensional nonlinear evolution equation generalizing the (1 + 1)-dimensional Korteweg-deVries equation. The inverse scattering method (ISM) is applied for numerical solution of the NV equation. It is the first time the ISM is used as a computational tool for computing evolutions of a (2 + 1)-dimensional integrable system. In addition, a semi-implicit method is given for the numerical solution of the NV equation using finite differences in the spatial variables, Crank-Nicolson in time, and fast Fourier transforms for the auxiliary equation. Evolutions of initial data satisfying the hypotheses of part I of this paper are computed by the two methods and are observed to coincide with significant accuracy.
Journal of Physics: Conference Series, 2008
The importance of the solution of the boundary integral equation for the exponentially growing so... more The importance of the solution of the boundary integral equation for the exponentially growing solutions to the Schrödinger equation arising from the 2-D inverse conductivity problems is demonstrated by a study of reconstructions of simple piecewise constant conductivities on a disk from two methods of approximating the scattering transform in the Dbar method and from Calderón's linearization method.
Journal of Computational Physics, 2004
Numerical solution method for the dbar-equation in the plane A fast method for solving∂-equations... more Numerical solution method for the dbar-equation in the plane A fast method for solving∂-equations of the form∂v = Tv is presented, where v and T are complex-valued function of two real variables. The multigrid method of Vainikko is adapted to the problem with a FFT implementation. Convergence with rate O(h) is proved for the method applied to equations of the form above. One-grid and two-grid versions of the method are implemented and their effectiveness is demonstrated on an application arising in electrical impedance tomography (EIT).
Inverse Problems, 2001
The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity... more The 2D inverse conductivity problem requires one to determine the unknown electrical conductivity distribution inside a bounded domain ⊂ R 2 from knowledge of the Dirichletto-Neumann map. The problem has geophysical, industrial, and medical imaging (electrical impedance tomography) applications. In 1996 A Nachman proved that the Dirichlet-to-Neumann map uniquely determines C 2 conductivities. The proof, which is constructive, outlines a direct method for reconstructing the conductivity. In this paper we present an implementation of the algorithm in Nachman's proof. The paper includes numerical results obtained by applying the general algorithms described to two radially symmetric cases of small and large contrast.
Inverse Problems, 2012
A direct reconstruction algorithm for complex conductivities in W 2,∞ (Ω), where Ω is a bounded, ... more A direct reconstruction algorithm for complex conductivities in W 2,∞ (Ω), where Ω is a bounded, simply connected Lipschitz domain in R 2 , is presented. The framework is based on the uniqueness proof by Francini [Inverse Problems 20 2000], but equations relating the Dirichlet-to-Neumann to the scattering transform and the exponentially growing solutions are not present in that work, and are derived here. The algorithm constitutes the first D-bar method for the reconstruction of conductivities and permittivities in two dimensions. Reconstructions of numerically simulated chest phantoms with discontinuities at the organ boundaries are included.
Inverse Problems, 2010
A direct three dimensional EIT reconstruction algorithm based on complex geometrical optics solut... more A direct three dimensional EIT reconstruction algorithm based on complex geometrical optics solutions and a nonlinear scattering transform is presented and implemented for spherically symmetric conductivity distributions. The scattering transform is computed both with a Born approximation and from the forward problem for purposes of comparison. Reconstructions are computed for several test problems. A connection to Calderón's linear reconstruction algorithm is established, and reconstructions using both methods are compared.
Uniqueness and numerical recovery of a potential on the real line
Inverse Problems, 1997
We consider the recovery of the potential q(x) in the singular problem Under suitable conditions ... more We consider the recovery of the potential q(x) in the singular problem Under suitable conditions the coefficient is uniquely determined from the set of flux data at the origin corresponding to the source terms , which constitute a basis for . To facilitate numerical recovery, the problem is formulated as an infinite-dimensional least-squares minimization problem. Tikhonov regularization is employed, and the resulting problem is discretized via sinc collocation. Several numerical examples are included.
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Papers by Jennifer Mueller