Bastian Harrach

On uniqueness in diffuse optical tomography

A prominent result of Arridge and Lionheart (1998 Opt. Lett. 23 882–4) demonstrates that it is in general not possible to simultaneously recover both the diffusion (aka scattering) and the absorption coefficient in steady-state (dc) diffusion-based optical tomography. In this work we show that it suffices to restrict ourselves to piecewise constant diffusion and piecewise analytic absorption coefficients to regain uniqueness. Under this condition both parameters can simultaneously be determined from complete measurement data on an arbitrarily small part of the boundary.

Monotonicity-Based Shape Reconstruction in Electrical Impedance Tomography

Current-voltage measurements in electrical impedance tomography (EIT) can be partially ordered with respect to definiteness of the associated self-adjoint Neumann-to-Dirichlet operators. With this ordering, a pointwise larger conductivity leads to smaller current-voltage measurements, and smaller conductivities lead to larger measurements. We present a converse of this simple monotonicity relation and use it to solve the shape reconstruction (a.k.a. inclusion detection) problem in EIT. The outer shape of a region where the conductivity differs from a known background conductivity can be found by simply comparing the measurements to that of smaller or larger test regions.

EXACT SHAPE-RECONSTRUCTION BY ONE-STEP LINEARIZATION IN ELECTRICAL IMPEDANCE TOMOGRAPHY ∗

For electrical impedance tomography (EIT), the linearized reconstruction method using the Frechet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finite-dimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of t...

DETECTING INCLUSIONS IN ELECTRICAL IMPEDANCE TOMOGRAPHY WITHOUT REFERENCE MEASUREMENTS

We develop a new variant of the factorization method that can be used to detect inclusions in electrical impedance tomography from either absolute current-to-voltage measurements at a single, nonzero frequency or from frequency-difference measurements. This eliminates the need for numerically simulated reference measurements at an inclusion-free body and thus greatly improves the methods robustness against forward modeling errors, e.g., in the assumed bodys shape.

Factorization Method and Its Physical Justification in Frequency-Difference Electrical Impedance Tomography

Time-difference electrical impedance tomography (tdEIT) requires two data sets measured at two different times. The difference between them is utilized to produce images of time-dependent changes in a complex conductivity distribution inside the human body. Frequency-difference EIT (fdEIT) was proposed to image frequency-dependent changes of a complex conductivity distribution. It has potential applications in tumor and stroke imaging since it can visualize an anomaly without requiring any time-reference data obtained in the absence of an anomaly. In this paper, we provide a rigorous analysis for the detectability of an anomaly based on a constructive and quantitative physical correlation between a measured fdEIT data set and an anomaly. From this, we propose a new noniterative frequency-difference anomaly detection method called the factorization method (FM) and elaborate its physical justification. To demonstrate its practical applicability, we performed fdEIT phantom imaging experiments using a multifrequency EIT system. Applying the FM to measured frequency-difference boundary voltage data sets, we could quantitatively evaluate indicator functions inside the imaging domain, of which values at each position reveal presence or absence of an anomaly. We found that the FM successfully localizes anomalies inside an imaging domain with a frequency-dependent complex conductivity distribution. We propose the new FM as an anomaly detection algorithm in fdEIT for potential applications in tumor and stroke imaging.

Resolution Guarantees in Electrical Impedance Tomography

Electrical impedance tomography (EIT) uses current-voltage measurements on the surface of an imaging subject to detect conductivity changes or anomalies. EIT is a promising new technique with great potential in medical imaging and non-destructive testing. However, in many applications, EIT suffers from inconsistent reliability due to its enormous sensitivity to modeling and measurement errors. In this work, we show that it is principally possible to give rigorous resolution guarantees in EIT even in the presence of systematic and random measurement errors. We derive a constructive criterion to decide whether a desired resolution can be achieved in a given measurement setup. Our results cover the case where anomalies of a known minimal contrast in a subject with imprecisely known background conductivity are to be detected from noisy measurements on a number of electrodes with imprecisely known contact impedances. The considered settings are still idealized in the sense that the shape of the imaging subject has to be known and the allowable amount of uncertainty is rather low. Nevertheless, we believe that this may be a starting point to identify new applications and to design and optimize measurement setups in EIT.

Recent Progress on the Factorization Method for Electrical Impedance Tomography

The Factorization Method is a noniterative method to detect the shape and position of conductivity anomalies inside an object. The method was introduced by Kirsch for inverse scattering problems and extended to electrical impedance tomography (EIT) by Brühl and Hanke. Since these pioneering works, substantial progress has been made on the theoretical foundations of the method. The necessary assumptions have been weakened, and the proofs have been considerably simplified. In this work, we aim to summarize this progress and present a state-of-the-art formulation of the Factorization Method for EIT with continuous data. In particular, we formulate the method for general piecewise analytic conductivities and give short and self-contained proofs.

Regularizing a linearized EIT reconstruction method using a sensitivity-based factorization method

For electrical impedance tomography (EIT), most practical reconstruction methods are based on linearizing the underlying non-linear inverse problem. Recently, it has been shown that the linearized problem still contains the exact shape information. However, the stable reconstruction of shape information from measurements of finite accuracy on a limited number of electrodes remains a challenge. In this work, we propose to regularize the standard linearized reconstruction method (LM) for EIT using a non-iterative shape reconstruction method (the factorization method). Our main tool is a discrete-sensitivity-based variant of the factorization method (herein called S-FM) which allows us to formulate and combine both methods in terms of the sensitivity matrix. We give a heuristic motivation for this new method and show numerical examples that indicate its good performance in the localization of anomalies and the alleviation of ringing artifacts.

A Unified Variational Formulation for the Parabolic-Elliptic Eddy Current Equations

Transient excitation currents generate electromagnetic fields which, in turn, induce electric currents in proximal conductors. For slowly varying fields, this can be described by the eddy current equations, which are obtained by neglecting the dielectric displacement currents in Maxwells equations. The eddy current equations are of parabolic-elliptic type: In insulating regions, the field instantaneously adapts to the excitation (quasistationary elliptic behavior), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behavior). For fixed conductivity, the equations are well studied. However, little rigorous mathematical results are known for the solutions dependence on the conductivity, in particular for the solutions sensitivity with respect to the equation changing from elliptic to parabolic type. In this work, we derive a new unified variational formulation for the eddy current equations that is uniformly coercive with respect to the conductivity. ...

Interpolation of missing electrode data in electrical impedance tomography

Novel reconstruction methods for electrical impedance tomography (EIT) often require voltage measurements on current-driven electrodes. Such measurements are notoriously difficult to obtain in practice as they tend to be affected by unknown contact impedances and require problematic simultaneous measurements of voltage and current. In this work, we develop an interpolation method that predicts the voltages on current-driven electrodes from the more reliable measurements on current-free electrodes for difference EIT settings, where a conductivity change is to be recovered from difference measurements. Our new method requires the a-priori knowledge of an upper bound of the conductivity change, and utilizes this bound to interpolate in a way that is consistent with the special geometry-specific smoothness of difference EIT data. Our new interpolation method is computationally cheap enough to allow for real-time applications, and simple to implement as it can be formulated with the standard sensitivity matrix. We numerically evaluate the accuracy of the interpolated data and demonstrate the feasibility of using interpolated measurements for a monotonicity-based reconstruction method.