Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particular, almost all reconstruction algorithms require the precise shape of the imaged body as an input. In this work, the need for prior geometric information is relaxed by introducing a Newton-type output least squares algorithm that reconstructs the admittivity distribution and the object shape simultaneously. The method is built in the framework of the complete electrode model and it is based on the Fr\'echet derivative of the corresponding current-to-voltage map with respect to the object boundary shape. The functionality of the technique is demonstrated via numerical experiments with simulated measurement data.Comment: 3 figure

A survey on inverse problems for applied sciences

The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting

Improvements to the acoustic pulse reflectometry technique for measuring duct dimensions

In the study of tubular structures such as pipeline sections, musical wind instruments and human airways, acoustic pulse reflectometry has become established as a useful tool for non-invasively measuring the input impulse response, from which the internal duct dimensions can be calculated. In this thesis, the theory describing wave propagation in a duct of varying crosssection is outlined, culminating in a discussion of the layer peeling algorithm used to reconstruct a duct’s bore profile from its input impulse response. Experimental measurements of the input impulse responses of various test objects, together with the subsequent bore reconstructions, are then presented. The problem of offset in input impulse response measurements is discussed and the effect on the bore reconstruction is shown. The offset is found to consist of both a DC component and a sinusoidal component. Methods for eliminating the two offset components are explored and the resultant improvement in the stability and reproducibility of the bore reconstructions is demonstrated. Two adaptations to the reflectometry technique, designed to extend the bandwidth of input impulse response measurements, are described. The improved high frequency content brought about by these adaptations is shown to lead to bore reconstructions of high axial resolution, allowing rapid changes in cross-sectional area to be more accurately reproduced. Finally, limitations of the acoustic pulse reflectometry technique (particularly those brought about by the bandwidth improvements) are discussed and potential future ways of overcoming the limitations are proposed

Detecting, locating and sizing leaks in gas-filled pipes using acoustical measurements

A single leak in a duct can be detected, located and sized by measuring the input impedance of the duct and then analytically solving an inverse problem. However, previously applied analytical methods break down when it comes to predicting smaller hole sizes. Results are presented which show that, by treating smaller holes as capillaries and applying appropriate theoretical approximations, accurate predictions of smaller hole sizes are possible. Extending the analytical methods to a duct containing multiple leaks is non-trivial as the resulting mathematical expressions are highly complex. In this thesis, an alternative approach which uses optimisation methodology to detect, locate and size multiple leaks in a duct is described. The optimisation algorithms are applied to a measurement of the duct’s input impedance but they are able to cope with the presence of multiple leaks. Results are presented which illustrate the success of the optimisation approach in detecting, locating and sizing multiple leaks in a duct. An objective function incorporating the theoretical input impedance of a model duct and experimental input impedance of the cylindrical pipe under investigation is designed. By studying the behaviour of the objective function and the application of different numerical optimisation methods, it is possible to determine those methods most suitable for investigating leaks. Results are presented showing that the Rosenbrock optimisation algorithm provides predictions of hole sizes and locations which are in good agreement with their actual values. The success of the Rosenbrock optimisation algorithm is attributed to function minimisation techniques incorporating non derivative based search directions and optimisation steps

A butterfly‐based direct solver using hierarchical LU factorization for Poggio‐Miller‐Chang‐Harrington‐Wu‐Tsai equations

A butterfly‐based hierarchical LU factorization scheme for solving the PMCHWT equations for analyzing scattering from homogenous dielectric objects is presented. The proposed solver judiciously re‐orders the discretized integral operator and butterfly‐compresses blocks in the operator and its LU factors. The observed memory and CPU complexities scale as O(N log2 N) and O(N1.5 log N), respectively. The proposed solver is applied to the analyses of scattering several large‐scale dielectric objects.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143676/1/mop31166.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/143676/2/mop31166_am.pd

Mathematics and Algorithms in Tomography

This is the eighth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, sonar, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, vector tomography, and texture analysis