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Master theses

Current and past ideas and concepts for Master Theses.

Structure preserving solution methods for non-linear inverse problems

Subject

EIT, which is concerned with the reconstruction of a spatially varying conductivity distribution inside a bounded domain from partial knowledge of the Neumann-to-Dirichlet or Dirichlet-to-Neumann map on its boundary, is a notoriously difficult inverse problem due to its nonlinear and severely ill-posed nature. Despite theoretical limitations and often disappointing performance, output least squares (OLS) based reconstruction methods retain a prominent role in EIT. ETRO researchers instead have derived a double-constrained variational formulation for the nonlinear boundary value problem of EIT and demonstrated its convergence. Whereas OLS can be regarded as minimizing an error norm, solutions are recovered here as the minimizers of a closely related residual norm problem arising directly from the governing differential equations. This key difference is found to have a profound effect on the numerical properties of the proposed method. Derivation of a nonlinear conjugate gradient-based solution scheme has been shown to lead to a sequence of structured sparse matrix problems, the conditioning of which appears to be far more favourable than typically observed in output least squares. The resulting sparse problem formulation defines sets of discrete subspaces to which admissible solutions necessarily are confined. Unlike OLS, these subspaces no longer are continuous, but instead assume a discontinuous form, whereby noise perturbations systematically are restricted in their effects.

Kind of work

In this thesis, we will further investigate the discrete sparsity structures that derive from a novel doubleconstrained variational formulation for the inverse problem of EIT and relate this distinguishing feature with a demonstrated superiority in terms of numerical robustness and noise resilience. Thesis work will combine numerical experiments (Matlab), with a further theoretical validation (linear algebra) building upon a thorough literature survey.

Framework of the Thesis

Back in the 1980s, it was an impressive achievement: The first Electrical Impedance Tomography (EIT) devices delivered cross-sectional views at a frame rate of one image per second, provided they were used in a proper interference-free electromagnetic surrounding. Today after being part of everyday clinical routine for several years, EIT is capable of generating a far more impressive 50 high-quality tomographic images every second. EIT visualizes internal structures using low-level alternating currents applied through a set of surface electrodes positioned around the body. Clinical changes inside the body result in alterations in electrical conductivity that manifest as minute voltage changes on the surface of the skin. In addition to dynamic imaging, EIT has other advantages as well: The examination does not expose patients or users to any ionising radiation, while it allows for a continuous monitoring of regional phenomena and thus provides a functional visualization. At intensive care, EIT visualises respiratory functions directly at the patient's bedside, providing continuous, regional information about the distribution of ventilation in the lungs. This opens up the option for completely new ventilation strategies. Using EIT based pulmonary monitoring, ventilation parameters and therapeutic measures can be individually adjusted to meet the patient's needs, resulting in a more protective ventilation.

Promotor

Prof. Dr. Ir. Johan Stiens

+32 (0)2 629 2397

jstiens@etrovub.be

more info

Supervisor

  Bart Truyen

+32 (0)2 629 3954

btruyen@etrovub.be

more info

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