finite difference immersed boundary method for an inverse problem in medical imaging: electrical impedance tomography.
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Electrical Impedance Tomography (EIT) is a non-invasive imaging method used to determine the internal electrical conductivity of an object by applying currents on its boundary and measuring the resulting voltages. Mathematically, this inverse problem, known as Calderón’s problem, is highly ill-posed, making accurate reconstruction difficult. In practical applications, currents are injected through surface electrodes, and voltage differences are measured. To realistically model these measurements, the Complete Electrode Model (CEM) is used, accounting for electrode geometry and the resistive interface (shunting effect) between electrodes and the object. This work introduces an immersed boundary method (IBM) for solving EIT problems numerically. The approach uses a Cartesian mesh and avoids the need for precise boundary discretization, making it well-suited for complex or changing geometries. The method’s convergence is proven, and its effectiveness is validated through two-dimensional simulations for both direct and inverse problems.
