On the Laplacian operation with applications in magnetic resonance electrical impedance imaging

Abstract

Consider the numerical computation of Laplacian operation from the noisy data of a given function f(x) defined in Ω ⊂ ℝ2. By expressing this ill-posed problem as a Fredholm integral equation of the first kind, we construct the approximate solution from a PDE boundary value problem directly, which can be considered as the Lavrentiev regularization in essence. The optimal error estimate of this regularization for an appropriate a posteriori choice strategy of the regularization parameter is given under some source conditions of the exact Laplacian operation. An iterative algorithm for computing this a posteriori regularization parameter under the framework of model function method is also given with its convergence analysis. The advantage of the proposed scheme is that we treat the Laplacian operation as a whole, rather than the partial derivatives of each variable, and therefore the Laplacian operation can be computed in a bounded domain with arbitrary boundary shape for noisy data specified at randomly scattered points. At last, we apply this scheme to the iterative algorithm of reconstructing the conductivity in magnetic resonance electrical impedance tomography, where the input data is the magnetic flux data for which the Laplacian operation must be carried out. The validity of the proposed scheme is shown numerically for this new medical imaging technology.