An adjoint field approach to Fisher information-based sensitivity analysis in electrical impedance tomography

Abstract

An adjoint field approach is used to formulate a general numerical framework for Fisher information-based sensitivity analysis in electrical impedance tomography. General expressions are given for the gradients used in standard least-squares optimization, i.e. the Jacobian related to the forward problem, and it is shown that these gradient expressions are compatible with commonly used electrode models such as the shunt model and the complete electrode model. By using the adjoint field formulations together with a variational analysis, it is also shown how the computation of the Fisher information can be integrated with the gradient calculations used for optimization. It is furthermore described how the Fisher information analysis and the related sensitivity map can be used in a preconditioning strategy to obtain a wellbalanced parameter sensitivity and improved performance for gradient-based quasi-Newton optimization algorithms in electrical impedance tomography. Numerical simulations as well as reconstructions based on experimental data are used to illustrate the sensitivity analysis and the performance of the improved inversion algorithm in a four-electrode measurement set-up.

Data Fusion for Electromagnetic and Electrical Resistive Tomography Based on Maximum Likelihood

“…where Jδθ, ξ d = δθ, J * ξ , see also [10][11][12]16]. Based on the scalar product (10), the negative loglikelihood function (7) can now be written…”

“…where I + denotes a truncated pseudoinverse of the Fisher information I, which is based on the singular value decomposition (SVD), see also [10,11,17]. It can be readily shown that the regularized maximum likelihood solution (18) is equivalent to a regularized pseudoinverse based directly on the linearized operator problem…”

“…provided that the scalar products are properly chosen as in (9) and (10). To this end, it is assumed that the operator J is compact and hence that the following singular system exists…”

“…The figure illustrates that ER tomography contributes mainly at the lower eigenvalue indices whereas EM tomography contributes at higher indices in this particular problem. It can be anticipated that this behavior is typical since the eigenvalue distribution of ER tomography usually drops of rapidly already from the first indices whereas with EM tomography there is typically a threshold in the distribution of eigenvalues after which the eigenvalues drop off in rapid descent, see, for example, [10,11,13,19,20]. …”

“…In this paper, the general approach to information fusion is to perform a multiphysics data fusion based on the maximum likelihood principle, and to exploit the elements of Fisher information analysis that has been developed in [9][10][11]. In particular, an infinite-dimensional Fisher information formulation is employed which is suitable for analytical Green's function and gradient-based approaches [11], as well as with adjoint field formulations [10].…”