An iterative EIT reconstruction repeatedly compares simulated with measured data and improves the conductivity. It combines five exchangeable pieces.
1. Data
Measurements are voltages for applied current patterns, or currents for applied voltages (see Measurement Protocols, Current Patterns). Real data contain noise and modelling errors (see Noise Models for EIT Data). Simulated test data must be generated with a different model than the reconstruction to avoid the Inverse Crime. How much information the data carry is limited by the Decay of Boundary Measurements.
2. Objective
A functional measures the disagreement between model and data:
- the least-squares misfit in a chosen metric (see Data Fidelity Terms);
- the Kohn-Vogelius Functional, which compares a current-driven and a voltage-driven solution in the energy norm.
The problem is ill-posed (see Well-Posedness, Stability of the Calderón Problem), so a regulariser is added, (see Variational Regularization). can be classical (see Tikhonov Regularization, Total Variation) or learned (see Classical and Learned Priors).
3. Gradient
The Adjoint State Method gives the gradient of the misfit at the cost of one additional linear solve, whatever the number of unknowns (see PDE-Constrained Optimization, Lagrangian Formulation). The Jacobian (see Linearized EIT and the Sensitivity Kernel) costs one solve per measurement. The gradient is a dual vector; turning it into a direction requires a choice of inner product (see Gradient Representation and the Riesz Map). Gradients should always be checked against finite differences (see Gradient Testing).
4. Optimiser
Gauss–Newton-type methods exploit the least-squares structure. Quasi-Newton methods need only gradients. Proximal splitting methods handle non-smooth or learned regularisers (see Choosing an Optimizer). Bounds keep the conductivity positive (see Box Constraints on Conductivity). The iteration stops by the discrepancy principle or other rules (see Stopping Criteria, Choosing the Regularization Parameter).
5. Linear algebra and discretisation
Each evaluation solves the forward problem and each gradient an adjoint problem, both with the same system matrix (see From Physics to Linear Algebra, Choosing a Linear Solver). The mesh can be adapted to the current conductivity and to the quantity of interest (see Adaptive Meshing in EIT).
The loop of these steps is described in Iterative Reconstruction Loop. Direct, non-iterative alternatives are the D-bar Method and other Reconstruction Algorithms with Guarantees.
References
- J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344
- C. R. Vogel (2002). Computational Methods for Inverse Problems. SIAM. doi:10.1137/1.9780898717570