Classical and Learned Priors

Because EIT is severely ill-posed (see Stability of the Calderón Problem), prior knowledge about the conductivity determines what a reconstruction can show. Priors range from explicit penalties to generative models learned from data.

Explicit penalties

Variational Regularization adds a functional $\alpha R(\sigma )$ to the misfit:

• Tikhonov Regularization favours small or smooth conductivities;
• Total Variation favours piecewise constant ones with sharp edges (Smoothed Total Variation makes it differentiable);
• Spectral Sobolev Norms on Rectangles interpolate between these smoothness classes.

The weight α trades data fit against prior (see Choosing the Regularization Parameter). Stopping an iteration early also regularises (see Implicit Regularization). In the Bayesian view, $R$ is a negative log-prior and the reconstruction a posterior mode (see Bayesian Inversion).

Priors from data

Learned Regularization replaces the hand-made $R$ by one fitted to example conductivities (see Synthetic Conductivity Data):

• Denoisers as proximal operators. A trained denoiser can replace the proximal step of a splitting method (see Plug-and-Play Priors, Regularization by Denoising).
• Explicit learned energies. These can be convex by construction (see Energy-Based Models, Input Convex Neural Networks).
• Implicit models. Reconstructions can also be defined as fixed points of learned maps (see Deep Equilibrium Models).

Generative priors

Diffusion Models learn the score of the prior distribution (see Score Function, Denoising Score Matching). They can be combined with the EIT data term during sampling (see Diffusion Posterior Sampling, DiffPIR, Diffusion Models for EIT). Instead of a single estimate they produce samples, and with them a measure of uncertainty (see Langevin Dynamics).

Learned priors can also produce plausible structures that are not in the data (see Hallucinations and Uncertainty). Symmetries of the problem can be built into the networks (see Symmetries of the EIT Problem).

References

1. S. Arridge, P. Maass, O. Öktem, C.-B. Schönlieb (2019). Solving inverse problems using data-driven models. Acta Numer. 28, 1–174. doi:10.1017/S0962492919000059
2. M. Benning, M. Burger (2018). Modern regularization methods for inverse problems. Acta Numer. 27, 1–111. doi:10.1017/S0962492918000016
3. J. Kaipio, E. Somersalo (2005). Statistical and Computational Inverse Problems. Springer. doi:10.1007/b138659