Synthetic Conductivity Data

Learned priors (see Learned Regularization) need samples from a distribution of plausible conductivities. Possible sources:

1. Real EIT reconstructions or anatomical atlases (thorax, head, breast). These are closest to the application, but scarce, and they carry the blur of the reconstruction method that produced them.
2. Parametric phantoms: random ellipses, circles, polygons, or smooth random fields such as Gaussian processes. They are cheap and controllable but lack realistic texture.
3. Natural grayscale images (for example Tiny ImageNet, landscape collections) resampled onto the conductivity grid. They are rich in edges, textures and multi-scale structure, which makes them a demanding stress test of whether a method can recover sharp features. They are not anatomical, however.

Mapping images to conductivities. Intensities in $[0,1]$ are mapped affinely to $[{\sigma }_{\min },{\sigma }_{\max }]$ with ${\sigma }_{\min }>0$, so the forward problem stays elliptic (see Box Constraints on Conductivity). They are then interpolated onto the finite element space of $\sigma$. On an $n\times n$ grid of $Q_1$ elements, nodal values correspond directly to pixels.

Held-out evaluation. Test conductivities must not appear in training. Test data should also be simulated on a different, preferably finer, mesh than the reconstruction mesh to avoid the Inverse Crime.

In ModularEIT.jl: random_inclusions, InclusionPhantom, gaussian_random_field, image_phantom, conductivity.

References

1. Tiny ImageNet (2017). Kaggle dataset. kaggle.com/c/tiny-imagenet
2. S. Arridge, P. Maass, O. Öktem, C.-B. Schönlieb (2019). Solving inverse problems using data-driven models. Acta Numerica 28, 1–174. doi:10.1017/S0962492919000059
3. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344