Parametrizations of the Conductivity

The unknowns of a reconstruction need not be the coefficients of the conductivity in the finite element space of the forward model. A parametrization $\sigma =P\theta$ describes the conductivity by parameters $\theta$ of a different space. It separates two choices: the resolution needed to simulate the measurements, and the resolution at which the conductivity is sought.

Chain rule

For a data objective $J(\sigma )$, the objective in the parameters is $\tilde{J}(\theta )=J(P\theta )$, with

$${\nabla }_{\theta }\tilde{J}=P^{\top }{\nabla }_{\sigma }J,\frac{\partial r}{\partial \theta }=\frac{\partial r}{\partial \sigma }P$$

for the residual $r$ of a least-squares misfit. Adjoint gradients (see Adjoint State Method), Gauss–Newton Jacobians and all optimisers carry over unchanged.

Pixels on any mesh

With $\theta$ the values of an image on a rectangle containing the domain (see Pixel Images and Finite Element Functions), a natural $P$ is the lumped L2 Projection of the pixel function onto the conductivity space:

$$P_{ak}=\frac{\int {\phi }_a{\chi }_k\mathrm{d}x}{\int {\phi }_a\mathrm{d}x},$$

with ${\chi }_k$ the indicator of pixel $k$. The weights are non-negative and sum to one. Constants are therefore reproduced, every coefficient is a weighted average of pixel values, and bounds on the pixels bound the conductivity (see Box Constraints on Conductivity). On meshes whose cells lie inside single pixels the map is exact.

The forward mesh can then be refined where the physics demands it, for example towards the boundary and the electrodes (see Adaptive Meshing in EIT). The unknowns stay pixels: the natural variables of image priors (see Learned Regularization), and of pixel-based penalties such as Total Variation.

Subspaces: regularisation by parametrisation

Restricting the pixels to a subspace, ${\theta }_{\text{pixels}}=Bc$, is itself a regulariser: only conductivities in the span of $B$ can be reconstructed (see Implicit Regularization).

• Low-frequency cosine modes (see Discrete Cosine Transform): a smooth, low-dimensional conductivity with few, well-determined unknowns. As the first stage of a coarse-to-fine reconstruction it provides a stable starting point for the full pixel problem.
• Cosine modes plus a boundary band: the measurements determine the conductivity near the boundary much better than in the interior (see Decay of Boundary Measurements, Linearized EIT and the Sensitivity Kernel). A basis of smooth modes everywhere, plus free pixels in a band along the boundary, matches the resolution of the parametrisation to that of the data.
• Singular vectors of the Jacobian: the data-optimal subspace, ordered by how well each direction is determined (see Truncated SVD Regularization).

Subspace coefficients cannot be bounded like pixels, since a combination of modes can become negative. Trial conductivities have to be checked for positivity instead.

In ModularEIT.jl: PixelParametrization, SubspaceParametrization, ParametrizedObjective, dct_basis, boundary_band_basis, jacobian_basis.

References

1. B. Kaltenbacher, A. Neubauer, O. Scherzer (2008). Iterative Regularization Methods for Nonlinear Ill-Posed Problems. de Gruyter. doi:10.1515/9783110208276
2. H. W. Engl, M. Hanke, A. Neubauer (1996). Regularization of Inverse Problems. Kluwer. doi:10.1007/978-94-009-1740-8
3. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344