Data Fidelity Terms

The data fidelity term $d(\mathcal{F}(\sigma ),y)$ in a variational reconstruction measures how far the predicted boundary data are from the measured data.

Output least squares (boundary $L^2$). For Neumann data $g_i$ and measured voltages $f_i$:

$$J(\sigma )=\sum _{i=1}^N\frac{1}{2}\Vert u_i(\sigma ){|}_{\partial \Omega }-f_i{\Vert }_{L^2(\partial \Omega )}^2,$$

where $u_i(\sigma )$ solves the Neumann Problem. This is the most common choice. Discretely, $\Vert v{\Vert }_{L^2(\partial \Omega )}^2=v^{\top }{M}_{\Gamma }v$ with the boundary mass matrix. The plain Euclidean norm of nodal values only approximates it on a uniform boundary mesh.

Noise-weighted. For Gaussian noise with covariance $\Gamma$, use $\frac{1}{2}\Vert \cdot {\Vert }_{{\Gamma }^{-1}}^2$ (see Bayesian Inversion).

Sobolev norms. Measuring the misfit in $H^{1/2}(\partial \Omega )$ or $H^{-1/2}(\partial \Omega )$ matches the natural function spaces of the Neumann-to-Dirichlet Map and weights frequencies differently (see Discrete Fractional Sobolev Norms).

Energy (Kohn–Vogelius). An interior mismatch between Dirichlet and Neumann solutions (see Kohn-Vogelius Functional). It is equivalent to a data fit in the energy norm.

Optimal transport. Wasserstein distances between boundary data, treated as densities after normalisation, give smoother landscapes with fewer spurious local minima (Bao & Zhang 2022).

Operator-level. When a whole discrete operator ${\Lambda }_{\text{meas}}$ is available, one can compare operators, for example in Frobenius or operator norm, possibly after a spectral truncation (see Truncated SVD Regularization).

Any differentiable $d$ fits the Adjoint State Method. Only the right-hand side of the adjoint equation, ${\partial }_{u}d$, changes.

In ModularEIT.jl: SquaredEuclidean, WeightedSquaredEuclidean.

References

1. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201
2. G. Bao, Y. Zhang (2022). Optimal Transportation for Electrical Impedance Tomography. arXiv:2210.16082
3. J. Kaipio, E. Somersalo (2005). Statistical and Computational Inverse Problems. Springer. doi:10.1007/b138659