Noise Models for EIT Data

Simulated EIT data must be corrupted with noise to mimic real measurements and to avoid overly optimistic results (see also Inverse Crime).

Additive Gaussian noise on voltages (most common). For each pattern:

$$f_i^{\delta }=f_i+{\eta }_i,{\eta }_i\sim \mathcal{N}(0,(\delta \Vert f_i\Vert /\sqrt{m}{)}^{2}I),$$

relative to the signal level ($m$ is the number of boundary values), or with a fixed absolute standard deviation from the instrument specification.

Noise on currents and voltages separately. With parameters ${\sigma }_g$ (current-source error) and ${\sigma }_f$ (voltmeter error):

$$f_i={\mathcal{R}}_{\gamma }(g_i+{\sigma }_g{\epsilon }_{g,i})+{\sigma }_f{\epsilon }_{f,i},\epsilon \sim \mathcal{N}(0,I).$$

The injected current is projected back to zero mean before the solve. The algorithm is given the nominal $g_i$ and the noisy $f_i$.

Operator-level noise. With an estimated Discrete Boundary Operator $\mathcal{R}$, add a random symmetric perturbation

$$\tilde{\mathcal{R}}=\mathcal{R}+s\mathcal{E},\mathcal{E}=\frac{1}{2}(\hat{\mathcal{E}}+{\hat{\mathcal{E}}}^{\top }),{\hat{\mathcal{E}}}_{jk}\sim \mathcal{N}(0,1)\text{i.i.d.}$$

This preserves symmetry but not positive semidefiniteness: for large $s$, small eigenvalues can become negative. New data pairs are then extracted from $\tilde{\mathcal{R}}$. Inverting $\tilde{\mathcal{R}}$ to get DtN data amplifies the noise strongly in the small-eigenvalue directions, giving heavy-tailed errors.

Beyond white noise. Real data also contain electrode contact-impedance errors, electrode position errors, drift and correlated noise. These modelling errors are often larger than instrument noise. The approximation error approach models them statistically.

In ModularEIT.jl: GaussianNoise, RelativeGaussianNoise, SourceMeterNoise, add_noise, perturb_boundary_operator, perturb_contact_impedance, electrode_angles.

References

1. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM, p. 197 ff. doi:10.1137/1.9781611972344
2. J. Kaipio, E. Somersalo (2005). Statistical and Computational Inverse Problems. Springer. doi:10.1007/b138659