Current Patterns

A current pattern is a boundary current $g$ with ${\int }_{\partial \Omega }g\mathrm{d}s=0$ (or electrode currents with ${\sum }_{\ell }{I}_{\ell }=0$). The data set is a collection of pairs $(g_i,f_i)$ with $f_i={\mathcal{R}}_{\gamma }{g}_i$ (see Neumann-to-Dirichlet Map). With $N$ linearly independent patterns one observes the NtD map on an $N$-dimensional subspace.

Common choices

• Adjacent / neighbouring: current between neighbouring electrodes. Simple hardware, but poor sensitivity in the centre.
• Opposite and skip-$k$ patterns: current between electrodes a fixed distance apart.
• Trigonometric (Fourier) patterns: on a boundary parametrised by the angle or arc length $\theta$, $g_k(\theta )=\cos (k\theta ),\sin (k\theta ),k=1,2,\dots$ They are orthogonal, zero-mean and easy to build.

Decay with frequency. For a homogeneous disc, ${\mathcal{R}}_1{e}^{ik\theta }=|k{|}^{-1}{e}^{ik\theta }$, so the voltage amplitude falls off like $1/k$. The information a pattern carries about the interior decays even faster. A harmonic of frequency $k$ behaves like $r^{|k|}{e}^{ik\theta }$ in polar coordinates, so high-frequency patterns only probe a thin layer near the boundary. With noise, only the first few frequencies are informative (see Decay of Boundary Measurements).

Optimal patterns. Isaacson (1986) showed that the patterns that best distinguish two conductivities ${\gamma }_1,{\gamma }_2$ are the eigenfunctions of ${\mathcal{R}}_{{\gamma }_1}-{\mathcal{R}}_{{\gamma }_2}$ with the largest eigenvalues. For rotationally symmetric perturbations of a disc, these are exactly the trigonometric patterns. In practice one can compute them from data through the SVD of an estimated boundary operator (see Truncated SVD Regularization).

In ModularEIT.jl: trigonometric_patterns, pattern_svd.

References

1. D. Isaacson (1986). Distinguishability of Conductivities by Electric Current Computed Tomography. IEEE Trans. Med. Imaging 5(2), 91–95. doi:10.1109/TMI.1986.4307752
2. D. Gisser, D. Isaacson, J. C. Newell (1990). Electric Current Computed Tomography and Eigenvalues. SIAM J. Appl. Math. 50(6), 1623–1634. doi:10.1137/0150096
3. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344