Decay of Boundary Measurements

Why can only a few current patterns be used effectively?

Voltage amplitude. For a homogeneous unit disc, a Fourier current $g_k=\cos (k\theta )$ produces the boundary voltage $f_k=\frac{1}{k}\cos (k\theta )$. The Neumann-to-Dirichlet Map has eigenvalues $1/|k|$. For general conductivities the NtD map is a pseudodifferential operator of order $-1$, so the same $1/k$ decay holds asymptotically. Plotting $\Vert f_k\Vert$ against $k$ for any conductivity shows this monotone decay: low frequencies carry most of the voltage.

Information content. More important is how the difference ${\mathcal{R}}_{\sigma }-{\mathcal{R}}_{{\sigma }_0}$, the part that carries information about the interior, decays. The harmonic extension of $\cos (k\theta )$ is $r^k\cos (k\theta )$. It is concentrated in a boundary layer of width $\sim 1/k$, so an inclusion at depth $d$ changes the $k$-th measurement by roughly $(1-d{)}^{2k}$. That is exponentially small in $k$. The singular values of the linearised map from conductivity to data therefore decay exponentially. This is the discrete face of the logarithmic stability.

Consequence. Once $(1-d{)}^{2k}$ falls below the relative noise level, higher patterns add only noise. Truncating to the leading patterns or singular pairs is a natural regulariser (see Truncated SVD Regularization and Current Patterns).

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. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344
3. N. Mandache (2001). Exponential instability in an inverse problem for the Schrödinger equation. Inverse Problems 17(5), 1435–1444. doi:10.1088/0266-5611/17/5/313