Radial perturbations give electrical impedance tomography a spherical-harmonic structure

What the study found: The study found that, for rotationally symmetric conductivity perturbations in a unit ball, the eigenfunctions of the linearized electrical impedance tomography operator are spherical harmonics, which are standard functions on a sphere. It also found an explicit formula for the corresponding eigenvalues.
Why the authors say this matters: The authors say this structure is favorable for further analysis of the operator in numerical algorithms, and they conclude that the Fréchet derivative can be approximated by finite-rank operators when restricted to rotationally symmetric perturbations.
What the researchers tested: The researchers analyzed the Fréchet derivative, which maps a conductivity perturbation to the linearized change in boundary measurements governed by the conductivity equation, on the unit ball in dimensions d ≥ 2. They considered perturbations from the Hilbert space L2(B) and focused on rotationally symmetric perturbations.
What worked and what didn't: The spherical-harmonic eigenstructure was established, and the authors showed that the eigenvalue decay is uniform with respect to the degree of the spherical harmonics for perturbations from any bounded subset. They also showed that the Fréchet derivative can be approximated by finite-rank operators under the rotational symmetry restriction.
What to keep in mind: The abstract does not describe experimental data or numerical tests, and it does not state limitations beyond the restriction to the unit ball, L2(B) perturbations, and rotational symmetry in the main results.

Key points

• For rotationally symmetric conductivity perturbations, the linearized operator's eigenfunctions are spherical harmonics.
• The authors give an explicit formula for the associated eigenvalues.
• For perturbations from any bounded subset, the eigenvalue decay is uniform across spherical-harmonic degree.
• The Fréchet derivative can be approximated by finite-rank operators when restricted to rotationally symmetric perturbations.
• The authors say these properties are favorable for further analysis in numerical algorithms.