Electric Impedance Tomography problem for surfaces with internal holes

TL;DR

This paper addresses the inverse problem of determining a Riemann surface with internal holes from boundary measurements, using an algebraic approach based on holomorphic functions and the Dirichlet-to-Neumann maps.

Contribution

It introduces an algebraic method leveraging the Gelfand spectrum of holomorphic functions to uniquely recover the surface from boundary data.

Findings

The algebra of holomorphic functions is determined by the DN maps.

The Gelfand spectrum constructs a conformally equivalent surface.

The method provides a solution to the inverse EIT problem on surfaces with holes.

Abstract

Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent boundary $\Gamma=\Gamma_0\cup\Gamma_1\cup\dots\cup\Gamma_m=:\Gamma_0\cup\tilde\Gamma$. Let $u=u^f$ obey $\Delta u=0$ in $M$, $u|_{\Gamma_0}=f,\,\,u|_{\tilde\Gamma}=0$ (the grounded holes) and $v=v^h$ obey $\Delta v=0$ in $M$, $v|_{\Gamma_0}=h,\,\,\partial_\nu v|_{\tilde\Gamma}=0$ (the isolated holes). Let $\Lambda_{g}^{\rm gr}: f\mapsto\partial_\nu u^f|_{\Gamma_{0}}$ and $\Lambda_{g}^{\rm is}: h\mapsto\partial_\nu v^h|_{\Gamma_{0}}$ be the corresponding DN-maps. The EIT problem is to determine $M$ from $\Lambda_{g}^{\rm gr}$ or $\Lambda_{g}^{\rm is}$. To solve it, an algebraic version of the BC-method is applied. The main instrument is the algebra of holomorphic functions on the ma\-ni\-fold ${\mathbb M}$, which is obtained by gluing two examples of $M$ along $\tilde{\Gamma}$. We show that this algebra is determined by $\Lambda_{g}^{\rm gr}$ (or $\Lambda_{g}^{\rm is}$) up to isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g',\Gamma_{0}')$ of $(M,g,\Gamma_{0})$. This copy is conformally equivalent to the original, provides $\Gamma_{0}'=\Gamma_{0},\,\,\Lambda_{g'}^{\rm gr}=\Lambda_{g}^{\rm gr},\,\,\Lambda_{g'}^{\rm is}=\Lambda_{g}^{\rm is}$, and thus solves the problem.