Boundary determination for hybrid imaging from a single measurement
For researchers in hybrid imaging, this work provides a theoretical foundation for boundary determination from minimal data, though the extension to variable exponents is incremental.
The paper recovers the boundary conductivity from a single measurement in hybrid imaging (AET or CDII) using an elementary local argument, and extends the result to variable exponent p(·)-Laplacian forward models, showing that single measurement specifies boundary conductivity when p−q ≥ 1, otherwise two alternatives remain. Numerical examples illustrate the results.
We recover the conductivity $σ$ at the boundary of a domain from a combination of interior and boundary data, with a single quite arbitrary measurement, in AET or CDII. The argument is elementary and local. More generally, we consider the variable exponent $p(\cdot)$-Laplacian as a forward model with the interior data $σ|\nabla u|^q$, and find out that single measurement specifies the boundary conductivity when $p-q \ge 1$, and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both $p$ and $q$ may depend on the spatial variable $x$, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples.