Thermal tomography with unknown boundary

TL;DR

This paper develops an adaptive sparse pseudospectral approximation method for thermal tomography that effectively reconstructs internal properties and shape of an object from boundary temperature data, even with unknown boundary shape.

Contribution

It introduces a novel polynomial surrogate approach for joint reconstruction of thermal and geometric parameters in time-dependent thermal tomography.

Findings

Successful numerical experiments with simulated data in 2D

Efficient reconstruction of thermal properties and shape

Demonstrates robustness with incomplete boundary information

Abstract

Thermal tomography is an imaging technique for deducing information about the internal structure of a physical body from temperature measurements on its boundary. This work considers time-dependent thermal tomography modeled by a parabolic initial/boundary value problem without accurate information on the exterior shape of the examined object. The adaptive sparse pseudospectral approximation method is used to form a polynomial surrogate for the dependence of the temperature measurements on the thermal conductivity, the heat capacity, the boundary heat transfer coefficient and the body shape. These quantities can then be efficiently reconstructed via nonlinear, regularized least squares minimization employing the surrogate and its derivatives. The functionality of the resulting reconstruction algorithm is demonstrated by numerical experiments based on simulated data in two spatial dimensions.