3D thermal tomography with experimental measurement data

Thermal Tomography with Unknown Boundary

“…We start this section by mathematically formulating our forward model for thermal tomography following [19,20]. Subsequently, a parametric extension is introduced for a certain two-dimensional measurement configuration.…”

“…Usually, the heat transfer coefficient c is large on those parts of the boundary that are covered by heaters and much smaller (but still positive) on the rest of the boundary (cf. [20]). Note that even though f vanishes outside the heaters, the heat still leaks to the surrounding air due to the non-vanishing Robin coefficient c. It is assumed that the boundary temperature of Ω is measured at R point-like sensors {s r } R r=1 ⊂ ∂Ω that are well-separated from the heaters.…”

“…We use the generic value δ = 10 −2 for the regularization parameter in all the following examples. Note that δ and ς couple in (20) in such a way that decreasing δ is equivalent to increasing ς. There is, however, a semi-heuristic Bayesian justification for our choices of δ and ς.…”

“…Figure 3 shows the target configuration which corresponds to the case ϑ = 0 except that the coefficient c takes two distinct values, namely 0.02 on the first four boundary segments between the heaters and 0.2 on the remaining ones. The second and third rows of the figure demonstrate that optimizing (20) only for ϑ (a) , ϑ (b) and ϑ (Ω) employing the new surrogate yields an inaccurate reconstruction, whereas using the full adaptively computed common-basis surrogate from Section 4.1 with |P(K)| ≈ 5565 and optimizing the whole vector ϑ results in a far superior outcome.…”

“…In addition, the boundary heat transfer coefficient between the heater elements is reconstructed. Such a setting was considered with simulated data in [19] and subsequently with experimental data in [20]. As a novelty of this work, the exterior shape of the imaged object is also reconstructed.…”