  Active thermography is a non-destructive testing technique to detect the
internal structure of a heat conductor, which is widely applied in industrial
engineering. In this paper, we consider the problem of identifying an unknown
cavity with Robin boundary condition inside a heat conductor from boundary
measurements. To set up the inverse problem mathematically, we first state the
corresponding forward problem and show its well-posedness in an anisotropic
Sobolev space by the integral equation method. Then, taking the
Neumann-to-Dirichlet map as mathematically idealized measured data for the
active thermography, we present a linear sampling method for reconstructing the
unknown Robin-type cavity and give its mathematical justification by using the
layer potential argument. In addition, we analyze the indicator function used
in this method and show its pointwise asymptotic behavior by investigating the
reflected solution of the fundamental solution. From our asymptotic analysis,
we can establish a pointwise reconstruction scheme for the boundary of the
cavity, and can also know the distance to the unknown cavity as we probe it
from its inside.
