Deterioration of stones is a complex problem and one of the main concern for people working in the field of conservation and restoration of cultural heritage. One important point in cultural heritage is to obtain information about the damage in a non-invasive way. By this paper, we propose a new non-invasive tool that permits evaluation of the thickness of (Formula presented.) (gypsum) grown (sulfation) on marble stones, using a mathematical model on data detected by pulsed infrared thermography.

Here we give mathematical support to the choice of the Helmholtz versus the Laplace equation for the formulation of an inverse problem in infrared thermography. Such a choice accounts for the values of physical parameters like the thermal conductivity and the Biot number. We check the effectiveness of the corresponding boundary value problems and compare the condition numbers. The Helmholtz choice is confirmed to be usually better, but there are cases in which Laplace (less expensive) works well.

A thin conducting plate has an inaccessible side in contact with aggressive external agents. On the other side, we are able to heat the plate and take temperature maps in laboratory conditions. Detecting and evaluating damages on the inaccessible side from thermal data requires the solution of a nonlinear inverse problem for the heat equation in presence of suitable boundary conditions for heat equation. We carry on this task using domain derivative and integral formulation of the corresponding boundary value problem. Under non restrictive hypothesis, we find explicit regularized schemes for Fourier reconstruction of damages.

A thin plate ? has an inaccessible side in contact with aggressive external agents. On the other side we are able to heat the plate and take temperature maps (thermal data) in laboratory conditions. Detecting and evaluating damages on the inaccessible side from thermal data requires the solution of a nonlinear inverse problem for the heat equation. To do this, it is extremely important to assign correct boundary conditions, in particular on the inaccessible boundary of ?. In several cases the boundary conditions must take account of heat exchange between ? and the environment. Here we discuss, from the quantitative point of view, the relation between the physical constants of the system (conductivity, width of the plate, ...) and the heat transfer through the boundary of ?.