## 2 Materials and Methods
### Measurement and Processing  
In LIT [27], the object is submitted to a frequency-modulated, sinusoidal thermal excitation. In the stationary regime, the thermal response of the object to this excitation is also sinusoidal, with a phase and amplitude that depends on the characteristics of the input wave. Higher modulation frequencies confine the thermal response to the surface of the object, while lower frequencies have a deeper depth range [28]. The minimum measurement \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Model** & **Spectral Range (\(\upmu\)m)** & **NETD (mK)** \\ \hline A-MWIR\_O & FLIR X6540 sc & 2.5–5.0 & \textless{}25 \\ B-MWIR\_F & FLIR X6540 sc & 4.1–5.0 & \textless{}25 \\ C-LWIR & FLIR A655 sc & 7.5–14.0 & \textless{}30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the used camera setups and their most important characteristics. Figure 2: (**a**) Schematic of the measurement setup. Either a short pulse (PPT) or a modulated sinusoidal wave (LIT) is sent to the light controller. The distance between the painting and the camera depends on the size of the painting. (**b**) RGB image showing the measurement setup during experiments, on the Saint-George icon. duration is one full period, but ideally the measurement lasts multiple modulation cycles, until the object reaches a stationary regime [27]. The paintings were measured at a modulation frequency of 0.2 Hz and an amplitude of approximately 1.4 kW lamp power per lamp. Decreasing the frequency further did not seem to reveal additional information, and eventually led to blurry images because of increased lateral heat transfer. The paintings were measured for only one full heating period, in order to limit the heating up of the painting as much as possible. The maximum temperature fluctuation during the measurement was 1.8 \({}^{\circ}\)C. The signal that was used to control the lamps is shown in Figure 2, while Figure 3 shows the pixel intensity fluctuation during the measurement. The intensity plot clearly indicates that the thermal response of the object is not entirely sinusoidal. There is also a slight delay between the lamps receiving the signal and the object heating up. Phase and amplitude images can be derived from just four thermograms, located equidistantly on the sinusoidal modulation cycle [27; 28]. However, because of the very limited amount of data points, this method produces noisy images. An alternative processing method was used [27; 29], where the entire measured image sequence was multiplied by two reference functions of known phases, differing by 90\({}^{\circ}\), and the products were then added, in order to produce the in-phase and out-of-phase components for each pixel:  
\[0^{\circ}=\sum_{t}\sin(2\pi\omega t+\frac{3\pi}{2})*T(t), \tag{1}\]  
\[90^{\circ}=\sum_{t}\cos(2\pi\omega t+\frac{3\pi}{2})*T(t). \tag{2}\]  
where \(\omega\) is the modulation frequency and _T(t)_ is the measured thermal response at a specific time. Both reference functions are phase-shifted by \(\frac{3\pi}{4}\), so the in-phase component has the  
Figure 3: Pixel intensity evolution during a measurement. For the LIT measurement, the entire measurement cycle is shown. For the PT measurement, the cool-down process is shown. same phase-shift as the sinusoidal wave that is used for thermal excitation. The amplitude and phase components for each pixel can then be calculated as follows:  
\[A=\sqrt{(0^{\circ})^{2}+(90^{\circ})^{2}}, \tag{3}\]  
\[\phi=\tan^{-1}(\frac{90^{\circ}}{0^{\circ}}). \tag{4}\]  
In PPT [28, 30, 31, 32], the object is thermally excited by a rectangular pulse, varying from several milliseconds to a few seconds, depending on the thermal properties of the object under inspection. When an object undergoes pulse excitation, thermal waves of various phases and amplitudes simultaneously excite the object in a transient regime. This is in contrast with LIT, where the object is excited by a single frequency and amplitude in a stationary regime.