## 3 Development of PT Signal Processing
### Thermographic Signal Reconstruction (TSR)  
[PERSON] et al. [40] introduced Thermographic Signal Reconstruction (TSR), which provides improved detection of low contrast and low aspect ratio features and the ability to detect and measure the characteristics of extended interfaces [41].  
A general TSR is used to qualitatively detect defects by selecting the best-derived images related to all depth ranges or to evaluate depth at a characteristic time quantitatively (Figure 4) [42]. Fitting and derivation of the thermogram depends on the time domain being considered. It is necessary to define the time window for characterization, considering only the part of the thermal image affected by the physical phenomenon. The defect image thus obtained has a better signal-to-noise ratio and sharpness than the raw thermal image. It may also be necessary to optimize the polynomial degree depending on the case.  
TSR is a method of estimating the through-plane thermal diffusivity \(\alpha_{z}\). This estimate of \(\alpha_{z}\) is referred to as \(\alpha_{cz}\), the effective through-plane thermal diffusivity. The TSR algorithm can be described in the following way. First, the ambient temperature \(T_{a}\) is subtracted from the temperature-time sequence, yielding \(\Delta T\). Then, least-squares polynomial regression is carried out in log-log scales. This polynomial is referred to as the TSR polynomial. Suggestions based on analytical studies [39] have been made, to determine the degree \(n\) of the TSR polynomial that should ideally be used. The optimal degree \(n\) depends on the sample under consideration and measurement acquisition parameters, but it is in the range of \(n=7\) to \(n=9\)[43]. The logarithmic transform of Equation (3) can be expressed by Equation (14) [43,44].  
\[ln(T)=ln\bigg{(}\frac{Q}{e}\bigg{)}-\frac{1}{2}ln(\pi t)=a_{0}+a_{1}ln(t)\ +a_{2}[ln(t)]^{2}+\ldots+a_{n}[ln(t)]^{n} \tag{14}\]  
where \(T\) is the increment in temperature as a function of time and \(a_{0}\), \(a_{1}\), \(\ldots\), \(a_{n}\) are the polynomial coefficients. In NDT, fifth- or sixth-order polynomial was found effective in temporal noise reduction, and the optimum degree of the polynomial in use was found up to ninth-order. The first log-time derivative \(dln(T)/dln(t)\) and the second derivative \(d^{2}ln(T)/dln^{2}(t)\) of the polynomial enhanced the defect detectability and SNR.  
The maximum of the second derivative can be calculated in two ways: either by evaluating the second analytic derivative or by deriving the TSR polynomial three times and finding and evaluating all real-valued roots. The second derivative is, in essence, the curvature of the temperature-time sequences, and one would also expect to find the depth  
Figure 4: Principle of signal processing by Thermography Signal Reconstruction (TSR).  
of the air pocket by the TSR method [45]. In practice, this is not the case because polynomials can only approximate the temperature-time sequences. The sequence obtained by plotting the second derivative of the TSR polynomial often shows multiple peaks, wherein it is often not clear which were caused by an air pocket and which were due to oscillations (Runge's phenomenon) of the polynomial [46].