# A Dataset of Pulsed Thermography for Automated Defect Depth Estimation
## 2 Related Work  
During a pulsed thermographic experiment, the specimen is subject to a controlled heat pulse followed by a cooling phase. This process is meticulously monitored using an infrared camera. The distinct thermal behaviors exhibited by defective regions at specific depths, as opposed to sound regions, facilitates the accurate determination of defect positions and depths. The temperature evolution on the specimen's surface is described by the following equation:  
\[\Delta T=\frac{Q}{e\sqrt{\pi t}} \tag{1}\]  
where \(\Delta T\) represents the temperature increase and \(Q\) represents the input energy. \(t\) is time, and \(e\) is the thermal effusivity, which can be calculated using the following formula:  
\[e=\sqrt{k\rho c_{p}} \tag{2}\]  
In this formula, \(k\), \(\rho\), and \(c_{p}\) represent the sample's thermal conductivity, the sample's density, and the heat capacity of the sample, respectively. In homogeneous material areas within the specimen, abnormalities are not anticipated in the PT data. This is attributed to the uniformity in thermal effusivity within these areas. However, the presence of defects, such as foreign objects or structural anomalies, causes variations in thermal effusivity relative to the surrounding intact areas. Such variations appear as the contrast in thermographic imagery. This highlights the efficacy of pulsed thermography as an NDE technique. Raw PT images often lack sufficient contrast to accurately identify the location of defects and, thus, require preprocessing for detailed analysis. Techniques such as PCT [19], Pulsed Phase Thermography (PPT) [20], and Thermographic Data Reconstruction (TSR) [21]are prevalent in PT image analysis. These techniques are instrumental to enhance the contrast of defects and to reduce the dataset dimension. With these methods, the crucial features of the raw dataset can be effectively extracted. PCT involves processing image sequences using principal component analysis along the time axis, typically utilizing the initial few Empirical Orthogonal Functions (EOFs) for in-depth analysis. Conversely, PPT applies a Fourier transform along the time axis of the PT sequence, generating amplitude and phase coefficients. In general, the phase coefficients are observed to provide enhanced contrast in defect visualization. TSR utilizes logarithmic polynomial fitting to model thermal profiles, as described by the following equation:  
\[ln(\Delta T)=a_{0}+a_{1}ln(t)+a_{2}[ln(t)]^{2}+a_{3}[ln(t)]^{3}+...+a_{n}[ln(t )]^{n} \tag{3}\]  
where \(\Delta T\) denotes the temperature increase and \(t\) is time. In infrared sequences, the frames index can be seen as a time representation. The coefficients of these polynomials are then used to reconstruct images, resulting in \(n+1\) frames containing TSR coefficients. For manual labeling, PCT is a suitable choice, as it generates images with sufficient contrast for defect identification. Nonetheless, deducing the exact defect depth by directly inspecting the images produced by these methods is challenging. To overcome this, various approaches have been investigated, for example a method based on mathematical modeling. In [22], a mathematical modeling method for defect depth predication was reported. This approach is based on the analysis of a one-dimensional solution of pulsed thermography, where the temperature decay is multiplied by the corresponding time to derive a function, denoted as f. It is subsequently demonstrated that f exhibits linearity with respect to the square of the defect depth. Additionally, another defect depth estimation method utilizing peak contrast information from thermographic data was proposed in [23]. This method aids in quantitatively determining the defects' depth within stainless steel. Ever since deep neural networks have emerged as state-of-the-art methods for data classification, efforts have been made to employ deep learning techniques to tackle the issue of depth estimation.