# Quaternion Processing Techniques for Color Synthesized NDT Thermography
## 2 Background on Thermographic NDT Processing Techniques  
Among the processing algorithms most widely used in thermographic applications, the following ones may be mentioned. Thermographic Signal Reconstruction (TSR)  
Thermographic signal reconstruction is a processing technique widely used in IRT NDT for enhancing detection of defects and filtering noise [22,23]. The application procedure consists of several steps including the logarithmic transformation of the captured data, the interpolation at early stages and the polynomial fitting, among others. The polynomial fitting stage consists of fitting the temperature-time history of each pixel to an n-degree polynomial through the expression (1), where \(\Delta T\) is the temperature variation, \(t\) is the time variable and \(a_{i}\) are the fitting coefficients. The resulting polynomial is differentiated to produce an increase in the signal-to-noise (SNR) compared to the original thermographic sequence. \[In(\Delta T)=a_{n}[ln(t)]^{n}+a_{n-1}[ln(t)]^{n-1}+\dots+a_{1}ln(t)+a_{0} \tag{1}\]  
Spectral Analysis  
This processing algorithm consists of applying the Fourier Transform (FT) to the raw thermographic data sequence [24]. The IRT sequence is transformed from time to frequency domain by the relation (2), where \(F_{n}\) is the \(n\)th complex Fourier coefficient with \(Re_{n}\) and \(Im_{n}\) as real and imaginary parts respectively, \(T\) is the temperature value of each pixel and \(N\) is the total number of elements in the time sequence. \[F_{n}=\sum_{k=1}^{N-1}T(k)e^{\frac{(2\pi\mathrm{i}nk)}{N}}=Re_{n}+Im_{n} \tag{2}\]Principal Component Analysis (PCA)  
Principal component analysis is a common technique used to synthesize data of high dimensionality. It is a classical multivariate analysis technique useful for data compression and detection of linear relationships based on the second order statistics of the initial data [25]. PCA is performed with thermographic data by the singular value decomposition of the temperature-time sequence of each pixel according to the relation:  
\[A_{p}=U_{S}^{T}A \tag{3}\]  
where \(U\) is the eigenvector matrix, \(A\) is the initial matrix and \(A_{p}\) is the final result taking into account the first \(s\) principal components whose quantity could be selected based on the desired amount of the variance proportion retained in the \(s\) eigenvalues. Partial Least Squares Regression (PLSR)  
Partial least squares regression is a processing technique based on the decomposition of the thermographic data sequence into a set of latent variables. The bilinear decomposition of the temperature-time sequence matrix and the observation time matrix produces a new set of thermal images and observation time vector of latent variables that consider only the most important variations [26]. PLS regression method is applied to IRT data by the expression:  
\[X=TP^{T}+E \tag{4}\] \[Y=TQ^{T}+F \tag{5}\]  
where \(X\) is the thermal sequence, \(Y\) is the time series, \(T\) is the scores matrix, \(P\) and \(Q\) are the loadings matrices and \(E\) and \(F\) are the residuals matrices. Higher Order Statistics (HOS)  
The calculation of the higher order statistical moments of the surface temperature distribution in thermographic NDT is a processing approach that provides a mapping of the surface temperature evolution through a unique image built with these statistical values [27]. The most efficient moments for IRT NDT applications were demonstrated to be the standardized central moments calculated by:  
\[M_{n}=\frac{E[(X-E[X])^{n}]}{\sigma^{n}} \tag{6}\]  
where \(E[X]\) is the mean value of the distribution, \(\sigma\) is the standard deviation and \(n\) is the order of the moment. In this study, TSR was not applied but only the polynomial fitting stage was performed for data processing. The other stages of TSR were not applied because of the lower effectiveness when using step heating, the particular thermal stimulation used in this study.