## 3 Development of PT Signal Processing
### Principal Component Thermography (PCT)  
The Principal Component Thermography (PCT) technique was proposed by [PERSON] and is a representative signal processing technique that is being applied to inspection and component inspection in various fields of NDT&E [36; 37]. Principal Component Thermography (PCT) facilitates the reduction of undesirable data while preserving the major features of thermal image sequences. Principal Component Thermography (PCT) is applied by Principal Component Analysis (PCA) and utilizes the singular value decomposition (SVD) of measured temperature characteristics [36]. This technology reduces the dimension of temporal and spatial thermal data from three-dimensional to two-dimensional. In addition, this technique reconstructs the input thermal image from a matrix where each row consists of a raster-like arrangement of each single image pixel. As a result, each column represents the temporal evolution of a particular pixel. Then, using eigenvalues and eigenvectors as the essential products of SVD, the input data as a reconstructed matrix will be decomposed to its empirical orthogonal functions (EOFs).  
Empirical orthogonal function (EOF) analysis ([PERSON] and [PERSON] (1998)) provides a framework for constructing a set of orthogonal statistical models that furnish a complete description of the variability in a set of observations [38]. The analysis involves the calculation of a covariance matrix or, alternatively, the application of a singular value decomposition. The latter is computationally more efficient and is the approach employed in the present work. This analysis produces a remarkably compact description of the salient spatial and temporal signal variations relating to the contrasts associated with underlying structural flaws when applied to thermographic data. In many cases, a richly informative description of these contrasts is furnished in a single spatial mode and its complementary characteristic time vector, known as the principal component.  
Data acquired in the course of a normal active thermal inspection are customarily organized in the manner depicted in Figure 3, a three-dimensional array, which, for convenience, is defined here as \(T(i,j,k)\),where \(i=1,2,\ldots,N_{x},j=1,2,\ldots,N_{y}\), and \(k=1,2\ldots,N_{t}\). The pixel values in each image frame are drawn out into a vector to condense the original data \"cube\" into a matrix \(A\) with dimension \(M\times N\), where \(M=N_{x}\times N_{y}\) with dimension \(M\times N\), where \(M=N_{x}\times N_{y}\) and \(N=N_{t}\). Each \(M\) column vectors are then subjected to a standardization process to ensure uniformity on a pixel-by-pixel basis. [33; 39],  
\[A(n,m)=\frac{A(n,m)-\mu_{m}}{\sigma_{m}} \tag{10}\]  
\[\mu_{m}=\frac{1}{N}\sum_{n=1}^{N}A(n,m) \tag{11}\]  
\[\sigma_{m}^{2}=\frac{1}{N-1}\sum_{n=1}^{N}(A(n,m)-\mu_{m})^{2} \tag{12}\]  
This standardized matrix \(A\) is then reduced by singular value decomposition as follows:  
\[A=USV^{T} \tag{13}\]  
where \(U\) is the orthogonal matrix of dimension \(M\times N\) and comprises a set of empirical orthogonal functions (EOFs) representing spatial variation. \(S\) is a diagonal matrix of dimension \(M\times N\) and possesses the singular values of matrix \(A\) on the diagonals. \(V^{T}\) is the transpose matrix of the \(N\times N\) orthogonal matrix representing characteristic time.  
Figure 3: Principle of signal processing by Principal Component Thermography (PCT): (**a**) Thermographic data conversion from a 3D sequence to a 2D A matrix in order to apply singular value decomposition (SVD); and (**b**) rearrangement of 2D U matrix into a 3D matrix containing the empirical orthogonal functions (EOFs).