# Extraction of Independent Structural Images for Principal Component Thermography
## 2 Methodology  
### Principal Component Thermography Approach  
The Principal Component Thermography (PCT) technique was proposed by [PERSON] in 2002 [17] and has found application in the inspection of a wide variety of materials and components from industrial materials to the analysis of works of art. The theory and appropriate application procedures are well established and discussed in the literature (e.g., [18; 19; 20; 21; 22; 23]). Unlike most other thermographic techniques, PCT is not based on a physical model, but rather utilizes a statistical approach to processing of raw data. In general terms, the procedure works as follows. The sample is brought out of thermal equilibrium, the thermal imager collects a series of thermal snapshots (Figure 2a), each of which is represented by a matrix of pixels (e.g., \(N_{x}\ \times\ N_{y}\)). Each of these images can be treated as an array of \(N_{x}\cdot N_{y}\) values denoting brightnesses of each pixel (e.g., a \(320\ \times\ 256\) image would contain 81,920 such values). If these values are treated as coordinates  
Figure 1: Basic thermographic setup scheme. in a fictitious \(N_{x}\cdot N_{y}\)-dimensional space, each image can be treated as a single point in this space, where each of the coordinates is determined by brightness value of each of the pixels (Figure 2b). In such approach, \(N_{t}\) images collected in an experiment can be treated as a cloud of \(N_{t}\) points in \(N_{x}\cdot N_{y}\)-dimensional space (e.g., if someone collects 500 thermographic images with dimensions of \(320~{}\times~{}256\), then such a sequence can be treated as 500 points in a fictitious 81,920-dimensional space formed by the brightness of each image pixel). The \"cloud\" of such points is suitable for extraction of the so-called principal components (PCs). PCs represent a set of basis vectors constructed in such a way that the first principal component (PC1) describes most of the variance of the initial \"cloud\", while each subsequent component describes most of the variance in the direction orthogonal to previous PCs (Figure 3). In many practical cases, a few first PCs determine the most variance of the initial data, and all remaining PCs determine only very small variations such as uncorrelated noises. This also allows for lossy compression of data by only preserving a few first principal components. Extraction of the PCs can be performed with the aid of a well-known procedure known as Singular Value Decomposition (SVD):  
\[A=U\cdot S\cdot V^{T} \tag{1}\]  
In Expression (1), A represents a 2D matrix with its columns being thermal snapshots collected by imager unwrapped into 1D arrays. Matrix \(U\) is the matrix of principal components, and its columns form an orthonormal basis. The rows of the matrix \(S\cdot V^{T}\) can be treated as the time evolution profile for each of the patterns (Figure 4). Figure 3: Explanation of Principal Components (only three dimensions are shown). Red arrows indicate the directions of the first three principal components [24]. Figure 2: (a) Explanation of stack of thermal images; (b) Illustration of multidimensional representation of each thermal image in stack (only first three of all \(N_{x}\cdot N_{y}\) dimensions are shown). Let us have a look at a simple example of PCT. Figure 5 contains a few principal component images collected in a flash heat experiment on a piece of art. The sample represents an unattributed tempera/wood icon, 28 cm by 20.5 cm. The icon depicts a common \"Madonna with the Child\" scene. A few degradations are visible on the surface of the icon, and delamination defects are assumed to be possible. Figure 4: Explanation of Singular Value Decomposition (SVD) [24]. Figure 5: _Cont._  
Upon observation, the PC images higher than the 7 th one demonstrate no visible image except for noise-like patterns. Higher PCs contain mostly white noise images (these images are not shown).