# Extraction of Independent Structural Images for Principal Component Thermography
## 3 Extraction of Independent Components  
### Proposed Approach  
From a practical point of view, it would be desirable to have a set of images which would represent individual structural features such as surface drawing, structure and defects. On the other hand, the principal component images extracted by (1) have an important limitation--namely, they must form an orthonormal basis. In a general case, two independent images--e.g., images of overlapping defects--are not orthogonal, and thus such patterns cannot be acquired by the extraction of principal components. At the same time, since PCs form an orthonormal basis, they can be used for construction of other images--including those which might not be orthogonal. This allows the possibility to use the extracted PCs for the construction of a set of (generally) non-orthogonal image patterns associated with the features of interest. The task is to find such a procedure which would allow for the construction of such independent patterns. The procedure proposed here allows for construction of independent image patterns (hereinafter, archetypes) constructed of the principal component images. The following assumptions are made in this approach:  
* Archetypes do exist. * Existing archetype images do not necessarily form an orthonormal basis. * Since principal components form a complete orthonormal basis, one can use this basis for constructing the archetypes. * High-order principal components represent noise-like images and provide little information in the original stack of thermographic images. Thus, meaningful archetypes can be constructed using only a few low-order PCs. Figure 5: Examples of principal components (PCs) extracted: (**a**) General view of the inspected object; (**b**) PC1; (**c**) PC2; (**d**) PC3; (**e**) PC4; (**f**) PC5; (**g**) PC6; (**h**) PC7; (**i**) PC8; (**j**) PC9. * More than one PC may contain image patterns which belong to same archetype (e.g., the wood structure may be seen in PC2-7 in Figure 5). Thus, extraction of a single archetype requires using available PCs in such a way that an individual independent feature is extracted while all (or most) others are suppressed. Let us assume that a stack of thermal images with sizes \(N_{x}\ \times\ N_{y}\) pixels has been acquired in an experiment, and each image was converted to 1D array of length \(N_{x}\cdot N_{y}\) in such a way that the data collected are represented by a 2D array \(A\). Assume that there is a set of 1D image arrays \(\{\overline{x}\}\), where \(\overline{x}_{i}\) represents an empty image with \(i\)-th pixel set to unit brightness. Thus, each pattern \(A_{t}\) can be expressed as  
\[A_{t}=k_{t1}\overline{x}_{1}+k_{t2}\overline{x}_{2}+\cdots+k_{tN_{x}N_{y}} \overline{x}_{1N_{x}N_{y}}=\sum_{j=1}^{N_{x}\cdot N_{y}}k_{tj}\overline{x}_{j}, \tag{2}\]  
where \(k_{tj}\) can be treated as components of the vector \(A_{t}\), or, in other words, as brightness of \(j\)-th pixel of \(t\)-th image. At the same time, set of archetypes \(\{P\}\) can be used to represent any thermal pattern as:  
\[A_{t}=\alpha_{t1}P_{1}+\alpha_{t2}P_{2}+\cdots+\alpha_{tN_{p}}P_{N_{p}}+ \widetilde{N}_{t}=\sum_{j=1}^{N_{p}}\alpha_{tj}P_{j}+\widetilde{N}_{t}, \tag{3}\]  
where \(N_{p}\) is the number of the archetype patterns present, and \(\widetilde{N}_{t}\) is the part of \(A_{t}\) which cannot be determined by archetype patterns (e.g., image noise patterns). One can think of \(\widetilde{N}_{t}\) as an image, which means it can be represented as \(\widetilde{N}_{t}=\sum_{m=1}^{N_{x}\cdot N_{y}}\widetilde{n}_{tm}\overline{x}_ {m}\).