# Extraction of Independent Structural Images for Principal Component Thermography

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## Abstract

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## 1. Introduction

## 2. Methodology

#### Principal Component Thermography Approach

## 3. Extraction of Independent Components

#### 3.1. Proposed 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.
- 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.

- Collect a stack of raw thermographic images and convert this stack to a 2D array
- Apply SVD to extract principal components.
- Leave only meaningful PCs (those which determine most of variance of raw data).
- Choose point(s) where only one independent pattern is present (e.g., only the pattern describing the defective area). These points are selected manually as it is important to find the point where only one independent pattern is present and thus Expression (6) is satisfied.
- Create a large number of random linear combinations from the PCs extracted. For research presented in this article, a set of 18,000 random combinations was constructed.
- Sort the new stack of images in such a way that the brightness of pixels in point(s) chosen change in a harmonic way.
- Find all pixels which happen to have similar modulation. These pixels belong to the same archetype which is present in the point(s) chosen.

#### 3.2. Synthetic Example

## 4. Experimental Application of the Proposed Approach

#### 4.1. Experimental Setup and Calculation Deta

#### 4.2. Non-Destructive Analysis of Works of Art

#### 4.3. Inspection of Composite Materials

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**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).

**Figure 3.**Explanation of Principal Components (only three dimensions are shown). Red arrows indicate the directions of the first three principal components [24].

**Figure 4.**Explanation of Singular Value Decomposition (SVD) [24].

**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.

**Figure 6.**Explanation of the algorithm. The red and blue graphs schematically indicate how brightnesses of two particular pixels change through the stack of the sorted images.

**Figure 7.**Two 1D archetype functions example: (

**a**) the general form of the archetypes; (

**b**) examples of noisy superpositions of the archetypes (raw patterns); (

**c**) the first three PCs for the set of raw patterns, (

**d**) examples of random superpositions of PC1 and PC2.

**Figure 8.**A set of random patterns constructed from the first principal components: (

**a**) unsorted patterns; (

**b**) sorted patterns. The dashed lines indicate the positions of the two reference points (denoted as A and B).

**Figure 9.**Reconstruction of 1D archetype shapes: (

**a**) amplitude spectrum; (

**b**) reconstructed profiles (different colours are used to distinguish between two reconstructed archetypes).

**Figure 10.**Examples of reconstructed structural images (archetypes): (

**a**) points chosen for “defect”; (

**b**) points chosen for “structure”; (

**c**) points chosen for “drawing”; (

**d**) reconstructed “defect”; (

**e**) reconstructed “structure”; (

**f**) reconstructed “drawing”.

**Figure 11.**PCs extracted from a set of thermographic images collected from a composite material piece (side opposite to loading): (

**a**) PC1; (

**b**) PC2; (

**c**) PC3; (

**d**) PC4; (

**e**) PC5; (

**f**) PC6.

**Figure 12.**Examples of reconstructed structural images (archetypes) for a damaged composite plate: (

**a**) points chosen for “defect #1”; (

**b**) points chosen for “defect #2”; (

**c**) reconstructed “defect #1”; (

**d**) reconstructed “defect #2”.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Gavrilov, D.; Maev, R.G.
Extraction of Independent Structural Images for Principal Component Thermography. *Appl. Sci.* **2018**, *8*, 459.
https://doi.org/10.3390/app8030459

**AMA Style**

Gavrilov D, Maev RG.
Extraction of Independent Structural Images for Principal Component Thermography. *Applied Sciences*. 2018; 8(3):459.
https://doi.org/10.3390/app8030459

**Chicago/Turabian Style**

Gavrilov, Dmitry, and Roman Gr. Maev.
2018. "Extraction of Independent Structural Images for Principal Component Thermography" *Applied Sciences* 8, no. 3: 459.
https://doi.org/10.3390/app8030459