# Extension of the Thermographic Signal Reconstruction Technique for an Automated Segmentation and Depth Estimation of Subsurface Defects

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## Abstract

**:**

## 1. Introduction

## 2. Thermographic Evaluation Method

#### 2.1. Signal Acquisition

#### 2.2. Heat Diffusion Equation

#### 2.3. Numerical Solution of the Heat Equation

^{®}can be used to simulate several temperature–time diagrams (Figure 2a), which were obtained by using the geometry depicted in Figure 2b, assuming adiabatic boundary conditions; i.e., the sample is thermally isolated from its environment. The temperature was measured at the surface, above the center of an air pocket, marked in Figure 2b. The geometry is made up of two different materials, material 1 the composite material, and material 2, an air pocket, which simulates a delamination. The physical parameters used are shown in Table 1. Here material 1 has an anisotropic diffusion character, which is typical for FRPs. The in-plane thermal conductivity ${k}_{xy}$ was three times larger than the through-plane thermal conductivity ${k}_{z}$. Equation (2) shows that the thermal conductivity impacts the thermal diffusivity linearly.

#### 2.4. Thermograpic Signal Reconstruction (TSR) Algorithm

## 3. Modification of Thermograpic Signal Reconstruction Method

#### 3.1. Interpreting the Characteristic Times

#### 3.2. Implementation of the Modification

- Fit a polynomial of the form of Equation (4) to the temperature–time sequence, but use $t+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${f}_{s}$}\right.$ for the x-axis for improved stability of the polynomial.
- Add the value $0.48$ to the polynomial coefficient ${\beta}_{1}$.
- Calculate the first derivative of this polynomial.
- Evaluate the real valued roots of this polynomial and discard roots caused by oscillation (i.e., real valued roots outside the measured time).
- If three or more roots are found this indicates a defect, and location of the first root roughly indicates the defect depth according to Equation (5).

## 4. Experimental Study

#### 4.1. Measurement and Computational Complexity

#### 4.2. Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FRP | Fiber reinforced polymer |

3D-XCT | 3D-Xray computed tomography |

UT | Ultrasonic testing |

AT | Active Thermography |

IR | Infrared |

PT | Pulse thermography |

PPT | Pulse phase thermography |

TSR | Thermographic signal reconstruction |

APST | Absolute peak slope time method |

DDT | Dynamic thermal tomography |

VWC | Virtual wave concept |

NDE | Non-destructive evaluation |

FBH | Flat bottomed hole |

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**Figure 1.**Pulsed thermography experiment: (

**a**) measurement setup in reflection and the (

**b**) data format of a thermographic measurement.

**Figure 2.**Numerical solution of the heat equation: (

**a**) simulated temperature–time sequences and (

**b**) the geometry used for the numerical simulations.

**Figure 3.**Normalized temperature–time sequences using simulation data multiplied by $\sqrt{t}$ (see text).

**Figure 4.**Interpreting the characteristic times: (

**a**) second logarithmic derivative and (

**b**) the first logarithmic derivative of simulation data.

**Figure 5.**3D view of the test specimen: (

**a**) front and (

**b**) rear side. The voxel size of the measurement was 80 μm; for the area indicated by the black rectangle a voxel size of 30 μm was available.

**Figure 6.**Comparison of active thermography with 3D-computed tomography: (

**a**) results of the proposed algorithm and (

**b**) the density map obtained by 3D-XCT at a depth of 1 mm from the front side.

**Figure 9.**Polynomial derivatives of the temperature–time sequences from Figure 7a: (

**a**) second logarithmic derivative of a polynomial of degree n = 6 (TSR method) and (

**b**) the first logarithmic derivative of a polynomial of degree n = 9 (proposed method).

Geometry Units in mm | Material 1, Composite | Material 2, Air | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

D | L | d | w | ${k}_{xy}\left[\frac{\mathrm{W}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}}\right]$ | ${k}_{z}\left[\frac{\mathrm{W}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}}\right]$ | $\rho \left[\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}}\right]$ | $c\left[\frac{\mathrm{J}}{\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}}\right]$ | ${k}_{xyz}\left[\frac{\mathrm{W}}{\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}}\right]$ | $\rho \left[\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}}\right]$ | $c\left[\frac{\mathrm{J}}{\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}}\right]$ |

4 | 3 | 1 | 0.2 | 1.92 | 0.64 | 1500 | 1200 | 0.0262 | 1.2 | 1 |

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**MDPI and ACS Style**

Schager, A.; Zauner, G.; Mayr, G.; Burgholzer, P.
Extension of the Thermographic Signal Reconstruction Technique for an Automated Segmentation and Depth Estimation of Subsurface Defects. *J. Imaging* **2020**, *6*, 96.
https://doi.org/10.3390/jimaging6090096

**AMA Style**

Schager A, Zauner G, Mayr G, Burgholzer P.
Extension of the Thermographic Signal Reconstruction Technique for an Automated Segmentation and Depth Estimation of Subsurface Defects. *Journal of Imaging*. 2020; 6(9):96.
https://doi.org/10.3390/jimaging6090096

**Chicago/Turabian Style**

Schager, Alexander, Gerald Zauner, Günther Mayr, and Peter Burgholzer.
2020. "Extension of the Thermographic Signal Reconstruction Technique for an Automated Segmentation and Depth Estimation of Subsurface Defects" *Journal of Imaging* 6, no. 9: 96.
https://doi.org/10.3390/jimaging6090096