# Dynamic Line Scan Thermography Parameter Design via Gaussian Process Emulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Generation

#### 2.2. Gaussian Processes

#### 2.3. Active Learning

- A small selection of data points is sampled uniformly from the dataset. Alternatively, those points could lay an n-dimensional grid, or be a Latin hypercube sampled or chosen from a Sobol sequence. In [12], a comparison between the different sampling methods is made. In this work, we restrict ourselves to uniform sampling, as it is the most simple method. For a more comprehensive study on this topic, we refer the reader to [16,17].
- The model (in our case, the Gaussian process) is trained on this initial small dataset.
- The point from the input space with the highest uncertainty (variance) in the GP’s posterior distribution is chosen and added to the dataset of the GP, which is then retrained. This method is called uncertainty sampling (US). Alternatively, the point which reduces the total variance of the posterior could be chosen. This method is called integrated variance reduction (IVR). We implement US because it is cheaper to compute [18].
- Step 3 is repeated until a certain criterion is met. When limited by a computational budget, this could be a fixed number of iterations. Another criterion is convergence in the posterior distribution, which means that adding new data points no longer has a significant result on the predictions of the GP.

Algorithm 1 Active learning with uncertainty sampling. | |

1: trainset ← dataset(n) | ▹ start with a training set of n random points |

2: testset ← dataset - trainset | ▹ put remaining points in test set |

3: GP.train(trainset) | ▹ train Gaussian process on trainset |

4: oldPosterior ← GP.predict(testset + trainset) | ▹ get the GP posterior |

5: while nrOfIterations < maxNrOfIterations | ▹ check computational budget |

6: or diffPosterior > minDiffPosterior do | ▹ check for convergence |

7: ActiveLearningIteration(GP) | ▹ perform one iteration |

8: newPosterior ← GP.predict(testset + trainset) | ▹ get the GP posterior |

9: diffPosterior ← newPosterior - oldPosterior | ▹ calculate the change |

10: oldPosterior ← newPosterior | ▹ store for next iteration |

11: end while | |

12: procedure ActiveLearningIteration(GP) | ▹ Active Learning iteration |

13: trainset ← dataset(US(testset)) | ▹ update training set |

14: testset ← dataset - trainset | ▹ update test set |

15: GP.train(trainset) | ▹ retrain GP |

16: end procedure | |

17: procedure US(testset) | ▹ Uncertainty Sampling |

18: for all ${\mathbf{x}}_{\mathrm{test}}\in \mathrm{testset}$ do | ▹ evaluate every test point |

19: if $\mathrm{var}\left({\mathbf{x}}_{\mathrm{test}}\right)>\mathrm{var}\left({\mathbf{x}}_{\mathrm{mostVar}}\right)$ then | |

20: ${\mathbf{x}}_{\mathrm{mostVar}}\leftarrow {\mathbf{x}}_{\mathrm{test}}$ | ▹ this point becomes new candidate |

21: end if | |

22: end for | |

23: return ${\mathbf{x}}_{\mathrm{mostVar}}$ | ▹ return test point with most variance |

24: end procedure |

## 3. Results

#### 3.1. Model Performance

- Distance between the heat source and the camera, range 50 to 600 mm;
- Heating power, range 50 to 800 W;
- Start depth of the defect, range 2 to 9.8 mm;
- Diameter of the defect, range 12 to 24 mm.

- The root mean square error between the posterior mean in each test point and the actual values from the simulations. This number serves as a measurement for the deviation of the model from the underlying truth.
- The average posterior standard deviation for all remaining test points. This is a measurement for how much uncertainty there still is in the system. The point with the highest variance, i.e., the highest uncertainty, becomes the point that is moved from the test set to the training set of the Gaussian process in the next iteration.

#### 3.2. Parameter Design

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Figure 4.**Visualization of the temperature difference for six different defect diameters. (

**a**) 12 mm, (

**b**) 14 mm, (

**c**) 16 mm, (

**d**) 18 mm, (

**e**) 20 mm and (

**f**) 22 mm. Red indicates temperature differences that might result in damaging the sample under inspection. These plots serve as a warning when designing a setup.

**Figure 5.**Visualization of the temperature difference for six different defect diameters. (

**a**) 12 mm, (

**b**) 14 mm, (

**c**) 16 mm, (

**d**) 18 mm, (

**e**) 20 mm and (

**f**) 22 mm. Red are temperature differences below 5 °C, yellow above 25 °C and green in between. Only the green regions are of practical value in real-world applications.

Parameter | Example 1 | Example 2 | Example 3 | Example 4 |
---|---|---|---|---|

Ambient Temperature [°C] | 20 | 20 | 20 | 20 |

Velocity [mm/s] | 10 | 10 | 10 | 10 |

Camera Height [mm] | 450 | 450 | 450 | 450 |

Diameter Hole [mm] | 14 | 12–24 | 22 | 12 |

Startdepth Hole [mm] | 6 | 2–9.8 | 2–9.8 | 2–9.8 |

Heating Power [W] | 50 | 500 | 200–400 | 600–800 |

Distance cam.–heat [mm] | 335–420 | 100 | 500–600 | 50–200 |

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Maldague, X. Nondestructive Evaluation of Materials by Infrared Thermography–Xavier P.V. Maldague; Springer: Berlin/Heidelberg, Germany, 1993. [Google Scholar]
- Verspeek, S.; Gladines, J.; Ribbens, B.; Maldague, X.; Steenackers, G. Dynamic Line Scan Thermography Optimisation Using Response Surfaces Implemented on PVC Flat Bottom Hole Plates. Appl. Sci.
**2021**, 11, 1538. [Google Scholar] [CrossRef] - Ibarra-Castanedo, C.; Servais, P.; Ziadi, A.; Klein, M.; Maldague, X. RITA–Robotized Inspection by Thermography and Advanced Processing for the Inspection of Aeronautical Components. In Proceedings of the 2014 Quantitative InfraRed Thermography, Bordeaux, France, 7–11 July 2014. [Google Scholar] [CrossRef]
- Peeters, J.; Verspeek, S.; Sels, S.; Bogaerts, B.; Steenackers, G. Optimised dynamic line scanning thermography for aircraft structures. Quant. InfraRed Thermogr. J.
**2019**, 16, 260–275. [Google Scholar] [CrossRef] - Peeters, J.; Ibarra-Castanedo, C.; Khodayar, F.; Mokhtari, Y.; Sfarra, S.; Zhang, H.; Maldague, X.; Dirckx, J.J.; Steenackers, G. Optimised dynamic line scan thermographic detection of CFRP inserts using FE updating and POD analysis. NDT E Int.
**2018**, 93, 141–149. [Google Scholar] [CrossRef] - Ringermacher, H.I.; Mayton, D.L.; Howard, D.R. Towards a flat-bottom hole standard for thermal imaging. Rev. Prog. Quant. Nondestruct. Eval.
**1998**, 17, 425–429. [Google Scholar] - Frendberg Beemer, M.; Shepard, S. Aspect Ratio Considerations for Flat Bottom Hole Defects in Active Thermography. Quant. InfraRed Thermogr. Conf.
**2016**, 15, 1–16. [Google Scholar] [CrossRef] - Paleyes, A.; Pullin, M.; Mahsereci, M.; McCollum, C.; Lawrence, N.D.; Gonzalez, J. Emulation of physical processes with emukit. arXiv
**2021**, arXiv:2110.13293. [Google Scholar] - Rasmussen, C.E.; Williams, C.K.I. Gaussian Processes for Machine Learning; The MIT Press: Cambridge, MA, USA, 2006; Volume 38. [Google Scholar]
- Ho, T.K. Random decision forests. In Proceedings of the 3rd International Conference on Document Analysis and Recognition, Montreal, QC, Canada, 14–16 August 1995; Volume 1, pp. 278–282. [Google Scholar] [CrossRef]
- Lim, T.; Wang, K. Comparison of machine learning algorithms for emulation of a gridded hydrological model given spatially explicit inputs. Comput. Geosci.
**2022**, 159, 105025. [Google Scholar] [CrossRef] - Noè, U.; Lazarus, A.; Gao, H.; Davies, V.; MacDonald, B.; Mangion, K.; Berry, C.; Luo, X.; Husmeier, D. Gaussian process emulation to accelerate parameter estimation in a mechanical model of the left ventricle: A critical step towards clinical end-user relevance. J. R. Soc. Interface
**2019**, 16, 20190114. [Google Scholar] [CrossRef] [PubMed] - Pan, T.; Cai, Y.; Chen, S. Development of an Engine Calibration Model Using Gaussian Process Regression. Int. J. Automot. Technol.
**2021**, 22, 327–334. [Google Scholar] [CrossRef] - Liu, D.C.; Nocedal, J. On the limited memory BFGS method for large scale optimization. Math. Program.
**1989**, 45, 503–528. [Google Scholar] [CrossRef][Green Version] - Kristjanson Duvenaud, D.; College, P. Automatic Model Construction with Gaussian Processes Declaration. Available online: https://www.cs.toronto.edu/~duvenaud/thesis.pdf (accessed on 10 February 2022).
- Fang, K.T.; Li, R.; Sudjianto, A. Design and Modeling for Computer Experiments; Chapman and Hall/CRC: Boca Raton, FL, USA, 2005. [Google Scholar]
- Jones, D.R.; Schonlau, M.; Welch, W.J. Efficient Global Optimization of Expensive Black-Box Functions. J. Glob. Optim.
**1998**, 13, 455–492. [Google Scholar] [CrossRef] - Sacks, J.; Welch, W.J.; Mitchell, T.J.; Wynn, H.P. Design and Analysis of Computer Experiments. Stat. Sci.
**1989**, 4, 409–423. [Google Scholar] [CrossRef] - Buisson-Fenet, M.; Solowjow, F.; Trimpe, S. Actively learning gaussian process dynamics. In Proceedings of the 2nd Conference on Learning for Dynamics and Control, Online, 11–12 June 2020; pp. 5–15. [Google Scholar]
- Pasolli, E.; Melgani, F. Gaussian process regression within an active learning scheme. In Proceedings of the 2011 IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada, 24–29 July 2011; pp. 3574–3577. [Google Scholar] [CrossRef]
- Gessner, A.; Gonzalez, J.; Mahsereci, M. Active multi-information source Bayesian quadrature. In Proceedings of the 35th Uncertainty in Artificial Intelligence Conference, Tel Aviv, Israel, 22–25 July 2019; pp. 712–721. [Google Scholar]
- Oladyshkin, S.; Mohammadi, F.; Kroeker, I.; Nowak, W. Bayesian3 Active Learning for the Gaussian Process Emulator Using Information Theory. Entropy
**2020**, 22, 890. [Google Scholar] [CrossRef] [PubMed] - Swiler, L.P.; Gulian, M.; Frankel, A.L.; Safta, C.; Jakeman, J.D. A survey of constrained Gaussian process regression: Approaches and implementation challenges. J. Mach. Learn. Model. Comput.
**2020**, 1, 119–156. [Google Scholar] [CrossRef]

**Figure 1.**Visualization of the finite element simulation consisting of a flat bottom hole plate (blue) and a line heater (yellow). The line heater moves above the sample in a linear motion. For a more detailed figure and explanation, we refer the reader to [2].

**Figure 2.**Response surface as generated in [2]. The surface is generated from 1000 finite element simulations, using eight input parameters. The simplified response surface has all input parameters fixed, except for the heat load and the source velocity. The fixed parameters are: ${d}_{heat-cam}$ = 425 mm, ${d}_{start}$ = 5.8 mm, ${D}_{hole}$ = 9 mm, ${d}_{height}$ = 430 mm, ${T}_{ambient}$ = 48 ${}^{\xb0}$C. Using this response surface, one can find the best temperature difference as a valley or top.

**Figure 3.**Graphical visualization of the learning process of the Gaussian process for 5 runs of 500 iterations. (

**a**) represents the root mean square error (RMSE) of the learned surrogate compared to the response surface created in [2]. (

**b**) shows the average standard deviation of the Gaussian process posterior prediction.

Run | ${\mathit{l}}_{1}$ [mm] | ${\mathit{l}}_{2}$ [W] | ${\mathit{l}}_{3}$ [mm] | ${\mathit{l}}_{4}$ [mm] | ${\mathit{\sigma}}_{\mathit{f}}^{2}\phantom{\rule{0.166667em}{0ex}}\left[\mathsf{\xb0}\mathbf{C}\right]$ |
---|---|---|---|---|---|

1 | 142.81 | 246.17 | 2.28 | 2.35 | 11.97 |

2 | 150.00 | 255.99 | 2.30 | 2.41 | 11.75 |

3 | 182.72 | 216.52 | 2.23 | 1.98 | 10.87 |

4 | 177.48 | 223.02 | 2.26 | 1.96 | 10.97 |

5 | 153.98 | 238.46 | 2.60 | 2.37 | 12.84 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Verspeek, S.; De Boi, I.; Maldague, X.; Penne, R.; Steenackers, G. Dynamic Line Scan Thermography Parameter Design via Gaussian Process Emulation. *Algorithms* **2022**, *15*, 102.
https://doi.org/10.3390/a15040102

**AMA Style**

Verspeek S, De Boi I, Maldague X, Penne R, Steenackers G. Dynamic Line Scan Thermography Parameter Design via Gaussian Process Emulation. *Algorithms*. 2022; 15(4):102.
https://doi.org/10.3390/a15040102

**Chicago/Turabian Style**

Verspeek, Simon, Ivan De Boi, Xavier Maldague, Rudi Penne, and Gunther Steenackers. 2022. "Dynamic Line Scan Thermography Parameter Design via Gaussian Process Emulation" *Algorithms* 15, no. 4: 102.
https://doi.org/10.3390/a15040102