# Hot Shoes in the Room: Authentication of Thermal Imaging for Quantitative Forensic Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Scenario

#### 2.2. Experimental Shoe Types

#### 2.3. Temperature and Image Recording

^{®}Core™ i7-6700HQ processor and 20 GB RAM. The ICT Combined Instrument Software v1.0.5.5 (ICT International PTY Ltd., Armidale, Australia) was used to set up the logging system to sample live and log data at 15 s intervals over 30 min. Pilot experiments showed that longer sampling times provided no additional information as the cooling process has almost completely stopped after 30 min.

#### 2.4. Cooling Function and Time Predictive Function from Camera Values

_{ij}measured at round i; β

_{0}and β

_{1}are the coefficients estimated for the fixed effects of the linear model; α

_{i}represents the random effect for the ith measurement; and ε

_{ij}is an error term.

_{0}and β

_{1}are the coefficients estimated for Equation (1), and ρ represents the pixel intensity value for any pixel in the image.

_{0}and β

_{1}were estimated as part of the regression analyses performed to obtain the functions defined by Equations (1) and (2) (Table 1 and Table 2, respectively), whilst the uncertainty introduced by the camera’s response was directly modelled from empirical data following methods in [8]: camera responses were first sorted in bins of five pixel intensity levels each starting from the minimum pixel intensity value recorded for each shoe ($\rho \approx 45)$ up to the maximum pixel intensity value recorded $\left(\rho \approx 100\right)$, and a beta distribution was subsequently fitted to each bin in order to model the distribution of the pixel intensity levels at each bin. The two shape parameters defining each beta distribution [27] were estimated by maximum likelihood using the package fitdistrplus v. 1.0-7 [28] available for the statistical package R. Confidence bounds for the shape parameters of each beta distribution were obtained using nonparametric bootstrapping with 10,000 iterations using the command bootdist available in the fitdistrplus package.

## 3. Results

## 4. Discussion

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Spectral emission from ideal Planckian radiators heated at different temperatures (solid black lines), and spectral bandwidths commonly used for forensic imaging (colour rectangles). Square markers indicate the maximum amplitude (λ

_{max}) for each spectrum: daylight (correlated colour temperature of 6500 K), halogen–tungsten filament (4000 K), 75 W house bulb (2800 K), wax candle (1900 K), hot plate for technical purposes heated at about 370 °C (644 K), human body at 37 °C (310 K), and a black body heated at 20 °C (293 K) and 0 °C (273 K). Note how some sources emit radiation across several spectral bands: ultraviolet (UV), visible (VIS), near infrared (NIR), and intermediate and far infrared. Spectral bands cover the sensitivity range of most common imaging devices used for forensic and technical applications in the ultraviolet, visible, and infrared regions of the electromagnetic spectrum [2,10,11,19,21].

**Figure 2.**(

**a**) Temperature sampling points on three adult male shoes and (

**b**) representation of shoe cooling in pilot studies. See text and Figure 3 for quantitative analysis. Points A–E represent temperature sampling points with Therm-Micro probes on points A (External Toe); B (External Heel); C (Internal Toe); D (Internal Heel); E (Internal Point). Circular boxes represent sampled point on the surface of shoe, square boxes represent points internally sampled.

**Figure 3.**Experimental data and modelling results for a method allowing the prediction of cooling time of three different shoes—Cumulus (first column), leather (middle column), and Fuji shoe (third column)—from pixel values of a thermal imaging device in typical room conditions after wearing three different types of shoes (refer to Materials and Methods section for details). Panels (

**a**–

**c**) show the relationship between camera response, expressed as pixel intensity values (ρ), and shoe temperature reading of the FLIR i50 30 min after shoe removal. Points represent experimental data, the solid black line represents the predicted function, and the shaded area represents the 95% confidence intervals for the function. The curves at the bottom of each panel represent the shape of the probability density distribution (pdf) of the camera responses at 11 different intensity levels modelled assuming a beta distribution (See Supplementary Materials for coefficients defining each distribution); Panels (

**d**–

**f**) show temperature as a function of time for the tested shoe types modelled assuming a bi-exponential function. Points represent the experimental data from the Temperature Sensor Meter, the solid black line represents the nonlinear regression function, and the shaded area represents the 95% confidence intervals. Panels (

**g**–

**i**) represent the cooling function for each shoe type: (

**g**) Cumulus, (

**h**) leather, and (

**i**) Fuji shoe. Error bars on the x-axis represent 95% confidence intervals for camera responses at different pixel intensity values, whilst error bars on the y-axis represent the 95% confidence intervals of the predicted cooling time. In panels (

**g**–

**h**), the green shaded area represents the temperature range for which time can be reliably predicted from camera responses. Blue and red areas represent areas of high uncertainty where data should be interpreted with caution.

**Table 1.**Coefficients and associated 95% confidence intervals for the linear function mapping from pixel intensity values into temperature (Equation (1)). Values in parentheses indicate the lower and upper limit of the 95% confidence intervals for each coefficient.

Shoe Type | Coefficient | Value (95% CI) | p-Value |
---|---|---|---|

Cumulus | β_{0} | 15.3 (15.1, 15.5) | <0.001 |

β_{1} | 0.125 (0.124, 0.126) | <0.001 | |

Leather | β_{0} | 15.5 (15.3, 15.7) | <0.001 |

β_{1} | 0.121 (0.120, 0.122) | <0.001 | |

Fuji | β_{0} | 15.4 (15.2, 15.6) | <0.001 |

β_{1} | 0.123 (0.122, 0.124) | <0.001 |

**Table 2.**Coefficients and associated 95% confidence intervals for the bi-exponential function used to model the cooling process of the three different shoe models used for the experiments (Equation (2)). Values in parentheses indicate the lower and upper limit of the 95% confidence intervals for each coefficient.

Shoe Type | Coefficient | Value (95% CI) |
---|---|---|

Cumulus | a (min) | 1.04 × 10^{6} (7.10 × 10^{5}, 1.49 × 10^{6}) |

b (°C)^{−1} | −4.29 × 10^{−1} (−4.45 × 10^{−1}, −4.14 × 10^{−1}) | |

c (min) | 1.78 × 10^{25} (5.74 × 10^{22}, 1.96 × 10^{27}) | |

d (°C)^{−1} | −2.52 (−2.74, −2.26) | |

Leather | a (min) | 9.58 × 10^{7} (1.33 × 10^{7}, 3.24 × 10^{8}) |

b (°C)^{−1} | −6.60 × 10^{−1} (−7.11× 10^{−1}, −5.78 × 10^{−1}) | |

c (min) | 8.34 × 10^{22} (4.85 × 10^{19}, 2.08 × 10^{25}) | |

d (°C)^{−1} | −2.52 (−2.26, −1.91) | |

Fuji | a (min) | 2.71 × 10^{6} (1.85 × 10^{6}, 4.02× 10^{6}) |

b (°C)^{−1} | −4.69 × 10^{−1} (−4.85 × 10^{−1}, −4.53 × 10^{−1}) | |

c (min) | 1.84 × 10^{26} (1.96 × 10^{23}, 2.15 × 10^{28}) | |

d (°C)^{−1} | −2.61 (−2.83, −2.29) |

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**MDPI and ACS Style**

Chua, J.H.J.; Dyer, A.G.; Garcia, J.E.
Hot Shoes in the Room: Authentication of Thermal Imaging for Quantitative Forensic Analysis. *J. Imaging* **2018**, *4*, 21.
https://doi.org/10.3390/jimaging4010021

**AMA Style**

Chua JHJ, Dyer AG, Garcia JE.
Hot Shoes in the Room: Authentication of Thermal Imaging for Quantitative Forensic Analysis. *Journal of Imaging*. 2018; 4(1):21.
https://doi.org/10.3390/jimaging4010021

**Chicago/Turabian Style**

Chua, Justin H. J., Adrian G. Dyer, and Jair E. Garcia.
2018. "Hot Shoes in the Room: Authentication of Thermal Imaging for Quantitative Forensic Analysis" *Journal of Imaging* 4, no. 1: 21.
https://doi.org/10.3390/jimaging4010021