# Assessing the Influence of Temperature Changes on the Geometric Stability of Smartphone- and Raspberry Pi Cameras

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hardware

#### 2.1. Smartphone Camera Application

## 3. Methods and Algorithms

#### 3.1. Single-Image Camera Calibration

#### 3.1.1. IOP Estimation

#### 3.1.2. Designing the 3D Test Field

#### 3.1.3. Data Acquisition and Processing

#### 3.2. Simulating the Impact of Differently Changing IOP at Measurements in 3D Object Space

## 4. Results and Discussion

#### 4.1. Self-Heating Temperature Impacts at Smartphone Cameras

#### 4.2. Temperature Impacts at the Stability of RPi Cameras

#### 4.3. Statistical Evaluation of Temperature Dependencies

#### 4.4. Temperature-Related Error Assessment in Object Space: Results of Monte-Carlo Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CMOS | complementary metal-oxide-semiconductor |

CPU | central processing unit |

DSLR | digital single-lens reflex |

EOP | exterior orientation parameters |

IOP | interior orientation parameters |

MEMS | micro-electro-mechanical systems |

MP | megapixel |

RPi | Raspberry Pi |

## Appendix A. Monte-Carlo Simulation

#### Appendix A.1. General Approach of the Monte-Carlo Simulation

#### Appendix A.2. Implementing the Monte-Carlo Simulation

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**Figure 1.**Transformation of a 3D object point $\overrightarrow{X}$ into a 2D image point $\overrightarrow{x{}^{\prime}}$ where $\overrightarrow{P}[\mathbf{R}|\overrightarrow{{X}_{0}}]$ is the camera projection center in a world coordinate system.

**Figure 2.**Measurement setup to investigate the relation between the IOP and temperature variations occurring from (

**a**) self-heating and (

**b**) ambient temperature variations using smartphone cameras and low-cost RPi cameras, respectively. Camera-to-object distance: about 90 cm. Test field dimensions: 70 × 50 cm, Marker diameter: 10 mm.

**Figure 3.**IOP-to-temperature assessment; x-axis: number of measurements per measurement series M1/M2; y-axis: estimated deviation per parameter $\Delta k$ compared to the respective initial value; color-code: temperature difference $\Delta t$ (measured at the battery) to the initial temperature value when the phone was started.

**Figure 4.**Visualisation of image point shifts and zooming effects between the first (red dots) and the last (heads of the blue arrows) measurement within measurement series M1 respectively for (from left to right) the LG Google Nexus 5 camera (cold-started, warm-started) and for the Samsung Galaxy S8 camera (cold-started, warm started). The arrow length is superimposed by a factor of 50.

**Figure 5.**IOP-to-temperature assessment with RPi camera v2.1, which was exposed to alternating temperature, evaluating two measurement series (M1, M2); x-axis: number of measurements per measurement series; y-axis: estimated deviation per investigated parameter compared to the initial value; color-code: temperature difference $\Delta t$ to the initial temperature value before the red-light radiation lamp was switched on for the first time.

**Figure 6.**Visualisation of image point shifts and zooming effects considering the turning points between the heating and cooling phases of the RPi camera investigation M1. The arrow length is superimposed by a factor of 50.

**Figure 7.**Visualisation of the linear dependencies between changing IOP and temperature on the example of principal distance c. Considering the RPi observations, the aggregated data is sorted by temperature change. Light colors refer to the respective measurement series M1 and darker colors refer to the respective measurement series M2.

**Figure 8.**Correlation between changing IOP and changing temperature given by Pearson’s correlation coefficient $\rho $, calculated from the measurement series M1 and M2 for each camera.

**Figure 9.**Median correlations $\stackrel{~}{\rho}$ between the individual IOP determined for each camera using the observations of measurement series M1 and M2. From left to right, LG Google Nexus 5 cold-started, warm-started; Samsung Galaxy S8 cold-started, warm-started, Raspberry Pi camera v2.1.

**Figure 10.**Projection of image points $p{{}^{\prime}}_{i}({x}^{\prime},{y}^{\prime})$ onto a virtual object plane in a distance of Z = 10 m with fixed EOP- and simulated IOP (visualisation of every 10th point ${P}_{i}(X,Y)$ ). LG Google Nexus 5, cold started (

**a**) and warm started (

**b**); Samsung Galaxy S8, cold started (

**c**) and warm started (

**d**) and Raspberry Pi v2.1 (

**e**). The generated object points were colourised by means of their Euclidean distance $d({P}_{i},{P}_{\mu})$ to the expected object point coordinates. The white ellipses are the confidence ellipses with 95% probability.

**Figure 11.**Principal standard deviations ${s}_{1}^{*}$ and ${s}_{2}^{*}$ giving the directional error in object space for each projected object point and for each investigated camera.

LG Google Nexus 5 | Samsung Galaxy S8 | RPi Camera v2.1 | |
---|---|---|---|

Release | October 2013 | March 2017 | 2016 |

Operation system | Android 6.0.1 | Android 8.0 | (-) |

Camera specifications | |||

CMOS Sensor | Sony IMX179 Exmor R | Sony IMX333 Exmor RS | Sony IMX219PQ |

Sensor size | 4.6 mm × 3.5 mm | 5.6 mm × 4.2 mm | 3.7 mm × 2.8 mm |

Total pixels | 3288 × 2512 (8.26 MP) | - | 3296 × 2512 (8.28 MP) |

Active pixels | 3264 × 2448 (7.99 MP) | 4032 × 3024 (12.2 MP) | 3280 × 2464 (8.08 MP) |

Pixel size | 1.40 µm × 1.40 µm | 1.40 µm × 1.40 µm | 1.12 µm × 1.12 µm |

Focal length | 3.97 mm | 4.25 mm | 3.0 mm |

**Table 2.**Changes in the IOP of the built-in smartphone cameras from LG Google Nexus 5 and Samsung Galaxy S8 between the last and first estimated variables. $\Delta {t}_{cpu}$ and $\Delta {t}_{batt}$ are the deviations between the temperatures of the device measured at the CPU and the battery, respectively. $\Delta k$ are the deviations of the estimated IOP.

LG Google Nexus 5, Cold Started | LG Google Nexus 5, Warm Started | Samsung Galaxy S8, Cold Started | Samsung Galaxy S8, Warm Started | |||||
---|---|---|---|---|---|---|---|---|

M1 | M2 | M1 | M2 | M1 | M2 | M1 | M2 | |

$\Delta {t}_{cpu}/\Delta {t}_{batt}$ [${}^{\circ}$C] | 23.1/22.9 | 23.6/25.0 | 9.9/7.9 | 13.4/9.9 | 21.9/21.3 | 25.9/26.9 | 8.1/8.6 | 8.1/8.6 |

$\Delta c$ [mm] | 0.0078 | 0.0082 | 0.0028 | 0.0045 | 0.0055 | 0.0075 | 0.0014 | 0.0018 |

[Px] | 5.60 | 5.85 | 2.03 | 3.19 | 3.95 | 5.37 | 1.00 | 1.27 |

$\Delta {x}_{0}{}^{\prime}$ [mm] | 0.0205 | 0.0200 | 0.0092 | 0.0102 | −0.0408 | −0.0460 | −0.0105 | −0.0138 |

[Px] | 14.62 | 14.32 | 6.56 | 7.28 | −29.17 | −32.87 | −7.48 | −9.87 |

$\Delta {y}_{0}{}^{\prime}$ [mm] | −0.0085 | −0.0071 | −0.0041 | −0.0045 | 0.0328 | 0.0367 | 0.0077 | 0.0118 |

[Px] | −6.11 | −5.11 | −2.96 | −3.19 | 23.46 | 26.22 | 5.53 | 8.41 |

$\Delta {a}_{1}\left[{\mathrm{mm}}^{-2}\right]$ | −5.7 × 10${}^{-4}$ | 5.3 × 10${}^{-4}$ | −5.0 × 10${}^{-4}$ | 3.8 × 10${}^{-4}$ | −1.6 × 10${}^{-4}$ | −9.8 × 10${}^{-4}$ | −4.2 × 10${}^{-4}$ | 8.4 × 10${}^{-4}$ |

$\Delta {a}_{2}\left[{\mathrm{mm}}^{-4}\right]$ | 3.6 × 10${}^{-4}$ | −4.1 × 10${}^{-4}$ | 2.4 × 10${}^{-4}$ | −1.2 × 10${}^{-4}$ | 1.4 × 10${}^{-4}$ | 2.8 × 10${}^{-4}$ | 1.1 × 10${}^{-4}$ | −2.5 × 10${}^{-4}$ |

$\Delta {a}_{3}\left[{\mathrm{mm}}^{-6}\right]$ | −7.2 × 10${}^{-5}$ | 7.3 × 10${}^{-5}$ | −3.6 × 10${}^{-5}$ | −4.9 × 10${}^{-5}$ | −2.0 × 10${}^{-5}$ | −2.0 × 10${}^{-5}$ | −9.1 × 10${}^{-6}$ | 2.2 × 10${}^{-5}$ |

$\Delta {b}_{1}\left[{\mathrm{mm}}^{-1}\right]$ | 5.9 × 10${}^{-4}$ | 5.8 × 10${}^{-4}$ | 2.5 × 10${}^{-4}$ | 2.7 × 10${}^{-4}$ | 6.9 × 10${}^{-6}$ | −6.7 × 10${}^{-5}$ | −2.6 × 10${}^{-5}$ | 4.8 × 10${}^{-5}$ |

$\Delta {b}_{2}\left[{\mathrm{mm}}^{-1}\right]$ | −1.1 × 10${}^{-4}$ | −1.6 × 10${}^{-4}$ | −2.9 × 10${}^{-5}$ | 6.4 × 10${}^{-5}$ | −6.0 × 10${}^{-5}$ | −5.0 × 10${}^{-5}$ | −2.1 × 10${}^{-5}$ | −3.3 × 10${}^{-5}$ |

$\Delta {\widehat{s}}_{0}$ [mm] | 2.7 × 10${}^{-5}$ | 1.8 × 10${}^{-5}$ | −3.6 × 10${}^{-5}$ | 2.7 × 10${}^{-5}$ | 1.7 × 10${}^{-4}$ | 3.0 × 10${}^{-4}$ | 8.0 × 10${}^{-5}$ | −1.2 × 10${}^{-6}$ |

[Px] | 0.02 | 0.01 | −0.03 | 0.02 | 0.12 | 0.21 | 0.06 | 0.00 |

**Table 3.**Extracts from a subset of images of measurement series M1 with temperature overlay (measured at the battery). All extracts were sampled at the same image position. They reveal temperature-induced camera sensor movements and out of focus appearances due to changing temperatures.

Measurement m | 1 | 25 | 50 | 75 | 100 | 125 | 150 |
---|---|---|---|---|---|---|---|

LG Google Nexus 5, cold start | |||||||

LG Google Nexus 5, warm start | |||||||

Samsung Galaxy S8, cold start | |||||||

Samsung Galaxy S8, warm start |

Null Hypothesis | Alternative Hypothesis | Test Statistic | Critic f-Value | Rejection Criteria |
---|---|---|---|---|

${H}_{0}:{\sigma}_{1}^{2}\le {\sigma}_{2}^{2}$ | ${H}_{1}:{\sigma}_{1}^{2}>{\sigma}_{2}^{2}$ | $Q=\frac{{s}_{x}^{2}}{{s}_{y}^{2}}({s}_{x}^{2}>{s}_{y}^{2})$ | ${f}_{{n}_{x}-1,{n}_{y}-1,1-\alpha}$ | $Q>f$ |

**Table 5.**F-test success ratios $\zeta (k)$ to assess whether the temperature-related variance ${s}_{k}^{2}$ of parameter k is significant compared to the measurement precision ${\widehat{s}}_{k}^{2}$.

$\mathit{\zeta}(\mathit{c})$ | $\mathit{\zeta}({\mathit{x}}_{0}{}^{\prime})$ | $\mathit{\zeta}({\mathit{y}}_{0}{}^{\prime})$ | $\mathit{\zeta}({\mathit{a}}_{1})$ | $\mathit{\zeta}({\mathit{a}}_{2})$ | $\mathit{\zeta}({\mathit{a}}_{3})$ | $\mathit{\zeta}({\mathit{b}}_{1})$ | $\mathit{\zeta}({\mathit{b}}_{2})$ | |
---|---|---|---|---|---|---|---|---|

LG Google Nexus 5, cold started | 0.94 | 0.90 | 0.88 | 1.00 | 1.00 | 1.00 | 0.86 | 1.00 |

LG Google Nexus 5, warm started | 1.00 | 0.71 | 0.85 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Samsung Galaxy S8, cold started | 0.87 | 1.00 | 1.00 | 0.81 | 0.71 | 0.72 | 0.50 | 1.00 |

Samsung Galaxy S8, warm started | 0.96 | 0.89 | 0.86 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

total | 0.94 | 0.87 | 0.90 | 0.95 | 0.93 | 0.93 | 0.84 | 1.00 |

**Table 6.**Extracts from a subset of images of both measurement series M1 and M2 using the RPi camera v2.1 with a fixed focal length of 3 mm. All extracts were sampled at the same image position and superimposed with information about the prevalent temperature measured by DHT 11 sensor.

Measurement m | 1 | 25 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225 | 250 |
---|---|---|---|---|---|---|---|---|---|---|---|

RPi camera v2.1, M1 | |||||||||||

RPi camera v2.1, M2 |

**Table 7.**Percentages of projected object points classified by their Euclidean distances to the expected object point coordinates. Clusters equal the classes used in Figure 10.

Clusters of Euclidean Distances $\mathit{d}({\mathit{P}}_{\mathit{i}},{\mathit{P}}_{\mathit{\mu}})$ [cm] | (0,1.5] | (1.5,5] | (5,10] | (10,20] |
---|---|---|---|---|

LG Google Nexus 5, cold started | 66.6 | 33.3 | 0.10 | 0.0 |

LG Google Nexus 5, warm started | 98.0 | 2.0 | 0.0 | 0.0 |

Samsung Galaxy S8, cold started | 19.5 | 69.7 | 10.8 | 0.0 |

Samsung Galaxy S8, warm started | 92.6 | 7.40 | 0.0 | 0.0 |

RPi camera, 3 mm | 79.4 | 20.6 | 0.0 | 0.0 |

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**MDPI and ACS Style**

Elias, M.; Eltner, A.; Liebold, F.; Maas, H.-G.
Assessing the Influence of Temperature Changes on the Geometric Stability of Smartphone- and Raspberry Pi Cameras. *Sensors* **2020**, *20*, 643.
https://doi.org/10.3390/s20030643

**AMA Style**

Elias M, Eltner A, Liebold F, Maas H-G.
Assessing the Influence of Temperature Changes on the Geometric Stability of Smartphone- and Raspberry Pi Cameras. *Sensors*. 2020; 20(3):643.
https://doi.org/10.3390/s20030643

**Chicago/Turabian Style**

Elias, Melanie, Anette Eltner, Frank Liebold, and Hans-Gerd Maas.
2020. "Assessing the Influence of Temperature Changes on the Geometric Stability of Smartphone- and Raspberry Pi Cameras" *Sensors* 20, no. 3: 643.
https://doi.org/10.3390/s20030643