# A Local 3D Voronoi-Based Optimization Method for Sensor Network Deployment in Complex Indoor Environments

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Methodology

_{1}allowing its displacement from other sensors. In Figure 5, Sensor 1 (S

_{1}) is moved to position MS

_{1}in the direction of ${\overrightarrow{V}}_{{s}_{1},v1}$, and is then projected on the nearest plane (position PS

_{1}). In the strategy, the sensor should be kept away from the obstacle’s surfaces based on a defined distance.

## 4. Formal Representation of the Proposed Optimization Algorithm

_{2}and O

_{3}in Figure 6) in which the angle between the normal vector of a back-face surface and the sensor direction is less than 90 degrees, (2) elimination of the surfaces that lie on the back side of sensor deployable surface (especially for the walls), and (3) elimination of the surfaces that lie outside the sensing distance (e.g., surface F

_{1}). Next, we project the remaining surfaces on a perspective plane which is defined parallel to the floor surface. Then, we overlay these surfaces in the projection plane (e.g., projected surface of O

_{1}= F

_{2}and F), in order to find the visible part of the floor (surface F-F

_{2}= F

_{3}) covered by the sensor. Finally, we calculate the covered area by the given sensor on the floor (F

_{3}).

Algorithm 1. Pseudo-code of the 3D Voronoi approach for sensor network optimization in an indoor environment | |||

input: n omnidirectional cameras Si(x_{i}, y_{i}, z_{i}) | |||

output: (X_{i}, Y_{i}, Z_{i}) optimal solution | |||

objective: Maximizing the coverage of the camera network | |||

1 | Initialize: Random distribution of the cameras on deployment planes | ||

2 | (walls/ceilings) Compute initial sensor network coverage; | ||

3 | 3D_Voronoi(S_{i},...,S_{n}); | ||

4 | step_size ← initial_step_size; | ||

5 | fori ← 1 to n do | ||

6 | S′_{i} ← Movement strategy(S_{i}); | ||

7 | { | ||

8 | 1- choose the farthest vertex in the same direction of path segments; | ||

9 | 2- amount of movement is step_size * vector’s length; | ||

10 | 3- project the movement vector on the nearest sensor deployed plane; | ||

11 | 4- if the movement vector has an intersection with an obstacle, keep a given distance | ||

12 | between sensor and obstacle; | ||

13 | } | ||

14 | g’S_{i} ← coverage (S_{1},...,S’_{i},...,S_{n})-coverage (S_{1},...,S_{i},...,S_{n}); | ||

15 | PQ ← add(g’S_{i},S’_{i}); | ||

16 | end | ||

17 | PQ ← Sort(PQ, highest gain); | ||

18 | K ← 1; | ||

19 | while (not terminated condition) do | ||

20 | step_size ← (initial_step_size) ∗ (Iteration − K)/Iteration; | ||

21 | S’u ←PQ(1).S; | ||

22 | Su ←S’u; | ||

23 | Update_3D_Voronoi(S_{i},...,S_{u}...,S_{n}); | ||

24 | S’_{u} ← Movement strategy(S_{u}); | ||

25 | g’S’_{u} ← coverage (S_{1},...,S’_{u},...,S_{n})-coverage (S_{1},...,S_{u},...,S_{n}); | ||

26 | PQ ← add(g’S’_{u},S’_{u}); | ||

27 | forj ← 1 to N_neighboringSu do | ||

28 | NS’_{i} ← Movement strategy(NS_{j}); | ||

29 | g’NS_{j} ← coverage (S_{1},...,NS’_{j},...,S_{n})-coverage (S_{1},...,NS_{j},...,S_{n}); | ||

30 | PQ ← add(g’NS_{j},NS’_{j}); | ||

31 | end | ||

32 | PQ ←Sort(PQ, highest gain); | ||

33 | K ←K + 1; | ||

34 | end |

_{si}) they produce. Hence, the first sensor in the list will have the priority to move before the others.

_{i}) to move to its new position (S′

_{i}) is based on the neighborhood relations of the sensor defined by the 3D Voronoi diagram as well as the configuration of the environment and the presence of the potential obstacles. The direction of movement is towards the farthest vertex of its Voronoi cell. The amount of the movement is defined by initial_step_size which is 80% of the vector’s length from the sensor to the farthest vertex.

_{S’u}) and its neighbors (g’

_{NSj}) are added in the PQ and sorted based on the highest coverage gain among the sensor. In the case of the presence of a permanent obstacle in the sensor movement direction, the sensor needs to be kept away from the obstacle. For this purpose, we define a distance constraint to avoid the obstruction of the sensor’s field of view. The while loop is stagnated with the terminated condition when the maximum coverage gain is less than a predefined coverage gain threshold (ε) for 10 iterations in a row.

## 5. Case Study

#### 5.1. Model of the Indoor Environment and Sensors

#### 5.2. Experimental Results and Discussions

#### 5.3. Comparison and Validation

#### 5.4. Sensitivity and Efficiency Analysis

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Arampatzis, T.; Lygeros, J.; Manesis, S. A survey of applications of wireless sensors and wireless sensor networks. In Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, Limassol, Cyprus, 27–29 June 2005; pp. 719–724. [Google Scholar]
- Zou, Y.; Chakrabarty, K. Sensor deployment and target localization based on virtual forces. In Proceedings of the INFOCOM 2003, Twenty-Second Annual Joint Conference of the IEEE Computer and Communications, San Francisco, CA, USA, 30 March–3 April 2003; Volume 2, pp. 1293–1303. [Google Scholar]
- Wang, G.; Cao, G.; La Porta, T.F. Movement-assisted sensor deployment. IEEE Trans. Mob. Comput.
**2006**, 5, 640–652. [Google Scholar] [CrossRef] - Cortes, J.; Martinez, S.; Karatas, T.; Bullo, F. Coverage control for mobile sensing networks. IEEE Trans. Robot. Autom.
**2004**, 20, 243–255. [Google Scholar] [CrossRef] - Huang, C.-F.; Tseng, Y.-C. The coverage problem in a wireless sensor network. Mob. Netw. Appl.
**2005**, 10, 519–528. [Google Scholar] [CrossRef] - Konda, K.R.; Conci, N. Optimal Configuration of PTZ Camera Networks Based on Visual Quality Assessment and Coverage Maximization. In Proceedings of the Seventh International Conference on Distributed Smart Cameras (ICDSC), Palm Springs, CA, USA, 29 October–1 November 2013. [Google Scholar] [CrossRef]
- Beni, L.H.; Mostafavi, M.A.; Pouliot, J.; Gavrilova, M. Towards 3D spatial dynamic field simulation within GIS using kinetic Voronoi diagram and Delaunay tetrahedralization. Int. J. Geogr. Inf. Sci.
**2011**, 25, 25–50. [Google Scholar] [CrossRef] - Akbarzadeh, V.; Gagné, C.; Parizeau, M.; Argany, M.; Mostafavi, M.A. Probabilistic sensing model for sensor placement optimization based on-line-of-sight coverage. IEEE Trans. Instrum. Meas.
**2013**, 62, 293–303. [Google Scholar] [CrossRef] - Argany, M.; Mostafavi, M.A.; Akbarzadeh, V.; Gagné, C.; Yaagoubi, R. Impact of the quality of spatial 3D city models on sensor networks placement optimization. Geomatica
**2012**, 66, 291–305. [Google Scholar] [CrossRef] - Kivimäki, T.; Vuorela, T.; Peltola, P.; Vanhala, J. A review on device-free passive indoor positioning methods. Int. J. Smart Home
**2014**, 8, 71–94. [Google Scholar] [CrossRef] - Guvensan, M.A.; Yavuz, A.G. On coverage issues in directional sensor networks: A survey. Ad. Hoc. Netw.
**2011**, 9, 1238–1255. [Google Scholar] [CrossRef] - Kumar, S.; Lai, T.H.; Balogh, J. On k-coverage in a mostly sleeping sensor network. In Proceedings of the 10th Annual International Conference on Mobile Computing and Networking, Philadelphia, PA, USA, 26 September–1 October 2004; pp. 144–158. [Google Scholar]
- Li, X.-Y.; Wan, P.-J.; Frieder, O. Coverage in wireless ad hoc sensor networks. IEEE Trans. Comput.
**2003**, 52, 753–763. [Google Scholar] [CrossRef] - Roy, S.; Das, G.; Das, S. Computing best coverage path in the presence of obstacles in a sensor field. Algorithms Data Struct.
**2007**, 13, 577–588. [Google Scholar] - Ghosh, A.; Das, S.K. Coverage and connectivity issues in wireless sensor networks: A survey. Pervasive Mob. Comput.
**2008**, 3, 303–334. [Google Scholar] - Ma, H.; Zhang, X.; Ming, A. A coverage-enhancing method for 3D directional sensor networks. In Proceedings of the INFOCOM 2009, Rio de Janeiro, Brazil, 19–25 April 2009; pp. 2791–2795. [Google Scholar]
- Wang, Y.; Cao, G. On full-view coverage in camera sensor networks. In Proceedings of the INFOCOM 2011, Shanghai, China, 10–15 April 2011; pp. 1781–1789. [Google Scholar]
- Akbarzadeh, V.; Gagné, C.; Parizeau, M.; Mostafavi, M.A. Black-box optimization of sensor placement with elevation maps and probabilistic sensing models. In Proceedings of the 2011 IEEE International Symposium on Robotic and Sensors Environments (ROSE), Montreal, QC, Canada, 17–18 September 2011; pp. 89–94. [Google Scholar]
- Romoozi, M.; Ebrahimpour-Komleh, H. A positioning method in wireless sensor networks using genetic algorithms. Phys. Procedia
**2012**, 33, 1042–1049. [Google Scholar] [CrossRef] [Green Version] - Kulkarni, R.V.; Venayagamoorthy, G.K. Particle swarm optimization in wireless-sensor networks: A brief survey. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev.
**2011**, 41, 262–267. [Google Scholar] [CrossRef] [Green Version] - Niewiadomska-Szynkiewicz, E.; Marks, M. Optimization schemes for wireless sensor network localization. Int. J. Appl. Math. Comput. Sci.
**2009**, 19, 291–302. [Google Scholar] [CrossRef] [Green Version] - Argany, M. Development of a GIS-Based Method for Sensor Network Deployment and Coverage Optimization; Université Laval: Quebec City, QC, Canada, 2015. [Google Scholar]
- Yaagoubi, R.; Yarmani, M.E.; Kamel, A.; Khemiri, W. HybVOR: A Voronoi-based 3D GIS approach for camera surveillance network placement. ISPRS Int. J. Geo-Inf.
**2015**, 4, 754–782. [Google Scholar] [CrossRef] [Green Version] - Afghantoloee, A.; Doodman, S.; Karimipour, F.; Mostafavi, M.A. Coverage Estimation of Geosensor in 3d Vector Environments. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2014**, 40, 1. [Google Scholar] [CrossRef] [Green Version] - De Rainville, F.-M.; Gagné, C.; Laurendeau, D. Automatic Sensor Placement for Complex Three-Dimensional Inspection and Exploration. In Proceedings of the International Symposium on Artificial Intelligence, Robotics, and Automation in Space, Montreal, QC, Canada, 17–19 June 2014. [Google Scholar]
- Potter, M.A.; De Jong, K.A. Cooperative coevolution: An architecture for evolving coadapted subcomponents. Evol. Comput.
**2000**, 8, 1–29. [Google Scholar] [CrossRef] [PubMed] - Kouakou, M.T.; Yasumoto, K.; Yamamoto, S.; Ito, M. Cost-efficient sensor deployment in indoor space with obstacles. In Proceedings of the 2012 IEEE International Symposium on a World of Wireless, Mobile and Multimedia Networks (WoWMoM), San Francisco, CA, USA, 25–28 June 2012; pp. 1–9. [Google Scholar] [CrossRef]
- Doodman, S.; Afghantoloee, A.; Mostafavi, M.A.; Karimipour, F. 3D Extention of the Vor Algorithm to Determine and Optimize the Coverage of Geosensor Networks. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2014**, XL-2/W3, 103–108. [Google Scholar] [CrossRef] [Green Version] - OGC. OGC IndoorGML Standard. OGC Member Approved International Standard. 2014. Available online: http://www.opengeospatial.org/ (accessed on 12 April 2017).
- Li, K.-J.; Conti, G.; Konstantinidis, E.; Zlatanova, S.; Bamidis, P. OGC IndoorGML: A standard approach for indoor maps. In Geographical and Fingerprinting Data to Create Systems for Indoor Positioning and Indoor/Outdoor Navigation; Elsevier: Amsterdam, The Netherlands, 2019; pp. 187–207. [Google Scholar]
- Gröger, G.; Kolbe, T.; Nagel, C.; Häfele, K.-H. OGC City Geography Markup Language (CityGML) Encoding Standard. Available online: https://www.ogc.org/standards/citygml (accessed on 24 November 2021).
- Ledoux, H. Computing the 3d Voronoi diagram robustly: An easy explanation. In Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007), Glamorgan, UK, 9–11 July 2007; pp. 117–129. [Google Scholar]
- Simpkins, A.; De Callafon, R.; Todorov, E. Optimal trade-off between exploration and exploitation. In Proceedings of the 2008 American Control Conference, Seattle, WA, USA, 11–13 June 2008; pp. 33–38. [Google Scholar]
- Argany, M.; Karimipour, F.; Mafi, F.; Afghantoloee, A. Optimization of wireless sensor networks deployment based on probabilistic sensing models in a complex environment. J. Sens. Actuator Netw.
**2018**, 7, 20. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**The proposed sensors deployment approach in a 3D indoor environment based on IndoorGML and a 3D Voronoi diagram.

**Figure 4.**(

**a**) A convex cell represented by its walls surfaces (green) and its floor surface (white). (

**b**) A non-convex cell that includes several wall surfaces as obstacles (red).

**Figure 5.**Movement strategy of sensors inside a 3D space based on a 3D Voronoi diagram (colored 3D cells) with consideration of projection and obstacle avoidance.

**Figure 6.**Coverage estimation based on overlapping visible surfaces projected onto the perspective plane. Surface F

_{1}is out of whole area of sensing distance, and surface F

_{2}is hidden by surface O

_{1}.

**Figure 8.**Indoor views of different spaces (corridors and the main hall) which show the complexity of surfaces in the Convention Center building.

**Figure 10.**The result of the proposed 3D Voronoi approach including the positions of sensors (blue points), and their coverage (red area) for the first four iterations.

**Figure 12.**Final locations of cameras obtained by the 3D Voronoi approach are shown in (

**a**) side view and (

**b**) top view.

**Table 1.**Results of 3D Voronoi, GA, and CMA-ES algorithms including coverage average (AVG-coverage) and its standard deviation (SD-coverage).

Method | AVG-Coverage (%) | SD-Coverage (%) | Time (s) |
---|---|---|---|

3D Voronoi | 98.86 | 0.76 | 646.45 |

GA | 98.1 | 1.64 | 3898.86 |

CMA-ES | 98.45 | 1.82 | 3264.92 |

Method | Sensing Range | AVG-Coverage (%) | SD-Coverage (%) | Time (s) |
---|---|---|---|---|

3D Voronoi | 30 | 67.44 | 5.69 | 176.62 |

50 | 96.17 | 1.94 | 304.92 | |

70 | 98.16 | 1.16 | 486.503 | |

90 | 98.61 | 0.88 | 654.734 | |

GA | 30 | 70.42 | 6.74 | 2112.72 |

50 | 93.50 | 2.76 | 2663.72 | |

70 | 98.19 | 2.32 | 3647.88 | |

90 | 97.55 | 1.56 | 3820.56 | |

CMA-ES | 30 | 70.74 | 5.98 | 2089.38 |

50 | 94.68 | 2.20 | 2509.52 | |

70 | 98.08 | 1.60 | 3511.1 | |

90 | 98.52 | 1.21 | 3858.34 |

**Table 3.**Results of 3D Voronoi, GA, and CMA-ES algorithms for different numbers of cameras with 30 m sensing range.

Method | Number of Cameras | AVG-Coverage (%) | SD-Coverage (%) | Time (s) |
---|---|---|---|---|

3D Voronoi | 4 | 67.44 | 5.69 | 176.62 |

6 | 86.48 | 5.07 | 263.02 | |

8 | 95.42 | 2.01 | 369.82 | |

10 | 98.08 | 1.90 | 494.09 | |

GA | 4 | 70.42 | 6.74 | 2112.72 |

6 | 78.93 | 6.32 | 3431.64 | |

8 | 89.68 | 4.24 | 4114.16 | |

10 | 86.70 | 2.89 | 6726.16 | |

CMA-ES | 4 | 70.74 | 5.98 | 2089.38 |

6 | 86.06 | 6.12 | 3252.98 | |

8 | 86.48 | 3.45 | 4434.94 | |

10 | 90.63 | 2.37 | 5867.38 |

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**MDPI and ACS Style**

Afghantoloee, A.; Mostafavi, M.A.
A Local 3D Voronoi-Based Optimization Method for Sensor Network Deployment in Complex Indoor Environments. *Sensors* **2021**, *21*, 8011.
https://doi.org/10.3390/s21238011

**AMA Style**

Afghantoloee A, Mostafavi MA.
A Local 3D Voronoi-Based Optimization Method for Sensor Network Deployment in Complex Indoor Environments. *Sensors*. 2021; 21(23):8011.
https://doi.org/10.3390/s21238011

**Chicago/Turabian Style**

Afghantoloee, Ali, and Mir Abolfazl Mostafavi.
2021. "A Local 3D Voronoi-Based Optimization Method for Sensor Network Deployment in Complex Indoor Environments" *Sensors* 21, no. 23: 8011.
https://doi.org/10.3390/s21238011