# 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

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**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