# An Entropy Analysis-Based Window Size Optimization Scheme for Merging LiDAR Data Frames

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Mobile LiDAR Systems

#### 2.2. Point Cloud Data Merging

## 3. Entropy Analysis based Window Size Optimization Scheme

#### 3.1. Application Scenario

#### 3.2. System Modeling

#### 3.3. Window Size Optimization Algorithm

_{I}), the set of referenced linear structures’ IDs (R), and the total number of extracted linear structures at each point of interest in the ideal result (N). First, the entropy indicator of the ideal result (E

_{R}(X)) is calculated using N and C

_{i}and initializes the minimum value of the difference between E

_{R}(X) and the entropy indicator of the actual result at window size i (E

_{i}(X)). Next, the entropy indicator at each point of interest is repeatedly accumulated by T

_{I}. When the absolute value of the current difference between E

_{R}(X) and E

_{i}(X) is less than V

_{min}, V

_{min}and the optimal window size (ω) are updated. Finally, when the loop ends, we can find the optimal window size, ω, and the algorithm returns the ω value.

Algorithm 1 Finding Optimal Window Size | |

Input: I, T_{I}, R, N | |

Output: Optimal window size ω | |

1 | Calculate E_{R}(X) |

2 | V_{min} = INF |

3 | for i = 2 to 10 do |

4 | E_{i}(X) = 0 |

5 | for j = 1 to T_{I} do |

6 |
E_{i}(X) +=$P\left({x}_{j}\right)\ast \mathrm{log}\frac{1}{P\left({x}_{j}\right)},where{x}_{j}\in I$ |

7 |
end for |

8 | if$\left|{E}_{R}\left(X\right)-{E}_{i}\left(X\right)\right|$ < V_{min} then |

9 |
V_{min} $=\left|{E}_{R}\left(X\right)-{E}_{i}\left(X\right)\right|$ |

10 | ω = i |

11 | end if |

12 | end for |

13 | return ω |

## 4. Evaluation Results

#### 4.1. Experimental Environment

#### 4.2. Effects of the Proposed Window Mechanism

#### 4.3. Entropy Indicator Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 5.**Extracted References at Point of Interest. (

**a**) Ideal result (

**b**) Actual result (F/P is False Positive).

**Figure 8.**Comparison between (

**a**) Without and (

**b**) With Window Mechanism. Red box areas noticeably improve the shape of linear structures.

**Figure 10.**Entropy Indicator Values Comparison among the Three Schemes at Different Speeds. (

**a**) Entropy indicator of single frame; (

**b**) Entropy indicator of static window size; (

**c**) Entropy Indicator of optimal window size; (

**d**) Entropy indicator comparison of each scheme.

**Figure 11.**Entropy Indicator Values Comparison among the Three Schemes during Movement. (

**a**) Entropy indicator of single frame; (

**b**) Entropy indicator of static window size; (

**c**) Entropy Indicator of optimal window size; (

**d**) Entropy indicator comparison of each scheme.

Velodyne Puck | Velodyne Ultra Puck | |
---|---|---|

# of channels | 16 | 32 |

Max range | 100 m | 200 m |

Accuracy | ±3 cm | ±3 cm |

FoV | 30° (−15° to +15°) | 40° (−25° to +15°) |

Rotation rate | 5~20 Hz | 5~20 Hz |

Vertical angular resolution | 2° | 0.33° |

Horizontal angular resolution | 0.1~0.4° | 0.1~0.4° |

# of frames | 10 | 10 |

Weight | 830 g | 925 g |

Notation | Description |
---|---|

E(X) | Indicator of system X’s entropy |

I | Set of point of interests $(\mathrm{e}.\mathrm{g}.,\text{}I=\left\{p1,p2,p3,p4\right\})$ |

T_{I} | Total number of point of interests |

R | Set of referenced linear structures’ ID $(\mathrm{e}.\mathrm{g}.,\text{}R=\left\{r1,r2,r3,\dots r15\right\})$ |

W | Window size (2 <= W <= 10) |

$N$ | Total number of extracted linear structure at each point of interest in ideal result (e.g., N = 16 in Figure 5a) |

${C}_{i}$ | The number of detected linear structure at point i (pi) |

${F}_{i}$ | The number of incorrectly detected linear structure at point i (pi) |

$P\left({x}_{i}\right)$ | The probability of correctly detected linear structure at point i (pi) |

E_{R}(X) | Entropy indicator of ideal result |

E_{W}(X) | Entropy indicator of actual result at window size W |

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**MDPI and ACS Style**

Kim, T.; Jung, J.; Min, H.; Jung, Y.-H.
An Entropy Analysis-Based Window Size Optimization Scheme for Merging LiDAR Data Frames. *Sensors* **2022**, *22*, 9293.
https://doi.org/10.3390/s22239293

**AMA Style**

Kim T, Jung J, Min H, Jung Y-H.
An Entropy Analysis-Based Window Size Optimization Scheme for Merging LiDAR Data Frames. *Sensors*. 2022; 22(23):9293.
https://doi.org/10.3390/s22239293

**Chicago/Turabian Style**

Kim, Taesik, Jinman Jung, Hong Min, and Young-Hoon Jung.
2022. "An Entropy Analysis-Based Window Size Optimization Scheme for Merging LiDAR Data Frames" *Sensors* 22, no. 23: 9293.
https://doi.org/10.3390/s22239293