# HOVE-Wedge-Filtering of Geomorphologic Terrestrial Laser Scan Data

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Preconditions for Filtering Terrestrial Laser Scanning Data

#### 2.2. Wedge Iterative (Modified Model)

_{li,re}is the left or right wedge angle; ∆

_{vert}is the difference of the vertical angle; and ∆

_{horz}is the difference of the horizontal angle.

_{li}∩ α

_{re}> λ

_{TLV}

_{li,re}is the left or right wedge angle; and λ

_{TLV}is the threshold limit value.

_{diff}< λ

_{TLV}

_{diff}is the difference of left (from right adjacent point) and right (from left adjacent point) wedge angle; and λ

_{TLV}is the threshold limit value.

#### 2.3. Theory of Filtering over Vertical Horizontal (HOVE) Angular Grid

- The output grid contains no gaps.
- It is easier to find terrain structures (e.g., terrain edges).
- Quick recalculation of adjacent points is easily computable.
- 3D filtering is possible.

#### 2.3.1. Horizontal and Vertical Angles within the HOVE Grid

_{wi}is the angle from examined point to adjacent point; L

_{puex}is the length from laser scanner to examined point; L

_{puad}is the length from laser scanner to adjacent point; and β

_{puex}is the angle from laser scanner to examined point; β

_{puad}: angle from laser scanner to adjacent point.

_{wi}below the point to be examined are determined (Figure 1, green points).

_{wi}will be used for calculation.

#### Find Distance Measurement Errors

_{dive}∩ θ

_{diho}> λ

_{TLV}

_{dive}is the difference of the vertical back-angles; θ

_{diho}is the difference of the horizontal back-angles; and λ

_{TLV}is the threshold limit value.

#### Reduction to a 2.5-Dimensional Terrain

_{wi}of such points is less zero. If the angle to the point below with the lowest α

_{wi}is less zero, the examined point will be eliminated:

_{wi vert below}> λ

_{TLV}

_{wi vert below}is the angle from examined point to adjacent point below; and λ

_{TLV}is the threshold limit value.

#### Eliminate Vegetation Points

#### 2.3.2. Determination of the Point Density

#### 2.3.3. Finding the Adjacent Points for Iterative Wedge Filtering

_{li,re}is the minimum value; ∆

_{vert}is the difference of the vertical angle (from laser scanner); ∆

_{horz}is the difference of the horizontal angle (from laser scanner); fakt is the factor for ∆

_{horz}; diwi is the average HOVE raster resolution; and ∆

_{len}is the difference of length (from laser scanner).

## 3. Results

#### 3.1. Description of the Test Area

#### 3.2. DEM Calculation of the Test Areas Using the Modified Wedge Filtering Method

#### Steps of Calculation

- Calculate the horizontal and vertical angles as well as the length distances from the recording centre to every single point.
- Determine the average raster spacing of the horizontal–vertical angle grid.
- Eliminate measuring errors, as described in Section 2.3.1, Point (a). An angle of 179° was chosen as the threshold value.
- Reduce to 2.5D, as described in Section 2.3.1, Point (b). With this step, approximately 40% of the points are eliminated from the further calculation.
- Perform HOVE-Wedge Filtering as described in Section 2.3. This process is carried out iteratively, until no further point is eliminated. The wedge angles (see Section 2.3.3), horizontal and vertical angles of the HOVE-grid (see Section 2.3.1) and point density (points within a certain angular segment, see Section 2.3.2) have influence in Equation (8):$$\begin{array}{c}(({\theta}_{wedge}-180)\times F+({\theta}_{vert}-180)\times F+({\theta}_{hori}-180)\times F)\times -density>{\lambda}_{TLV}\\ \to classification\text{}as\text{}non-ground\text{}point\end{array}$$
_{wedge}is the difference of wedge angle; θ_{vert}is the difference of vertical angle from HOVE-Raster; θ_{hori}is the difference of horizontal angle from HOVE-Raster; F is the weight factors; λ_{TLV}is the threshold limit value; and density is the value of point-density from HOVE-Raster.

#### 3.3. Calculation and Accuracy of Digital Elevation Model (DEM)

_{i},y

_{i}) is the observed value at (x

_{i},y

_{i}); and w

_{i}is the weight associated with f(x

_{i},y

_{i}).

#### 3.4. Results of the Test Area

#### Detail of Test Area with a High Point Density

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 7.**Test area (inclination map); image source: © Land Tirol, tiris, www.tirol.gv.at/tiris.

**Figure 8.**Screenshot of the computer program with vegetation points (

**a**); and screenshot of the computer program only with ground points (

**b**).

**Figure 9.**DoDs of different filtering methods and image source: © Land Tirol, tiris, www.tirol.gv.at/tiris.

Statistic | Modified Wedge-Filtering | Wedge Absolute-Filtering (60°) | IDWMO Method |
---|---|---|---|

Groundpoints | 23,135 | 29,229 | 12,326 |

Mean error | −0.070 | 2.908 | 1.032 |

RMSE | 1.610 | 5.284 | 3.333 |

Standard deviation | 1.609 | 4.412 | 3.169 |

Median | −0.138 | 1.277 | 0.166 |

NMAD | 1.498 | 5.085 | 3.411 |

68.3% quantile | 0.242 | 3.819 | 1.258 |

95% quantile | 2.707 | 11.765 | 7.718 |

Statistic | Modified Wedge-Filtering | Wedge Absolut-Filtering (60°) | IDWMO Method |
---|---|---|---|

Groundpoints | 2703 | 3014 | 1346 |

Mean error | −0.142 | 1.296 | 0.361 |

RMSE | 0.555 | 1.855 | 1.366 |

Standard deviation | 0.536 | 1.537 | 1.366 |

Median | −0.061 | 0.456 | −0.054 |

NMAD | 0.510 | 1.529 | 1.057 |

68.3% quantile | 0.053 | 0.828 | 0.125 |

95% quantile | 0.509 | 4.390 | 1.876 |

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**MDPI and ACS Style**

Panholzer, H.; Prokop, A. HOVE-Wedge-Filtering of Geomorphologic Terrestrial Laser Scan Data. *Appl. Sci.* **2018**, *8*, 263.
https://doi.org/10.3390/app8020263

**AMA Style**

Panholzer H, Prokop A. HOVE-Wedge-Filtering of Geomorphologic Terrestrial Laser Scan Data. *Applied Sciences*. 2018; 8(2):263.
https://doi.org/10.3390/app8020263

**Chicago/Turabian Style**

Panholzer, Helmut, and Alexander Prokop. 2018. "HOVE-Wedge-Filtering of Geomorphologic Terrestrial Laser Scan Data" *Applied Sciences* 8, no. 2: 263.
https://doi.org/10.3390/app8020263