# Estimating the Rut Depth by UAV Photogrammetry

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## Abstract

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## 1. Introduction

- a canopy height model to focus the point cloud on the logging trail. The model is produced by a solid angle filtering (SAF) of point clouds [25];
- a surface model as a triangularized irregular network (TIN) for trail detection; the model is produced by applying SAF with a different parameter setting, followed by mean curvature flow smoothing (MCF) [26];
- ground model vectorization (orientation of possible ruts and likelihood of having a trail through any given point;
- the height raster of a straightened harvesting trail; this phase uses a histogram of curvatures (HOC) method;
- a collection of 2D rut profile curves.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Point Cloud Data

#### 2.3. Methodology

#### 2.4. Step 0: Point Cloud Generation

#### 2.5. Steps 1, 2: Point Cloud Preprocessing

- (a)
- Eliminating point pairs with horizontal distance less than $5\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$. This makes the further processing code simpler. The point above/below gets removed when computing the ground/canopy model, respectively. The minimum distance was decided by the 0.5% data loss criterion. See Figure 4. This process can be done in linear $O(|P|)$ time, where $\left|P\right|$ is the size of the point cloud.
- (b)
- Building the ground TIN model by applying SAF two times in order to have a structured reduction of the canopy noise at the narrow stripes at the further steps. The first SAF run builds (a) the canopy TIN (green in the middle detail of Figure 5). The model is based on controlled triangularization unlike usual canopy models based on local windows; see e.g., [31]. The second run builds (b) the ground TIN. The spatial angle limit parameters used can be found in Section 2.9. Two models built are used in Steps (c) and (d).
- (c)
- The mean curvature flow (MCF) [26] was applied to the ground TIN model to smoothen it. The MCF procedure contains one method parameter $\lambda \in \left[0,1\right]$, which controls the aggressiveness of the smoothing effect. MCF is a TIN smoothing method, which resembles the mean filtering of the raster data. It has a typical trade-off between smoothing the noise and possibly deteriorating a useful signal.
- (d)
- Thinning of the point cloud. Thinning is dictated by the mean natural neighbor distance $\overline{l}$ (see Figure 4), which increases during the process from the preliminary average $\overline{l}=0.07\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ to the target value ${l}_{b}=0.20\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ in the red zone of Figure 5. The limit value still enables a rather good imprint of the rut shape on the resulting surface triangularization. The green canopy area was subjected to thinning to ${l}_{c}=5.0\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$. This was to eliminate the noise at the border areas of the ground model. Parameters ${l}_{b}$ and ${l}_{c}$ were deemed the most suitable for the further process. See the listing of the actual Algorithm 1.

Algorithm 1: Thinning the point iteratively making the mean neighborhood distance reach the target value. ${l}_{target}$ is increased incrementally during the process. Note: The algorithm is applied twice: first, as it is presented, and the second time with the stopList being initiated with the canopy front. |

#### 2.6. Steps 3, 4: Trail Detection

#### 2.7. Manual Selection of Trails

#### 2.8. Steps 4, 5: Profile Evaluation

#### 2.9. Parameterization

## 3. Results

## 4. Discussion

## 5. Conclusions

- before-after type of data collection
- GPS data of harvesting routes
- geo-referencing for utilizing the digital elevation maps (DEM)

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ALS | aerial laser scan |

CAN-bus | controller-area network bus |

CV | cross-validation |

DEM | digital elevation map |

DDG | discrete-differential geometry |

ETRS-TM35FIN | co-ordinates based on European Terrestrial Reference System 1989. The central meridian is 27° and false easting 500 km. |

FSC | (national) Forest Stewardship Council |

GIS | geographic information system |

GPS | geo-positioning system |

GNSS/ISS | global navigation satellite + inertial system |

GSD | ground sample distance |

HOC | histogram of oriented curvatures |

HOG | histogram of gradients |

LiDAR | light detection and ranging |

MCF | mean curvature flow |

NLS | National Land Survey of Finland |

NN | natural neighbors |

PEFC | Programme for the Endorsement of Forest Certification Schemes |

SAF | solid angle filtering |

SFM | structure-from-motion |

TIN | triangular irregular network |

UAV | unmanned aerial vehicle |

## Appendix A. Directed Curvature on TIN

- The directed curvature $H-t(\alpha )$ of Equation (A3) ${H}_{t}(\alpha )$ with orientation $\alpha $ is constant over the triangle t.

**Figure A1.**

**Left**: Projection of vertex normals on the directional plane spanned by vectors $\overline{\alpha}$ and ${e}_{3}$.

**Right**: Splitting of the raster rectangle $(a,b,c,d)$ to four triangles.

## Appendix B. Profile Analysis

#### Appendix B.1. The Convolution Filter

**Figure A2.**The shape of the convolution filter.

**Top**: The left rut profile construction.

**Bottom**: The length profile construction. See also Figure 8 for the general view.

i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

${v}_{i}$ or ${u}_{i}$ | $-w/2$ | 0 | $w/2$ | w | $7w/2$ | ${r}_{rut}$ | $0.8r$ | $0.6r$ |

${z}_{i}$ | 0 | −1 | 0 | 0.35 | 0 | 0 | −2.8 | 0.8 |

${z}_{i}^{\prime}$ | 0 | 0 | 2 | −0.1 | 0 | 0 | 0 | 0 |

#### Appendix B.2. Forming the Height Raster

#### Appendix B.3. Convolution Fit of the Trail Center Line

**Figure A3.**A schematic presentation of the trail center line matching. A convolution filter positioned at $q(t)\in S(a)$ has been outlined. A sub-segment $[{t}_{j},{t}_{j+1}]\in \mathcal{T}$ is shown with a thick line. The local coordinate frame $(u,v)$ can also be seen in Figure A2.

**Figure A4.**The trail central line detection by height convolution maximization. The local coordinates $(u,v)$ are the same as in Figure A2.

#### Appendix B.4. Profile Adjustment

**Figure A5.**Top two: Trail profiles in the u direction after the trail center line has been fixed with the height convolution adaptation. Bottom: Trail profiles in the u direction after the local profile adjustment. Note: the height scale given concerns all height plots.

**Figure A6.**

**Left**: Effect of the profile adjustment to the mean rut profile.

**Right**: The TIN model before the final adjustment.

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**Figure 1.**

**Top**and

**middle**: Trail 1 and Trail 2 areas as orthophotos and the drone flight plan schematics;

**bottom**: the harvester, forwarder and drone used in the field campaign are shown with the map location (with the ETRS-TM35FIN coordinate system, see Abbreviations section) of the site.

**Figure 2.**Trail 1 (below) and Trail 2 (above) are separated by approximately 1 km. The geo-reference markers are depicted by circles, the flight path is shown with the yellow cross-line and the trails by thick white curves. Trail 2 was not fully covered by the UAV photogrammetry. The starting points of the flight paths are indicated by the double circle.

**Figure 3.**The process chart of the computational steps (1)–(6), which are detailed later in the text. SAF is used twice to produce two TIN models. The thinning is done first to all ground points with no limitation, then to that part of the remaining canopy front points, which are overlapping the canopy points. HOC produces a curvature state, which is visualized in order to choose manual control points. The control points are inserted manually, and this phase has potential for further automation (orange box). The end product is the rut depth profile distribution.

**Figure 4.**

**Left**: The TIN model after SAF.

**Middle**: The TIN model after the thinning process where the average triangle edge length has been forced to 20 cm.

**Right**: Initial and final NN distance distributions. The mean point distance $\overline{l}$ has been shifted from initial $0.07\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ to $0.20\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$.

**Bottom right**: Definition of the NN distance as an average of the TIN triangle side lengths.

**Figure 5.**

**Left**: The canopy detection is based on two separate SAF runs with different parameters to detect the canopy top surface (green) and the normal ground model (red).

**Right**: A vertical slice of the point cloud depicted at the left. Blue vertical bars indicate where the extra thinning of the ground model is being applied. Note the ground model profile arising at the canopy front due to the lack of photogrammetric ray penetration.

**Figure 6.**

**Left**: Trail 1;

**Right**: Trail 2. The directional curvature at the locally-dominant direction. The visualization is formed from 20 different directional curvature images by tiling them using 3.8 m image tiles.

**Left**: Trail 1 is in the center.

**Right**: Trail 2 traverses horizontally across the lower part of the image. Co-ordinates are in ETRS-TM35FIN (m).

**Figure 7.**

**Top**, from left to right: The height map of a 5.3-m square spot with two parallel ruts, the direction with the smoothest curvature, the direction with the most drastic curvature distribution and the corresponding histograms ${f}_{1}$ and ${f}_{2}$.

**Bottom**: A spot with a ditch beside a road having a sharp V shape. beside a road having a sharp V shape. The curvature is rather isotropic. Actual samples are circular and centered on squares depicted with a radius $r=2.1\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$.

**Figure 8.**

**Left**: The left and the right rut have mirror filters within a distance D, which can be set for each machine type. See Appendix B for the exact definitions. The ideal profile has an average nature, which is more pronounced at the average profile cumulated over the trail length.

**Right**: The filter ${g}_{left}(p,\alpha )$ of the left rut in an orientation $\alpha ={19}^{\circ}$. The shape is formed by cubic splines and has a theoretically correct shape, e.g., to detect constant, gradient and unit impulses.

**Bottom**: The result of the height convolution of Trail 1 in ETRS-TM35FIN co-ordinates. The image is a combination of the strongest responses of each orientation $\alpha $.

**Figure 9.**Trail 1 and 2 rut depth distributions over the rut length. The distributions of manually-measured rut depths and the rut depths extracted from the UAV photogrammetric point cloud. Each rut (left and right) of each trail has been plotted separately. Trail 1 has occasional deep depressions, whereas Trail 2 has a dominantly moderate rut depth.

**Figure 10.**Correlation of the left and right rut depths measured manually and by the UAV method. Trail 1 at the top and Trail 2 at the bottom. The UAV values are extracted from the ground TIN model at the manual measurement points by the standard barycentric interpolation.

**Figure 11.**

**Top**and

**middle**: The depth profiles of the left and right rut of both trails extracted from the UAV photogrammetric point cloud.

**Bottom**: The 3D trail cross-profile averaged over the length. The left and right ruts of each trail are visible. The reference height is set to $z=0$ m. The narrow bottom of the ruts is an artificial result of the final rut profile rectification.

**Table 1.**Details of the photogrammetric data. GSD is provided by the GeoDrone manual. Vertical noise is an approximation of the vertical accuracy of the point cloud points.

Trail | Flight Height (m) | GSD (cm) | # Photos | Cloud size | Route Length (m) | Vert.Noise (std.cm) |
---|---|---|---|---|---|---|

1 | 100 | 2.0 | 42 | $8.1\times {10}^{6}$ | 208 | 4.0 |

2 | 150 | 3.0 | 34 | $7.1\times {10}^{6}$ | 280 | 3.5 |

Phase | Param. and Value | Explanation |
---|---|---|

thinning | ${l}_{target}=0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | thinning limit |

thinning | $0.1\left|P\right|$ | stopList size limit |

thinning | $0.5{l}_{target}$ | triangle edge length limit |

vectorization | $r=2.1$ m | sample radius |

$\delta =3.5$ m | sample grid interval | |

SAF canopy | ${\omega}_{1}=\pi $ | pike limit |

${\omega}_{2}=3\pi $ | hole limit | |

SAF ground | ${\omega}_{1}=4.4$ | |

${\omega}_{2}=8.1$ | ||

MCF | $\lambda =0.7$ | smoothing degree |

thinning | ${l}_{b}=0.2$ m | NN length at ground |

${l}_{c}=5.0$ m | NN length at canopy front | |

trail detection | ${n}_{\alpha}=20$ | number of curvature directions |

profile evaluation | $D=2.8$ m | rut distance |

$w=1.2$ m | rut width | |

$d=0.6$ m | rut depth (no actual effect) | |

${r}_{rut}=\phantom{\rule{4pt}{0ex}}3.8$ m | convolution sample length | |

medium | ||

$\Delta L=0.2$ m | rut profile sampling interval | |

7 free shape params. | see Table A1 | |

convolution fit | ${\lambda}_{1}=0.7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ | regularization weight |

$\Delta l=0.6\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | convolution sampling frequency |

**Table 3.**The point cloud properties. Horizontal densities $\rho $ are in m${}^{-2}$. The abbreviations TP, TN, FP and FN correspond to true and false positives and negatives given as percentages.

Trail | Initial $\mathit{\rho}$ | $\mathit{\rho}$ TIN | $\mathit{\rho}$ after Thinning | TP (%) | TN (%) | FP (%) | FN (%) |
---|---|---|---|---|---|---|---|

1 | 190 | 120 | 16 | 19 | 51 | 26 | 4 |

2 | 180 | 117 | 15 | 38 | 24 | 24 | 14 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nevalainen, P.; Salmivaara, A.; Ala-Ilomäki, J.; Launiainen, S.; Hiedanpää, J.; Finér, L.; Pahikkala, T.; Heikkonen, J.
Estimating the Rut Depth by UAV Photogrammetry. *Remote Sens.* **2017**, *9*, 1279.
https://doi.org/10.3390/rs9121279

**AMA Style**

Nevalainen P, Salmivaara A, Ala-Ilomäki J, Launiainen S, Hiedanpää J, Finér L, Pahikkala T, Heikkonen J.
Estimating the Rut Depth by UAV Photogrammetry. *Remote Sensing*. 2017; 9(12):1279.
https://doi.org/10.3390/rs9121279

**Chicago/Turabian Style**

Nevalainen, Paavo, Aura Salmivaara, Jari Ala-Ilomäki, Samuli Launiainen, Juuso Hiedanpää, Leena Finér, Tapio Pahikkala, and Jukka Heikkonen.
2017. "Estimating the Rut Depth by UAV Photogrammetry" *Remote Sensing* 9, no. 12: 1279.
https://doi.org/10.3390/rs9121279