# Highly Accurate Tree Models Derived from Terrestrial Laser Scan Data: A Method Description

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{b}

- the destructive and time-consuming nature of the method;
- logs must be transported to the xylometer;
- logs might absorb water;
- the only obtained parameter is volume

_{i}determining the cross-sectional areas at relative lengths i and l being the total length of the tree:

- terrestrial laser scanning (TLS); the scanning system is stationary mounted on a tripod
- airborne laser scanning (ALS); the scanning system is mounted under an aircraft
- vehicle-based laser scanning (VLS); the scanning system is mounted on a ground vehicle

#### 1.1. Related Work

^{3}, while the sum of cylinders was determined to be 74 m

^{3}.

## 2. Rationale

## 3. Materials and Methods

#### 3.1. Terrestrial Laser Scanner

Parameter | Value |
---|---|

Beam divergence | <0.3 mrad (fullangle) |

Beam diameter (at a 0.1-m distance) | ∼3.5 mm |

Range | 187.3 m |

Resolution range | 0.1 mm |

Vertical field of view | 320° |

Horizontal field of view | 360° |

Vertical resolution | 0.0004° |

Horizontal resolution | 0.0002° |

Vertical accuracy | 0.007°rms |

Horizontal accuracy | 0.007°rms |

Scan mode “high” | 10,000 pixel/360° |

Scan mode “superhigh” | 20,000 pixel/360° |

Scan mode “ultrahigh” | 40,000 pixel/360° |

#### 3.2. Description of the Scanning Campaign

#### 3.3. Artificial Point Cloud

Group I | Group II | Group III | |
---|---|---|---|

Species | Prunus avium | Prunus avium | Prunus avium |

Acer pseudoplatanus | |||

Quercus robur | |||

Fraxinus excelsior | |||

Number of trees/branches | 14 branches | 2 trees | 24 trees |

Spacing, within and between rows | — | 7.5 m × 15 m | 1.5 m × 7.5 m |

Scan mode | superhigh | ultrahigh | high |

Average scans per tree/branch | 4 | 3 | 1 |

**Figure 1.**Group III: map of tree rows in the agroforestry system with marked scan positions. TLS, terrestrial laser scanning.

#### 3.4. Software

#### 3.5. Computer Specification

#### 3.6. Ground Truth Measurements

## 4. Method Description

- filtered scan points representing the surface of the target tree;
- noise points, which mainly represent ground artifacts and branches of other trees;
- height above the ground of the lowest z-coordinate, manually measured with Z+F LaserControl [41];
- optional point coordinates of two points, p
_{r}_{1}and p_{r}_{2}, visually identified as non-ambiguous in natural environment and scan data, and the bearing between p_{r}_{1}and p_{r}_{2}, measured with a compass.

#### 4.1. De-Noising

#### 4.2. Referencing

^{T}, represents the north direction in the data. Let p

_{r}

_{1}and p

_{r}

_{2}be two identified points in the input point cloud and α the bearing between p

_{r}

_{1}and p

_{r}

_{2}; Equations (4)–(6) can then be applied:

_{r}

_{1y}− p

_{r}

_{2y}, p

_{r}

_{1x}− p

_{r}

_{2x}) + atan2(p

_{r}

_{1y}− p

_{r}

_{3y}, p

_{r}

_{1x}− p

_{r}

_{3x})

#### 4.3. Cylinder Model Creation

#### 4.3.1. Cylinder Detection with the Usage of a Sphere (Refer to Figure 2(a)–2(c))

_{i}, with P

_{i}, representing the i − th cross-sectional area (refer to Figure 2(a)). As cross-sectional areas of stem and branches can be approximated by circles, into each P

_{i}, a circle, c

_{i}, with radius cr

_{i}and center point cc

_{i}is fitted (refer to Section 4.6). New spheres, s

_{i}, are generated, with cc

_{i}as the center point and cr

_{i}× x (x ∈ ℝ ∧ x > 1) as the radius, in accordance with a lower limit for the radius. Additionally, cylinders with start point sc, endpoint cc

_{i}and radius cr

_{i}are stored in a list, with an optional threshold set for the radius (refer to Figure 2(b)). For all spheres, s

_{i}, the described procedure can be repeated (refer to Figure 2(c)).

_{i}are detected.

#### 4.3.2. Initialization and Termination of the Method; the Usage of FIFO-Queues F1 and F2 to Follow the Stem and Branches According to Their Order ( Refer to Figure 2(d)–2(g))

_{1}, is initialized with a sphere located at the stem base. Since the de-noising procedure ensures that no compartments of other trees or ground artifacts are included in the point cloud, a thin slice at the height of the minimum z-coordinate contains only a cross-sectional area of the target tree stem. A circle is fitted into the slice and transformed into a sphere, which is then stored in F

_{1}(refer to Figure 2(d)).

**Figure 2.**Visualization of the method in detailed steps; a branch of Group I is used: (a) The cross-sectional area, P

_{i}, derived from the ϵ-neighborhood of the visualized sphere with sphere center sc and radius r; (b) A circle, c

_{i}, is fitted into P

_{i}; circle parameters cr

_{i}(radius) and center point cc

_{i}are used as the radius and endpoint of a new cylinder with its start point being the sphere center, sc; (c) A new sphere, si, is generated with cc

_{i}as the center point and a radius larger than cr

_{i}; (d) Initialization at the base of the tree/branch; (e) Terminus of branch with order zero (stem for a real tree) is reached (FIFO queue F

_{2}is transferred to FIFO queue F

_{1}the first time); (f) All branches of order one have been processed (F

_{2}is transferred to F

_{1}the second time); (g) The cylinder following terminates; F

_{1}and F

_{2}are both empty; (h) All detected cylinders after termination of the cylinder following; (i) The final cylinder model after post-processing steps; branches of order one are colored differently; the fit quality has been improved.

_{2}, until no more cross-sectional areas along the followed branch or stem are detected (i.e., the main branch or stem has been processed to its tip). This procedure is repeated for all spheres stored in F

_{1}sequentially. When F

_{1}is empty, all spheres contained in F

_{2}are transferred to F

_{1}(refer to Figure 2(e) and Figure 2(f)). Up to five thresholds can be adjusted at this time, namely two clustering parameters (refer to Appendix A), the minimum sphere size, the minimum cylinder radius and the size of the ϵ-neighborhood of the sphere surface. The reason for the adjustment is an expected change in the order of processed branches by one. Branches of order n + 1 have a diameter smaller than their n-th-order predecessors. At the same time, the frequency of branches becomes greater after the order change, leading to a denser neighborhood.

_{1}and F

_{2}are empty (refer to Figure 2(g) and Figure 2(h)). Termination is assured, as during every generation of child-spheres, points are deleted from the input point cloud, and the number of points is finite.

Algorithm 1: Cylinder model creation. |

Data: A list of denoised and referenced TLS points located on a single tree surface |

Result: A hierarchical tree-like data structure containing cylinders representing the tree |

Create two empty FIFO-queues, F_{1} and F_{2}, to store spheres; |

Create an empty list, L_{Cyl}, to store the cylinders; |

Compute a starting sphere on base of the tree and enqueue to F_{2}; |

start from first cylinder to build parent child relation in L_{Cyl} to gain an hierarchical tree-like data |

structure; |

allocate points to each cylinder ; |

improve the fit of each cylinder by proposed method; |

#### 4.3.3. Post-Processing Steps (Refer to Figure 2(i))

#### 4.3.3.1. Hierarchical Tree-Like Data Structure

#### 4.3.3.2. Improvement of Branch Junctions

#### 4.3.3.3. Improvement of Fit Quality

#### 4.3.3.4. Extraction of Tree Parameters

#### 4.4. Implemented Search Structure

^{2}) or more, though, is unpractical. Often, search-operations in 3D data either rely on the usage of either kD-trees [25] or octrees [20].

^{2/3}) for a balanced tree and (n) for the unbalanced version. To assure (n

^{2/3}) at all times, deletion operations require time costly re-balancing. While in an octree worst case, the time complexity for a range search is always (n), the expected runtime is sublinear; although, being less efficient regarding runtime complexity the octree provides other advantages over the kD-tree. It can contain other geometric objects than points (i.e., cylinders), and the deletion of objects never requires re-balancing. In an octree, the test for an octree cell intersecting with a sphere is straightforward, enabling the efficient search for points inside a cylinder, as these are approximately represented by their bounding spheres.

Parameter | Parameter |
---|---|

Tree length | Sum of all stem cylinder lengths |

Tree length | Maximum z-coordinate of all cylinder end points |

Total above ground volume | Sum of all cylinder volumes |

Stem volume | Sum of all stem cylinder volumes |

Solid volume | Sum of all cylinder volumes with cylinder diameter larger than 7 cm |

DBH | Diameter of the cylinder at 1.3 m |

Crown space occupation | Volume of the convex hull of crown cylinders’ endpoints |

Crown projection area | Area of the 2D convex hull of projected crown cylinders’ endpoints |

Branch azimuth | Angle between north vector and direction vector of a branch segment |

Branch length | Length along the longest path in a branch |

Branch volume | Sum of all branch cylinders volume |

Branch height | Z-coordinate of first branch cylinders start point |

#### 4.5. Imperfect Point Clouds

_{i}between the start point, sp, of c and all cylinder endpoints, ep

_{i}, in the primary list are determined. Let ep′ be the endpoint with the minimum distance; then, the cylinder with endpoint ep′ is chosen as c′.

#### 4.6. Circle Fitting

_{0}, y

_{0}, r), into a subset, P, of the TLS points, representing a cross-sectional area of a branch or stem.

_{i}(x

_{i}, y

_{i}) ∈ ℝ

^{2}, using least squares [50,51,52]. After a transition of P into ℝ

^{2}, the Gauss–Newton algorithm [53] is applied. For implementation details, refer to Appendix B.2.

_{0}, y

_{0}) of c is the orthogonal projection of the sphere center, sc, onto the plane. With distance d between sc and cc and sr being the sphere radius, we can apply Equation (8) to compute the circle radius, cr:

_{0}, y

_{0}) and the median distance of all allocated points to cc as radius cr.

#### 4.7. Cylinder Fitting

_{0}, y

_{0}, z

_{0}, a, b, c, r) can be fitted with least squares [8]; here, p

_{0}(x

_{0}, y

_{0}, z

_{0}) being a point on the axis, ax, = (a, b, c)

^{T}the direction vector of ax and r the cylinder radius. The implementation details of the applied Gauss–Newton algorithm [53] are given in Appendix B.1.

- incorrectly fitted cylinders are often located in components representing very small branches or in regions with low point cover;
- in each iteration of the fit, the diameter of the cylinder grows rapidly.

_{i}to the cylinder axis is computed for each allocated point. The cylinder radius is updated according to a quantile of the set of all d

_{i}(refer to Section 5.1 for the quantile adjustment procedure).

#### 4.8. Distance Analysis between TLS Cloud and Cylinder Model

_{ij}, between a point, p

_{i}, and a cylinder, c

_{j}, is defined according to distance d

_{i}in Equation (A4) within Appendix B.1, if the projection, , of p

_{i}onto the cylinder axis is between start point c

_{s}and end point c

_{e}of the cylinder. Else, we define in Equation (9):

_{i}the distance defined in Equation (A4). The minimum of all possible distances between p

_{i}and every cylinder in the model is the distance between p

_{i}and the cylinder model. With n being the number of input points and n

^{*}being the number of points with a distance of less than 3 cm to the model, we define the cover of our model in Equation (11) as a measure for the quantity of the completeness of the fit.

^{*}nearest points to the cylinder model to obtain different measures for the quality of the fit, i.e., the standard deviation, sd, and the mean, x, of the original distance data and the standard deviation, σ, and the mean, μ, of the fitted normal distribution.

## 5. Results and Discussion

#### 5.1. Group I Results

_{GT}and TLS results VOL

_{TLS}for this group. First, we chose to use the median as the quantile for the cylinder fitting method, as outlined in Section 4.7, which is performed whenever a least squares fit is not applied. The linear model in Equation (12) (adjusted coefficient of determination ( ) = 0.95) points to a slight underestimation of the volume within the TLS results.

_{GT}= 0.99V O L

_{TLS}+ 129

_{GT}= 1.01V O L

_{TLS}− 44

**Figure 4.**Group I results: (a) normalized root mean square error (NRMSE) in dependency of the quantile used for the quantile cylinder fitting method; (b) Linear regression between TLS results and manual volume measurements.

#### 5.2. Group II Results

**Figure 5.**Impact on the selection of different circle fitting methods of tree #1 (height ≈ 12 m): (a) the input point cloud of Tree 1; colors represent (unused) intensity values; (b) a model derived by the NLS method with large failure cylinders; (c) a model derived by the plane fitting method with failure cylinders; (d) a model derived by the quantile method covering the complete input point cloud without errors, with successfully detected stem and branches colored differently.

**Figure 6.**Distance histogram with fitted normal distribution for both Group II point clouds: (a) distance histogram for Tree 1; (b) distance histogram for Tree 2.

Tree ID | Number of Points | sd (mm) | x(mm) | σ (mm) | μ(mm) | Cover (%) |
---|---|---|---|---|---|---|

1 | 19,669,123 | 6.04 | 0.64 | 4.45 | 0.16 | 99.03 |

1 | 1,688,509 | 6.09 | 0.79 | 4.47 | 0.28 | 97.04 |

2 | 20,362,484 | 5.94 | 0.31 | 3.91 | 0.03 | 99.38 |

2 | 2,618,278 | 6.10 | 0.68 | 3.99 | 0.22 | 97.74 |

**Figure 7.**Analysis between TLS-derived data and ground truth data for 28 branches with respect to: (a) branch height; (b) azimuthal direction of branches; (c) TLS-volume against biomass (fresh weight); (d) diameter at branch collar.

**Figure 8.**Crown projection analysis for Tree 1: (a) comparison to manual results; (b) in the x,y-plane projected point cloud as visual validation.

Tree ID | Method | DBH (cm) | Length (m) | Crown projection area (m^{2}) |
---|---|---|---|---|

1 | manual | 18.6 | 11.52 | 8.59 |

1 | TLS | 18.1 | 11.57 | 8.54 |

2 | manual | 20.0 | 12.60 | 10.70 |

2 | TLS | 21.4 | 12.62 | 12.00 |

**Figure 9.**Visualization of both Group II trees (height ≈ 12 m): (a) TLS point clouds; (b) cylinder models with branches colored differently; (c) extracted stems of the cylinder models; (d) model of the crowns, stems and crown projections; (e) enlarged view into the lower crown point cloud of 2; (f) enlarged view into the lower crown cylinder model of 2.

#### 5.3. Artificial Point Cloud Results

**Table 6.**Comparison between original data and the proposed method results for the artificial point cloud.

Data Set | DBH (cm) | height (m) | Volume (L) | Crown Projection Area (m^{2}) |
---|---|---|---|---|

original | 22.4 | 17.73 | 666.5 | 30.81 |

program results | 22.4 | 17.74 | 683.6 | 29.70 |

Tree ID | Number of Points | sd (mm) | x (mm) | σ (mm) | μ (mm) | Cover (%) |
---|---|---|---|---|---|---|

artificial | 3,442,699 | 5.00 | 2.134 | 2.13 | 0.07 | 99.48 |

#### 5.4. Group III Results

ID | Points | DBH_{TLS(GT)1} (cm) | Vol(l) | sd (mm) | x (mm) | σ (mm) | μ (mm) | Cover (%) |
---|---|---|---|---|---|---|---|---|

3 | 503,794 | 10.69 (11.3) | 64.24 | 5.39 | 1.11 | 3.61 | 0.30 | 81.01 |

4 | 1,750,783 | 16.25 (16.5) | 164.30 | 5.80 | 1.35 | 4.51 | 0.57 | 94.24 |

5 | 1,033,913 | 8.96 (9.0) | 54.40 | 5.24 | 1.31 | 3.45 | 0.41 | 97.99 |

6 | 1,901,388 | 14.38 (14.5) | 139.96 | 5.79 | 1.69 | 4.34 | 0.78 | 86.84 |

7 | 1,091,585 | 14.35 (13.3) | 122.11 | 7.36 | 1.96 | 5.77 | 1.01 | 78.03 |

8 | 722,546 | 10.41 (10.5) | 65.21 | 6.27 | 1.77 | 4.52 | 0.78 | 86.16 |

9 | 1,289,349 | 15.08 (15.1) | 140.82 | 5.39 | 1.19 | 3.65 | 0.31 | 78.82 |

10 | 604,092 | 7.83 (7.9) | 34.53 | 4.90 | 1.07 | 2.97 | 0.23 | 88.05 |

11 | 2,414,504 | 15.70 (14.9) | 159.42 | 5.93 | 1.73 | 3.99 | 0.67 | 82.12 |

12 | 1,332,867 | 13.80 (14.3) | 131.83 | 6.41 | 1.77 | 4.47 | 0.73 | 84.05 |

13 | 1,241,908 | 12.84 (12.9) | 109.74 | 6.04 | 1.57 | 4.08 | 0.48 | 90.00 |

14 | 645,269 | 10.09 (9.9) | 62.17 | 5.84 | 1.44 | 3.64 | 0.38 | 88.04 |

15 | 1,289,349 | 10.11 (9.8) | 49.28 | 6.58 | 1.95 | 3.78 | 0.62 | 71.80 |

16 | 523,777 | 7.36 (8.1) | 32.74 | 5.58 | 1.44 | 3.55 | 0.48 | 97.38 |

17 | 1,743,865 | 12.18 (12.2) | 102.83 | 5.35 | 1.47 | 3.69 | 0.57 | 92.16 |

18 | 1,316,364 | 12.71 (12.7) | 96.06 | 5.51 | 1.64 | 3.73 | 0.68 | 79.26 |

19 | 649,161 | 10.95 (11.4) | 58.93 | 5.38 | 1.06 | 3.71 | 0.31 | 93.10 |

20 | 933,400 | 13.21 (12.8) | 91.17 | 5.40 | 1.48 | 3.63 | 0.54 | 88.62 |

21 | 266,787 | 7.30 (7.4) | 22.85 | 5.30 | 1.39 | 2.85 | 0.34 | 84.19 |

22 | 355,830 | 7.99 (7.8) | 27.65 | 5.27 | 1.29 | 2.84 | 0.26 | 77.52 |

23 | 1,074,614 | 12.34 (12.5) | 80.72 | 5.92 | 1.81 | 3.85 | 0.72 | 89.03 |

24 | 674,759 | 11.90 (11.2) | 74.51 | 5.77 | 1.27 | 4.24 | 0.40 | 96.51 |

25 | 373,704 | 7.39 (7.4) | 28.16 | 5.21 | 1.36 | 3.33 | 0.49 | 92.30 |

26 | 410,720 | 9.42 (9.8) | 40.33 | 5.61 | 1.62 | 3.46 | 0.50 | 77.60 |

mean | 1,006,255 | 11.36 (11.43) | 81.41 | 5.72 | 1.49 | 3.82 | 0.52 | 86.45 |

^{2}equaling 0.965 and 0.968, respectively. A visualization of a group of eight neighboring trees can be seen in Figure 3, depicting overlapping crowns. Low scan resolution together with occlusion effects causes a low density of data points, leading to incorrectly modeled branches located in the upper crowns.

**Figure 10.**Visualization of the artificial tree results (height ≈ 18 m): (g) artificial point cloud; (h) resulting cylinder model with branches colored differently; (i) volume distribution in both the original data and program results.

**Figure 12.**Two allometric models for volume estimation derived from 24 Group III models using TLS-derived DBH as the input variable: (a) total above-ground volume modeled as a function of DBH; (b) solid volume modeled as a function of DBH.

## 6. Conclusions

## 7. Future Work

## Appendix

#### **A. Clustering**

^{2}) with an expected runtime of (n log(n)).

#### **B. Least Squares Fit for Cylinders and Circles**

_{i}from a point, p

_{i}, to the approximated circle or cylinder, one can solve Equation (A1), with being the Jacobian matrix.

#### B.1. Cylinder Least Squares Fit

_{s}, an end point, c

_{e}, and a radius, r, is already known. The initial parameters of a cylinder, c(x

_{0}, y

_{0}, z

_{0}, a, b, c, r), can be derived from c

_{s}, c

_{e}and r.

_{0}, y

_{0}, a, b, r) by:

- a normalization of d
_{a}, so c can be computed from a and b - a translation of p
_{0}to the origin of the coordinate system and a rotation of the input data, so d_{a}is along the z-axis; therefore, z_{0}can be determined from a, b, x_{0}, y_{0}.

**U**defined as in Equation (A7), replacing with .

_{i}of a point, , to the cylinder in Equation (A4):

- The standard deviation of the points to the fitted cylinder falls below a threshold of 3 mm.
- A maximum number of 30 iterations is reached.

#### B.2. Circle Least Squares Fit

^{2}. As a cross-sectional cut through the point cloud is coplanar, first, the best fit plane procedure (refer to Section C) is applied to calculate plane Pl( , p). A rotation of P is applied, so that is parallel to the z-axis. Rotation-matrix

**U**is computed by a multiplication of a rotation,

**U**

_{1}, around the x-axis to bring into the x, z plane and a second rotation,

**U**

_{2}, around the y-axis directing along the z-axis.

**U**is described in Equation (A7), with α being the angle between and the x, z-plane and β the angle between

**U**

_{1}and the z-axis.

^{r}is now translated into ℝ

^{2}by using only x, y-coordinates of ∈ P

^{r}. For an initial estimation of x

_{0}, y

_{0}and r of a circle, c(x

_{0}, y

_{0}, r), we first minimize the distance function given in Equation (A8), with r

_{i}being the distance of p

_{1}to p

_{0}(x

_{0}, y

_{0}):

_{0}, y

_{0}, r, we can minimize , with d

_{i}being defined in Equation (A11):

_{i}leads to the linear least squares system in Equation (A12):

_{0}:= x

_{0}+ p

_{x0}

y

_{0}:= y

_{0}+ p

_{y0}

r := r + p

_{r}

- The standard deviation of P
^{r}to the fitted circle falls below a threshold of 3 mm. - A maximum number of 30 iterations is reached.

#### **C. Plane Fitting Using Principal Component Analysis**

_{i}|p

_{i}∈ P ∧ i ∈ {1, ..., n}}

**C**) is formed according to Equation (A15).

**C**is a 3 × 3, real, positive, semi-definite matrix [59].

**C**can be decomposed into the eigenvectors, , satisfying the linear Equation (A16), where λ

_{i}are the eigenvalues corresponding to each .

_{1}> λ

_{2}> λ

_{3}. Then, is defined as the first principal component of P, being the vector oriented along the direction of the greatest variance of P. All principal components are orthogonal to each other. The fitted plane in point-normal form is defined by p and .

#### **D. Required Software Interaction**

## Acknowledgments

## Author Contributions

^{1,2,3,5,6,7,8}, Christopher Morhart

^{1,2,3,7,8}, Jonathan Sheppard

^{1,2,3,7,8}, Heinrich Spiecker

^{1,8,9}and Mathias Disney

^{4,8}.

## Conflicts of Interest

## References

- Picard, N.; Saint-André, L.; Henry, M. Manual for Building Tree Volume and Biomass Allometric Equations: From Field Measurement to Prediction; Food and Agricultural Organization of the United Nations: Rome, Italy; Centre de Cooperation Internationale en Recherche Agronomique pour leDeveloppement: Montpellier, France, 2012. [Google Scholar]
- Van Laar, A.; Akça, A. Forest Mensuration; Springer London, Limited: London, UK, 2007; Volume 13. [Google Scholar]
- GlobeAllomeTree. Available online: http://www.globallometree.org/ (accessed on 20 November 2013).
- Zianis, D.; Seura, S.M. Biomass and Stem Volume Equations for Tree Species in Europe; Finnish Society of Forest Science, Finnish Forest Research Institute: Vantaa, Finland, 2005; Volume 4. [Google Scholar]
- Leeuwen, M.; Nieuwenhuis, M. Retrieval of forest structural parameters using LiDAR remote sensing. Eur. J. For. Res.
**2010**, 129, 749–770. [Google Scholar] [CrossRef] - White, J.; Wulder, M.; Varhola, A.; Vastaranta, M.; Coops, N.; Cook, B.; Pitt, D.; Woods, M. A Best Practices Guide for Generating Forest Inventory Attributes From Airborne Laser Scanning Data Using an Area-Based Approach; Information Report FI-X-010; Natural Resources Canada, Canadian Forest Service, Canadian Wood Fibre Centre: Victoria, Canada, 2013. [Google Scholar]
- Simonse, M.; Aschoff, T.; Spiecker, H.; Thies, M. Automatic determination of forest inventory parameters using terrestrial laser scanning. In Proceedings of the ScandLaser Scientific Workshop on Airborne Laser Scanning of Forests, Umea, Sweden, 19 September 2003; pp. 252–258.
- Thies, M.; Pfeifer, N.; Winterhalder, D.; Gorte, B.G. Three-dimensional reconstruction of stems for assessment of taper, sweep and lean based on laser scanning of standing trees. Scand. J. For. Res.
**2004**, 19, 571–581. [Google Scholar] [CrossRef] - Bienert, A.; Scheller, S.; Keane, E.; Mullooly, G.; Mohan, F. Application of terrestrial laser scanners for the determination of forest inventory parameters. In Proceedings of the International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, Dresden, Germany, 25–27 September 2006; Volume 36.
- Eysn, L.; Pfeifer, N.; Ressl, C.; Hollaus, M.; Grafl, A.; Morsdorf, F. A practical approach for extracting tree models in forest environments based on equirectangular projections of terrestrial laser scans. Remote Sens.
**2013**, 5, 5424–5448. [Google Scholar] [CrossRef] - Moskal, L.M.; Zheng, G. Retrieving forest inventory variables with terrestrial laser scanning (TLS) in urban heterogeneous forest. Remote Sens.
**2011**, 4, 1–20. [Google Scholar] [CrossRef] - Fritz, A.; Kattenborn, T.; Koch, B. UAV-Based Photogrammetric Point Clouds—Tree Stem Mapping in Open Stands in Comparison to Terrestrial Laser Scanner Point Clouds. In Proceedings of the ISPRS—International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Rostock, Germany, 4–6 September 2013; Volume XL-1/W2, pp. 141–146.
- Litkey, P.; Liang, X.; Kaartinen, H.; Hyyppä, J.; Kukko, A.; Holopainen, M. Single-scan TLS methods for forest parameter retrieval. In Proceedings of SilviLaser 2008, 8th International Conference On LiDAR Applications in Forest Assessment and Inventory, Edinburgh, UK, 17–19 September 2008; pp. 295–304.
- Liang, X.; Hyyppä, J.; Kaartinen, H.; Holopainen, M.; Melkas, T. Detecting Changes in Forest Structure over Time with Bi-Temporal Terrestrial Laser Scanning Data. ISPRS Int. J. Geo-Inf.
**2012**, 1, 242–255. [Google Scholar] [CrossRef] - Liang, X.; Litkey, P.; Hyyppä, J.; Kaartinen, H.; Vastaranta, M.; Holopainen, M. Automatic stem mapping using single-scan terrestrial laser scanning. IEEE Trans. Geosci. Remote Sens.
**2012**, 50, 661–670. [Google Scholar] [CrossRef] - Pfeifer, N.; Gorte, B.; Winterhalder, D. Automatic reconstruction of single trees from terrestrial laser scanner data. In Proceedings of the 20th ISPRS Congress, Istanbul, Turkey, 12–23 July 2004; pp. 114–119.
- Bucksch, A.; Lindenbergh, R. {CAMPINO} A skeletonization method for point cloud processing. ISPRS J. Photogramm. Remote Sens.
**2008**, 63, 115–127, Theme Issue: Terrestrial Laser Scanning. [Google Scholar] [CrossRef] - Bucksch, A.; Lindenbergh, R.; Menenti, M. SkelTre. Vis. Comput.
**2010**, 26, 1283–1300. [Google Scholar] [CrossRef] - Bucksch, A. Revealing the Skeleton from Imperfect Point Clouds. Ph.D. Thesis, Technische Universiteit Delft, Delft, The Netherlands, 2011. [Google Scholar]
- Meagher, D. Geometric modeling using octree encoding. Comput. Graph. Image Process.
**1982**, 19, 129–147. [Google Scholar] [CrossRef] - Xu, H.; Gossett, N.; Chen, B. Knowledge and Heuristic-based Modeling of Laser-scanned Trees. ACM Trans. Graph.
**2007**, 26. [Google Scholar] [CrossRef] - Dijkstra, E. A note on two problems in connexion with graphs. Numer. Math.
**1959**, 1, 269–271. [Google Scholar] [CrossRef] - Livny, Y.; Yan, F.; Olson, M.; Chen, B.; Zhang, H.; El-Sana, J. Automatic reconstruction of tree skeletal structures from point clouds. ACM Trans. Graph.
**2010**, 29, 151:1–151:8. [Google Scholar] - Yan, D.M.; Wintz, J.; Mourrain, B.; Wang, W.; Boudon, F.; Godin, C. Efficient and robust reconstruction of botanical branching structure from laser scanned points. In Proceedings of the 11th IEEE International Conference on Computer-Aided Design and Computer Graphics, Huangshan, China, 19–21 August 2009; pp. 572–575.
- Bentley, J.L. Multidimensional binary search trees used for associative searching. Commun. ACM
**1975**, 18, 509–517. [Google Scholar] [CrossRef] - Cohen-Steiner, D.; Alliez, P.; Desbrun, M. Variational shape approximation. ACM Trans. Graph.
**2004**, 23, 905–914. [Google Scholar] [CrossRef] - Bayer, D.; Seifert, S.; Pretzsch, H. Structural crown properties of Norway spruce (Picea abies [L.] Karst.) and European beech (Fagus sylvatica [L.]) in mixed versus pure stands revealed by terrestrial laser scanning. Trees
**2013**, 27, 1035–1047. [Google Scholar] [CrossRef] - Edelsbrunner, H.; Mücke, E.P. Three-dimensional alpha shapes. ACM Trans. Graph.
**1994**, 13, 43–72. [Google Scholar] [CrossRef] - Belton, D.; Moncrieff, S.; Chapman, J. Processing Tree Point Clouds Using Gaussian Mixture Models. In Proceedings of the ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Antalya, Turkey, 11–13 November 2013; Volume II-5/W2, pp. 43–48.
- Reynolds, D. Gaussian Mixture Models. In Encyclopedia of Biometrics; Springer Science + Business Media: New York, USA, 2009; pp. 659–663. [Google Scholar]
- Dassot, M.; Colin, A.; Santenoise, P.; Fournier, M.; Constant, T. Terrestrial laser scanning for measuring the solid wood volume, including branches, of adult standing trees in the forest environment. Comput. Electron. Agric.
**2012**, 89, 86–93. [Google Scholar] [CrossRef] - Raumonen, P.; Kaasalainen, M.; Åkerblom, M.; Kaasalainen, S.; Kaartinen, H.; Vastaranta, M.; Holopainen, M.; Disney, M.; Lewis, P. Fast automatic precision tree models from terrestrial laser scanner data. Remote Sens.
**2013**, 5, 491–520. [Google Scholar] [CrossRef] - Jayaratna, S. Baumrekonstruktion aus 3D-Punktwolken. Diploma Thesis, Institut für Informatik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany, 2009. [Google Scholar]
- Schnabel, R.; Wahl, R.; Klein, R. Efficient RANSAC for point-cloud shape detection. Comput. Graph. Forum
**2007**, 26, 214–226. [Google Scholar] [CrossRef] - Dassot, M.; Constant, T.; Fournier, M. The use of terrestrial LiDAR technology in forest science: Application fields, benefits and challenges. Ann. For. Sci.
**2011**, 68, 959–974. [Google Scholar] [CrossRef] - Data Sheet Z + F IMAGER 5010. Available online: http://www.zf-laser.com/fileadmin/editor/Datenblaetter/Datasheet_Z_F_IMAGER_5010_E_kompr_01.pdf (accessed on 18 November 2013).
- Web-based Weather Request and Distribution System (WebWerdis). Available online: http://www.dwd.de/webwerdis (accessed on 18 November 2013).
- Morhart, C.; Sheppard, J.; Spiecker, H. Above Ground Leafless Woody Biomass and Nutrient Content within Different Compartments of a P. maximowicii x P. trichocarpa Poplar Clone. Forests
**2013**, 4, 471–487. [Google Scholar] [CrossRef] - Disney, M.; Kalogirou, V.; Lewis, P.; Prieto-Blanco, A.; Hancock, S.; Pfeifer, M. Simulating the impact of discrete-return lidar system and survey characteristics over young conifer and broadleaf forests. Remote Sens. Environ.
**2010**, 114, 1546–1560. [Google Scholar] [CrossRef] - Disney, M.; Lewis, P.; Gomez-Dans, J.; Roy, D.; Wooster, M.; Lajas, D. 3D radiative transfer modelling of fire impacts on a two-layer savanna system. Remote Sens. Environ.
**2011**, 115, 1866–1881. [Google Scholar] [CrossRef] - Data Sheet Laser Control. Available online: http://www.zf-laser.com/fileadmin/editor/Broschueren/Z_F_LaserControl_kompr.pdf (accessed on 18 November 2013).
- Java Standart Edition Development Kit. Available online: http://www.oracle.com/technetwork/java/javase/downloads/jdk7-downloads-1880260.html (accessed on 18 November 2013).
- Java 3D. Available online: http://www.oracle.com/technetwork/java/javase/tech/index-jsp-138252.html (accessed on 19 May 2014).
- Jama 1.0.3. Available online: http://math.nist.gov/javanumerics/jama/ (accessed on 18 November 2013).
- CyberVRML97 for Java. Available online: http://sourceforge.net/projects/cv97java/ (accessed on 20 November 2013).
- MeshLab. Available online: http://sourceforge.net/projects/meshlab/ (accessed on 20 November 2013).
- CloudCompare. Available online: http://www.danielgm.net/cc/ (accessed on 18 November 2013).
- Chand, D.R.; Kapur, S.S. An Algorithm for convex polytopes. J. ACM
**1970**, 17, 78–86. [Google Scholar] [CrossRef] - De Berg, M.; van Kreveld, M.; Overmars, M.; Schwarzkopf, O.C. Computational Geometry; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
- Gander, W.; Golub, G.; Strebel, R. Least-squares fitting of circles and ellipses. BIT Numer. Math.
**1994**, 34, 558–578. [Google Scholar] [CrossRef] - Pratt, V. Direct least-squares fitting of algebraic surfaces. SIGGRAPH Comput. Graph.
**1987**, 21, 145–152. [Google Scholar] [CrossRef] - Ahn, S.J.; Rauh, W.; Warnecke, H.J. Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola. Pattern Recognit.
**2001**, 34, 2283–2303. [Google Scholar] - Marquardt, D. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math.
**1963**, 11, 431–441. [Google Scholar] [CrossRef] - Zobel, B.J.; Buijtenen, J.V. Wood Variation: Its Causes and Control; Springer-Verlag: Berlin/Heidelberg, Germany, 1989. [Google Scholar]
- SimpleTree Project. Available online: http://www.simpletree.uni-freiburg.de/ (accessed on 19 May 2014).
- Disney, M.; Raumonen, P.; Lewis, P. Testing a new vegetation structure retrieval algorithm from terrestrial lidar scanner data using 3D models. In Proceeding of the SilviLaser 2012, Vancouver, BC, Canada, 16–19 September 2012.
- Ester, M.; Kriegel, H.P.; Sander, J.; Xu, X. A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise. In Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, Portland, OR, USA, 2–4 August 1996; pp. 226–231.
- SphereFitting. Available online: http://www.caves.org/section/commelect/DUSI/openmag/pdf/SphereFitting.pdf (accessed on 27 November 2013).
- El-Halawany, S.; Lichti, D. Detection of Road Poles from Mobile Terrestrial Laser Scanner Point Cloud. In Proceedings of the 2011 International Workshop on Multi-Platform/Multi-Sensor Remote Sensing and Mapping (M2RSM), Xiamen, China, 10–12 January 2011; pp. 1–6.

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**MDPI and ACS Style**

Hackenberg, J.; Morhart, C.; Sheppard, J.; Spiecker, H.; Disney, M.
Highly Accurate Tree Models Derived from Terrestrial Laser Scan Data: A Method Description. *Forests* **2014**, *5*, 1069-1105.
https://doi.org/10.3390/f5051069

**AMA Style**

Hackenberg J, Morhart C, Sheppard J, Spiecker H, Disney M.
Highly Accurate Tree Models Derived from Terrestrial Laser Scan Data: A Method Description. *Forests*. 2014; 5(5):1069-1105.
https://doi.org/10.3390/f5051069

**Chicago/Turabian Style**

Hackenberg, Jan, Christopher Morhart, Jonathan Sheppard, Heinrich Spiecker, and Mathias Disney.
2014. "Highly Accurate Tree Models Derived from Terrestrial Laser Scan Data: A Method Description" *Forests* 5, no. 5: 1069-1105.
https://doi.org/10.3390/f5051069