Abstract
Modeling branch taper curve and form factor contributes to increasing the efficiency of tree crown reconstruction: the branch taper, defined as the sequential measure of diameters along the course of the branch, is pivotal to accurately estimate key branch variables such as biomass and volume. Branch diameters or volumes have commonly been estimated from terrestrial laser scanning (TLS) based on automatized voxelization or cylinder-fitting approaches, given the whole branch length is sufficiently covered by laser reflections. The results are, however, often affected by ample variations in point cloud characteristics caused by varying point density, occlusions, and noise. As these characteristics of TLS have been difficult to be sufficiently controlled or eliminated in automatized-techniques, we proposed a new branch-based taper curve model and form factor (BFF), which can be employed directly from the laser reflections and under variable point cloud characteristics.
In this paper, the approach is demonstrated on primary branches using a set of TLS-derived diameter datasets from a sample of 20 trees of 6 species. The result shows an improvement in the accuracy of the diameter estimates and, at best, enabled for predicting encompassing finer branch scales (<10 cm), with R2 of 0.86 and a mean relative absolute error of 1.03 cm (29%) when validated with field-measured diameters. This approach was also capable of retrieving branch diameters for a large percentage of explicitly identified primary branches (>85%) directly from the filtered points when validated with panoramic images acquired concurrently with laser scanning. Frequently used automatized crown reconstructions from the quantitative structural model (QSM), on the other hand, was largely obscured by discrepancies in the point clouds, with the crown-tops and finer branches being the most critical.
Furthermore, our approach provides mean BFF of 0.35 and 0.49 with the diameters determined from 5% and 10% of the total branch length, respectively, which may have the potential to produce branch volume information with reasonable accuracy from only knowing the length and respective diameter.
Although our model can be regarded as a first approximation to the taper curve and form factor for the primary branches on a relatively small set of samples, the approach can further our understanding of alternative ways to improve the accuracy of the assessment of branch diameter and volume. The approach may also be extended to other branch orders. This could expand the horizon for volumetric calculations and biomass estimates from non-destructive TLS proxies in tree crowns.

Keywords: Branch diameter; Branch form factor; Branch taper curve; Branch volume; QSM; TLS

Methods
Terrestrial laser scanning (TLS) was used to acquire point cloud datasets of individual trees during the leaf-off season. Scanning was performed for six tree species using Trimble TX5 3D Laser Scanner (Trimble Inc., Sunnyvale, CA, USA). The point clouds are collected based on the scanner setup with 43 million points (mpts) per scan with a quality setting 4x. With this setup, the scan duration was ~11 minutes. We scanned each tree from six scan positions, distributed circularly at a regular interval of 60 from each other.
The processed datasets comprise comma-delimited files (.csv), Excel (.xlsx), Rscript (.R), sample tree point clouds (.asc + .las), quantitative structural model (QSM) pipeline scripts (.xsct2), and the corresponding fitted cylinders (.ply).
Data1.csv file provides the list of tree species with the details of stem and crown variable metrics. All the variables were determined and computed directly from the laser reflections using CloudCompare software version 2.12.0 (CloudCompare, 2021). Data2.csv is the dataset used for the modeling of the branch taper curve. The series of diameters (D) along the branch was determined from the point slice of TLS laser reflections. Initially, a series of thin point slices (~10 mm long cross-sections) at 0.50 m intervals along the course of the branch were cross-sectioned using CloudCompare software by flattening the branch point clouds horizontally. Each slice was then transformed to vertical straightness to correct the effect of the branch inclination. We performed the correction based on a three-dimensional rotation matrix using CloudCompare software by picking up three points after leveling the slice horizontally. After inclination correction, we projected the slices horizontally in the xy plane and then applied the Random Sample Consensus (RANSAC) circle-fitting algorithm (Fischler and Bolles, 1981). The package TreeLS (de Conto, 2020) in R language (RStudio Team, 2021) was used for RANSAC circle-fitting. This served as a non-destructive TLS-based dataset used to fit the branch taper curve and calculate the respective branch form factor (BFF). Data3.xlsx comprises the model prediction and validation datasets. The ground-truth manual diameter measurements were collected in the field using a caliper for the few branches in the lower parts of the crown that were reached from a standing position. Data4.csv comprises the BFF determined at two reference diameters: 5% (BFF0.05) and 10% (BFF0.10) of the branch relative lengths. The BFF is computed based on the ratio of the actual branch volume to the volume of the cylinder over the reference diameters. The actual branch volumes were calculated based on fitted taper curves. The Taper curve of individual isolated sample branches was fitted based on the cubic spline model. From the smoothed cubic spline model, the volume was calculated at every 0.1 m section. The volume of each section was calculated based on Hubers formula. The total actual branch was obtained by summing the volumes of overall sections from the base (0 m) up to the total branch length.
TaperCurve-BFF.R is the Rscript for the taper curve using the multivariate spline regression model. The optimal sub-model was determined using the generalized cross-validation (GCV) approach. The modeling process was performed using the Earth package in R language (Milborrow, 2021). The script also comprises the BFF calculations.
PSP28.asc; PSP28.las; APS30.asc; and APS30.las are two examples of filtered tree point clouds (Platanus species and Acer platanoides). MN-QSM-asc.xsct2 and MN-QSM-las.xsct2 are QSM cylinder pipeline model scripts developed for asc and las points, respectively, following Hackenberg et al. (2015) using the Computree software version 5.0.221b (Computree, 2021) with a plugin of SimpleForest version 5.3.2 (SimpleForest, 2021). PSP28.ply and APS30.ply are the corresponding QSM cylinder fits.

Reference
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