Tree Trunk Curvature Extraction Based on Terrestrial Laser Scanning Point Clouds

Abstract

The degree of tree curvature exerts a significant influence on the utilization of forestry resources. This study proposes an enhanced quantitative structural modeling (QSM) method, founded upon terrestrial laser scanning (TLS) point cloud data, for the precise extraction of 3D curvature characteristics of tree trunks. The conventional approach operates under the assumption that the tree trunk constitutes an upright rotating body, thereby disregarding the tree trunk’s true curvature morphology. The proposed method is founded on the classical QSM algorithm and introduces two zoom factors that can dynamically adjust the fitting parameters. This improvement leads to enhanced accuracy in the representation of tree trunk curvature and reduced computational complexity. The study utilized 146 sample trees from 13 plots in Jixi, Anhui Province, which were collected and pre-processed by TLS. The study combines point cloud segmentation, manual labeling of actual curvature and dual-factor experiments, and uses quadratic polynomials and simulated annealing algorithms to determine the optimal model factors. The validation results demonstrate that the enhanced method exhibits a greater degree of concordance between the predicted and actual curvature values within the validation set. In the regression equation, the coefficient of the two-factor method for fitting a straight line is 0.95, which is substantially higher than the 0.75 of the one-factor method. Furthermore, the two-factor model has an R² of 0.21, indicating that the two-factor optimization method generates a significantly smaller error compared to the one-factor model (with an R² of 0.12). In addition, this study discusses the possible reasons for the error in the results, as well as the shortcomings and outlook. The experimental results demonstrate the augmented method’s capacity to accurately reconstruct the 3D curvature of tree trunks in most cases. This study provides an efficient and accurate method for conducting fine-grained forest resource measurements and tree bending studies.

1. Introduction

Forests represent a vital component of terrestrial ecosystems, distinguished by their rich biodiversity, intricate structures and diverse functions. In recent years, there has been a proliferation of studies that have employed remote sensing techniques for the purpose of monitoring forest carbon emissions [1]. Furthermore, forest ecosystems are closely linked to human health, as they contribute to air purification, the improvement of living environments and the provision of various natural and social benefits, including resources and habitats [2]. Precise measurement of tree structures is a fundamental technical step in bridging large-scale forest carbon sequestration assessments with physiological studies of individual trees [3]. It is also a key process in forestry economics, improving timber yield and supporting forest cultivation practices to mitigate pest infestations [4]. Furthermore, forest inventories play a vital role in strategic forest management planning and daily forestry operations, such as thinning and harvesting. Accurate inventory data have been shown to maximize the value of harvested timber and enhance operational efficiency [5].

The advent of remote sensing technology and TLS has precipitated a paradigm shift in forestry science, ecology and environmental monitoring. The utilization of point cloud data for forest resource inventory has emerged as a pivotal tool, eclipsing conventional measurement methods in terms of precision and accuracy. TLS has been demonstrated to achieve millimeter-level accuracy in delineating the intricate three-dimensional structure of trees, a feat unattainable by traditional methods [6,7]. Its non-destructive nature and exceptional precision render it a formidable instrument for acquiring three-dimensional spatial information. The utilization of point cloud data for 3D modeling facilitates not only a more precise analysis of forest spatial distribution but also provides fundamental data for the study of tree growth patterns, mechanical properties and ecological functions [8,9].

In recent years, considerable research has been conducted on forest inventory using TLS, demonstrating significant advantages in estimating tree trunk parameters. For instance, Chen et al. [10] estimated tree height using TLS technology, while other studies have focused on parameters such as trunk diameter and crown width [11]. Traditional standing tree measurement methods directly determine the DBH (Diameter at Breast Height) and tree height, typically assuming the tree trunk as a straight rotational body. Consequently, this has led to the development of various methods for volume calculation and trunk-related equations [12]. However, it should be noted that tree trunks exhibit varying degrees of curvature at different positions, which makes it difficult for two-dimensional trunk axes to accurately represent their true characteristics [13]. Trunk curvature provides a more precise morphological description, especially in cases where the trunk is irregular, bent, or has branches. The influence of curvature becomes even more significant under such conditions. Conventional tree volume calculation methods frequently neglect variations in trunk axes in three-dimensional space, relying exclusively on three-dimensional fitting techniques to extract the central axis of a cylinder [14] to determine parameters such as trunk diameter. However, research on three-dimensional trunk curvature remains limited, particularly with regard to the inversion algorithm of tree trunk curvature.

The current literature delineates two-dimensional (2D) and three-dimensional (3D) approaches as the prevailing methods for estimating DBH using TLS data, with each approach exhibiting distinct advantages and limitations [15]. The most common approach involves 2D fitting of the trunk, with the least squares algorithm being demonstrated as the most accurate among several circle-fitting methods in [16]. However, this method disregards the influence of trunk curvature. Conversely, researchers have employed 3D methods, such as cylinder fitting, to achieve more accurate results for inclined trees. The QSM algorithm is a widely used approach for 3D reconstruction of trees based on TLS point cloud data [16]. It effectively addresses the challenge of fitting tree models with consecutive cylinders. The 3D cylindrical model derived from QSM fitting provides an intuitive representation of the tree’s actual growth conditions. Furthermore, even in cases where some point cloud data are missing, this method can still establish the cylindrical model and extract relevant parameter factors [17]. Due to its effectiveness, QSM has been extensively applied in the construction of tree 3D models, which are subsequently used to extract tree parameters. For instance, De et al. (2018) [18] reconstructed the 3D structure of trees from TLS-acquired point cloud data using the QSM algorithm and then extracted structural parameters to estimate tree volume. Similarly, Muumbe et al. (2021) [19] applied QSM to model the 3D structure of tropical rainforest trees using TLS data and subsequently estimated the aboveground biomass of individual trees. Sheppard et al. (2016) [20] utilized TLS data from 21 wild cherry trees in southern Germany to construct QSM models, from which tree height, DBH, trunk volume and branch volume were extracted. QSM has undergone multiple refinements to improve accuracy, leading to the development of various models such as TreeQSM [21], PypeTree [22] and SimpleTree [23], all of which have demonstrated high precision in trunk cylinder fitting tasks. However, these 3D methods incur higher computational costs, limiting their practicality for large-scale applications. There have been several studies emphasizing the need to balance accuracy and computational efficiency [24], highlighting the necessity of developing more efficient methods for extracting tree trunk parameters. In terms of parameter selection, the QSM algorithm employs an empirical model based on upright trees, requiring eight iterations to determine the optimal solution. However, the modeling results obtained by this method are not necessarily optimal, and it also has a high computational complexity.

Furthermore, the majority of extant studies on tree structure are predicated on the assumption that tree trunks are upright rather than curved [25], thus failing to account for the actual bending observed in trees. However, under natural conditions, tree bending inevitably occurs [26], leading to discrepancies between the fitted trunk parameters and the true values. Currently, research on tree trunk curvature remains limited. Existing studies have primarily computed trunk curvature in a two-dimensional plane [27]. From a spatial perspective, Paradis et al. (2013) [28] investigated stress wave velocity measurements in a plantation of hybrid poplars and mentioned the calculation of curvature for trunk sections above 10 cm from the ground. However, this method relies heavily on constructing the three-dimensional trunk axis curve, which is highly dependent on the quality of the point cloud data. When point cloud data exhibit irregularities, severe occlusions, or missing information, the accuracy of curvature estimation is significantly reduced.

The present study proposes a novel algorithm for retrieving tree trunk curvature based on quantitative structure modeling (QSM). In this algorithm, two zoom factors are introduced to dynamically optimize the input parameters of the traditional QSM algorithm for each individual tree. The method under consideration here calculates curvature in full three-dimensional space by fitting orthogonal profiles, and is validated using a 70:30 training–validation split on 146 sample trees across 13 forest plots. The primary innovations of this study are enumerated below: (1) The introduction of dual zoom factors is proposed as a means of enhancing the adaptability of QSM parameterization to bent trunks. It is hypothesized that this will result in improved fitting flexibility and reduced computational complexity. (2) The extension of the QSM framework for curvature estimation is a significant development in the field, enabling accurate extraction of 3D trunk bending features even under conditions of reduced point cloud quality. (3) The integration of curvature modeling into the QSM-based pipeline represents a novel and scalable approach for high-resolution quantification of tree structural traits. To mitigate measurement errors caused by missing data, this study used the maximum curvature value measured in the XOZ plane (laser emission direction) and the YOZ plane (perpendicular direction) as the overall trunk curvature index. This dual-projection curvature extraction approach effectively reduces errors caused by deficiencies in the 3D point cloud while providing a more comprehensive representation of the trunk’s actual three-dimensional bending characteristics.

2. Materials and Methods

2.1. Study Area

The study site is located in Jixi County, Xuancheng City, Anhui Province, China, and includes a total of 146 sample trees across 13 different plots, each of which is square-shaped with an area of 10 × 10 m². The distribution of the study plots is shown in Figure 1. The dominant tree species in this region include Chinese fir (Cunninghamia lanceolata), Masson pine (Pinus massoniana), Huangshan pine (Pinus hwangshanensis) and Mongolian oak (Quercus mongolica). As illustrated in Figure 1b, the forest is densely distributed, and each study plot is labeled using the format Plot Number—Number of Sample Trees in the Plot. While species-level sampling was not differentiated in this study, the average height of all sampled trees was approximately 19.6 m. According to regional forestry data, the typical height ranges for these species are as follows: Chinese fir (18–22 m), Masson pine (17–21 m), Huangshan pine (20–23 m) and Mongolian oak (15–18 m).

2.2. TLS Measurements

The TLS measurements were conducted in April 2024 utilizing a handheld sensor (GoSlam, manufactured by Beijing Tianke Zhizao Aviation Technology Co., Beijing, China) and a tripod-mounted sensor (Faro, manufactured by FARO Technologies, Inc., Lake Mary, FL, USA). The specific parameters are enumerated in

Table 1. Plots topography and sample information.

Plot_ID	Aspect	Elevation	Average Treeheight	Slope
Plot01	229	167	13.55	3
Plot02	113	194	8.25	5
Plot03	155	254	13.51	20
Plot04	247	202	8.86	2
Plot05	62	194	13.44	27
Plot06	342	204	8.95	7
Plot07	23	270	8.84	16
Plot08	203	176	9.89	16
Plot09	328	252	15.77	12
Plot10	69	209	10.83	18
Plot11	234	253	17.25	15
Plot12	242	424	15.24	15
Plot13	335	196	13.67	17

, with additional parameters available in the user manuals (GoSlam: (https://www.goslam.com/IndustryApp/T100-MT (accessed on 21 March 2025)) and Faro: (https://knowledge.faro.com/Hardware/Focus/Focus/User_Manuals_and_Quick_Start_Guides_for_the_Focus_Laser_Scanner?mt-learningpath=focus_downloads (accessed on 21 March 2025))). Multiple reference targets were placed in each plot, and their positions were used as reference points for scanning to ensure a complete capture of the tree trunks. Due to hardware limitations and the lack of manufacturer disclosure, precise beam density in cubic meters could not be obtained. Nonetheless, the captured point clouds were visually inspected to ensure sufficient coverage and quality for 3D modeling, and no issues in QSM reconstruction were observed due to data sparsity.

2.3. Data Preprocessing

In this study, LiDAR360 v8.1 was utilized for data preprocessing, encompassing point cloud extraction, preprocessing, trunk segmentation and vector point plotting. For the point cloud data from the 13 study plots, procedures such as noise removal, ground filtering employing the Cloth Simulation Filter (CSF) and normalization were implemented to eliminate interference points surrounding tree trunks and mitigate the impact of ground undulations. Segmentation and trunk extraction were conducted based on the Comparative Shortest Path (CSP) algorithm. This algorithm primarily consists of two key steps: trunk detection and crown segmentation. The final segmentation results achieved a kappa coefficient ranging from 0.83 to 0.93, with an error rate between 5% and 16%. When trees were free of leaf occlusion, the segmentation error was only 6.1%, whereas for trees with leaf occlusion, the segmentation error increased to 13.2% [29]. Additionally, manual refinement was performed to remove individual points that were clearly outside the tree crown. These erroneous point clouds mainly resulted from errors in individual tree segmentation and reflections from airborne objects.

In this study, the authors obtained tree trunk curvature vectors by manually selecting trunk contour points on the point cloud projection surface, thereby capturing the edge curvature of the trunk. Simultaneously, they extracted the trunk center-line features in a direction perpendicular to the projection surface. These two sets of data served as references for the ground-truth curvature. However, due to noise, occlusion and partial data loss in the 3D point cloud, directly locating precise surface points in 3D space is challenging. Moreover, performing 3D curve fitting based on inaccurate vector points can introduce significant errors. To overcome these limitations, this study reconstructed trunk contours on two key planes: Laser emission plane (Figure 2a): This plane reflects the edge characteristics of the trunk in the direction of laser projection. Plane perpendicular to the laser emission plane (Figure 2b): This plane approximately represents the trunk centerline in the corresponding direction. On both planes, curvature curves were manually drawn, capturing the bending characteristics of the trunk in different directions. Due to the reduced point cloud density on the posterior aspect of the trunk and the consequent severe data loss, direct 3D curvature estimation was deemed unreliable.

2.4. Quantitative Structure Modeling (QSM) Algorithm Based on a Single Zoom Factor

Single-tree QSM is a method proposed by Pasi Raumonen et al. (2013) [30] to rapidly and automatically generate accurate tree models from TLS point clouds of trunks and branches. The method employs a hierarchical set of cylinders to reconstruct the quantitative structural model of trees, thereby estimating the topological, geometric and volumetric details of tree woody structures. The primary steps involved in QSM are outlined below: (1) Noise and leaf removal: This involves the elimination of noise points and leaf points surrounding the tree. (2) Topological reconstruction of branch structures: This step involves the segmentation of the point cloud into trunks and branches. (3) Geometric reconstruction of individual tree surfaces: Constructing a cylinder-based model. The original QSM code can be accessed via Supplementary Materials. This study is founded on the implementation of customized modifications to version 2.4.1, with the objective of facilitating curvature inversion tasks. The objective of this research was to develop a QSM-based tree trunk curvature retrieval algorithm capable of accurately capturing trunk bending characteristics along the vertical direction. Building on the traditional QSM approach, this study introduces a multi-segment cylinder fitting strategy, wherein the trunk is modeled as a series of short, consecutive cylinders. This strategy facilitates the estimation of local orientation changes along the trunk axis, which are subsequently utilized to compute curvature.

In order to enhance the adaptability of the model and to control the level of geometric detail, the algorithm integrates the QSM optimization strategy proposed, which introduces a zoom factor as a tuning parameter. While earlier work proposed a general single-parameter optimization framework, it did not include any curvature-related modeling or profile-based fitting. The present study is the first to extend that framework to three-dimensional trunk curvature extraction. This zoom factor adjusts the spatial resolution of cylinder fitting, specifically by scaling the parameters PatchDiam2Min and PatchDiam2Max, which define the minimum and maximum patch sizes in the cylinder fitting process. The following equations were used to modify these parameters:

$$PatchDiam2Min=0.02\times zoo{m}_{factor}$$

(1)

$$PatchDiam2Max=0.07\times zoo{m}_{factor}$$

(2)

In the course of the optimization process for identifying the most suitable parameters, a multitude of single-tree models were generated by inputting an array of divergent parameter values. In this study, the discrepancy between the estimated curvature and the ground-truth curvature was utilized as an evaluation metric for the QSM algorithm’s fitting performance. The zoom factor was set within a range from 0.5 to 2, meaning that the minimum value was half of the initial value, while the maximum value was twice the initial value. The interval step was set to 0.1, resulting in the generation of 16 single-tree QSM models with different zoom factor values. The curvature estimated from each model was then compared to the ground-truth curvature to analyze the relationship between the zoom factor and curvature estimation accuracy.

2.5. QSM Trunk Curvature Inversion Algorithm Based on Dual Zoom Factors

In the classical TreeQSM algorithm, the parameter PatchDiam, which is employed in the generation of the covering set, is pre-set based on empirical values. The employment of multiple different empirical values results in the generation of eight single-tree QSM models, and the selection of the one with the highest fitting accuracy is made using an accuracy evaluation function. However, this empirical approach is primarily designed for upright trees and requires a significant computation time. The present study proposes an alternative approach, which involves the introduction of two zoom factors to independently regulate the values of the PatchDiam2Min and PatchDiam2Max parameters within the algorithm. The computational model is constructed as follows:

$$\left\{\begin{array}{c}PatchDiam1=0.08\\ PatchDiam2Min=0.02\times {factor}_1\\ PatchDiam2Max=0.07\times {factor}_2\end{array}\right.$$

(3)

The values of ${factor}_1$ and ${factor}_2$ are both in the range of 0.5–2.0, i.e., relative to the initial value, the minimum value is half of the initial value, and the maximum value is twice of the initial value, and the step size is set to 0.1. In a similar manner, the difference between the estimated curvature and its true curvature is utilized as the evaluation criterion for each model. For the initial value, the minimum value is half of the initial value, the maximum value is twice of the initial value, and the step size is set to 0.1. The difference between the estimated curvature and the true curvature of each model is also used as the evaluation standard. The optimal parameters for each tree are determined by the simulated annealing algorithm to fit the surface to determine the optimal parameter curvature of each tree.

2.6. Trunk Bending Calculations

The study calculated the spatial curvature of tree trunks. CEN (1997) [31] defined log bending curvature as the maximum inclination angle per meter of log length. To achieve automated computation in QSM, following the modifications by Thies et al. (2004) [32], the axes of the fitted cylinders were aligned to approximate the central axis of the log. Specifically, the center points of the fitted cylinders were connected from the tree base to the lowest branching height. The first point ($P_f$) was determined by subtracting the radius vector coordinate from the center of the first fitted cylinder, while the last point ($P_e$) was determined by adding the radius vector coordinate to the center of the last fitted cylinder. The chord length was then defined as the distance between these two endpoints. Within the segment of chord length a, there were n fitted cylinders. The perpendicular distance ($b_n$) from the center point of each fitted cylinder ($P_n$, n = 1, 2, …, n) to the chord was recorded. The curvature c was then computed as

$$c={\displaystyle \frac{max\left(b_n\right)}{a}}$$

(4)

As shown in Figure 3, the black straight line denotes the chord along the axis of the entire fitted cylinder, whilst the black bi-directional arrow signifies the corresponding distance from the center point of the cylinder to the chord. Due to the inherent limitations in accurately selecting the contour points of the trunk in three-dimensional space, it is challenging to account for the position of points perpendicular to the direction of the view plane when plotting the contour points. Consequently, a projection in space is employed as an alternative.

3. Results

3.1. QSM-Based Inverse Algorithm for Trunk Curvature

The study employed a training–validation split approach, where 70% (103 trees) of the 146 sample trees were randomly selected as the training set to establish a relationship model between the zoom factor and curvature error, determining the optimal zoom factor parameter value. The remaining 30% (43 trees) of the total sample trees were used as the validation set. The optimal parameter value obtained from the training process was then applied to the validation set in order to compute the corresponding optimal curvature values. These were then compared with the ground-truth curvature values.

The findings suggest that the substantial discrepancy in the left portion of the curve, where curvature calculation errors are more pronounced, is attributable to overfitting resulting from the utilization of overly brief fitted cylinder segments. This phenomenon leads to an increased number of outliers in the curvature calculation, thereby compromising the overall accuracy. Conversely, underfitting manifests on the right side of the curve, where excessively lengthy cylinder segments prove ineffective in capturing curvature variations. Furthermore, previous studies proposed a moving average method to preprocess the initial data, mitigating errors introduced by randomness in the seed point. In this study, a window size of 3 was utilized for smoothing, where each point’s value was replaced by the average of itself and its two adjacent points. It is important to note that the boundary points were not subject to modification and thus retained their original values.

In order to ascertain the optimal zoom factor for curvature calculation, the study implemented a quadratic polynomial fitting method and augmented the sampling density in order to identify the optimal value within a continuous range. Initially, the interval points were densified through uniform interpolation within the 0.5 to 2 range. The pervious study demonstrated that as the zoom factor varies, the average curvature error initially decreases and then increases, following an approximately quadratic function relationship. Consequently, a quadratic polynomial was selected as the fitting function. Given the limited number of parameters, the least squares method was employed to determine the quadratic polynomial that best approximates all discrete points. The results demonstrated that after averaging across 103 sample trees, the relationship between the QSM zoom factor and curvature was plotted. The red dashed line represents the trend where the difference between the estimated and true curvature first decreases and then increases as the zoom factor increases. The green point indicates the optimal zoom factor for the 103 trees, where zoom_factor = 1.7042, achieving the best performance, as shown in Figure 4.

The present study utilized a total of 43 trees from the sample set, with 30% of these designated for the validation set. This approach was adopted to facilitate a comparative analysis between the optimal curvature calculation results and the true curvature, as illustrated in Figure 5. The blue scatter points in the figure represent the validation set data, the black solid line signifies the optimal curvature calculation results for the validation set, and the red dashed line denotes the reference line under ideal conditions. The cylinder-fitting QSM algorithm is inherently random, and thus multiple calculations were performed, with the optimal solution from five runs selected for regression analysis. The regression equation established a reliable curvature calculation model using the training samples, with the results indicating that the predicted curvature values tend to be larger than the true values. This discrepancy arises due to the QSM algorithm fitting the tree using consecutive cylinders, and due to surface protrusions and the uneven distribution of point clouds, the fitted cylinders tend to shift outward from the actual trunk. Additionally, since the maximum curvature point may not lie within the projection plane, other planes in space may contain regions with greater curvature, leading to an underestimation of the true curvature value.

3.2. Optimized QSM-Based Inverse Algorithm for Trunk Curvature

The study employed a training–validation split approach, utilizing the same training and validation sets as in the previous method to ensure comparability between different methods. Different scaling parameters were distinguished to determine the optimal values for ${factor}_1$ and ${factor}_2$. The training set comprised 70% (103 trees) of the total sample trees, while the remaining 30% (43 trees) constituted the validation set. The optimal parameter values obtained from the training process were then applied to the validation set in order to compute the corresponding optimal curvature values. These were then compared with the true curvature values. The training data results are visualized in Figure 6, in which the X and Y axes represent the zoom factors ${factor}_1$ and ${factor}_2$, respectively. The Z-axis represents the difference between the estimated curvature obtained from the QSM method under different parameter settings and the true curvature in the training samples. As demonstrated in Figure 6, the relationship between the estimated curvature error and the zoom factors approximately forms a paraboloid surface.

However, due to the large number of surface parameters, the traditional least squares method struggles to accurately describe surface characteristics, making it difficult to establish an accurate surface function. To determine the optimal fitting surface, the study employed the simulated annealing algorithm to search for the optimal solution. The simulated annealing algorithm is a stochastic global optimization algorithm inspired by the thermodynamic annealing process. The fundamental idea is to simulate the gradual reduction of system energy from a high-temperature to a low-temperature state during metal annealing. Specifically, when generating a new solution x′ from the current solution x, the change in the objective function value is given by

$$∆E=f\left(x^{\prime }\right)-f\left(x\right)$$

(5)

The probability that the new solution is accepted is then

$$P\left(∆E\right)=e^{-{\displaystyle \frac{∆E}{T}}}$$

(6)

The current temperature is indicated by T. As the temperature, T, gradually decreases in accordance with a predetermined cooling schedule, the algorithm will progressively converge to the global optimal solution. The study involved the modification of the paraboloid surface equation in order to serve as a nonlinear optimization objective function. The standard equation for a paraboloid surface is as follows:

$$z=x^2 OR z=y^2$$

(7)

In order to ensure that the dependent variable z adapts to the separate effects of the two parameters and that the optimal fit can be found through iterative adjustments, the following modification is made to equation:

$$z\prime =p\left(1\right)\times {(x+p(2\left)\right)}^{\left(p\right(3\left)\right)}+p\left(4\right)\times {(y+p(5\left)\right)}^{\left(p\right(6\left)\right)}+p\left(7\right)\times x\times y+p$$

(8)

In this equation, p(n) represents the parameters of the function that change with iterations. Equation (9) indicates that on the fitted surface, the dependent variable z′ can be predicted using a p(3)-order polynomial in terms of the independent variable x and a p(6)-order polynomial in terms of the independent variable y. When the sum of squared errors between the predicted values z′ and the true values z on the surface reaches its minimum, the surface achieves optimal fitting. The objective function and constraint conditions are established as shown in Equation (9):

$$\begin{array}{c}\min \left(\sum {{(z}^{\prime }-z)}^2\right)\\ s.t.\left\{\begin{array}{c}lb<p\left(n\right)<ub\\ p\left(3\right)\in N^{+}\\ p\left(6\right)\in N^{+}\end{array}\right.\end{array}$$

(9)

In this study, lower and upper bounds of the function parameters p(n) in the constraint conditions are denoted by lb and ub, respectively. Given that the focus is on real-number solutions, the parameters p(3) and p(6), which are positioned in the exponent, are constrained to positive integers. Utilizing these objective functions and constraint conditions, the simulated annealing method was employed for surface fitting. The outcomes of this fitting process are presented in Figure 7.

The findings indicate that when the parameters are relatively small, the curvature calculation error is amplified, resulting in overfitting and substantial computational errors. Conversely, when both zoom factors are relatively large, the optimal solution is identified in the expanding direction, whereas parameters that exceed the optimal value manifest underfitting characteristics. Following the averaging of 103 sample trees, the relationship between the two QSM zoom factors and curvature in this region was plotted, as illustrated in Figure 6. As both parameters increase, the fitted surface firstly decreases and then increases. The difference between the estimated and true curvature shows a decreasing trend as the two parameters increase, which is particularly evident in the PatchDiam2Min direction. In contrast, the PatchDiam2Max direction exhibits a clear pattern of initially decreasing and then increasing curvature estimation error.

The results of the fitted Equation (10) are as follows:

$$\left\{\begin{array}{c} p\left(1\right)=0.0270 (p=0.000) \\ p\left(2\right)=-2.0324 (p=0.799)\\ p\left(3\right)=2\\ p\left(4\right)=-0.0622 (p=0.000)\\ p\left(5\right)=-1.2081 (p=0.130)\\ p\left(6\right)=3\\ p\left(7\right)=0.0135 (p=0.000)\\ p\left(8\right)=0.0123 (p=0.000)\end{array}\right.$$

(10)

Following the extraction of the surface equation, the DFP Quasi-Newton Method was utilized to ascertain the lowest point on the surface. This method is a widely employed optimization algorithm that plays a pivotal role in the resolution of unconstrained optimization problems by iteratively seeking the minimum value of the surface. The findings suggest that when PatchDiam2Min = 1.8206 and PatchDiam2Max = 0.845, the surface attains its lowest point. It should be noted that in the surface equation, the parameters p(3) and p(6) appear as exponents and were intentionally rounded to integer values during model fitting. This rounding process was introduced to facilitate curve surface fitting while maintaining computational stability. As a result, these parameters function as discrete hyperparameters rather than continuous regression coefficients. Due to their discrete and non-differentiable nature, they were determined through grid search rather than statistical estimation, and thus, statistical significance measures (such as standard errors or p-values) are not applicable to these parameters.

The study validated the results using the same validation samples as previously used, comparing the differences between the optimal curvature calculations and the true curvature, as illustrated in Figure 8. The blue scatter points represent the validation dataset, and the repeated calculation method is consistent with that employed in the previous results. The findings suggest that the enhanced QSM method generates a curvature distribution that is more aligned with the true values. However, it is important to note that when both the QSM and its improved method are employed, there is a possibility of significant underestimation for trees with higher curvature, resulting in an overall decrease in accuracy.

4. Discussion

4.1. Regional Differences in Trunk Curvature

In order to quantitatively assess whether trunk curvature differed significantly among the 13 sampling plots, a non-parametric Kruskal–Wallis H test was applied to the full set of curvature values, with trees grouped according to their plot of origin. The selection of this approach was made with the intention of circumventing any presumptions concerning the normality and homogeneity of variance in curvature distributions. The statistical analysis yielded a chi-square statistic of 29.89 (χ² = 29.89) with 12 degrees of freedom, and a corresponding p-value of 0.0029 (p < 0.05), indicating a statistically significant difference in curvature among the plots. In order to identify the specific group pairs contributing to this significance, post hoc pairwise comparisons were conducted using Dunn’s multiple comparison test with Bonferroni correction. The results revealed statistically significant differences between Plot1 and Plot8 (adjusted p = 0.0464), and between Plot8 and Plot12 (adjusted p = 0.0219), suggesting that the spatial distribution of trunk curvature is not uniform across the study area.

The results obtained provide evidence that regional-level environmental or ecological variability may influence trunk bending patterns. The curvature distribution across plots is illustrated in Figure 9, and the descriptive statistics for each plot, including sample size, mean, standard deviation and maximum curvature, are outlined in

Table 2. Comparison of tree structure modeling results obtained using GoSlam (a mobile handheld LiDAR system) and Faro (a terrestrial laser scanning system, FARO Focus 3D).

Sensor	GoSlam	Faro
Scanning range	360° × 285°	360° × 150°
Beam divergence angle	-	0.3 mrad
Wavelength	-	1550 nm
Beam diameter	-	2.12 mm

.

Table 3. Statistical information on the curvature of different trunks.

Plot	Tree Count	Mean Curvature	Std. Deviation	Max Curvature
Plot1	17	0.12	0.08	0.38
Plot2	28	0.08	0.06	0.28
Plot3	15	0.11	0.07	0.26
Plot4	13	0.06	0.05	0.16
Plot5	6	0.11	0.07	0.20
Plot6	10	0.06	0.08	0.26
Plot7	14	0.07	0.03	0.11
Plot8	4	0.10	0.06	0.18
Plot9	7	0.14	0.08	0.30
Plot10	10	0.17	0.12	0.46
Plot11	4	0.12	0.04	0.17
Plot12	8	0.12	0.08	0.27
Plot13	10	0.17	0.08	0.32

shows the curvature of the trunks in each sample plot. In order to further evaluate the hypothesis that there is a correlation between terrain steepness and trunk curvature, a Pearson correlation analysis was performed between the average slope and mean trunk curvature across the 13 sample plots. The result demonstrated a moderate positive correlation (r = 0.43), thus indicating that plots with steeper slopes tend to have higher curvature values. However, the correlation was not statistically significant (p = 0.145), likely due to the limited number of plots. Despite the absence of statistical significance, this trend lends support to the hypothesis that terrain may exert an influence on the shaping of tree architecture. The investigation of this hypothesis is recommended, with the undertaking of such research being advised at the level of a larger sample size or higher-resolution slope measurements. The correlation is shown in Figure 10.

4.2. Explanation of Results

This study employs an enhanced QSM method, which optimizes the calculation of tree trunk three-dimensional curvature. This is achieved by introducing dual correction factors and a simulated annealing strategy. The experimental results demonstrate that, in the single-factor model, the curvature fitting error initially decreases and then increases as the zoom factor increases. In the dual-factor model, the employment of the simulated annealing algorithm for global optimization has been shown to be an effective method of reducing errors and enhancing the stability of the results. The analysis of parameter sensitivity has revealed that the use of excessively short fitting segments can lead to the phenomenon of overfitting, whilst the use of excessively long segments can cause underfitting, ultimately resulting in an underestimation of curvature values. This phenomenon aligns with previous findings in the field of fruit tree branch bending detection, where researchers have observed that the parameter tuning process significantly influences the accuracy of the fitting process. They demonstrated that selecting an appropriate fitting segment length is crucial for improving detection precision [27]. Similarly, Pfeifer et al. (2004) emphasized in their study on automatic tree reconstruction that global optimization strategies play a significant role in overcoming local optima and enhancing the stability of 3D reconstruction models [33,34]. The findings of this study demonstrate that the overestimation of curvature values using QSM is consistent with the observations reported by Guzmán et al. (2017) [26]. The aforementioned researchers reported biases in tree trunk morphological parameters in tropical tree species research. This underestimation may result from missing TLS point cloud data in the rear regions of tree trunks and the limitations of fitting algorithms in capturing subtle curvatures. To address this issue, this study integrates a global optimization approach, employing simulated annealing and the DFP quasi-Newton method for fine-tuning parameters, which partially mitigates the problem. Moreover, the global optimization method proposed by Wang et al. (2014) [35] has demonstrated excellent performance in tree model reconstruction based on TLS data. Their study confirmed that leveraging global search strategies can better adapt to the heterogeneity of point cloud data, thereby improving model accuracy.

Despite the moderate to low correlation between estimated and true curvature values (R² = 0.21 and 0.12 in Figure 5 and Figure 8, respectively), this is largely attributable to the limited variance in the dataset, with the majority of trees exhibiting curvature values below 0.1. Such a narrow distribution has been shown to restrict the model’s learning capacity and weaken correlation-based metrics. This predicament is a recurring theme in the context of regression analysis applied to low-variance targets, wherein accurate predictions often result in low R² values.

In the dataset under consideration, only two samples exhibited strong curvature (>0.1), which hinders reliable performance evaluation in the high-curvature domain. However, within the prevailing range of the data, the model exhibited minimal absolute errors and followed the anticipated trend. This finding indicates that the proposed method is effective in realistic forest scenarios where strong trunk bending is uncommon. It is evident that further studies utilizing a more diverse array of samples are required to substantiate the model’s capacity to generalize to more extreme cases.

4.3. Limitations and Prospects

The findings suggest that this approach offers a more precise depiction of tree trunk three-dimensional bending morphology. This novel method provides a valuable technical framework for forest resource measurement and tree growth pattern analysis. However, it is important to note that practical applications of this method are still subject to challenges and limitations that necessitate further investigation and optimization. Firstly, the study relies on manually extracting trunk contour points as a reference for comparison, which ensures accurate edge and centerline information in specific projection planes, but is difficult to use to describe trunk curvature when it exists in other spatial planes. Secondly, point cloud data are affected by occlusion, noise and uneven density, leading to significant data loss in the backside of tree trunks and certain regions, which introduces uncertainty in curvature extraction accuracy. Consequently, future research should focus on the development of automated extraction techniques that can effectively address these issues. Secondly, the enhanced QSM model deploys quadratic polynomial fitting and simulated annealing algorithms for parameter optimization, thereby reducing overall computation time and enhancing fitting accuracy. However, given the random nature of the QSM algorithm with respect to initial point selection and parameter settings, results may vary across different experiments. Future research could explore more stable and robust automatic parameter adjustment methods, such as machine learning-based parameter prediction models, to further enhance the model’s stability and generalizability. Furthermore, in the context of practical forest resource surveys [36], variations in forest structure and point cloud acquisition conditions have been shown to affect the accuracy of the algorithm. Therefore, expanding the sample size, conducting comparative experiments on diverse samples and integrating multi-sensor data fusion techniques will facilitate a comprehensive evaluation of the adaptability and application prospects of this method.

Although slope and aspect were not explicitly included in the trunk curvature modeling framework, the observed variability among plots suggests that topographic conditions may influence tree form. This aligns with previous findings that terrain features can affect growth orientation and mechanical stress. Future studies may benefit from incorporating these factors to better explain variation in trunk architecture.

The enhanced methodology proposed in this study possesses potential for application in the domain of forestry, particularly with regard to the precise measurement of forest resources, the analysis of tree growth patterns and the estimation of timber volume. The approach under discussion facilitates the more accurate capture of tree trunk bending characteristics, thereby enhancing the precision of biomass and carbon storage estimation. In addition, it provides a scientific foundation for the evaluation of tree wind resistance and mechanical properties. In future research, further exploration of the correlations between curvature and other tree structural parameters could lead to the development of a more comprehensive tree growth model, thereby promoting forest ecosystem research and contributing to the sustainable development of forest management practices [37,38,39,40].

5. Conclusions

This study proposes a novel method for retrieving tree trunk curvature based on TLS point cloud data and QSM, introducing dual correction factors to optimize the parameters of the traditional QSM model. This approach effectively addresses the limitations of conventional methods, which often overlook the actual three-dimensional morphology of tree trunks in curvature measurements. The experimental results demonstrate that this method not only reduces computational complexity but also accurately reconstructs the three-dimensional curvature characteristics of tree trunks. The method maintains high measurement accuracy even in the presence of noise, uneven distribution or partial missing data in the point cloud data. Although slight underestimation occurs under extreme bending conditions, the overall improved model provides a robust technical foundation for precise tree structure measurement, forest resource assessment and research on tree growth patterns and ecological functions, supporting sustainable forestry development. While the proposed method performs reliably in estimating curvature across trees with low to moderate bending, its evaluation on highly curved trunks remains limited due to the scarcity of such samples in the dataset. The observed low correlation values can be attributed largely to the low variance of the curvature measurements, rather than to systematic model errors. Notwithstanding this finding, the method exhibited commendable stability and accuracy within the prevailing curvature range.

In addition to the methodological improvements, the study also analyzed trunk curvature patterns across 13 distinct forest plots. Results revealed substantial spatial variation in curvature, with significant differences detected among plots. Tree height and terrain conditions varied markedly across sites, with mean tree height ranging from 108 to 223 m and plot-level slopes spanning from 2° to 27°. A moderate but statistically non-significant correlation (r = 0.43, p = 0.145) was observed between slope and trunk curvature, suggesting that topographic conditions may influence curvature development to some extent. These contextual findings enhance the ecological interpretation of the results and provide valuable references for field application. Nevertheless, the central contribution of this study remains the development of a more accurate and generalizable curvature estimation framework. Future research could further expand the applicability of the model by exploring its performance across diverse samples, multiple tree species and complex terrains. Additionally, the integration of real-time data acquisition technologies may enhance the operational efficiency of forest monitoring and structural analysis in dynamic environments. Future research could further expand the applicability of the model by exploring its performance across diverse samples, multiple tree species and complex terrains. Additionally, the integration of real-time data acquisition technologies has the potential to enhance the efficiency of forest monitoring and management.