Exploring trees in three dimensions: VoxR, a novel voxel-based R package dedicated to analysing the complex arrangement of tree crowns

Abstract

Background

Interest in tree form assessments using the terrestrial laser scanner (TLS) has increased in recent years. Yet many existing methods are limited to small-sized trees, principally due to noise and occlusion phenomena. In this paper, a novel voxel-based program that is dedicated to the analyses of large tree structures is presented. The method is based on the assumption that architectural trait variations (i.e. branching angle, bifurcation ratio, biomass allocation, etc.) influence the way a tree explores space. This method uses the concept of space exploration that considers a voxel as a portion of space explored by the tree. Once the TLS scene is voxelized, the program provides tools that extract qualitative (geometrical) and quantitative (volumetric) metrics. These tools measure (1) voxel dispersion in three dimensions (3-D), (2) projections of the voxel cloud in 2-D and (3) multi-temporal changes within a single tree crown.

Scope

To test algorithm capabilities of measuring larger tree architectural traits, two application studies were conducted using point clouds that were either generated by a tree growth simulation model, thereby allowing algorithm application in a perfectly controlled environment, or acquired in the field with a TLS device. The space exploration concept makes it possible to take advantage of the volumetric nature of voxels to compensate for occlusion. The hypothesis that large-sized voxels can be used to reduce occlusion in the original point cloud was tested, as well as the consequences of voxel size on quantification of tree volume and on precision of derived metrics.

Conclusions

Results show that space exploration is well adapted to highlight architectural differences among trees. They also suggest that large-sized voxels are efficient for occlusion compensation at the expense of metrics precision in some cases. The best resolution to choose depending on the research objectives and quality of the TLS scan is discussed.

INTRODUCTION

Assessing tree form in situ is important to be able to infer the mechanisms that drive tree growth. Tree development results from the complex assemblage of botanical ‘objects’ (i.e. woody structure and leaves), which are governed by genetic rules and dynamic processes controlled by environmental conditions (Barthélémy and Cariglio, 2007). In recent decades, interest in and application of terrestrial laser scanning (TLS) to tree and forest studies have increased due to its ability to acquire accurate information rapidly on objects in a three-dimensional (3-D) space. Yet point clouds that are acquired from TLS in 3-D ‘are unstructured data that must be reconstructed by dedicated programs to provide information’ (Dassot et al., 2011, p. 961).

Many tools exist to derive tree traits from TLS data: leaf Area Index (Koetz et al., 2006; Kwak et al., 2007), leaf Area Density (Hosoi and Omasa, 2006; Béland et al., 2014), leaf Area Distribution or vegetation density (Durrieu et al., 2008; Van der Zande et al., 2010; Béland et al., 2011), crown volumes (Lefsky and McHale, 2008; Vonderach et al., 2012), crown form (Martin-Ducup et al., 2016) or tree overall dimensions (Hopkinson et al., 2004; Henning and Radtke, 2006; Bucksch and Fleck, 2009; Côté et al., 2011; Delagrange and Rochon, 2011; Delagrange et al., 2014).

Among these methods, a first set is based on perennial tissue reconstruction (Verroust and Lazarus, 2000; Gorte and Pfeifer 2004; Gorte and Winterhalder, 2004; Bucksch and Lindenbergh, 2008; Binney and Sukhatme, 2009; Bucksch and Fleck, 2011; Côté et al., 2011; Delagrange and Rochon, 2011; Delagrange et al., 2014; Hackenberg et al., 2015). Reconstruction is generally used to quantify woody structure volume or to produce realistic tree models. When needed, the foliage is generally added through architectural-based modelling (e.g. Côté et al., 2011) or the use of allometric relationships (e.g. Delagrange and Rochon, 2011). However, occlusion and noise phenomena (Van der Zande et al., 2010; Bucksch et al., 2010; Dassot et al., 2011) limit these methods to small trees or require modelling of smaller branches. This generally prevents the application of this method to large-dimension or complex tree structures.

A second set of methods focus on extracting tree traits independent of tree reconstruction by analysing the 3-D point clouds using spatially explicit algorithms (Hosoi and Omasa, 2006; Durrieu et al., 2008; Park et al., 2010; Van der Zande et al., 2010; Béland et al., 2011; Vonderach et al., 2012). These methods usually employ voxels to locally sample the TLS point cloud and derive vegetation density or related traits while correcting for occlusion or point density heterogeneity. Voxels also enable TLS point clouds to be consolidated into a 3-D volumetric object that may be used to simplify the TLS point cloud geometry and simplify the subsequent analyses (Bienert et al., 2010; Vonderach et al., 2012; Kato et al., 2013). Voxel-based methods are thus powerful tools for extracting meaningful variables from TLS data that aim to describe architecture and functions of potentially large trees.

In this paper, we present a novel method to easily derive qualitative (geometrical) and quantitative (volumetric) metrics from TLS data. We have two main objectives: (1) to derive consistent metrics that describe tree form and structure, and (2) to investigate the potential use of voxels to deal with occlusion and noise phenomena incurred when studying large tree architectures. To achieve these two objectives, we utilized TLS point cloud voxelization to develop the VoxR package (Lecigne et al., 2014), an R package that is dedicated to derive metrics that describe the way a tree explores space. First, we present the VoxR concept and the main advantage it provides and describe the functions and algorithms. Secondly, we use synthetic (i.e. simulated) data to evaluate (1) the metrics derived from TLS point clouds with VoxR to detect the consequences of contrasted tree architecture on space exploration, (2) voxel capabilities in compensating for occlusion and (3) the consequence of voxel size on estimates of the volume of space explored by a tree. Finally, we use TLS data (1) to provide an example of VoxR potential in evaluating the effect of pruning for power line clearance on tree space exploration in an urban area and (2) to evaluate the effect of voxel size on metric precision.

PACKAGE DESCRIPTION

Concept and consequences

The VoxR package is based on the conceptually new assumption that a voxel can be understood as a portion of space explored by the tree (i.e. containing a least an arbitrary number of TLS points). All analyses and algorithms provided in the VoxR package are thus based on the concept of space exploration. This approach enables multiple metrics to be derived from TLS data that describe the way a tree occupies the 3-D space quantitatively (i.e. estimate the volume of space explored by the tree) or qualitatively (i.e. localize the biomass location within a tree crown). These analyses are thus performed at the expense of quantifying more usual traits such as woody biomass, wood volume or leaf area. Thanks to the space exploration concept, and contrary to some other current methods (Béland et al., 2014), voxel size does not need to be defined as a function of physical or computational considerations and is thus driven only by the data quality and the research hypothesis (see Discussion). This notably enables one to take advantage of the volumetric nature of voxels in order to compensate for point cloud discontinuity caused by occlusion (see further analyses). To some extent (see Discussion), this approach is more robust to occlusion and can thus be used to analyse large tree structures while most of the currently available methods are generally limited to smaller trees. The space exploration concept also allows one to ignore empty voxels and (1) quickly complete the voxelization process, and (2) remove voxels potentially arising from noise by filtering the data based on the number of TLS points per voxel (Vonderach et al., 2012).

Voxelization algorithm

The vox function is a simple voxelization algorithm (Fig. 1). Voxelization algorithms are usually based on an iterative process that aims to classify points in a three-dimensional regular grid of voxels (Fernández-Sarría et al., 2013a). Here, according to the space exploration concept, only filled voxels (i.e. voxels containing TLS points) need to be considered. A simple voxelization can thus be achieved through rounding the points coordinates on both three axis of the Cartesian coordinate system: round(coord_(x,y,z)*res)/res, where res is the voxel resolution. The vox function output is the discrete X, Y, Z coordinates of the voxels’ centre and the number of points that are present within each voxel. This function is computationally rapid, which is an advantage, given that long processing times are generally associated with voxelization methods (Fernández-Sarría et al., 2013a).

Fig. 1.

Examples of voxelized point clouds using voxel resolutions of 0.1 m (A), 0.5 m (B) and 1 m (C).

Voxel geometric dispersion

The VoxR package provides three functions to derive variables describing the 3-D voxel dispersion in the 3-D and to some extent 2-D space. First, the point.distance function computes the euclidean distance of each voxel from a user-defined point (see Table 1 for equations). Secondly, the axis.distance function computes the euclidean distance of each voxel from a user-defined X-, Y- or Z-axis of the 3-D Cartesian coordinate system (see Table 1 for equations). Finally, the axis.angle function computes the angle that is formed by each voxel from a user-defined X-, Y- or Z-axis of the 3-D Cartesian coordinate system and the system’s origin (see Table 1 for equations). Note that for the axis.angle function, the voxel angles can be calculated using voxel cloud projection in a 2-D space to obtain the radial dispersion of voxels.

Table 1.

Equations used for computation of voxel geometric dispersion

Function	Equation
point.distance	$d\left(v_{i=1}^n,p\right)=\sqrt{{\left(x_{\left(v_i\right)}-x_{\left(p\right)}\right)}^{\text{2}}+{\left(y_{\left(v_i\right)}-y_{\left(p\right)}\right)}^{\text{2}}+{\left(z_{\left(v_i\right)}-z_{\left(p\right)}\right)}^{\text{2}}}$, where v is a voxel of an n row voxel matrix and p is the reference user-defined point.
axis.distance	$d\left(v_{i=\text{1}}^n,a\right)=\sqrt{x_{\left(v_i\right)}^{\text{2}}+y_{\left(v_i\right)}^{\text{2}}+z_{\left(v_i\right)}^{\text{2}}}$, where v is a voxel of an n row voxel matrix and a is the reference user-defined axis and $if\left(a=X\right)\Rightarrow x_{\left(v_i\right)}=\text{0}$ or $if\left(a=Y\right)\Rightarrow y_{\left(v_i\right)}=\text{0}$ or $if\left(a=Z\right)\Rightarrow z_{\left(v_i\right)}=\text{0}$
axis.angle	$\alpha \left(v_{i=\text{1}}^n,a\right)=\text{arcc}o\text{s}\left(\frac{\theta }{{\beta }_{\text{1}}{\beta }_{\text{2}}}\right)\frac{\text{180}}{\pi }$, where v is a voxel of an n row voxel matrix and a is the reference user-defined axis and ${\beta }_{\text{1}}=\sqrt{x_v^{\text{2}}+y_v^{\text{2}}+z_v^{\text{2}}}$, ${\beta }_{\text{2}}=\text{100}$ and $if\left(a=X\right)\Rightarrow \theta ={\beta }_{\text{2}}{x}_{\left(v_i\right)}$ or $if\left(a=Y\right)\Rightarrow \theta ={\beta }_{\text{2}}{y}_{\left(v_i\right)}$ or $if\left(a=Z\right)\Rightarrow \theta ={\beta }_{\text{2}}{z}_{\left(v_i\right)}$

Voxel densities in 2-D space

The project function provides projection of the voxel cloud in a 2-D space. To do so, the user must define a plane formed by two axes of the Cartesian coordinate system in which the voxels will be projected. The function returns a 2-D matrix containing pixel coordinates and the number of points and voxels that are contained in the corresponding voxel column (i.e. the voxels located above one single pixel), together with the ratio of the number of points/number of voxels. To ease visualization of the projections, the raster.proj function uses the default R device to visualize a raster image of the projection, including standard captioning capabilities (title, axis names, colours, density class names and surfaces). Colours, density class and surface are computed using two subfunctions: level, which is used to compute discrimination levels of density class using quantiles or percentage methods; and surf, which is used to compute the surface (or number of pixels) present in each density class.

Changes in space exploration in a multi-temporal measurement context

Multi-temporal TLS scanning enables one to observe changes occurring in 3-D space over time (Srinivasan et al., 2014). To do so, the VoxR package provides the sub.obj function that is devised for comparing two voxelized TLS scenes that have been taken at two different times (Fig. 2). It allows for the comparison of two TLS scenes (s₁ and s₂), based on voxel coordinates, in order to isolate voxels that are unique to s₁ (no voxels from s₂ are conserved). To do so, the sub.obj function conserves all voxels of s₁ isolated from any voxels of s₂ (i.e. with no neighbour in s₂) within a larger voxel (V_r) of user-defined resolution (see Supplementary Data Table S1 for a more complete description of the algorithm). Thus, this function can be used to analyse changes in space exploration related to tree growth or branch loss after biotic (i.e. branch death or herbivory) or abiotic (i.e. freezing rain, wind, pruning) stresses and may help to detect new growth occurring on both young and old structures (Fig. 2). The main assumption of this algorithm is that the two TLS scenes (s₁ and s₂) must be perfectly aligned. This can be easily achieved through registration of the two TLS scenes in external dedicated software, similar to registering TLS scans to compose a TLS scene. Particular attention is required regarding the choice of the parameter that sets the size of V_r (nvox.reaserch, expressed as a multiple of the original voxel resolution), due to possible voxel identification errors (see differences between A, B, C and D in Fig. 2). A risk of under-sampling (i.e. suppression of valuable voxels) can be associated with high values of nvox.reaserch. In contrast, small values may result in an over-sampling risk (i.e. conservation of inconsistent voxels due to branch movement during registration) within the final output. The choice of this parameter is user-dependent and thus needs a trial-and-error approach to be determined appropriately. In some cases, such as branch loss (Fig. 2), manual removal of inconsistent voxels can be performed, which would enable the analyst to use small values of nvox.reaserch and maintain precision in subsequent evaluations.

Fig. 2.

Examples of results using the sub.obj function. Images A, B, C and D consist of points that are obtained from subtracting the tree after pruning from the tree before pruning, and indicate branch losses incurred by pruning, using nvox.reaserch = 2 (A), 7 (B), 10 (C) or 15 (D). E: this image represents the point clouds (in black) of a tree, with the red points indicating the branches that were pruned (similar to C).

TESTING VOXR IN A CONTROLLED ENVIRONMENT: APPLICATIONS ON SIMULATED DATA

Methods

To evaluate the ability of the algorithms in VoxR to detect differences in tree architecture through changes in space exploration patterns, 3-D trees were generated using the plant development model AmapSim (Barczi et al., 2008; http://amapstudio.cirad.fr/amap). Three simulated trees were generated that possessed similar values in terms of branch length, branching points and height but with different branch insertion angles. Thus, the three simulated trees represent a gradient from plagiotropism, namely plant growth at an oblique angle to a stimulus, to heliotropism, namely plant growth in response to the direction of the sun (Fig. 3, from sim1 to sim3). Branching angle was selected as a target architectural trait to test VoxR algorithms due its influence on both tree crown architecture and branch arrangements in the 3-D space of the whole-tree crown. These simulated trees were then transformed into 3-D point clouds, with each point representing the centre of an internode, upon which VoxR algorithms could be applied (Fig. 3). The generated point clouds thus differ from TLS data on two main aspects: (1) no volumetric information was recorded (i.e. no internode diameter), and (2) no occlusion or noise phenomena were simulated. However, these virtual tree point clouds presented the advantage of recording aerial material localization, which enables comparisons between tree(s) space exploration patterns only and to test VoxR algorithms without bias due to quantitative differences amongst trees (i.e. trees have similar biomass, branch numbers, branch lengths, and insertion points) or data acquisition (i.e. noise and occlusion). To evaluate differences between trees, skewness and kurtosis coefficients, and minimum and maximum values of the variable distributions that were created by VoxR functions were studied. It should be remembered that negative skewness implies an asymmetrical distribution of the values to the right (i.e. high values), whereas positive skewness characterises an asymmetrical distribution of values to the left (i.e. small values); a skewness of 0 implies a symmetrical distribution. Furthermore, the greater the kurtosis parameter (positive, leptokurtic), the greater is the concentration of the distribution around the average value. In contrast, the lower kurtosis parameter values (negative, platykurtic) the more homogeneous the distribution.

Fig. 3.

Point clouds of simulated trees. Trees were simulated using AmapSim software (Barczi et al., 2008, http://amapstudio.cirad.fr/amap). In the text, A is referred to as sim1, B as sim2 and C as sim3.

As explained previously, the concept of using voxels to analyse space exploration enables the use of any voxel size as soon as this parameter matches the research hypothesis. This results in a potential to take advantage of the volumetric nature of voxels to reduce point cloud discontinuity caused by occlusion. To test the hypothesis that voxels may help in reducing occlusion influence, a short study based on simulated cylinders was conducted. TLS scans of a tree can be viewed as a complex layout of cylindrical objects, of which two types can be distinguished: empty cylinders and filled cylinders. Empty cylinders appear on multi-scan alignments of large-diameter objects (i.e. trunk or large branches), while filled cylinders appear on small-diameter objects (i.e. small branches) when the diameter of the object is smaller than the point density or the precision of the TLS laser beam (i.e. due to distance measurement imprecision between the branch and the TLS device). In this study, we randomly simulated TLS-like cylindrical point clouds of known length and diameter in which an occlusion was simulated by creating a gap of known length (see Fig. 4B). The sampling process (i.e. cylinder simulation and data registration) consists of an iterative random selection of a set of parameters that fixes cylinder diameter (cylinder length was set to 1 m in all simulations), cylinder position and orientation (rotations and translations), and occlusion zone length (see Table 2 for the ranges of parameters that were used). These simulated cylinders were then voxelized with the vox function using a defined resolution. Voxel resolutions that were used in the present study ranged from 0.01 to 0.15 m, and from 0.05 to 0.2 m (in increments of 0.01 m in both cases) for filled and empty cylinders, respectively. For each resolution, 500 random cylindrical point clouds were generated for both the filled and the empty cylinders. Occlusion compensation was expressed in terms of remaining occlusion (i.e. non-compensated occlusion, see Fig. 4C–E), which was calculated as: $R\left(r_{i=\text{1}}^{\text{500}}\right)=Nvo{x}_i-Noc{c}_i$ where R is the occlusion remaining for the ith cylinder, N_vox is the number of voxels of r resolution that were created by the ith cylinder without occlusion, and N_occ is the number of voxels that were created by the ith cylinder after the occlusion zone was created. The remaining occlusion was examined against the ratio voxel resolution/occlusion length (res/occ) that could be attributed to the presumed importance of voxel resolution for occlusion compensation. The average and maximum occlusion size that could be compensated for by each voxel resolution was separately investigated for filled and empty cylinders. To do so, occlusion length was recorded during the sampling process.

Fig. 4.

Example of the process to test voxels’ ability to compensate for occlusion (i.e. transform a discontinuous point cloud into a continuous voxel cloud). In A a cylinder of known diameter and length is generated (in the example the diameter was set to 0.1 m and the length to 1 m). In B a ‘gap’, which corresponds to occlusion, of known length is generated (in the example the gap length was 0.2 m). The resulting point cloud is then voxelized using a predefined resolution (C, D and E are examples of different voxel resolutions, r). In the example, the voxels in red correspond to non-compensated occlusion. Here the occlusion was fully compensated with a 0.15-m resolution only (E).

Table 2.

Parameter ranges used to randomize the generation of cylindrical point clouds to test voxel efficiency to compensate for occlusion

	Voxel resolution (m)	Length (m)	Diameter (m)	Point density (m)	Rotation (°)		Translation on X, Y and Z axes (m)	Occlusion length (m)
					on X	on Y
Filled	0.01–0.15	1	0.01–0.05	0.005–0.01	0–90	0–360	−0.5 to 0.5	0.001–0.4
Empty	0.05–0.2	1	0.2–0.5	0.005–0.01	0–90	0–360	−0.5 to 0.5	0.001–0.5

Finally, the total volume of space that was explored by the simulated trees (sim1, sim2, sim3) was observed to examine the differences and sensitivity of voxels to extracted volumes. The explored volume (V_sim) was calculated as follows: V_sim=N_vox× res³, where N_vox is the number of voxels, and res is voxel resolution.

Results

3-D voxel dispersion.

Voxel dispersions in the 3-D space functions were applied to three simulated trees. The distributions resulting from the axis.distance, point.distance and angle.distance (Fig. 5) were studied using skewness, kurtosis, minimum and maximum values (Table 3).

Fig. 5.

3-D voxel dispersion analyses for simulated trees (sim1, sim2 and sim3). From top to bottom: algorithm for voxel distances from an axis computation (axis.distance), algorithm for voxel distances from a point computation (point.distance), algorithm for voxel angles with an axis computation using projection (e.g. radial dispersion of voxels, axis.angle with projection) and algorithm for voxel angles with an axis (axis.angle) computation without using projection. Left panel: images of 3-D plots of simulated trees; the variable of interest is represented by a colour gradient. Black arrays and crosses represent the axis or point used as reference. Right panel: the Kernel density curves of the variable for each simulated tree. The colour scale is identical for the left- and right-hand panel graphics.

Table 3.

Distributional characteristics of 3-D voxel densities for sim1, sim2 and sim3; see Fig. 5 for graphics and visualizations

	Distance from Z axis			Distance from the origin			Radial dispersion			Angle from X axis
	sim 1	sim2	sim3	sim 1	sim2	sim3	sim 1	sim2	sim3	sim 1	sim2	sim3
Skewness	0.3	0.18	0.183	0.544	0.32	0.061	−0.034	−0.038	−0.034	0.139	−0.322	0.031
Kurtosis	−0.789	−1.069	−1.043	−0.141	−0.398	0.273	−1.203	−1.162	−1.198	0.806	0.946	0.328
Min.	0	0	0	0	0	0	0.02	0	0.004	22.949	20.299	53.605
Max.	5.878	5.538	4.463	12.7	12.7	12.7	359.99	359.99	359.99	170.1	143.85	131.99

The axis.distance function was applied to compute the euclidean distance of voxels from the Z-axis (i.e. the distance of voxels from the trunk centre). In sim1, the voxel distribution is more strongly left-skewed and less homogeneous than in sim2 and sim3 (skewness and kurtosis parameters, respectively; Table 3). This shows that plagiotropism results in the accumulation of material close to the trunk, whereas heliotropism and the intermediate architecture lead to more homogeneous material dispersion relative to the trunk. Moreover, heliotropism reduces the maximum voxel distance from the trunk, while plagiotropism increases it (Table 3). In a plagiotropic situation, trees adopt a pyramidal form involving the superposition of horizontal branches that exhibit a length gradient (i.e. along the trunk), which leads to a lower superposition of material at the proximal end of the longest branches. In contrast, heliotropism tends to increase material superposition (i.e. same amount of material within a smaller projected area) and to reduce the portion of longest branch lengths without material above it, which leads to a more homogeneous distribution of voxels relative to the trunk.

The point.distance function was used to compute voxel distance relative to the origin (the 0, 0, 0 coordinates, which correspond to the trunk base for all simulated trees). In sim3, the distribution of voxels is more symmetric and less homogeneous than in sim1 and sim2 (skewness and kurtosis parameters, respectively; Table 3), which both have an asymmetrical distribution to the left. This result demonstrates that the heliotropic simulated tree (i.e. sim3) exhibits a global elevation of material above the base of the trunk compared to the other two situations, which is a logically expected result given their smaller branching angles.

Finally, the axis.angle function was applied to calculate voxel angles relative to the X-axis, with and without projection of the point clouds leading to radial or semicircular voxel dispersions. No major difference appeared in the first case (Table 3, Fig. 5) due to the controlled plant development and the homogeneous environment that was created by the model. Indeed, this function was created to provide a standard of plant developmental asymmetry that is induced by environmental constraints, which were not present in the simulation model that was used in this study. However, the axis.angle function provided consistent results with respect to differences between simulated trees. In sim3, the distribution of voxels is more homogeneous and centred (around the average value), while in sim1 and sim2, the distribution is asymmetrical to the left and to the right, respectively; both of these latter simulations exhibited less homogeneous voxel distributions compared to sim3. These results further demonstrate that material distributions within heliotropic tree crowns are more homogeneous than in the two other architectures, even if the range of the distribution is smaller (Table 3; maximum and minimum). Yet, in all three simulated trees, the trunk constitutes the main accumulation of material (centred leptokurtic distributions).

2D voxel densities.

The project function was used to rasterize simulated trees into a 2-D space that was formed by the X- and Y-axes of the Cartesian coordinate system (i.e. projection to the ground). Two variables were extracted: the number of voxels per pixel and the number of points per pixel (Table 4, Fig. 6). To remove high values within tree density layers (of voxels or points separately), the variable of interest was merged for all three simulated trees and values greater than the 99th percentile were removed and replaced by the 99th percentile value. This process harmonizes the density values for all simulated trees and eases analysis, interpretation of results and visualization.

Table 4.

Distributional characteristics of 2-D voxel densities for sim1, sim2 and sim3; see Fig. 6 for graphics and visualizations. Note that only values lower than the 99th percentile for the variables of interest for all the simulated trees are considered in the calculation

	Voxel density			Point density
	sim1	sim2	sim3	sim1	sim2	sim3
Skewness	2.029	1.691	1.359	1.864	1.61	1.365
Kurtosis	5.518	2.866	1.751	4.406	2.516	1.61
Min.	1	1	1	1	1	1
Max.	10	10	10	16	16	16
No. of pixels	12^604	10^449	8804	12^604	10^449	8804

Fig. 6.

2-D voxel density analyses performed on simulated trees (sim1, sim2 and sim3) based on voxel cloud projections (project). Top: the number of voxels per pixel. Bottom: the number of points per pixel. Left panel: raster images of projected simulated trees; the variable of interest is represented by a colour gradient. Right panel: kernel density curves of the variable for each simulated tree. Colours on the graphics provide the scale for raster images. Note that to facilitate visualization of the results, the highest values of voxels and point densities have been removed from the graphs (above 10 or 16, respectively) and replaced by 10 or 16 in the raster images. These values were the 99th percentiles of the variable distributions for all simulated trees.

For both voxel and point densities, the project function allows the analyst to detect differences between simulated trees; the two variables provide approximately the same diagnostic: in sim3, the density distribution of points or voxels is less asymmetric to the left while being more homogeneous (skewness and kurtosis parameters respectively, Table 4) than in sim1 and sim2. These examples indicate that voxel or point density increments resulting in heliotropism lead to crown densification; this can be expected due to the presence of the same amount of material (in a controlled environment) within a smaller projected area (see Table 4, number of pixels), which is consistent with results obtained in the analyses of distances from trunk centres (Table 3). In contrast, the distributions are more asymmetrical to the left and leptokurtic (skewness and kurtosis parameters, respectively, Table 4) in sim1 than in sim2 and sim3. This shows that in a vertical projection, plagiotropism leads to lower material superposition than heliotropism, while sim2 appears to be an intermediate situation between sim1 and sim3.

Consequences of resolution choice for occlusion compensation and tree volume estimation.

The ability of voxels to reduce occlusion length has been explored using remaining occlusion as a function of res/occ (Fig. 7A and B) for filled or empty simulated cylindrical point clouds. In both cases, the remaining occlusion dramatically decreases between res/occ values of 0 and 1 (for res/occ = 1: voxel size = occlusion length), decreasing on average from 42 to 3 % and from 59 to 2 % of remaining occlusion for filled and empty cylinders, respectively. The first completely compensated occlusion appears when res/occ values are 0.49 and 0.58 for filled and empty cylinders, respectively. In both cases, an average of 10 % of remaining occlusion is observed for these fully compensated minimum values. For res/occ values that are greater than one, average remaining occlusion varies from 2 to 0 % for both filled and empty cylinders.

Fig. 7.

Voxel occlusion compensation. A and B: graphs of remaining occlusion (in % of voxels in the occlusion zone) vs. res/occ ratio (light grey dots) for filled (A) and empty (B) cylinders. The solid line represents the average percentage of remaining occlusion as a function of res/occ (increments of 0.1 units). Dashed lines represent the smallest values of res/occ for which the occlusion zone of at least one sample was fully compensated. C: graph of compensated occlusion length (average) and maximum (max) compensated occlusion length vs. voxel resolutions that have been adjusted by linear regression models for filled and empty cylinders.

The maximum and average occlusion lengths that can be compensated, depending on voxel resolution, were studied using linear regression models (Fig. 7C). The maximum fully compensated occlusion (FC_occ) generally corresponds to twice the voxel resolution for filled cylinders (linear model equation: FC_occ = 2 × res + 0.01) and less than 1.8 times (linear model equation: FC_occ = 1.87 res – 0.06) for empty cylinders. Average compensated occlusion (AC_occ) corresponds to 0.67 times the voxel resolution (linear model: AC_occ = 0.67 res) for filled cylinders and less than 0.59 times (linear model: AC_occ = 0.59 res – 0.02) for empty cylinders. These results show that the ability of voxels to reduce occlusion is more efficient for filled than for empty cylinders.

Lastly, the volume of space explored by the three simulated trees was calculated (Fig. 8, left-hand panel). Even if the three trees have the same quantity of biomass and branch length, the volume that was explored differs among them (Fig. 8A). Sim1 explores more space than sim2 and sim3, and sim2 explores more space than sim3. To estimate differences between the simulated trees in terms of explored volume, we calculated the deviation of sim2 and sim3 relative to sim1 (Fig. 8B). With increasing resolution, the resulting deviation between trees increases from −0.6 to −16 % for sim2 and from −2 to −37 % for sim3. These results reveal that the quantification of explored volume varies depending upon tree form and voxel resolution. These differences can be explained by the fact that the larger the voxels are, the more the different botanical objects (i.e. points corresponding to these objects) can be grouped within a single voxel. In this context, a small branch insertion angle increases this phenomenon due to a shorter distance between woody structures. As a consequence, we can deduce that the smaller the voxel, the less influence tree architecture has on the quantification of explored volume.

Fig. 8.

Explored volume calculation. A: calculated explored volume as a function of the resolution of the voxels (from 0.05 to 0.5 m) for each simulated tree. B: deviation of calculated volume of sim2 and sim3 compared to sim1.

TESTING VOXR IN THE REAL WORLD: APPLICATION ON TLS DATA

Method

VoxR algorithms were tested on TLS data obtained from an Optech ILRIS-3-D (Optech Inc., Ontario, Canada) terrestrial LiDAR. Data are from a study that aims to highlight changes in tree structure, and the subsequent changes in space exploration, induced by pruning under a power line in the urban area of Montreal, Canada. Two TLS scenes, each resulting from the alignment of three TLS scans, were used. The first scene corresponds to a pruned tree (P, Fig. 9A and C) and the second to a non-pruned tree (NP, Fig. 9B and D). In both cases, tree orientation in the scene was normalized:

Fig. 9.

Point clouds used in the application section. Lateral and zenith views are shown for a pruned tree in A and C and a non-pruned tree in B and D, respectively.

• The Y-axis of the scenes was aligned parallel to the street and/or the power lines.

• Tree top was orientated toward the zenith

• Crown base (junction between the trunk and the first branch) was set in a 0, 0, 0 coordinate.

LiDAR returns corresponding to the tree environment were then manually removed in order to only keep the returns corresponding to the tree structure. For both scenes, scan registration (i.e. alignment), tree orientation in 3-D space and non-tree returns removal were done manually in the Pointstream 3-DImageSuite (Arius 3-D Inc., Ontario, Canada). Point cloud density was then reduced using distance filters: (1) in a sphere of 2 cm in radius only one point was conserved and (2) points located at >5 cm from all other points of the scene were removed. These procedures resulted in the data shown in Fig. 9. Finally, the trunk was removed in order to consider only the tree crown.

To provide a comprehensive overview of VoxR algorithms and to evaluate the effect of voxel resolution, two contrasted voxel resolutions were used, 0.1 and 0.4 m, for each tree resulting in a total of four voxel clouds to be compared: P0.1, P0.4, NP0.1 and NP0.4. Four VoxR algorithms were then applied (Fig. 10):

• point.distance was used to evaluate voxel dispersion from the crown centre (i.e. the average coordinates of all voxels of the voxel cloud);

• axis.distance was used to evaluate voxel dispersion from the Z-axis (i.e. the symmetry axis of the tree);

• Project was used to evaluate voxel density

• axis.angle was used to evaluate the radial (azimuthal) distribution of voxels (i.e. voxel angle with the Y-axis using the projection option of the voxel cloud in the XY plane).

Fig. 10.

Kernel density of voxels for the four variables derived from the voxel clouds (i.e. P0.1, P0.4, NP0.1 and NP0.4) using some VoxR functions. A: distance from the crown centre (point.distance); B: distance from the Z-axis (axis.distance); C: voxel density (project); and D: voxel radial dispersion (axis.angle with projection in the XY plane).

In all cases, except for axis.angle, the resulting variables were reported as a proportion of the 99th percentile in order to ignore potential non-comparable points. Kurtosis, skewness, mean and median parameters of variable distribution were then used to evaluate differences among tree type and voxel resolution (Table 5).

Table 5.

Distribution parameters for the voxel dispersion variables calculated using VoxR functions: distance from the crown centre (point.distance), distance from the Z axis (axis.distance) and voxel density (project)

	Distance from the crown centre				Distance from Z axis				Voxel density
	P0.1	NP0.1	P0.4	NP0.4	P0.1	NP0.1	P0.4	NP0.4	P0.1	NP0.1	P0.4	NP0.4
Skewness	−0.25	−0.323	−0.27	−0.272	−0.225	−0.241	−0.177	−0.195	0.823	0.895	0.258	0.342
Kurtosis	−0.419	−0.17	−0.376	−0.268	−0.727	−0.84	−0.764	−0.825	0.16	0.406	−0.987	−0.35
Mean	0.645	0.613	0.645	0.604	0.56	0.537	0.557	0.53	0.306	0.3	0.423	0.401
Median	0.652	0.623	0.652	0.612	0.569	0.568	0.563	0.557	0.261	0.25	0.391	0.375

Results

For both the P and the NP tree, distance from the crown centre (DC) and from the Z-axis (DZ) distributions are slightly negatively skewed (asymmetry to the right) and platikurtic (negative kurtosis, Table 5). For DC, the P tree had a less asymmetrical (skewness 0 > P > NP) and a more homogeneous distribution (kurtosis P < NP < 0) than NP for both voxel resolutions. However, the P tree had a higher mean and median than NP, showing that voxels are generally located further from the crown centre in P compared to the NP tree (Table 5). For DZ, the P tree exhibits a more symmetrical (skewness 0 > P > NP) and a less homogeneous distribution (kurtosis P > NP > 0) than the NP tree. Higher mean and median parameters of the P tree suggest that it explores space further from the crown symmetry axis (i.e. the trunk axis) than the NP tree, which confirms observations made with DC. Although the voxel resolution did not affect interpretation of the results for DC and DZ when comparing the two trees, the differences in distribution parameters were higher while using a distribution of 0.1 m compared to 0.4 m (compare P0.1 with P0.4 and NP0.1 and NP0.4 in Table 5). This corroborates results obtained with simulated data, suggesting that smaller voxel sizes are better suited to observe fine changes in space exploration than large voxel sizes.

For voxel density, the results were strongly influenced by voxel resolution. There were dramatic changes in distribution parameters between 0.1 and 0.4 m voxel resolution (Fig. 10, Table 5). This suggests that analyses that use voxel density should be done carefully in order to avoid possible misinterpretation. This strong change in voxel density, however, did not affect interpretation of the results in this example comparing the P and NP tree. The P tree exhibits a less left skewed (skewness 0 < P < NP) and a more homogeneous distribution (kurtosis P < NP) than NP. P also exhibits higher means and medians than NP at both resolutions. This suggests that the P tree may have a more regular space exploration density than NP, which may result from the densification and/or a reduction in density of some parts of the crown compared to the NP tree.

Voxel radial dispersion shows great changes in the orientation of space exploration between the two trees: the P tree explores space more toward a lateral orientation relative to the power line (i.e. the 90°–270° axis) while the NP tree seems to preferentially explore space toward an approx. 140° axis corresponding to a south/west orientation. Although voxel resolution does not affect the visual interpretation, some ‘high-density’ orientations can be observed for a resolution of 0.4 m. This suggests that some errors may appear when using a large voxel size. This phenomenon occurs in particular orientations (i.e. 90°, 135°, 180°, etc) in which many voxels may align with the crown base (i.e. 0, 0, 0 coordinates), inducing potentially unexpectedly high density values.

DISCUSSION AND CONCLUDING REMARKS

VoxR package approaches and limitations

The VoxR package is based on the assumption that voxels can be viewed as the portion of space explored by the tree. This allows one to use a very simple, but computationally efficient, voxelization method that considers only filled voxels. A recent study performed in an urban tree pruning context presented strong correlations during the leaf-on period between biomass and volume estimation while considering filled voxels only (Fernández-Sarría et al., 2013b). However, extrapolation of these results to the leaf-off sampling period remains uncertain due to changes in voxel filling by leaves vs. woody structure alone. Moreover, we think that using such a voxelization process to evaluate tree biomass or woody volume may constitute a mis-utilization of VoxR, and it is thus preferable to limit VoxR utilization to the analyses of tree space exploration. However, the voxelization function that is provided by VoxR as well as the space exploration concept make it possible to eliminate voxels that may contain points only corresponding to noise in two different ways. First, the vox function records the number of points that are present within each single voxel, which enables the user to remove voxels containing a small number of points through observation of histograms (Vonderach et al., 2012). Secondly, the ability of the voxel to reduce occlusion can be extrapolated by combining the reduction of the original point cloud density (i.e. by the conservation of one point within a sphere of defined radius) and the suppression of isolated points. This approach is intended to reduce noise with minimal structural information loss (i.e. when voxel size is at least equal to the sphere radius used to remove points), given that voxels will compensate for the loss in point density.

Finally, this paper shows that a simple voxel-based method is a relevant tool for highlighting the consequences of tree architecture on space exploration and occupancy. The relevance of this approach for comparing different tree architectures, based on their geometric and quantitative description, was also demonstrated using both simulated and real TLS data.

VoxR functions

The VoxR package provides tools (functions) to discriminate space exploration patterns or material distribution. The use of these functions on both computer-generated and real TLS-acquired data has revealed the strong capability of VoxR to extract accurate and meaningful variables from contrasting tree forms. In this paper, we have shown that these functions can be used to highlight differences in space exploration between trees possessing different branch insertion angles or pruned vs. non-pruned trees. The ability of VoxR to provide simple parameters (e.g. skewness and kurtosis of distribution) that varies among contrasted architectures suggests that it may help in species recognition based on TLS data. In a similar way, we show that VoxR is well suited to highlighting architectural acclimation at the whole-tree scale to different growing conditions in urban areas. Such geometric parameters, which describe tree crown space occupancy, have been shown to be useful for characterizing tree acclimation to neighbouring competition in the 3-D space of forest stands (Martin-Ducup et al., 2016). This highlights the fact that the space exploration concept is a relevant approach that may be useful in tree architecture studies, and that it should receive more attention from tree biologists and forest ecologists.

The VoxR package also provides a function that is dedicated to the analysis of time changes occurring between two TLS scenes of the same object. This function aims to isolate differences in space exploration, based upon the identification of voxels that are unique to one of the two TLS scenes representing contrasting time sequences. This function could prove useful in tree studies that involve branch loss or growth. However, parameterization of this function can be difficult due to possible object movement or displacement between the two time frames that are used. Thus, selection of adequate settings in this function requires close monitoring, especially through a visual survey of output quality.

Choice of voxel size

Furthermore, results obtained from real data show that voxel dispersion-based analyses are well suited to highlighting architectural differences among trees growing in different conditions (i.e. experiencing pruning cycles). However, voxel size may influence the results in such analyses. Our results suggest that small voxels may help to more efficiently detect differences in space exploration than large voxels, which corroborates results of explored space volume estimation obtained with synthetic data (see discussion below). Although voxel size did not strongly affect distribution parameters for distance-based metrics (i.e. distance from an axis or from a point), voxel size has been shown to induce potentially important changes in absolute values of voxel density estimations. Voxel size also affected the voxel radial dispersion estimates by inducing potential artefact high-density values for some particular directions.

Voxel size also influences occlusion compensation and quantification of the explored volume. A low resolution is associated with low occlusion compensation but a more accurate and architecture-independent quantification of explored volume. In contrast, large voxels are generally more efficient in reducing the influence of occlusion, which increases with consolidation of different botanical objects into one voxel (as reported by Vonderach et al., 2012); consequently, quantification of explored space volume is more strongly influenced by tree architecture. In summary, the choice of the resolution to be used appears to be a compromise between occlusion compensation and accurate estimation of explored space volume. This choice must be made according to the objectives of the study and the quality of TLS datasets. However, we highly recommend using a small voxel size for studies that use good quality scans (accurate measurements and low occlusion), which aim to compare trees of different species or trees experiencing different environments. In contrast, a large voxel size should be restricted to constraining measurement contexts (i.e. dense forest stands) involving the strong influence of occlusion zones (Bucksch, 2011). However, one should keep in mind that occlusion compensation using a large voxel size may induce other problems such as point cloud distortion or fake adhesions (i.e. the grouping of different botanical objects within a unique voxel). If research interests need to take into account finer architectural traits, it is thus preferable to use a small voxel size with higher quality TLS data. Conversely, studies focusing only on tree space occupancy can use a large voxel size and thus reduce TLS sampling effort.

Using a different voxel resolution means that explored space volume estimates will vary depending on author choice. To overcome this problem, authors could provide a model that is able to estimate explored space volume for different voxel resolutions. An example of a model that could provide good estimates of tree volume, as a function of voxel size, is shown in Supplementary Data Table S2.

SUPPLEMENTARY DATA

Supplementary data are available online at www.aob.oxfordjournals.org and consist of the following. Table S1: detailed description of the sub.obj function algorithm. Figure S2: graphics of adjusted models for tree volume estimation and estimates derivation as a function of voxel size, performed on a set of nine TLS point clouds of urban trees.