Estimating tree stem diameters and volume from smartphone photogrammetric point clouds

Methods for the three-dimensional (3D) reconstruction of forest trees have been suggested for data from active and passive sensors. Laser scanner technologies have become popular in the last few years, despite their high costs. Since the improvements in photogrammetric algorithms (e.g. structure from motion—SfM), photographs have become a new low-cost source of 3D point clouds. In this study, we use images captured by a smartphone camera to calculate dense point clouds of a forest plot using SfM. Eighteen point clouds were produced by changing the densiﬁcation parameters (Image scale, Point density, Minimum number of matches) in order to investigate their inﬂuence on the quality of the point clouds produced. In order to estimate diameter at breast height (d.b.h.) and stem volumes, we developed an automatic method that extracts the stems from the point cloud and then models them with cylinders. The results show that Image scale is the most inﬂuential parameter in terms of identifying and extracting trees from the point clouds. The best performance with cylinder modelling from point clouds compared to ﬁeld data had an RMSE of 1.9 cm and 0.094 m 3 , for d.b.h. and volume, respectively. Thus, for forest management and planning purposes, it is possible to use our photogrammetric and modelling methods to measure d.b.h., stem volume and possibly other forest inventory metrics, rapidly and without felling trees. The proposed methodology signiﬁcantly reduces working time in the ﬁeld, using ‘non-professional’ instruments and automating estimates of dendrometric parameters.


Introduction
Traditionally, foresters use mechanical or optical instruments, such as calipers, hypsometers and measuring tapes, to measure biometric parameters in field plots. These traditional methods are time-consuming and cannot directly measure some tree attributes, such as stem volume and biomass components (total, stem and branches). Assessing the later parameters in the field typically requires felling the trees (Liang et al., 2016) or relying on allometries of earlier studies that felled and measured trees to establish these allometries. Numerous studies applied mathematical functions for representing the longitudinal profile of stems of certain species. These equations were used to estimate the diameters of cross sections of the stem at different heights from the ground, and the volume of stem portions, or to represent the particular shape of the stem (Brink and Gadow, 1986;Tarp-Johansen et al., 1997;Gaffrey et al., 1998;Dhôte et al., 2000;Kublin, 2003). These functions are useful for describing the stem profile at varying heights from the ground. However, this type of approach does not consider the particular morphology and inclination of individual stems, which are difficult parameters to evaluate on standing trees.
In the last two decades, introduction of terrestrial laser scanning (TLS) technologies has increased the set of available methods for assessing forest attributes in the field ( (Liang et al., 2014) but also possible to measure other biometric parameters useful for forest management and planning purposes including biomass (Van Aardt et al., 2008) and stem locations. Furthermore, some metrics that have less frequently been addressed so far, including lower crown height, or crown depth may potentially be extracted. The latter could be estimated by combining point clouds collected from the ground and from the air.

Forestry
The reconstruction of trees from 3D point clouds has been mostly accomplished using cylinder fitting algorithms so far. Although the cross sections of tree stems are generally ellipsoidal, the ratio between the major and the minor axes of these ellipses is close to 1 (Patrone, 1963), particularly if they are not located on heavily sloping lands or in areas subject to strong dominant winds. Therefore, since the diameters do not change very quickly along the stem, the use of cylinders enables good estimates of the volume on a relatively short vertical section (Åkerblom et al., 2015). Corresponding approaches were for example presented in the work of Dassot et al. (2012), Hackenberg et al. (2014), Calders et al. (2015), Hackenberg et al. (2015) and Stovall and Shugart (2018).
There is an ongoing discussion whether the efficiency of forest inventories could be improved using TLS in combination with automatic procedures extracting the desired tree metrics (Liang et al., 2016;Giannetti et al., 2018). Besides a potential improvement in efficiency, TLS data provide a permanent detailed spatial record of the forest structure of a given area. It could thus be possible to carry out temporal monitoring of measured parameters through additional acquisitions over time (Liang et al., 2012;Srinivasan et al., 2014;Liang et al., 2016).
However, TLS data acquisition is limited by the time and costs involved (Mikita et al., 2016;Wilkes et al., 2017), limited software and a possible lack of trained personnel (Liang et al., 2016). The mobility of instruments is still relatively low and hardware costs, although decreasing, are still rather high (Giannetti et al., 2018).
Considering particularly the latter points, photogrammetric approaches may be an interesting alternative to TLS based approaches. Given the improvements in photogrammetric algorithms, image-based techniques are now able to produce 3D point cloud data that are similar to the data obtained with TLS technologies (Cavegn et al., 2014;Liang et al., 2015;Liang et al., 2016). The most commonly applied algorithm is structure from motion (SfM) (Ullman, 1979;Szeliski, 2011). Similar to traditional photogrammetry, SfM uses images acquired from multiple viewpoints and dense image-matching algorithms, such as the Scale Invariant Feature Transform (SIFT) (Lowe, 2004) and the Speeded-Up Robust Features (SURF) (Bay et al., 2008), to create the 3D geometry of an object or surface present in multiple images (Remondino and (Liang et al., 2014). The main advantage of image-based techniques is that they produce similar outputs to TLS systems, but use less expensive hardware (Mokroš et al., 2018), i.e. small and light instruments that are affordable and easy for non-professional users to operate. As stated by Liang et al. (2016), in the future, an untrained person may be capable of carrying out forest inventories using photogrammetry after a short instruction. This method could therefore significantly reduce the cost of field plot inventories, thus increasing the number of field plots and enhancing forest field inventories (Liang et al., 2016).
Approaches to collect image-based point clouds in forestry applications vary in terms of the image acquisition system and accuracy assessment of the point cloud obtained. The image acquisition phase is very important. Mokroš et al. (2018) present seven data collection methods, which differ in terms of camera orientation, shooting mode and photographic path. These authors found that the most accurate method used vertical camera orientation, stop-and-go walking rhythms and a path around the plot with two diagonal paths through the plot. Liang et al. (2015) proposed an application of terrestrial image-based point clouds acquired with a handheld camera for forest plot inventories. They used two photographic paths to take photos: a circular path inside the plot (30 × 30 m) with the camera pointing out of the plot and a closed loop outside the plot. Panagiotidis et al. (2016) took 350 photos in a circular plot with a 6 m radius, following the photographic path along the perimeter of the plot and pointing the camera at eye level, at the base and at the top of the tree. Mikita et al. (2016) used aerial and terrestrial images to obtain a dense point cloud of a 0.8 ha research plot. The terrestrial images were captured by walking through the forest at 5 km h −1 , with one captured image per second. Forsman et al. (2016) developed a method for terrestrial mapping of tree structures; they used a multi-camera rig with five cameras, taking all images from the centre of the field plot, looking outwards in different directions. Piermattei et al. (2019) estimated diameter at breast height (d.b.h.) and stem curves up to 2.8 m above the ground using a cylinder fitting method in four forest plots with a tree density ranging from 390 to 875 stems ha −1 .
In most cases, the accuracy of the reconstructed 3D models was assessed by comparing the d.b.h. and heights measured in the field with those obtained using the 3D model and analysing the root mean square error (RMSE) or percentage error values for each parameter (Liang et al., 2014;Morgenroth and Gomez, 2014;Liang et al., 2015;Mikita et al., 2016;. In Panagiotidis et al. (2016), forest inventory metrics (d.b.h. and height) obtained from the terrestrial photogrammetry point clouds were compared with those measured on the TLS point clouds.
In this study, we build upon the findings of these earlier studies but examine several new aspects. First, we derived our 3D point clouds from images taken with a smartphone camera, which has rarely been examined so far. Considering a future operationalization of the presented approach, we developed and tested an image acquisition grid (walking pattern) that fits most practical cases. This is an important step to standardize the image acquisition phase. Further, we tested the influence of the different densification parameters (Image scale, Point density, Minimum number of matches) in the SfM algorithm to produce the point clouds to analyse which kind of set-up leads to the best results. Finally, we also developed an automatic cylinder fitting method to extract and model individual tree stems from the image-based point clouds. Although cylinder stem modelling has been applied to TLS point clouds, to our knowledge there were no previous applications on image-based data. Therefore, the overall aim of this study was to estimate the accuracy and feasibility of using SfM and cylinder stem modelling techniques for d.b.h. and volume estimates of forest stands.
Estimating tree stem diameters and volume from smartphone photogrammetric point clouds

Study area
The forest plot used in our study is located in Puglia, southern Italy ( Figure 1). It is part of the Mercadante Forest (1300 ha) in the Alta Murgia National Park. The forest was created by hydrogeological afforestation work to counteract the descent of rainwater from the Murgia uplands after floods struck the Bari area in the early 1900s. The plot is characterized by an even-aged forest stand, and the main tree species are Aleppo pine (Pinus halepensis) and cypress (Cupressus sempervirens). The plot ( Figure 1) is 30 × 30 m with a buffer zone of 10 m around the plot to enable photographic surveys. It consists of 45 trees (∼500 stems ha −1 ) and was chosen for the scarce presence of undergrowth combined with a good density of arboreal vegetation. Starting from the base of the trees, diameters were measured every 1 m using Wheeler's pentaprism integrated with a hypsometer to estimate the volume of the standing tree stems. These measurements were taken for 15 'volume model trees' out of the total of 45 trees, which were randomly selected among the most visible trees. Upper stem diameters were measured up to a height of 12 m, but for some trees it was limited to 6 m. At higher heights, it became increasingly difficult to measure stem diameters as the visibility in the pentaprism decreased. Hence, volumes by section were estimated up to a height of 6 m. Tree heights of the volume model trees were measured by a Häglof VL5 Vertex Laser hypsometer (max = 20.6 m, min = 12 m, mean = 17.80 m).

Volume estimates in the field
Diameters measured at different heights in the field enabled the areas of tree cross sections to be calculated at each height. The volume of each tree stem portion was calculated using Smalian's formula (Loetsch et al., 1973): where S 1 and S 2 are the areas of the two last sections of the stem portion, and h is the height of each stem section, which was selected as equal to 1 m. The data from field measurements were compared with those obtained using cylinder modelling.

Terrestrial photogrammetry
Images were taken on the same day using a smartphone camera with a focal length of 28 mm (35 mm equivalent) and a lens aperture of f/1.9. The ISO sensitivity of the camera was set to 80 and the exposure time to 1/250 s, which matched the local light conditions. The smartphone used was a Samsung Galaxy S6 and it was mounted on a DJI OSMO Mobile smartphone stabilizer. A total of 686 photos were taken using the stop-and-go method and following the photographic path shown in Figure 2. The images were captured at a distance of 2.5 m from each other and at two different height levels (2 and 4 m above ground), using a telescopic bar. The choice of the image acquisition grid was based on our experience with previous surveys and aimed to optimize the percentage overlap between the images. Photos were taken in different light conditions due to the position of the sun in relation to the study plot. Light conditions were optimal in the forward survey images, unlike those in the return survey, which were taken under backlighting conditions, with the camera pointed at the sun. Image resolution was 5312 × 2988 pixels. It took ∼30 min to acquire the photos on a single path, and a total of ∼3 h to complete the full photographic survey. In addition, in order to allow scaling and georeferencing of the 3D point clouds obtained with SfM processing, a Trimble PROXH GPS receiver was used to collect the GPS coordinates of five ground control points (GCPs) that were well distributed in the plot area. Five black and yellow squared targets (20 × 20 cm) were used to identify the positions of the five GCPs in the images in which they were visible.

Data processing
The images captured in the terrestrial survey were processed using Pix4Dmapper (Pix4D SA). This software implements a fully automated photogrammetric workflow for indoor, terrestrial, oblique and nadir imagery. Detailed descriptions of the Pix4D workflow can be found in the study of Küng et al. (2011).
Data were processed using a 64-bit system with an Intel Core i7-7700 CPU at 3.6 GHz, 6 GB GPU and 64.0 GB RAM, and the process took ∼1 h for each point cloud produced.
We tested the influence of the following parameters on the resulting point cloud densification: • Image scale (1, 1/2, 1/4, 1/8), which defines the scale of the images at which additional 3D points are computed. The software default parameters are 1/2 Image scale, Optimal point density and 3 as the Minimum number of matches. Varying the parameters resulted in different numbers of 3D points and average densities (per m 3 ).
In total, 18 point clouds were produced, combining the parameters only in combinations which resulted in sufficiently dense point clouds (Table 1).

Point cloud processing
Before the identification and modelling of the tree stems, the plot point cloud produced by photogrammetry was filtered ( Figure 3) to remove non-tree points, such as points from the ground, understory and noise (incorrectly identified points). This process contains five steps. First the point cloud was clipped to the area of interest to exclude areas outside the plot. Then, it was downsampled by partitioning the space containing the point cloud into small cubes (voxels-here we used 1 cm edge length) and keeping only one point from each non-empty cube Estimating tree stem diameters and volume from smartphone photogrammetric point clouds ( Figure 3a). Next, the ground level was approximated as a grid of horizontal squares covering the point cloud. The edge length of the squares was 1 m. The height of the squares was the height (z-coordinate) of the lowest point inside the squares ( Figure 3b). Then, based on this ground level model, all points below a height of 20 cm were removed, and points up to a height of 2 m were removed if the third meter above ground level contained no points ( Figure 3b). Finally, some noise was filtered out based on local point densities: for this, the space was divided into cubes measuring 5 × 5 × 5 cm, and the points were kept inside a cube if there were at least five points (for some very low-density point clouds, thresholds below five were used; Figure 3c).

Stem extraction and cylinder modelling from point clouds
In this section, we describe how the stems were extracted from the point cloud and how they were modelled with two different cylinder models. Figure 4 shows the workflow of this process, and the details are given in the following subsections.

Covering the point cloud with patches
Potential stem sections are considered to be large vertically continuous subsets of the point cloud above the ground level. To extract them, firstly the point cloud was covered with surface patches, which are small subsets of the point cloud. These patches were used as the smallest 'units' or 'building blocks' for segmenting the point cloud into stems. Their location and diameter (between 10 and 20 cm) were randomly generated as the branching structures of the trees were, at this stage of the methodology, still unknown. Also, the neighbours of the patches were identified and used for region growing. The patch generation process is explained in the study of Raumonen et al. (2013). Figure 5a shows examples of the surface patches.

Locating potential stem sections
Then, based on the height of the patches, layers of the patches were iteratively processed from the highest to the lowest layer: all patches between 2.5-4.0, 2.0-3.5, 1.5-3.0, 1.0-2.0 and 0.3-1.0 m above the ground were iteratively selected (Figure 5a). For each layer, the connected components of the patches in the layer were identified: each component is the collection of all the patches that are connected to each other in the sense of the neighbour-relation of the patches. Each component that had more than 15 patches was a potential section containing a stem (Figure 5b). When all the potential stem sections from one iteration had been identified, they were processed one at a time, from the section containing the largest number of patches to the section containing the smallest number of patches, in order to see if they could be extended to the whole stem. When the stems from one layer had been identified, those stem points were no longer used in other layers for locating stems. In addition, potential stem sections were disregarded after expansion if they contained points from already identified stems.

Cylindrical surface coverage
The potential stem sections were processed to find the best possible stem subset from the point cloud. A subset is considered a real stem section based on how completely the surface of the section is covered with points or how high is the so-called cylindrical surface coverage. The cylindrical surface coverage was estimated using two lines, an axis line inside the stem section and another line orthogonal to it, as a reference for partitioning the points close to the line into subsets that were intersections of sectors and horizontal layers. A random reference direction orthogonal to the axis and a given point in the axis make it possible to compute a single angle and projected height for each point (see Figure 6a-c). Practical estimation of surface coverage was performed as follows: using an axis line in space (and a point in the line) and a random orthogonal direction as a reference, the points close to the line were divided into 20 equal-angle sectors and 10 equal-height layers. The intersection of these sectors and layers produced a partition of 200 subsets (see the example in Figure 6) and the relative number of non-empty subsets gave the surface coverage.

Selecting a proper stem section
In order to correctly select only one stem from multi-stem trees or sections containing points from two neighbouring stems, a potential stem section was first expanded with five layers of neighbours (patches outside the layer defining the section) and then divided into layers ∼30 cm high. With this step, the connected components of the layers were determined. Considering each component as a stem section, its axis line was estimated and applied to estimate the surface coverage on the whole expanded stem section. The axis line giving the highest surface coverage was then chosen to define the final stem section. The extended stem section was once again partitioned into 200 subsets as above, and the points closest to the axis were selected from each subset (Figure 6d). Then, a cylinder was fitted into this filtered section, and the cylinder was accepted if it was reasonable, i.e. a radius of <1 m, axis nearly vertical, and surface coverage over 40 per cent.

Expanding stem sections into full stems
Once a stem section had been defined, it was then expanded into the whole stem by first expanding it towards the ground and then upwards, as follows. The cylinder axis and radius were used to extract 3D subsets from the point cloud below the cylinder (or above it, when moving upwards), so that the length of the subset was increased iteratively and so that the distance of the points from the axis was less than three times the radius. The length of the subsections was increased from ∼3 times the radius up to ∼16 times the radius. The subsets were filtered according to the surface coverage, as above. They were then fitted with cylinders to estimate their axis and radius. The subset with the Estimating tree stem diameters and volume from smartphone photogrammetric point clouds greatest surface coverage was selected as the one that continues the stem. This process was then repeated until the ground was reached. The process continued upwards as long as there were sufficient points (at least 20), and the axis and radius were similar to the cylinder below. The result of the process was a subset of the point cloud containing the stem and cylinders that model its surface.

Extraction Model and Section Model
The above process of extracting stem point clouds used cylinders and produced a cylinder model for each extracted stem. We call this the 'Extraction Model'. Subsequently, the stems were modelled again with cylinders, but now the locations and lengths of the cylinders were fixed for a comparison with the manual diameter measurements. First, a cylinder was fitted around a height of 1.3 m (section 1.0-1.6 m) to accurately estimate d.b.h.. Then, the entire stem was fitted with cylinders so that they covered the sections 0-0.5, 0.5-1.5, 1.5-2.5 m, etc. up to the top. Apart from the first mentioned, the midpoints of these cylinders were at a height of 1, 2, 3 m, etc., which were the same heights as the diameter measurements. The section volumes thus match the volume estimated from the diameter measurements using equation (1). We call this the 'Section Model'.
The surface patch generation needed in stem extraction has a random seed; therefore, when repeated, it always produces a different partition of the point cloud into patches. Thus, repeating the extraction process and cylinder modelling will always generate small variations and in some sensitive cases larger differences; for example, a stem may be found and extracted in one iteration but not in another. Extraction and cylinder modelling were repeated 10 times for each point cloud to capture the uncertainty in the process. The averages (i.e. volumes) from the 10 iterations were used as the result, and standard deviation estimates the uncertainty in the results.

Statistics to evaluate the modelling results
To evaluate our modelling results, we calculated the RMSE and bias as defined in equations (2) and (3): Bias = n estimated − measured n trees In both formulas, estimated values were calculated as the average value of d.b.h. or volume from the 10 iterations for each tree. The percentage values of these parameters were obtained by comparing each one with the average measured value.
The uncertainty and accuracy of the modelling results were estimated by analysing the standard deviation and absolute error of d.b.h. and section volume for each tree of the 18 point clouds from the 10 iterations. Then for each point cloud, the mean standard deviation and mean absolute error values were calculated as the average value of the 45 single-tree standard deviations and absolute errors.

Stem extraction and cylinder modelling from point clouds
Results obtained for each point cloud varied in the number of reconstructed trees and stems found in different iterations. Table 2 shows the number of trees found by applying our algorithm to each of the 18 point clouds produced with the photogrammetric method. In addition to the total number (trees found in at least one iteration), the table also shows the number of trees found in all 10 iterations and those found in at least 9 iterations. All 45 trees were found only in the point clouds generated with Image scale 1 / 4 or 1 / 2 . At the same time, with Image scale 1 / 2 , it was possible to find all trees with the Point density set to high and optimal. When our algorithm was used on the point cloud generated with the Image scale 1 / 2 and low density (point cloud 18), only 36 trees were found. The most problematic point cloud is n. 3, which was produced with the lowest density parameters (Image scale: 1/8; Point density: low; Minimum number of matches: 3). However, despite these problems, the algorithm was able to reconstruct 32 trees from the different iterations. All the trees were found in all the 10 iterations in point clouds 10, 11, 14 and 16. Interestingly, all these point clouds were produced with an Image scale of 1 / 2 , except for n. 10 (Image scale: 1/4; Point density: low; Minimum number of matches: 3).
We also investigated in how many point clouds each tree in the plot was reconstructed by applying the algorithm. A total of 12 of the 45 trees were reconstructed in all 10 iterations of the 18 point clouds and another 20 in at least 9 iterations of them.
Mapping these data for each tree (Figure 7) shows that the most problematic trees were those photographed only in the first two stretches of the forward survey path and in all the photos taken during the return survey. These photos were taken under non-optimal light conditions of backlighting due to the position of the sun and most had significant lighting problems.

Cylinder-based d.b.h. estimates from different point clouds
The RMSE and d.b.h. bias estimated by the cylinder-fitting approach compared with the field data of those point clouds with all the 45 trees reconstructed are summarized in Table 3. Mean standard deviations and absolute errors were also calculated for the 18 point clouds (Table 4).
D.b.h. RMSE values ranged from a minimum of 1.6 cm (5.58 per cent) to a maximum of 4.6 cm (16.09 per cent). The best result in terms of RMSE was achieved by point cloud 10 (Image scale: 1/4; Point density: low; Minimum number of matches: 3). This point cloud also had the lowest bias value. A strong correlation was also found between the d.b.h. measured in the field and the d.b.h. derived with the cylinder-fitting approach based on this point cloud (R 2 = 0.95) (Figure 8). This point cloud also had the lowest mean absolute error (0.58 cm), low uncertainty and high precision as indicated by the low mean standard deviation in the estimation of the d.b.h. (0.23 cm) (Table 4). However, the best result in terms of mean standard deviation was shown by point cloud 14 (Image scale: 1/2; Point density: optimal; Minimum number of matches: 3-the default settings of the software). The d.b.h. RMSE value of this point cloud was also good (RMSE = 2.14 cm; RMSE% = 7.72). Negative bias values indicate that the model tends to slightly underestimate d.b.h. Figure 9 compares the average absolute error and standard deviations of d.b.h. estimates, both from the 10 modelling iterations, for each tree and for each point cloud. Point clouds 1 and 2 have a high level of error and standard deviation, showing poor results in terms of accuracy and uncertainty. The best accuracy and uncertainty results were obtained from point cloud 10, with a maximum average error value of 2.9 cm and standard deviation of <0.5 cm for all but three of the trees. Figure 9a shows the standard deviation of the d.b.h. estimates obtained during the 10 iterations and for each tree in the different point clouds. The comparison of these values between the number of trees and number of point clouds highlights how well the algorithm worked in most cases. In point clouds 1 and 2, the standard deviation is large for almost every tree. In the other point clouds, except for point cloud 18, trees 37-45 usually have large standard deviations. Tree 27 had some problems in most of the point clouds, with standard deviation values of ∼3-4 cm. Of note is the peak in the standard deviation values at tree 10 for point clouds 11 and 12. As shown in Figure 9b, absolute error values are bigger than standard deviations, in particular in the first four point clouds. In addition, in most cases, the most problematic trees in terms of accuracy are trees 1-10 and trees 35-45, which were those captured in a smaller number of images because they were located at the beginning of the two photographic paths.

Volume estimates by sections from different point clouds
Volumes by sections, estimated using diameters measured at different heights of up to 6 m with a Wheeler pentaprism, were compared with those obtained using the Section Model. These data were also compared to section volumes given by the Extraction Model. Depending on the quality of the point clouds, it was not possible to use all 15 reference trees to calculate section volumes in all the point clouds. Table 5 summarizes the RMSE of the section volumes and the bias for the 9 point clouds in which the model managed to compute section volumes of up to 6 m for all the 15 reference trees. The minimum RMSE value (22.67 per cent) was obtained from point cloud 10 (Image scale: 1/4; Point density: low; Minimum number of matches: 3). Estimated section volumes obtained from point cloud 10 correlated strongly with those calculated by field measurements (R 2 = 0.872, Figure 10). Extraction Model RMSE and bias values were considerably smaller than those obtained with the Section Model (but see standard deviation results below).  Estimating tree stem diameters and volume from smartphone photogrammetric point clouds  (Table 6). Mean absolute error values for each tree also showed that volume estimates for tree 10 (dark grey column in the table) were the least accurate, while those for tree 17 (light grey column in the table) were the most accurate, with mean absolute error values higher than 0.04 m 3 only from point clouds 1 and 2. Table 7 summarizes the standard deviation values of the section volumes from each point cloud obtained using the Section Model for all the 15 trees measured in field. These values were highest for tree 42 (dark grey column) and lowest for tree 25 (light grey column) in most point clouds. Therefore, the volume estimates of tree 42 were the most uncertain, while those of tree 25 were the most precise. The most problematic point clouds were 1 and 13, with values up to 0.05593 m 3 . These point clouds produced results with the highest uncertainty, while estimates from point cloud 10 were the most precise in the Section modelling.
A comparison of the Section Model and Extraction Model results highlights that the Extraction Model gave more accurate volume estimates, but that the higher standard deviations indicate more uncertainty in these estimates ( Figure 11).

Correlation coefficients
An initial analysis of the results seems to indicate that the densification parameters have a significant influence on the results. To estimate the influence more quantitatively, we calculated the Estimating tree stem diameters and volume from smartphone photogrammetric point clouds

Discussion
The use of photogrammetric methods to estimate biometric parameters in the forest is still a developing field. Forests often have complex light conditions, high tree densities and occlusions by branches and shrubs can occur. This makes photogrammetric measurements challenging, and investigations are needed to for example identify optimal acquisition paths and processing  parameters for deriving dendrometric inventory data from these data.
In this study, we examined several of these parameters using photogrammetric surveys based on a smartphone camera. The use of terrestrial image-based point clouds from a smartphone camera-a non-specialized instrument that almost everyone owns-is a new and straightforward way to investigate the structure and features of forest plots. To our knowledge, in the published literature so far only professional cameras were used to produce point clouds of forest stands. This paper has explored the feasibility of using this low-cost photogrammetric method for the collection of forest inventory data and has compared the quality of the terrestrial photogrammetric point clouds obtained with different densification parameters of the point clouds.
In this section, we present a comparison of our results with those obtained in previous studies, although the comparison must take into account the presence of random errors in the estimation of the parameters in the field in the various studies, as well as general differences in the species composition and the complexity of the investigated forest stands.   The forest plots studied in these published articles are broadly comparable with that of the present study. It is hence interesting that the RMSE values found in this study (d.b.h. RMSE = 1.61 cm and stem volume RMSE = 0.094 m 3 ) are better than those presented in all the previous studies, except for Mikita et al. (2016). The latter is the only study that estimated stem volumes, in addition to the d.b.h., but the volume estimates were based on allometric relationships. The better performances observed in the study of Mikita et al. (2016) could be due to the lower density of trees in their forest plot compared to that of our study and due to the availability of well-suited allometries. Our experiment highlighted strengths and weaknesses of the applied image acquisition grid. It is evident that the photogrammetric process has given better results for trees located towards the centre of the plot. This is probably because these trees are captured by a larger number of images in both the forward and the return survey paths. It is therefore important to maximize the number of images in which each element of the scene is present, in order to obtain good results with the photogrammetric reconstruction of the trees.

Forestry
Light conditions are another factor to take into account when setting the image acquisition grid, because they greatly affect the quality of the point clouds and consequently the performance of the model. It is advisable to avoid lighting problems during the photographic survey. For example, images captured when the sun is higher or during the early afternoon (when there is a softer light in the forest) prevent the acquisition of images against the light. Similarly, an overcast sky also prevents lighting problems (Liang et  Recent studies demonstrated that the accuracy of photogrammetric point clouds varies with the resolution of the image sensor, the camera configuration and the equipment used to support the camera (Piermattei et al., 2019). In this study, we also investigated the influence of densification parameters on the 3D reconstruction and, hence, on d.b.h. and volume estimates. The results showed that Image scale is the parameter that most influences the model's performance in finding and extracting trees from the point clouds. All the trees were found in all the 10 modelling iterations in point clouds produced with an Image scale of 1 / 2 . The only point cloud produced with an Image scale different from 1 / 2 that allowed to find all the 45 trees was the number 10 (Image scale: 1/4; Point density: low; Minimum number of matches: 3). Details in the images increase as the image scale increases, which improves the production of the point cloud during the photogrammetric process. Therefore, using a higher Image scale value (increasing the image size during the photogrammetric process for the production of the point cloud and therefore the possibility to obtain more details) leads to a better reconstruction of the 3D structure of the trees, reducing the presence of noise caused by incorrectly identified points. Nevertheless, it is important to make the right combination of this parameter with the other parameters. The results show that the best performance for extracting d.b.h. and volume estimates for the individual tree was obtained with point cloud 10 (Image scale: 1/4; Point density: low; Minimum number of matches: 3). This indicates that a low density of the point cloud can positively influence the volume estimates, probably due to reduced noise. It is therefore possible to state that the presence of incorrectly positioned points (noise) and an excessively high Point density lead to incorrect fitting of cylinders to the stems and consequently cause errors in the estimates of biometric forest inventory parameters. Point cloud 10 also gave the best results Estimating tree stem diameters and volume from smartphone photogrammetric point clouds Besides assessing the influence of the densification parameters of the SfM algorithm, we also examined two different approaches to estimate the stem volume: the Extraction Model and the Section Model. The former models the stem with cylinders of variable height, while the latter uses 1 m tall stem sections (to compare model estimates with those obtained from data collected in the field). We found that volumes estimated by the Extraction Model generally gave lower RMSE values than the Section Model, and lower values of absolute error, but high values of standard deviation. This is probably because on the one hand the cylinders in the Extraction Model are generally shorter and can thus more accurately fit the local diameter. On the other hand, standard deviation is larger with the Extraction Model because the lengths and sections of the fitted cylinders can change from one iteration to the other, in contrast to the Section Model, where the sections are always the same. Thus, it could be possible to estimate volumes using only the Extraction Model, given its good correspondence with the total stem volume up to a certain height, as estimated with the traditional dendrometric method.
Concerning the efficiency of the presented approach, we found that the traditional method of estimating the examined forest inventory metrics, particularly the diameters of stem sections at multiple heights, is more time-consuming than the proposed method. The suggested approach takes approximately half the time that the traditional methods require (estimated time based on experience) ( Figure 12). Furthermore, it is possible to obtain other dendrometric parameters useful for forest management and planning purposes (in addition to d.b.h. and volume) including lower crown height, or crown depth estimated by combining images from the ground with aerial images, simply by implementing the estimation algorithm. Furthermore, it could be possible to use the photographic surveys as presented in this study: (1) to estimate all the parameters needed to produce volume tables rapidly and without felling trees; (2) to carry out thinning simulations, estimating the possible withdrawable assortments of wood and the residual volume; repeated photographic surveys would additionally allow for, (3) carrying out multi-temporal surveys in order to estimate stand growth over time and to monitor the evolution of forest stands.
In summary, our results show that photogrammetric point clouds can be useful for the reconstruction of 3D tree models to derive diameter and volumes estimates. Although d.b.h. and volumes were underestimated in comparison with those obtained using data collected in the field, both d.b.h. and volume up to a height of 6 m were estimated with good accuracies. Since it is not possible to assess the error in the estimates produced by the field measurements conducted here (attempting to measure irregular shapes, i.e. defined by the presence of ribs or irregularity of the bark) and the photogrammetric 3D model (dependent on many factors, including the image calibration process), it is not possible to define which method is more reliable. This would require to fell some of the trees and to measure the targeted parameters, which was out of the scope of this study. Nevertheless, it is encouraging to see that the estimates produced by the photogrammetric approach agreed very well with the measurements of the traditional approach and that the data collection was less time-intensive. Even though certain hardware and software requirements have to be fulfilled for subsequent data processing, a further optimization of the process may also result in a significant cost-saving.

Conclusions
We presented a rapid and automatic image-based, photogrammetric method for estimating diameters from multiple heights of a stem as well as the stem's volume. These parameters are difficult to assess in the field, and the suggested approach is a real alternative to other non-destructive measurements of stem volume and diameters at multiple heights and allows the structure of forest stands to be studied in a more objective way.
The possibility to obtain a 3D reconstruction of forest stands makes it possible to acquire the nearly complete structure of