# Stem Measurements and Taper Modeling Using Photogrammetric Point Clouds

^{*}

## Abstract

**:**

_{1.3}) and the error of photogrammetric rendering reduced the bias to 1.4 mm. The usability of the PPC measurements in taper modeling was assessed with four models: Max and Burkhart [1], Baldwin and Feduccia [2], Lenhart et al. [3], and Kozak [4]. The evaluation revealed that the data fit well with all the models (R

^{2}≥ 0.97), with the Kozak and the Baldwin and Feduccia performing the best. The results support the replacement of taper with PPC, as faster, and more accurate and precise product estimations are expected.

## 1. Introduction

_{1.3}) and total height [18]. However, mixed results were obtained when a taper was used in product identification and estimation, some arguing for [19], and some against [20,21].

_{1.3}, in many instances with precision superior to terrestrial lidar [34]. The encouraging SfM results recommended an expansion from d

_{1.3}to the diameter along the stem. Therefore, the objective of this research is an assessment of the accuracy of diameter measurements executed from the PPC obtained with the SfM technique. A secondary objective is a comparison between the diameters measured from PPC and the diameters estimated from taper equations.

## 2. Methods

#### 2.1. Field Data Collection

_{1.3}was on average 306.1 mm (variance 68.5), ranging from 213.0 mm to 449.6 mm. The total height (H) varied from 15.9 m to 26.8 m, with an average of 22.2 m and a variance of 12.8. The trees grew on productive sites, with site indices ≥60 at base age 25. Each tree was photographed with a Nikon D3200 (i.e., a complementary metal–oxide–semiconductor sensor of 23.2 mm × 15.4 mm) equipped with a Nikkor AF-S DX VR 18–55 mm zoom lens (aperture 3.5–5.6). To capture as much as possible from the tree, the images were acquired at the focal length of 18 mm. For calibration, each tree had the d

_{1.3}painted circularly, and on opposite sides of the d

_{1.3}two metal rods of 304.8 mm (i.e., 1 foot) were freely hanged. Shortly after the images were captured, the trees were felled and the diameter along the stem was measured every meter with a Spencer D-tape starting from d

_{1.3}, which was marked at 1.3 m. The accuracy of measurements was 1 mm for diameters and 10 mm for lengths along the stem (i.e., height). To verify the total length after the tree was felled, the total height was extracted from lidar data. The lidar flight scanning the area was executed in March 2012, at most four months before the trees were photographed and cut. To accommodate for the growth between flight time and field measurements, we considered that trees could have increased their height with at most 1 m. The point density was on average 30 points/m

^{2}.

#### 2.2. Photorammetric Point Cloud Generation and Diameter Measurements

^{2}. To ensure precise measurements, we reconstructed the surface of the trunk with at least 500,000 faces (Figure 2b). The faces were built with a ratio of 1:5 to the number of points. Compared with previous studies of image-based forest inventories [34,35,42], the selected parameters for SfM in Agisoft (Table 1) were either similar or provided superior solutions.

_{1.3}was assigned to the corresponding segment from the PPC. The scale on the vertical plane was carried out by allocating the known length of the metal bar (i.e., 304.8 mm) to the distance between the points delineating the bar inside the PPC (Figure 2c). After the length of the metal bar was introduced, the d

_{1.3}on two approximately perpendicular diameters was entered. Based on the tree values, Agisoft computes the scaling error by subtracting the estimated value from the inputted value [30]. If the scaling error was larger than 5% of the field measured d

_{1.3}, then the two diameters were re-measured on different positions. Depending on the size of the tree, the root mean square of the error ranged from 6.9 mm to 22.1 mm. Considering that the relationship among points inside the PPC are correct (i.e., are similar to reality) and unchanged, the maximum error that is expected for any linear measurement is 22.1 mm. Scaling is one of the three main sources of errors when PPC are used for actual measurements, and usually increases the magnitude of the investigated attributes with at least one order of magnitude (in our case three orders, from 1 to 1000).

_{1.3}is <5% [43]. The average d

_{1.3}measured in the field is 306 mm. Because the calibrating metal bar was 304.8 mm (i.e., close to the d

_{1.3}of measured trees), we considered that a vertical error of 5% is also admissible, even when field measurements for heights accept errors of <10% [43]. Therefore, the accepted total error for accurate field measurements is 21.6 mm (i.e., $0.05\sqrt{{306}^{2}{}_{average\text{\hspace{0.17em}\hspace{0.17em}}dbh}+{304}^{2}{}_{metal\text{\hspace{0.17em}\hspace{0.17em}}bar}}$). For consistency, a similar value could have been used for PPC accuracy, but we decided to tighten the requirements. Consequently, we considered that scaling will have a limited impact on measurements if the total error (i.e., horizontal and vertical) is <10 mm (less than half of the field accepted accuracy).

_{1.3}as the middle of the colored band marking the d

_{1.3}in the PPC, then measure all diameters starting from the identified d

_{1.3}(i.e., 1.3 m), or (2) identify the ground in the PPC, then measure the diameters with respect to the ground. Both ways offer a check, as either the ground should be at a height (length) of 0 (i.e., former), or d

_{1.3}should fall inside the colored band (i.e., later). Even when the difference between the d

_{1.3}identified using the two ways should be minor, its propagation could have a significant impact, particularly for the upper section of the stem. Therefore, diameters were measured in both ways.

#### 2.3. Assessment of Measurements and Bias Correction

_{field}) and its correspondent from AutoCAD (i.e., diameter@h

_{PPC}):

_{i,h}is the PPC-based error at height h for tree i that has diameter measured to maximum height H

_{i}, and n is the number of trees. It should be noticed that H

_{i}being an integer number acts as a count, besides being a linear measurement.

_{1.3}or total height) or during processing (e.g., software scaling errors). Furthermore, considering that the images were recorded from the ground, the upper sections of a tree will be described by fewer points than the lower sections, which will render the measurement process less accurate close to the terminal bud. Therefore, we expect that bias will change with height. Consequently, we will be using the following linear model for bias correction (Equation (5)):

_{h}is the bias correction at height h, RH is the relative height, RH = h/H d

_{1.3 based variable}and scaling

_{based}

_{variable}are linear variables derived from d

_{1.3}and PPC scaling, and b

_{i}, I = 0, … 4, are coefficients to be estimated.

_{i}0 or 1, which are easy to implement. However, this simplistic approach will likely not remove the bias. Nevertheless, if bias is reduced to ≤1% while the root mean square error (RMSE) is larger, the simplification becomes operationally justified.

#### 2.4. Taper Modeling

_{i,h}, and the estimated dimeter, ${\widehat{d}}_{i,h}$, at height h for tree i. Besides the previous three fit statistics, we have included the coefficient of determination R

^{2}to mirror other taper studies [64,65]. Because errors have different signs, bias is usually smaller than the MAB and the RMSE, which are always non-negative. MAB and RMSE are similar in their evaluation power [66], with the observation that RMSE is slightly higher than MAB, a direct result of the Cauchy–Bunyakovsky–Schwarz inequality [67].

_{i,h}is the PPC–based diameter of tree i at height ht, H

_{i}is the total height of tree i, ${\widehat{d}}_{i,h}$ is the diameter predicted from taper equations for tree i at height h, and ${\overline{d}}_{i}$ is the average diameter of tree i. (A1) was added to the denominator of each statistic to account for the d

_{1.3}measurement.

## 3. Results

#### 3.1. Tree Construction and Diameter Measurement

^{2}[68], on which five to seven trees are measured, the total time to acquire the data is approximately two hours. The acquisition of images for SfM reconstruction is less than 2 min/tree, with a total time of at most 15 min/plot. The PPC processing time for one tree with the parameters from Table 1 on a Dell Precision workstation 7910 CPU E5-2630 v3 @ 2.40 GHz and 32 Gb RAM was on average 15 min (i.e., ranging from 11 min to 18 min). Therefore, the total processing time for one plot would have been approximately 2 h, the same as for the ground measurements. However, the advantages of using PPC over ground data are tremendous, as a snapshot of the trees is obtained that can be used for subsequent investigations, including audit. Furthermore, while the field measurement time has remained almost unchanged for the last 50 years, the technological advances will most likely reduce the computation time. Therefore, the desired results will likely be obtained faster than by ground measurements.

_{1.3}>35 cm). Therefore, in further analyses, only the results for heights ≤12 m were considered.

_{i}either 0 or 1. Coefficients different than 0 or 1 require field measurements, which in most instances are not only not available but preclude the remote sensing approach advocated by the paper. Multiple trials revealed that bias can be reduced with a linear function derived from Equation (4) (i.e., b

_{1}and b

_{4}are 1, the rest are 0):

_{1.3}(i.e., d

_{1.3 field}–d

_{1.3 PPC}), and scaling error is the horizontal calibration error estimated by Agisoft.

_{1.3}, which is commonly recorded anyway, Equation (7) delivered the intended results: measurement bias is operational and statistically insignificant. Nevertheless, a formal assessment of the residual error, error

_{ht},

_{residual}, is required:

_{h},

_{residual}= error@h − BC

_{h}

_{1.3}error, at most 1% of the diameter (proof in the Appendix A). For the 18 trees, the bias reduction was almost 10 times (i.e., 17.2/1.8 = 9.5 times), which proved that biased corrected PPC-measurements are accurate and precise.

#### 3.2. Taper Equations

^{2}> 0.97, bias < 1 mm, MAB and RMSE around 10 mm (Table 5). The models refitted from the PPC-based measurements supplied comparable fit statistics with the ground-based data (Table 6), except for bias, which was twice as large (i.e., 1.4 mm for Baldwin and Feduccia and 2.2 mm for Max–Burkhart vs. 0.6 mm and 0.9 mm, respectively). It should be noted that because the upper portion of the stem could not be rendered from the PPC, the upper inflection point of the Max–Burkhart equation was not identified. Therefore, a simplified version of the best Max–Burkhart equation (Table 2) was used for the PPC-based values (Equation (2) in the original article) Irrespective of the equation, the fit statistics are slightly higher for PPC-derived models than the ground-based models. Overall, the Kozak model performed the best, with the smallest bias, MAB, RMSE, and largest R

^{2}.

_{2}and b

_{3}). The lack of significance is noticed for relative heights >0.5 (Figure 7d).

^{2}≥ 0.97 irrespective of the model type, bias < 4 mm, MAB < 10 mm, and RMSE < 15 mm. Even though Kozak’s model proved to be most suited to represent diameter variation along the stem, the Baldwin and Feduccia model, which is simpler, supplied similar fit statistics with a significant increase in parsimony.

## 4. Discussion

_{1.3}. The measurement error expands along the stem as the distance becomes increasingly further from the projection’s center. Consequently, we select the relative height as the correcting term. A higher order polynomial of relative height, or even a rotation correction term, could lead to better results than the proposed affine transformation (Equation (7)), as the images are restructured with a nonlinear process. It is possible that accuracy is influenced by errors occurring from multiple directions [73]. Even so, the largest error, relative to diameter, was 4% (i.e., at 12 m), the rest being ≤2%. The accuracy and precision of diameter measurements from PPC are not the only advantages of using 3D reconstruction from images. The PPC allows for diameter measurements at any height, not only at preset ones (e.g., every meter). Furthermore, high-density PPC supplies information for the detection and estimation of trunk defects, such as catface or sweep.

_{1.3}< 33 cm). Nevertheless, our study results show that the selection of model form would not significantly influence the model fit. Among the four models selected for taper assessment, we consider the Baldwin and Feduccia [2] approach to be the most trustworthy, as it is intuitive and relatively simple. The Kozak [4] model 02 outperformed all other models, but its lack of realism and low parsimony (e.g., six parameters compared to two for Baldwin and Feduccia) is not appealing. In fact, the difference between Kozak’s model 02 and the Baldwin and Feduccia model is minute, as bias, MAB, and RMSE are almost the same (Table 7). Therefore, the PPC measurements can be used not only for the direct estimation of diameter (bias was <2 mm), but also for taper modeling (bias < 4 mm and R

^{2}≥ 0.97).

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Estimation of Residual Bias after Application of Equation (7)

_{h}, becomes:

_{h}= error

_{h}− BC

_{h}= d

_{h,field}− d

_{h}

_{,PPC}− d

_{1.3,field}− d

_{1.3,PPC}− RH × error

_{Agisoft}

_{Agisoft}= d

_{PPC}− d

_{operator}

_{h}= −1.36 − 1.42 × RH

_{1.3}error, 1.20 mm, (i.e., intercept in Equation (7)) and to reported error by Agisoft, 1.22, (i.e., slope in Equation (7)). In fact, there was no significant difference between the coefficients of regression A3 and the two statistics. The choice of which one will be intercept and which one slope was driven by the magnitude of the bias reduction as well as the variation. The model with d

_{1.3}error as slope has a residual bias of 0.15 mm, while the alternative was 0.18 mm. However, even though the bias was smaller when d

_{1.3}error was slope, the variance was larger (i.e., 2.97 vs. 2.88), which suggested selection of the Equation (7) to reduce the bias.

_{h}:

_{d1.3}is the smallest among all heights (except for height 9 m, which likely is a random occurrence), a direct result of closeness to the operator, and consequently to the camera, of that section of the tree. Figure 6 supports the existence of a linear relationship between expectation of the error anywhere on the stem and d

_{1.3}error:

_{1.3}]/(total height). Considering that field measurements are executed for stands that require immediate attention, such as thinning or regeneration harvests, it can be assumed that the total height of the trees is larger than 13 m, which mean that the expected residual bias is less than 10% of E[error d

_{1.3}]. According to Table 3, E[error d

_{1.3}] is the smallest among errors at all heights (except for 12 m), therefore the residual bias is expected to be approximately 1 mm.

## References

- Max, T.A.; Burkhart, H.E. Segmented polynomial regression applied to taper equations. For. Sci.
**1976**, 22, 283–289. [Google Scholar] - Baldwin, V.C.; Feduccia, D.P. Compatible tree-volume and upper-stem diameter equations for plantation loblolly pines in the west gulf region. South. J. Appl. For.
**1991**, 15, 92–97. [Google Scholar] - Lenhart, J.D.; Hackett, T.L.; Laman, C.J.; Wiswell, T.J.; Blackard, J.A. Tree content and taper functions for loblolly and slash pine trees planted on non-old-fields in east texas. South. J. Appl. For.
**1987**, 11, 147–151. [Google Scholar] - Kozak, A. My last words on taper equations. For. Chron.
**2004**, 80, 507–515. [Google Scholar] [CrossRef] - Avery, T.E.; Burkhart, H. Forest Measurements; Mcgraw-Hill Ryerson: New York, NY, USA, 2001; p. 480. [Google Scholar]
- Husch, B.; Beers, T.W.; Kershaw, J.A. Forest Mensuration, 4th ed.; Wiley: New Yourk, NY, USA, 2002; p. 456. [Google Scholar]
- Burkhart, H.E.; Tome, M. Modeling Forest Trees and Stands; Springer: New York, NY, USA, 2012; p. 460. [Google Scholar]
- Lee, J.-H.; Ko, Y.; McPherson, E.G. The feasibility of remotely sensed data to estimate urban tree dimensions and biomass. Urban For. Urban Green.
**2016**, 16, 208–220. [Google Scholar] [CrossRef] - Lu, D.; Chen, Q.; Wang, G.; Liu, L.; Li, G.; Moran, E. A survey of remote sensing-based aboveground biomass estimation methods in forest ecosystems. Int. J. Dig. Earth
**2016**, 9, 63–105. [Google Scholar] [CrossRef] - Williams, M.S.; Cormier, K.L.; Briggs, R.G.; Martinez, D.L. Evaluation of the barr & stroud fp15 and criterion 400 laser dendrometers for measuring upper stem diameters and heights. For. Sci.
**1999**, 45, 53–61. [Google Scholar] - Shimizu, A.; Yamada, S.; Arita, Y. Diameter measurements of the upper parts of trees using an ultra-telephoto digital photography system. Open J. For.
**2014**, 4, 316–326. [Google Scholar] [CrossRef] - Nunes, M.H.; Görgens, E.B. Artificial intelligence procedures for tree taper estimation within a complex vegetation mosaic in brazil. PLoS ONE
**2016**, 11, e0154738. [Google Scholar] [CrossRef] [PubMed] - Cushman, K.C.; Muller-Landau, H.C.; Condit, R.S.; Hubbell, S.P. Improving estimates of biomass change in buttressed trees using tree taper models. Methods Ecol. Evol.
**2014**, 5, 573–582. [Google Scholar] [CrossRef] - Saarinen, N.; Kankare, V.; Vastaranta, M.; Luoma, V.; Pyörälä, J.; Tanhuanpää, T.; Liang, X.; Kaartinen, H.; Kukko, A.; Jaakkola, A.; et al. Feasibility of terrestrial laser scanning for collecting stem volume information from single trees. ISPRS J. Photogramm. Remote Sens.
**2017**, 123, 140–158. [Google Scholar] [CrossRef] - Olofsson, K.; Holmgren, J. Single tree stem profile detection using terrestrial laser scanner data, flatness saliency features and curvature properties. Forests
**2016**, 7, 207. [Google Scholar] [CrossRef] - You, L.; Tang, S.; Song, X.; Lei, Y.; Zang, H.; Lou, M.; Zhuang, C. Precise measurement of stem diameter by simulating the path of diameter tape from terrestrial laser scanning data. Remote Sens.
**2016**, 8, 717. [Google Scholar] [CrossRef] - Henning, J.G.; Radtke, P.J. Detailed stem measurements of standing trees from ground-based scanning lidar. For.Sci.
**2006**, 52, 67–80. [Google Scholar] - Coble, D.W.; Hilpp, K. Compatible cubic-foot stem volume and upper-stem diameter equations for semi-intensive plantation grown loblolly pipe trees in east Texas. South. J. Appl. For.
**2006**, 30, 132–141. [Google Scholar] - Farrar, R.M. Stem-profile functions for predicting multiple-product volumes in natural longleaf pines. South. J. Appl. For.
**1987**, 11, 161–167. [Google Scholar] - Westfall, J.A. Modifying Taper-Derived Merchantable Height Estimates to Account for Tree Characteristics; USDA Forest Service: Washington, DC, USA, 2006; p. 126. [Google Scholar]
- Newnham, R.M. Variable-form taper functions for four Alberta tree species. Can. J. For. Res.
**1992**, 22, 210–223. [Google Scholar] [CrossRef] - Lowe, D.G. Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vis.
**2004**, 60, 91–110. [Google Scholar] [CrossRef] - Smith, R.C.; Cheeseman, P. On the representation and estimation of spatial uncertainty. Int. J. Robot. Res.
**1986**, 5, 56–68. [Google Scholar] [CrossRef] - Harris Geospatial Solutions. Envi; Exelis Visual Information Solutions: Boulder, CO, USA, 2016. [Google Scholar]
- Hexagon Geospatial. Erdas Imagine; Hexagon AB: Stockholm, Switzerland, 2016. [Google Scholar]
- Schindler, K.; Bischof, H. On robust regression in photogrammetric point clouds. In Proceedings of the Pattern Recognition: 25th Dagm Symposium, Magdeburg, Germany, 10–12 September 2003; Michaelis, B., Krell, G., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; pp. 172–178. [Google Scholar]
- Soule, S.; Maurice, K.; Walcher, W.; Szabo, J. Advanced Point Cloud Generation for Photogrammetric Modeling of Complex 3d Objects. In Proceedings of the International Conference on Image Processing, Rochester, NY, USA, 24–28 June 2002; IEEE: New York, NY, USA, 2002; Volume 523, pp. 529–532. [Google Scholar]
- Lucas, B.D.; Kanade, T. An Iterative Image Registration Technique with an Application to Stereo Vision. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), Vancouverm, BC, Canada, 24–28 August 1981; pp. 674–679. [Google Scholar]
- Haralick, R.M. Digital step edges from zero crossing of second directional derivatives. IEEE Trans. Patt. Anal. Mach. Intell.
**1984**, 1, 58–68. [Google Scholar] [CrossRef] - Agisoft LLC. Agisoft Photoscan; Agisoft: St. Petersburg, Russia, 2014. [Google Scholar]
- Pix4d; Pix4D: Lausanne, Switzerland, 2014.
- Wu, C. Visualsfm v0.5.26. Available online: http://ccwu.me/vsfm/ (accessed on 11 July 2017).
- Fritz, A.; Kattenborn, T.; Koch, B. Uav-based photogrammetric point clouds—Tree stem mapping in open stands in comparison to terrestrial laser scanner point clouds. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2013**, 40, 141–146. [Google Scholar] [CrossRef] - Forsman, M.; Börlin, N.; Holmgren, J. Estimation of tree stem attributes using terrestrial photogrammetry with a camera rig. Forests
**2016**, 7, 61. [Google Scholar] [CrossRef] - Mikita, T.; Janata, P.; Surový, P. Forest stand inventory based on combined aerial and terrestrial close-range photogrammetry. Forests
**2016**, 7, 165. [Google Scholar] [CrossRef] - Mensah, S.; Glèlè Kakaï, R.; Seifert, T. Patterns of biomass allocation between foliage and woody structure: The effects of tree size and specific functional traits. Ann. For. Res.
**2016**, 1, 59. [Google Scholar] [CrossRef] - Garber, S.M.; Maguire, D.A. Modeling stem taper of three central oregon species using nonlinear mixed effects models and autoregressive error structures. For. Ecol. Manag.
**2003**, 179, 507–522. [Google Scholar] [CrossRef] - Valentine, H.T.; Gregoire, T.G. A switching model of bole taper. Can. J. For. Res.
**2001**, 31, 1400–1409. [Google Scholar] [CrossRef] - Nyland, R.D. Silviculture. Concepts and Applications; McGraw-Hill: New York, NY, USA, 1996; p. 633. [Google Scholar]
- Smith, D.M. The Practice of Silviculture: Applied Forest Ecology, 9th ed.; Wiley: New York, NY, USA, 1997; p. 537. [Google Scholar]
- Turner, D.; Lucieer, A.; Wallace, L. Direct georeferencing of ultrahigh-resolution uav imagery. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 2738–2745. [Google Scholar] [CrossRef] - Liang, X.L.; Jaakkola, A.; Wang, Y.S.; Hyyppa, J.; Honkavaara, E.; Liu, J.B.; Kaartinen, H. The use of a hand-held camera for individual tree 3d mapping in forest sample plots. Remote Sens.
**2014**, 6, 6587–6603. [Google Scholar] [CrossRef] - Robertson, F.D. Timber Cruising Handbook; USDA Forest Service: Washington, DC, USA, 2000; p. 268. [Google Scholar]
- Autodesk. Autocad Civil 3d; Autodesk: SanRafael, CA, USA, 2016. [Google Scholar]
- Barber, C.B.; Dobkin, D.P.; Huhdanpaa, H. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw.
**1996**, 22, 469–483. [Google Scholar] [CrossRef] - Maas, H.G.; Bienert, A.; Scheller, S.; Keane, E. Automatic forest inventory parameter determination from terrestrial laser scanner data. Int. J. Remote Sens.
**2008**, 29, 1579–1593. [Google Scholar] [CrossRef] - The Math Works Inc. Matlab R2017a; The Math Works Inc.: Natick, MA, USA, 2017. [Google Scholar]
- Daniilidis, K.; Spetsakis, M.E. Understanding noise sensitivity in structure from motion. In Visual Navigation; Aloimonos, Y., Ed.; Lawrence Erlbaum Associates: Hillsdale, NJ, USA, 1996; pp. 61–88. [Google Scholar]
- Stängle, S.M.; Sauter, U.H.; Dormann, C.F. Comparison of models for estimating bark thickness of picea abies in southwest germany: The role of tree, stand, and environmental factors. Ann. For. Sci.
**2017**, 74, 16. [Google Scholar] [CrossRef] - Montealegre, A.; Lamelas, M.; Riva, J. Interpolation routines assessment in als-derived digital elevation models for forestry applications. Remote Sens.
**2015**, 7, 8631–8654. [Google Scholar] [CrossRef] - Powell, S.L.; Cohen, W.B.; Healey, S.P.; Kennedy, R.E.; Moisen, G.G.; Pierce, K.B.; Ohmann, J.L. Quantification of live aboveground forest biomass dynamics with landsat time-series and field inventory data: A comparison of empirical modeling approaches. Remote Sens. Environ.
**2010**, 114, 1053–1068. [Google Scholar] [CrossRef] - Bilskie, M.V.; Hagen, S.C. Topographic accuracy assessment of bare earth lidar-derived unstructured meshes. Adv. Water Resour.
**2013**, 52, 165–177. [Google Scholar] [CrossRef] - Schabenberger, O.; Pierce, F.J. Contemporary Statistical Models for the Plant and Soil Sciences; CRC Press: Boca Raton, FL, USA, 2002; p. 730. [Google Scholar]
- Neter, J.; Kutner, M.H.; Nachtsheim, C.J.; Wasserman, W. Applied Linear Statistical Models; WCB McGraw-Hill: Boston, MA, USA, 1996; p. 1408. [Google Scholar]
- Grubbs, F.E. Sample criteria for testing outlying observations. Ann. Math. Stat.
**1950**, 21, 27–58. [Google Scholar] [CrossRef] - Thode, H.C. Testing for Normality; Marcel Dekker: New York, NY, USA, 2002; p. 368. [Google Scholar]
- Hollander, M.; Wolfe, D.A. Nonparametric Statistical Methods; John Wiley and Sons: New York, NY, USA, 1973; p. 503. [Google Scholar]
- SAS Institute. Sas 9.1; SAS Institute: Cary, NC, USA, 2010. [Google Scholar]
- Cao, Q.V. Calibrating a segmented taper equation with two diameter measurements. South. J. Appl. For.
**2009**, 33, 58–61. [Google Scholar] - Trincado, G.; Burkhart, H.E. A generalized approach for modeling and localizing stem profile curves. For. Sci.
**2006**, 52, 670–682. [Google Scholar] - Williams, M.S.; Reich, R.M. Exploring the error structure of taper equations. For. Sci.
**1997**, 43, 378–386. [Google Scholar] - Grothendieck, G. Nls2: Non-Linear Regression with Brute Force, Version 0.2. The Comprehensive R Archive Network (CRAN), 2013. Available online: https://cran.r-project.org/web/packages/nls2/nls2.pdf (accessed on 11 July 2017).
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2016. [Google Scholar]
- Jiang, L.-C.; Liu, R.-L. Segmented taper equations with crown ratio and stand density for dahurian larch (larix gmelinii) in Northeastern China. J. For. Res.
**2011**, 22, 347–352. [Google Scholar] [CrossRef] - Sharma, M.; Zhang, S.Y. Variable-exponent taper equations for jack pine, black spruce, and balsam fir in eastern canada. For. Ecol. Manag.
**2004**, 198, 39–53. [Google Scholar] [CrossRef] - Grimmett, G.D.; Stirzaker, D.R. Probability and Random Processes; Oxford University Press: New York, NY, USA, 2002; p. 600. [Google Scholar]
- Poole, D. Linear Algebra; Thomson Brooks/Cole: Toronto, ON, Canada, 2005; p. 712. [Google Scholar]
- Vidal, C.; Alberdi, I.A.; Mateo, L.H.; Redmond, J.J. National Forest Inventories: Assessment of Wood Availability and Use; Springer: Cham, Switzerland, 2016; p. 845. [Google Scholar]
- Lejeune, G.; Ung, C.-H.; Fortin, M.; Guo, X.; Lambert, M.-C.; Ruel, J.-C. A simple stem taper model with mixed effects for boreal black spruce. Eur. J. For. Res.
**2009**, 128, 505–513. [Google Scholar] [CrossRef] - Newberry, J.D.; Burkhart, H.E. Variable-form stem profile models for loblolly pine. Can. J. For. Res.
**1986**, 16, 109–114. [Google Scholar] [CrossRef] - Koutsoudis, A.; Vidmar, B.; Ioannakis, G.; Arnaoutoglou, F.; Pavlidis, G.; Chamzas, C. Multi-image 3d reconstruction data evaluation. J. Cult. Herit.
**2014**, 15, 73–79. [Google Scholar] [CrossRef] - Nikolov, I.; Madsen, C. Benchmarking close-range structure from motion 3d reconstruction software under varying capturing conditions. In Digital Heritage. Progress in Cultural Heritage: Documentation, Preservation, Proceedings of the Protection: 6th International Conference, Euromed 2016, Nicosia, Cyprus, 31 October–5 November 2016, Part I; Ioannides, M., Fink, E., Moropoulou, A., Hagedorn-Saupe, M., Fresa, A., Liestøl, G., Rajcic, V., Grussenmeyer, P., Eds.; Springer International Publishing: Cham, Switzerland, 2016; pp. 15–26. [Google Scholar]
- Weng, J.; Huang, T.S.; Ahuja, N. Motion and structure from two perspective views: Algorithms, error analysis, and error estimation. IEEE Trans. Patt. Anal. Mach. Intell.
**1989**, 11, 451–476. [Google Scholar] [CrossRef] - Li, R.; Weiskittel, A.R. Comparison of model forms for estimating stem taper and volume in the primary conifer species of the north american acadian region. Ann. For. Sci.
**2010**, 67, 302. [Google Scholar] [CrossRef]

**Figure 1.**Area showing the location of the trees: (

**a**) general position of the trees within Louisiana; (

**b**) locations of all the trees photographed and cut; (

**c**) the four most southern trees. The yellow arrows show the size of a rectangular box (highlighted) containing the southern four trees; (

**d**) lidar point cloud of the rectangle containing the southern four trees from (

**c**). The arrows from (

**c**) to (

**d**) indicate the top of each tree in the point cloud.

**Figure 2.**Workflow of the photogrammetric-based stem reconstruction and diameter measurement. (

**a**) Field photographs for an individual tree; (

**b**) structure from motion (SfM) process of reconstruction (tie points, densified points, surface); (

**c**) scaling the photogrammetric point clouds (PPC) with reference bar and d

_{1.3}; (

**d**) diameter measurements in AutoCAD.

**Figure 3.**An example of a reconstructed stem, with the lower part continuous, (i.e., measurable surface), and the upper part fragmented (i.e., unsuitable for accurate measurements).

**Figure 4.**Diameter measurements on cross sections of the stem (

**a**) successfully identified by the convex hull algorithm at height 4 m, (

**b**) unsuccessful identified by the convex hull algorithm at height 11 m. The red line is the circumference of the tree as computed by the convex hull algorithm.

**Figure 5.**Variation with height of diameters measured in the field and from the PPC for the side-view measurements.

**Figure 6.**The PPC-based error vs. the stem height; (

**a**) uncorrected (

**b**) after bias correction with Equation (7). The dots represents outliers, which are estimated using the interquartile range approach.

**Figure 7.**Comparisons of the models developed with the PPC-based measurement and the ground-based measurement. (

**a**) Max and Burkhart (

**b**) Baldwin and Feduccia (

**c**) Lenhart et al. (

**d**) Kozak.

Photo Alignment | Point Cloud Densification | Mesh Building | |||
---|---|---|---|---|---|

Accuracy | Medium & High | Quality | High | Face count | High |

Key points | 100,000 | Depth filtering | Disabled | Interpolation | Disabled |

Tie points | 60,000 |

Model | Equation |
---|---|

Max–Burkhart Equation (4) in original paper | $\begin{array}{l}{d}_{h}={d}_{1.3}\times {\left[{b}_{1}(\frac{{h}_{d}}{H}-1)+{b}_{2}\left(\frac{{h}_{d}{}^{2}}{{H}^{2}}-1\right)+{b}_{3}{\left({u}_{1}-\frac{{h}_{d}}{H}\right)}^{2}{I}_{1}+{b}_{4}{\left({u}_{2}-\frac{{h}_{d}}{H}\right)}^{2}{I}_{2}\right]}^{0.5}\text{}\\ where\text{\hspace{0.17em}}{I}_{1}=\{\begin{array}{c}1\text{\hspace{0.17em}\hspace{0.17em}}if\text{\hspace{0.17em}\hspace{0.17em}}{h}_{d}/H\le {a}_{1}\\ 0\text{\hspace{0.17em}\hspace{0.17em}}if\text{\hspace{0.17em}\hspace{0.17em}}{h}_{d}/H>{a}_{1}\end{array}\\ {I}_{2}=\{\begin{array}{c}1\text{\hspace{0.17em}\hspace{0.17em}}if\text{\hspace{0.17em}\hspace{0.17em}}{h}_{d}/H\le {a}_{2}\\ 0\text{\hspace{0.17em}\hspace{0.17em}}if\text{\hspace{0.17em}\hspace{0.17em}}{h}_{d}/H>{a}_{2}\end{array}\\ {a}_{1}<{a}_{2}\end{array}$ |

Baldwin-Feduccia Equation (2) in original paper | ${d}_{h}={d}_{1.3}\times \left\{{b}_{1}+{b}_{2}\mathrm{ln}\left[1-\left(1-{e}^{-{b}_{1}/{b}_{2}}\right)\times {\left({h}_{d}/H\right)}^{1/3}\right]\right\}$ |

Lenhart et al. Equation (26) in original paper | ${d}_{h}={d}_{1.3}\times {\left(\frac{H-{h}_{d}}{H-1.3}\right)}^{b}$ |

Kozak Equation (3) in original paper | $\begin{array}{l}d={a}_{0}\times {d}_{1.3}{}^{{a}_{1}}{X}_{i}^{q},\\ where\text{\hspace{0.17em}}{X}_{i}=\left[1-{\left({h}_{d}/H\right)}^{1/4}/(1-{0.01}^{1/4})\right]\\ q={b}_{0}+{b}_{1}\times \left(1/{e}^{{h}_{d}/H}\right)+{b}_{2}\times {d}_{1.3}{}^{{X}_{i}}+{b}_{3}\times {X}_{i}^{d1.3/H}\end{array}$ |

**Table 3.**Variation along the stem of diameter measurement error from PPC. Diameter is the diameter measured in the field. RMSE, root mean square error.

Height | Diameter | Bias | Mean Absolute Error | RMSE | |||
---|---|---|---|---|---|---|---|

[m] | [mm] | [mm] | [%] | [mm] | [%] | [mm] | [%] |

1 | 312 | −12.1 | −3.9 | 14.8 | 4.8 | 19.4 | 6.2 |

1.3 | 306 | −12.0 | −3.9 | 13.5 | 4.4 | 17.1 | 5.6 |

2 | 296 | −13.5 | −4.6 | 16.1 | 5.4 | 20.4 | 6.9 |

3 | 287 | −16.4 | −5.7 | 19.2 | 6.7 | 22.5 | 7.8 |

4 | 278 | −17.2 | −6.2 | 18.2 | 6.6 | 22.4 | 8.0 |

5 | 270 | −20.3 | −7.5 | 20.3 | 6.5 | 23.3 | 8.6 |

6 | 263 | −19.5 | −7.4 | 20.3 | 7.7 | 23.3 | 8.9 |

7 | 256 | −20.0 | −7.8 | 20.0 | 7.8 | 23.8 | 9.3 |

8 | 247 | −20.0 | −8.1 | 20.5 | 8.3 | 24.2 | 9.8 |

9 | 244 | −22.4 | −9.2 | 23.8 | 9.8 | 26.7 | 11.0 |

10 | 242 | −18.2 | −7.5 | 20.9 | 8.6 | 22.9 | 9.4 |

11 | 249 | −19.9 | −8.0 | 19.9 | 8.0 | 24.1 | 9.7 |

12 | 229 | −8.1 | −3.5 | 18.1 | 7.9 | 22.1 | 9.6 |

Total | - | −17.2 | −6.3 | 18.8 | 6.9 | 22.5 | 8.2 |

**Table 4.**Performance of existing taper models on field and PPC-based measurements. Max–Burkhart Model 4 was used for assessment. MAB, mean absolute bias.

Equation | Coeff | Original | Bias [mm] | MAB [mm] | RMSE [mm] | |||
---|---|---|---|---|---|---|---|---|

Field | PPC | Field | PPC | Field | PPC | |||

Max–Burkhart | b_{1} | −3.0257 | 5.7 | 12.4 | 14.1 | 19.4 | 18.1 | 24.4 |

b_{2} | 1.4586 | |||||||

b_{3} | −1.4464 | |||||||

b_{4} | 39.1081 | |||||||

a_{1} | 0.7431 | |||||||

a_{2} | 0.1125 | |||||||

Baldwin–Feduccia | b_{1} | 1.22467 | −6.2 | 0.1 | 14.7 | 17.9 | 18.6 | 21.9 |

b_{2} | 0.3563 | |||||||

Lenhart et al. | b | 0.841837 | 16.0 | 23.2 | 19.5 | 25.4 | 27.3 | 34.9 |

Equation | Coeff. | Data | p-Value | R^{2} | Bias [mm] | MAB [mm] | RMSE [mm] | RMSE [%] |
---|---|---|---|---|---|---|---|---|

Max–Burkhart | b_{1} | −0.48 | <0.001 | 0.98 | −0.9 | 8.7 | 12.2 | 4 |

b_{2} | −0.41 | 0.01 | ||||||

b_{3} | 2.72 | 0.02 | ||||||

a | 0.29 | <0.001 | ||||||

Baldwin–Feduccia | b_{1} | 1.11 | <0.001 | 0.98 | −0.6 | 8.6 | 11.7 | 5 |

b_{2} | 0.24 | <0.001 | ||||||

Lenhart et al. | b | 0.5288 | 0.01 | 0.97 | −2.0 | 9.8 | 12.9 | 5 |

Kozak | a_{0} | 1.35 | <0.001 | 0.98 | −0.01 | 8.4 | 11.9 | 5 |

a_{1} | 0.94 | <0.001 | ||||||

b_{0} | 0.19 | <0.001 | ||||||

b_{1} | 0.3 | 0.02 | ||||||

b_{2} | 0.01 | 0.25 | ||||||

b_{3} | −0.05 | 0.19 |

Equation | Coeff. | Estimates | p-Value | R^{2} | Bias [mm] | MAB [mm] | RMSE [mm] | RMSE [%] |
---|---|---|---|---|---|---|---|---|

Max–Burkhart | b_{1} | −0.64 | <0.001 | 0.95 | −2.2 | 13.5 | 18.4 | 6 |

b_{2} | −0.33 | <0.001 | ||||||

b_{3} | 3.53 | 0.37 | ||||||

a | 0.2 | <0.001 | ||||||

Baldwin–Feduccia | b_{1} | 1.12 | <0.001 | 0.95 | −1.4 | 13.3 | 18.3 | 6 |

b_{2} | 0.24 | <0.001 | ||||||

Lenhart et.al. | b | 0.519 | <0.001 | 0.94 | −0.7 | 13.6 | 18.6 | 7 |

Kozak | a_{0} | 1.81 | <0.001 | 0.95 | −0.1 | 12.5 | 16.7 | 5 |

a_{1} | 0.84 | <0.001 | ||||||

b_{0} | 0.15 | <0.001 | ||||||

b_{1} | 0.42 | 0.02 | ||||||

b_{2} | −0.01 | 0.54 | ||||||

b_{3} | −0.04 | 0.77 |

Equation | R^{2} | Bias [mm] | MAB [mm] | RMSE [mm] |
---|---|---|---|---|

Max-Burkhart | 0.98 | −3.6 | 9.7 | 14.0 |

Baldwin-Feduccia | 0.97 | −2.9 | 9.4 | 13.7 |

Lenhart et al. | 0.97 | −2.5 | 10 | 14.2 |

Kozak | 0.98 | −2.0 | 9.4 | 13.2 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fang, R.; Strimbu, B.M.
Stem Measurements and Taper Modeling Using Photogrammetric Point Clouds. *Remote Sens.* **2017**, *9*, 716.
https://doi.org/10.3390/rs9070716

**AMA Style**

Fang R, Strimbu BM.
Stem Measurements and Taper Modeling Using Photogrammetric Point Clouds. *Remote Sensing*. 2017; 9(7):716.
https://doi.org/10.3390/rs9070716

**Chicago/Turabian Style**

Fang, Rong, and Bogdan M. Strimbu.
2017. "Stem Measurements and Taper Modeling Using Photogrammetric Point Clouds" *Remote Sensing* 9, no. 7: 716.
https://doi.org/10.3390/rs9070716